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Contents
1 Introduction to interfacial flows 1
1.1 A brief history of Navier-Stokes equations . . . . . . . . . . . . . . . 1
1.2 Navier-Stokes bulk equations and equilibrium conditions at interfaces 6
1.2.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Simplified expressions . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Mass and momentum balance at interfaces . . . . . . . . . . . 23
1.3 Surface tension, Laplace pressure and Marangoni effect . . . . . . . . 25
1.3.1 Where does surface tension come from ? . . . . . . . . . . . . 25
1.3.2 Laplace pressure . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.3 Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.4 Momentum equilibrium at interfaces with surface tension . . 30
1.4 Useful dimensionless numbers and illustrations . . . . . . . . . . . . 31
1.4.1 The Reynolds number . . . . . . . . . . . . . . . . . . . . . . 31
1.4.2 The Capillary number . . . . . . . . . . . . . . . . . . . . . . 33
1.4.3 The Weber number . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.4 The Bond Number . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Interfaces and vibrations 37
2.1 Bubbles and the Rayleigh-Plesset equation . . . . . . . . . . . . . . . 38
2.1.1 Why are bubbles outstanding resonators ? . . . . . . . . . . . 39
2.1.2 Static equilibrium of bubbles . . . . . . . . . . . . . . . . . . 42
2.1.3 Modeling of the liquid phase: the Rayleigh-Plesset equation . 45
2.1.4 Modeling of the gas phase . . . . . . . . . . . . . . . . . . . . 53
2.1.5 From bubbles to metamaterials . . . . . . . . . . . . . . . . . 60
2.2 Inertio-capillary “Rayleigh-Lamb” modes of vibration . . . . . . . . . 61
2.2.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 61
2.2.2 Complete solution of the problem . . . . . . . . . . . . . . . . 61
2.2.3 Sessile droplets and possible use of these surface vibrations. . 65
3 Dynamics of bubbles or plugs in confined geometries 67
3.1 Two phase flow at small scales . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Bretherton law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.1 Semi infinite bubble . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.2 Long bubble motion . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 Large capillary number and rectangular channels . . . . . . . . . . . 80
3.3.1 Large capillary numbers . . . . . . . . . . . . . . . . . . . . . 80
3.3.2 Rectangular channels . . . . . . . . . . . . . . . . . . . . . . . 81
Bibliography 83
Chapter 1
Introduction to interfacial flows
Contents
1.1 A brief history of Navier-Stokes equations . . . . . . . . . . 1
1.2 Navier-Stokes bulk equations and equilibrium conditions
at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Simplified expressions . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Mass and momentum balance at interfaces . . . . . . . . . . 23
1.3 Surface tension, Laplace pressure and Marangoni effect . . 25
1.3.1 Where does surface tension come from ? . . . . . . . . . . . . 25
1.3.2 Laplace pressure . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.3 Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.4 Momentum equilibrium at interfaces with surface tension . . 30
1.4 Useful dimensionless numbers and illustrations . . . . . . . 31
1.4.1 The Reynolds number . . . . . . . . . . . . . . . . . . . . . . 31
1.4.2 The Capillary number . . . . . . . . . . . . . . . . . . . . . . 33
1.4.3 The Weber number . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.4 The Bond Number . . . . . . . . . . . . . . . . . . . . . . . . 34
The main objective of this workshop is to display basic knowledge of the physics
of interfaces along with real time experiments and numerical simulations. The range
of topics is expected to promote an exchange between students and scientists working
on contemporary issues of interfacial phenomena.
1.1 A brief history of Navier-Stokes equations
Interest in the behavior of fluids flow has been raised in early civilizations by the
fundamental need of transporting water for irrigation and supply. During the 3rd
century BC, Archimedes (287-212 BC) formulates “Archimedes principle”, which is
at the origin of hydrostatics (the study of fluids at rest). Afterwards, many tech-
nical treatises have been written during Roman empire describing the hydraulics
of aqueducts. One can cite the extensive work (“De aquaeductu”) of Sextus Julius
Frontinus (40-103 AD), a Roman aristocrate, whose treatise aims at describing the
water-supply of Rome. In 1643, Evangelista Toricelli (1608-1647), an Italian mathe-
matician sets out Toricelli’s law relating the speed of fluid flowing out of a container
2 Chapter 1. Introduction to interfacial flows
Figure 1.1: From left to right: Daniel Bernoulli, Jean le Rond d’Alembert, Leonhard
Euler, Henri Navier, George Gabriel Stokes
to the square root of the height of the fluid. Despite this early work on the dynamics
of fluids, Jean le Rond D’Alembert (1717-1783) dates back the origin of hydrodynam-
ics to the publication of “Hydrodynamica” in 1738 by Daniel Bernoulli (1700-1782)
[3] (Fig. 1.2). The work of Daniel Bernoulli differs from the one of his predecessors
since he establishes the relation between the velocity of a flow and gravity effects
from a general mechanics principle: the conservation of “vis viva” (living forces) .
This concept, proposed by Gottfried Leibniz (1646-1716), served as an elementary
formulation of the principle of conservation of energy. While Bernoulli evokes some
pressing [nisus], compression [compressio] and pressure [pressio], the concept of in-
ternal pressure does not appear directly in his equations and Bernoulli’s principle
is not formalized yet in its modern form.
Figure 1.2: First page of Daniel Bernoulli’s book “Hydrodynamica”
Using differential calculus introduced by Gottfried Leibniz and Isaac Newton
(1643-1727) and applying Newton’s laws of classical mechanics to a small amount of
fluid, Jean le Rond d’Alembert and Leonhard Euler (1707-1783) derive the so-called
Euler equations for an inviscid flow (frictionless fluids). Indeed, d’Alembert intro-
1.1. A brief history of Navier-Stokes equations 3
duces in 1749 the notions of “Eulerian” velocity field and partial derivatives [6]. He
also derives the incompressibility condition in an axially symmetric case. However
he fails to identify the role of internal pressure in the dynamics of incompressible
fluids. Between 1750 and 1755, Euler clarifies and generalizes the notions introduced
by d’Alembert, and includes both the notion of internal pressure and compressibility
in the equations. He obtains the current form of Euler equations [9] (see Fig. 1.3)
and also formalizes Bernoulli’s law for potential flow. It is interesting to see how
quickly new mathematical and physical concepts (differential calculus, Newton’s
laws of mechanics) have been assimilated by d’Alembert and Euler. The difficulties
at that time were to adapt them to a continuous medium with no reference state,
and to identify the role played by internal pressure.
Figure 1.3: Euler equations as written by Euler in 1755. In these equations, q is
the density, u, v, and w are the projections of the velocity on x, y and z axes, t is
the time, p the internal pressure, and P , Q, R the components of an external force
field (such as gravity) applied to the fluid.
However, Euler’s formulation of the equations suffers a contradiction with ex-
periments, the so-called d’Alembert paradox raised by this latter in 1752: the drag
force applied to an elliptical body moving with constant velocity in an inviscid flow
is zero. To solve this paradox, Henri Navier (1786-1836), a French scientist from
Ecole Polytechnique, introduces for the first time the “viscosity” of the flow in 1822
[24] (while the term “viscosity” does not appear in Navier’s manuscript). Navier
postulates the existence of a force between adjacent molecules which depends on
both the distance between them (repulsion force decreasing with the distance) and
their relative velocity (friction force). From this basic principle and by integrating
momentum on all molecules, he obtains the equations predicting the evolution of a
viscous incompressible fluid (Fig. 1.4). He also expresses the condition of adhesion
to a wall and the friction appearing between two fluid layers. Because of the use
of molecular theory and its initial hypotheses, some contemporary scientists remain
skeptical about Navier’s equations. In fact, Navier attributes to molecules a prop-
erty that should be attributed to fluid particles (that is to say fictitious particles
4 Chapter 1. Introduction to interfacial flows
containing a large amount of molecules): the stress between adjacent fluid particles
depends linearly on their relative velocity and thus on the flow velocity gradient.
d~PCd~-i-u~y+~d~wdz~âx'~'a~+âz'
-n ~p <~ < /M'ddw.
dw dw
dw daw
daw
d2wRidz-PCdt+udx+vdy-I-dz~ E.~dxx+dy~+dz'
En second lieu à l'égard desconditions qui se rapportent
aux points de la surface du fluide, si l'on désigne., comme
on l'a fait plus haut, par 7,/M,Kles angles que le plan tan-Figure 1.4: Navier’s equations for a viscous incompressible flow obtained by Navier
in 1822. The notations are similar to Fig. 1.3, with ε the fluid viscosity.
A few years later (1834), Adhémar Barré de Saint-Venant derive again these
equations without the help of molecular theory, by considering the friction between
adjacent “fluid particles” (in the modern sense of continuum mechanics). While
this work is achieved in 1834, it is only published in 1843. In the meanwhile,
Georges Gabriel Stokes [34] (1819-1913) develops a continuum mechanics theory of
compressible viscous flow. He identifies the rate of dilatation, shifting and rotation,
and the corresponding tensors (of course without tensorial notation). Then, by
postulating a linear relation between stress and elongation and with the help of
symmetry considerations, he obtains the “almost” modern formulation of Navier-
Stokes equations (Fig. 1.5).
Figure 1.5: Navier-Stokes equations for a viscous compressible flow obtained by
Stokes in 1845. Here, ρ is the density, X the x-component of the external force field
and µ the shear viscosity.
The only difference between Stokes mass and momentum equations and current
formulation of these equations is the existence of a second coefficient of viscosity
in analogy with the two Lamé coefficients in solid mechanics. While some early
references to this coefficient appear in Lamb’s Hydrodynamics, it will be properly
introduced much later, in 1942, by L. Tisza from Massachusetts Institute of Technol-
ogy. Stokes theory predicts that this second viscosity coefficient is equal to −2µ/3,
where µ is the shear viscosity. At that time, Stokes is well aware of the deficien-
1.1. A brief history of Navier-Stokes equations 5
cies of his argument: he writes in 1851 “... I have always felt that the last term of
this equation (the term which includes the dilatational viscosity) does not rest on
as firm a basis as the correctness of the equation of motion of an incompressible
fluid”. However, since the second coefficient of viscosity plays no role in incom-
pressible hydrodynamics, no experiment contradicts this theory before the study of
the attenuation of acoustical waves. Indeed, this latter depends on both shear and
dilatational viscosity. At the beginning of the 20th Century, measured attenuation
coefficients in liquid differ from Stokes theory by factors 3 to 1000 (Stokes theory
is indeed only appropriate for ideal monoatomic gases). Different theories are ad-
vanced to explain this discrepancy (see Karim and Rosenhead [15]). Following solid
mechanics, Tisza introduces a second coefficient of viscosity [37].
Finally, to complete compressible Navier-Stokes equations, it is necessary
to introduce some law of energy conservation and state equations to establish
relations between pressure, density and temperature. But these laws inherited
from equilibrium thermodynamics are another part of history, which will not be
developed here.
Although Navier-Stokes equations have been formulated in current actual form
by Euler, Navier and Stokes in the 18th and 19th centuries, the existence and
smoothness of their solution has not been demonstrated yet in 3 dimensions. It is
one of the 7 Millenium problems proposed by Clay Mathematics Foundation (see
http://www.claymath.org/millennium/). “The challenge is to make substantial
progress towards a mathematical theory, which will unlock the secrets hidden in the
Navier-Stokes equations”. In the following manuscript, we will show that interfaces
can still add some complexity to these “mysterious” equations.
All the historical manuscripts evoked in this course can be down-
loaded from http://films-lab.univ-lille1.fr/michael/michael/History_of_
Navier_Stokes_equations.html.
6 Chapter 1. Introduction to interfacial flows
1.2 Navier-Stokes bulk equations and equilibrium con-
ditions at interfaces
In this section, we will recall the different formulations of Navier-Stokes equations,
the physical interpretation of the various terms and possible simplifications of the
equations. This manuscript adopts a continuum mechanics point of view. Thus we
will often refer to the notion of “fluid particles”. A fluid particle is a volume of fluid
containing a large number of molecules, but whose size remains small compared to
the characteristic size of flow variations.
1.2.1 Navier-Stokes equations
Historical non-conservative form of the equations
From the historical derivation described in the first section, the following equations
have been obtained:
Mass conservation∂ρ
∂t+ div(ρ−→v ) = 0 (1.1)
Momentum conservation
ρ
(
∂−→v∂t
+−→v .−→∇(−→v ))
= −−→∇p+ µ∆−→v + (µ + µ′)−→∇(div(−→v )) +−→
fv (1.2)
with ddt =
∂∂t +
−→v .−→∇() the material derivative.
In these equations, ρ, −→v , p designate respectively the density, velocity and
internal pressure of the flow, µ and µ′ the shear dynamic viscosity and the second
coefficient of viscosity. We can note that the so-called “bulk viscosity” ξ = µ+2µ′/3
is sometimes introduced instead of the second coefficient of viscosity. We will now
try to understand the physical meaning of the different terms which appear in the
momentum conservation equation.
Unsteady term ∂−→v∂t
This term corresponds to the temporal variation of the flow. It appears only if
the velocity field evolves in time and is equal to zero otherwise.
Convective term −→v .−→∇(−→v )
To understand the physical meaning of the term −→v .−→∇(−→v ), we can consider a 1D
field represented by a function f(x, t), which maintains its shape while translating
along x-axis at a given velocity c. The equation describing the evolution in space
and time of this function can be obtained by considering two times separated by an
increment dt (see Fig. 1.6). Since this function propagates without deformation, we
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 7
Am
plitu
de f(x,t) f(x+dx,t+dt)
x x+dx
dx = c dt
Space
Figure 1.6: Function f(x, t) translating at a velocity c.
have:
f(x+ dx, t+ dt) = f(x, t) as long as dx = c dt
Thus:
f(x+ dx, t+ dt)− f(x, t) =∂f
∂xdx+
∂f
∂tdt = 0
Since, dx = c dt, we obtain:∂f
∂t+ c
∂f
∂x= 0
Now, if we consider a 1D velocity field (f(x,t) = v(x,t)) and suppose that it is
convected by its own velocity (c = v(x,t)), we obtain the so-called inviscid Burgers’
equation:∂v
∂t+ v
∂v
∂x= 0
Finally, if we generalize this formula to a 3D velocity field, we obtain:
∂−→v∂t
+−→v .−→∇(−→v ) = 0
Thus the term −→v .−→∇(−→v ) corresponds to the convection of the velocity field by
itself. A fundamental issue is that this term is nonlinear since it is quadratic with
the velocity field −→v . The complexity of turbulent flow is intimately related to the
nonlinearity of this term.
Viscous diffusion term µ∆−→v
To understand the meaning of the viscous diffusion term, we will follow its
original derivation. Let’s consider two adjacent incompressible fluid particles flowing
along x-axis and separated by a distance dy along y-axis (see Figure below). If
the fluid particles have the same velocity (vx(y + dy) = vx(y)), then there is no
friction between them. However if they slide one over another, some friction (viscous
8 Chapter 1. Introduction to interfacial flows
x−axisy+dy
y
v (y+dy)
v (y)
x
x
z−axis
y−axis
resistance) will appear at their interface. The shear stress (force per surface unit)
τxy exerted on the xz plane between the two fluid particles will thus depend on their
relative velocity δv = vx(y + dy) − vx(y) = ∂v∂ydy. If we suppose a linear relation
between these two quantities, we obtain:
τxy = µ∂vx∂y
(1.3)
We can note that viscous friction may have different microscopic origins depending
on the considered fluid. In gases, it is mainly due to the exchange of momentum
through “collisions” between molecules moving at different velocities. Indeed the
molecules in a fluid particle do not have the same velocity because of molecular
agitation (whose temperature is a measure). Thus, there are collisions between
molecules and some momentum is transmitted from regions where the average ve-
locity (and thus momentum) of the molecules is higher to regions where it is lower
(see Fig 1.7). An increase in the temperature contributes to raise the agitation of the
fluid and thus momentum exchanges, explaining why the viscosity of gas increases
with temperature. Sutherland’s law (which is valid for a large range of gases and
temperatures) indeed predicts this increase in viscosity µ with the temperature T :
µ(T )
µ(To)=
(
T
To
)3/2 To + S
T + S,
with To = 273.15K a reference temperature and S = 110.4K the Sutherland tem-
perature.
In liquids however, molecules are closer to each other and the attractive forces
between them (Van der Waals forces and hydrogen bounds), play a more important
role (see Fig 1.7). This attraction is at the origin of the friction which appears
between two fluid particles. In this case, when the temperature is raised, the
agitation of the molecules is increased and thus the molecules move more freely.
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 9
Thus for most liquids, an increase in temperature results in a decrease in its
viscosity (see oil in a warm pan).
vx(y+dy)
vx(y)
vx(y+dy)
vx(y)
Figure 1.7: Left: exchange of momentum between molecules in gas through collisions
leading to viscous friction between two fluid particles. Right: friction in liquid due
to attractive Van der Waals forces and hydrogen bounds (thick blue lines on the
graph).
Now, from relation 1.3, we can compute the volume force fµ applied to an
infinitesimal volume δV and resulting from the viscous friction which appears at its
top and bottom surfaces (see Fig. 1.2.1):
−→fµ = [(τxz(y + dy)dxdy − τxz(y)dxdy)]/dxdydz = µ
∂2vx∂y2
(y)
τ xz(y+dy)
y−axis
z−axis
y+dy
y τ xz
If we generalize this formula to a 3D velocity field depending on all directions
(−→v = vx(x, y, z)−→x + vy(x, y, z)
−→y + vz(x, y, z)−→z ), we obtain:
−→fµ = ∆−→v
An interesting point, is that, if we consider only the unsteady and diffusion terms
of Navier-Stokes momentum equation, we get:
∂−→v∂t
= ν∆−→v
10 Chapter 1. Introduction to interfacial flows
with ν = µ/ρ, the kinematic viscosity. This equation is the 3D analogous of
Fick’s law of molecular diffusion ∂φ∂t = D∆φ or heat diffusion equation ∂T
∂t = κ∆T ,
where φ and T are respectively the concentration of a given component, T the
temperature, and finally D and κ the diffusion coefficient of species and heat. All
these phenomena share the same diffusion equation and all refer (at least for gases)
to the transmission of momentum between molecules.
Viscous dilatation term (µ+ µ′)−→∇(div(−→v ))
This term represents the viscous dissipation due to the dilatation of fluid par-
ticles. Indeed, it depends on the divergence of the velocity div(−→v ), which vanishes
when the flow is incompressible (see section 1.2.2). The velocity divergence div(−→v )gives a measure of the local variation of fluid particles volume.
An interesting point is that the viscous dilatation force is proportional to the
gradient of this local change of volume. It means that if all the fluid particles were
dilating the same way, there will be no resulting volume force.
External force field−→fv
This term represents any volume force deriving from an external force field. In
most cases, this term simply corresponds to the effect of gravity ρ−→g . However,
other force fields can play a role in the flow dynamics such as magnetic or electric
ones (see e.g the behavior of a ferrofluid submitted to a magnetic field or in
astrophysics, the study of flows at the surface of the sun, Fig. 1.2.1).
Figure 1.8: Left: Ferrofluid moved by a magnetic field. Right: NASA image of the
flow at the sun surface during two simultaneous solar eruptions.
General considerations
Equations 1.1 and 1.2 do not form a closed set of equations since there are 5
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 11
unknown variables (the density ρ, the 3 projections of the velocity field ~v and the
pressure p) and only 4 equations (one equation of continuity and 3 projections of
the equation of momentum conservation). One equation is missing; a state equation
giving the relation between the pressure and the density of the fluid is required.
However, according to Gibb’s phase rule (introduced by Josiah Williard Gibbs in
the 1870s), the number of degrees of freedom of a single component, in a single
state (phase) and in absence of chemical reaction is equal to 2. It means that the
pressure depends on both the density ρ and another thermodynamical parameter
such as the temperature or the entropy: p = p(ρ, s). Thus another equations (the
equation of energy conservation) is required to close the system. In this course, we
will not develop further these thermodynamic considerations.
Conservative form of the equations
Navier-Stokes equations can be re-written under the following form by combining
properly the mass and momentum conservation equation and introduce the stress
tensor⇒
σ= −p⇒
I +2µ⇒
D +µ′div(−→v )⇒
I , with⇒
D= 1/2(
⇒
∇ −→v +T⇒
∇ −→v)
the rate of
strain tensor:
Mass conservation∂ρ
∂t+ div(ρ−→v ) = 0 (1.4)
Momentum conservation∂ρ−→v∂t
+ div(ρ−→v ⊗−→v ) = div(⇒
σ ) +−→fv (1.5)
with ⊗ the tensor product.
Demonstration
We will demonstrate this equation by using Einstein notation (summation con-
vention):
∂ρ−→v∂t
+ div(ρ−→v ⊗−→v ) = ∂ρvi∂t
+∂ρvivj∂xj
= vi
[
∂ρ
∂t+∂ρvj∂xj
]
+ ρ
[
∂vi∂t
+ vj∂vi∂xj
]
= −→v[
∂ρ
∂t+ div(ρ−→v )
]
+ ρ
[
∂−→v∂t
+−→v .−→∇(−→v )]
The first term in square bracket is null because of mass conservation and we therefore
obtain:
∂ρ−→v∂t
+ div(ρ−→v ⊗−→v ) = ρ
[
∂−→v∂t
+−→v .−→∇(−→v )]
12 Chapter 1. Introduction to interfacial flows
For the second part of the equations, we have:
div(⇒
σ ) =∂
∂xj
[
−pδij + 2µ∂
∂xj
[
1
2
(
∂vi∂xj
+∂vj∂xi
)]
+∂
∂xj
[
µ′∂vk∂xk
δij
]]
= − ∂p
∂xi+ µ
∂vi∂x2j
+ (µ′ + µ)∂
∂xi
∂vk∂xk
= −−→∇p+ µ∆−→v + (µ′ + µ)−→∇(div(−→v ))
where δij is Kronecker delta equal to 1 when i = j and 0 otherwise.
General considerations
While this form of the momentum equation, called conservative form, has
been obtained (here and historically) after the non-conservative form, it derives
directly from momentum conservation principle. The non-conservative form of the
momentum equation requires the mass conservation to be fulfilled. Of course, this
condition is most of the time verified. But for complex two-phase flow with phase
change or fluids with nuclear transformation, this momentum equation can no
longer be used. In addition, it plays a fundamental role in numerical simulations:
numerical schemes are generally obtained from this formulation of the equations.
Integral form of the equations
By integrating the conservative form of the equations over a volume Ω following the
movement of the fluid, we obtain:
Mass conservationd
dt
∫∫∫
ΩρdV = 0 (1.6)
Momentum conservationd
dt
∫∫∫
Ωρ−→v =
∫∫
∂Ω
⇒
σ .−→n dS +
∫∫∫
Ω
−→fvdV (1.7)
These equations describe balance of mass and momentum over an arbitrary volume
Ω.
Demonstration
To obtain this result, we first integrate the conservative form of local Navier-
Stokes equations and then use Green-Ostrogradski formula:∫∫∫
Ωdiv(
−→ψ )dV =
∫∫
∂Ω
−→ψ .−→n dS,
which relates the integral over a volume Ω of the divergence of any continuous
vectorial field−→ψ to an integral over the surrounding surface ∂Ω, −→n being the normal
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 13
Ω
−→n
∂Ω
vector. In that way, we obtain the following formula:
Mass conservation∫∫∫
Ω
∂ρ
∂t+ div(ρ−→v )dV = 0 (1.8)
Momentum conservation∫∫∫
Ω
∂ρ−→v∂t
+ div(ρ−→v ⊗−→v ) =∫∫
∂Ω
⇒
σ .−→n dS +
∫∫∫
Ω
−→fvdV (1.9)
Then, we use the expression of the material derivative ddt of the integral of a field
ψ over a volume Ω following the motion (that is to say that the volume Ω moves
according to the velocity field −→v ):
d
dt
∫∫∫
ΩψdV =
∫∫∫
Ω
∂ψ
∂t+ div(ψ−→v )dV
General considerations
In classical mechanics of continuum, these equations are in fact the starting
point since they derive directly from balance laws (mass and momentum) and do
not require the knowledge of constitutive equations, which are specific to a medium.
1.2.2 Simplified expressions
Incompressible flow: Navier equations
Derivation of the incompressibility condition
There are different ways to demonstrate the mathematical expression of this
condition. A first way is to consider an infinitesimal volume of fluid δV and state that
during a time interval dt, the same volume of fluid enters and exits this infinitesimal
volume. Through an infinitesimal surface dS, the amount of fluid which crosses the
surface is the quantity −→v .−→n dSdt. Thus, if we consider all the infinitesimal surfaces
surrounding δV , we get:
14 Chapter 1. Introduction to interfacial flows
n=z
n=y
n=−y
n=−z
v(x+dx,y,z)v(x,y,z)
v(x,y,z)
v(x,y+dy,z)
n=xn=−xv(x,y,z)
v(x,y,z+dz)
[vx(x+ dx, y, z) − vx(x, y, z)] dydzdt+ [vy(x, y + dy, z) − vy(x, y, z)] dxdzdt
+ [vz(x, y, z + dz)− vz(x, y, z)] dxdydt =
[
∂vy∂x
+∂vy∂y
+∂vz∂z
]
dxdydzdt =
div(−→v )δV dt = 0
that is to say:
div(−→v ) = 0 (1.10)
This equation can also be obtained by noticing that, since the flow is incom-
pressible, any domain of fluid Ω keeps its volume constant when it is transported
by the flow, that is to say if we use the material derivative of integrals:
d
dt
∫∫∫
Ω1 dV =
∫∫
∂Ω
∂1
∂t+ div(−→v )dV =
∫∫
∂Ωdiv(−→v )dV = 0
Since it is valid for any domain Ω, we find again div(−→v ) = 0.
Incompressible Navier-Stokes equations
As a consequence, the conservation equations adopt the following form:
Mass conservation
div(−→v ) = 0 (1.11)
Momentum conservation
ρ
(
∂−→v∂t
+−→v .−→∇(−→v ))
= −−→∇p+ µ∆−→v +−→fv (1.12)
In this case, since the density remains constant on fluid particle trajectories
(and therefore is no more an unknown variable), the system is closed. It means that
for an incompressible flow, the equations of energy conservation and the equations
of dynamics (mass+momentum) are uncoupled.
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 15
When can a flow be considered as incompressible ?
While the incompressibility condition is rather simple to obtain mathematically,
it is harder to define which flow can be considered as incompressible. Several situa-
tions can lead to an influence of the compressibility of the flow on its dynamics.
• Supersonic flow:
When the characteristic speed of the flow U is larger than the sound speed
co, that is to say when the Mach number M = U/co ≥ 1, the fluid can
experience some brutal variations of density, temperature and velocity. In
this case, the compressibility of the fluid cannot be neglected. The first
question that one can ask is: why is the sound speed the reference velocity ?
The sound speed can be viewed as the speed of propagation of information in
a fluid. For example, when you turn off the tap at home, the movement of
the fluid does not stop instantaneously in the whole pipe. The information
that the pipe is closed at its extremity propagates in the pipe at sound speed.
Thus imagine the following situation: a plane is moving in the air at a speed
smaller than the sound speed. In this case, the information that the plane is
arriving travels more rapidly than the plane. Thus fluid particles at rest are
“aware” that the plane is arriving and the flow can “adapt” to the arrival of
the plane. But if the plane is traveling at a larger velocity than the sound
speed, then fluid particles learn about its arrival at the last moment and the
flow will adapt brutally to the boundary conditions imposed by the plane.
These brutal variations of temperature, pressure and velocity are called a
shock and have been observed for the first time around a projectile by P.
Salcher and S.Riegler in 1887 [31], following E. Mach suggestion (see Fig. 1.9).
Figure 1.9: Left: first observation of shock waves around a projectile by P. Salcher
and S. Riegler following Mach’s suggestion. The shock is visible since the large
variations of density and temperature modify the light refraction index. Right:
shock wave around an airplane made visible by the condensation of water induced
by the variations of temperature and density.
16 Chapter 1. Introduction to interfacial flows
• Natural convection:
Natural convection is a flow produced by variations of temperature in a fluid,
leading to variations of density and hence buoyancy forces. Indeed, the hottest
regions of the fluid are less dense, and thus buoyancy forces induce a rise of
these regions. Natural convection is commonly observed in a room heated in
winter. In this case, heating leads to recirculation of air inside the room. An
academic example is the appearance of Benard cells in a fluid with a sufficient
viscosity submitted to up and down temperature gradients (see Fig. 1.10).
When natural convection is involved, the change of density due to tempera-
Figure 1.10: Left: natural convection flow due to heaters in an house. Right: Benard
cells appearing in gold paint due to the cooling of the top surface by evaporation of
a solvent (acetone).
ture gradients is at the origin of the fluid motion and thus the compressibility
cannot be neglected. In some cases however (Boussinesq approximation for
weak variations of density), the incompressibility condition div(−→v ) = 0 re-
mains valid to describe mathematically natural convection at first order. The
compressibility of the fluid appears only through a buoyancy force in the mo-
mentum equation. It does not mean that the compressibility plays no role in
the equations but only that at first order, the compressible terms appearing
in the mass balance can be neglected.
• Acoustic waves
A last example in which the fluid compressibility plays a fundamental role
is the propagation of acoustic waves. In fluid, acoustic waves propagate as
disturbance of pressure and thus density. These pressure perturbations can be
extremely weak (the threshold of human hearing is about 20 µPa). To give
a point of comparison, this corresponds to the variation of pressure (due to
gravity field) that you experience when you climb a ladder of ... “20 microns”.
However, these variations can be extremely rapid since the audible frequency
range (for average people) lies between 20 Hz to 20 kHz.
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 17
Incompressible creeping flow: Stokes equations
Stokes equations are valid for a steady incompressible creeping flow, that is to say a
flow with a low Reynolds number. The Reynolds number compares the convective
term to the diffusive one (see definition in section 1.4.1). When the Reynolds number
is small, convection can be neglected compared to viscous diffusion and the equations
become linear:
Mass conservation
div(−→v ) = 0 (1.13)
Momentum conservation−→∇p = µ∆−→v (1.14)
The linearity of these equations implies many physical and mathematical interesting
properties:
• Instantaneity: There is no time derivative in the Stokes equations. Thus
the flow depends only on instantaneous boundary conditions and there is no
memory of previous velocity fields.
• Superposition principle: If the fields (−→v 1, p1) and (−→v 2, p2) are solutions of
Stokes equations and satisfy respectively the boundary conditions BC1 and
BC2, then the fields (λ1−→v 1+λ2
−→v 2, λ1p1+λ2p
2) (with λ1, λ2 two constants)
are solutions of Stokes equations with the boundary conditions λ1BC1 +
λ2BC2.
• Reversibility: If the boundary conditions are reversed, then the inverse flow
−−→v ,−−→∇P is solution of the problem.
Figure 1.11: Motion of microorganisms using cilia (left) and flagella (right)
This last property has many important implications. For example, it is very com-
plicated to mix some fluids using creeping flow since “complicated patterns” do not
appear naturally like in turbulent flows. Second, some specific strategies must be
18 Chapter 1. Introduction to interfacial flows
developed by microorganism for their locomotion, since the Reynolds number is gen-
erally small at these scales. Because of the reversibility property, symmetric motions
as birds wings flap does not produce any propulsion. Thus the symmetry must be
broken see e.g. the collective motion of cilia or the rotative motion of flagella (Fig.
1.11).
To conclude, we can underline that analytical solutions of Stokes equations exist
since the pressure field verifies Laplace equation :
∆p = 0,
whose analytical solutions are called harmonic functions. Thus, creeping flow
with complex boundary conditions can be computed very rapidly by using fast
summation of harmonic functions (see Fast Multipole Method [11]).
Demonstration:
Laplace equation is obtained by taking the divergence of equation (1.14) and
using the double-curl formula:−→∇ × (
−→∇ × (−→v )) = −→∇div(−→v )−∆−→v :
∆p = µdiv(∆−→v ) = µ div(−→∇ × (
−→∇ × (−→v ))− div(−→∇(div−→v )))
Since div(−→∇ × (ψ)) = 0 for any field ψ and div(−→v ) = 0, we get:
∆p = 0.
Inviscid flow: Euler equations
Euler’s equations describe the motion of a fluid with no viscosity. Of course, all
fluids have a viscosity, except superfluids (like superfluid helium-4). However, these
equations remain valid when the magnitude of the convection term largely overcomes
the magnitude of the viscous diffusion term, that is to say when Re ≫ 1. The
equations become:
Mass conservation∂ρ
∂t+ div(ρ−→v ) = 0 (1.15)
Momentum conservation
ρ
(
∂−→v∂t
+−→v .−→∇(−→v ))
= −−→∇p+−→fv (1.16)
Since Euler equations are valid for compressible flows, the density ρ is not con-
stant, and as for the Navier-Stokes equations, state and energy conservation equa-
tions are required to close the system.
Euler’s equations are widely used to describe the dynamics of flow in supersonic
(1 < M = Uco< 5) and hypersonic regimes (5 < M = U
co< 10), see e.g. Fig. 1.12.
Acoustic waves equations
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 19
Figure 1.12: Hypersonic flow around a space shuttle simulated with the commercial
code ANSYS. One can see the shock waves (in green) appearing in front of the
different parts of the shuttle.
From Euler’s equations, one can easily obtain the equations describing the prop-
agation of acoustic waves by considering an adiabatic evolution of a small pertur-
bation field (p1,−→v 1) around a steady state (po,
−→v o =−→0 ):
p = po + εp1−→v = 0 + ε−→v 1
ρ = ρo + ερ1 (1.17)
with ε≪ 1. If we replace these expressions in equations (1.15) and (1.16), we obtain
at first order:
Mass conservation∂ρ1∂t
+ ρodiv(−→v 1) = 0 (1.18)
Momentum conservation
ρo∂−→v∂t
= −−→∇p1 (1.19)
Since the evolution is adiabatic, we have c2o = ∂p∂ρ = p1
ρ1, with co the sound speed.
Now if we subtract ∂∂t (1.18) - div(1.19) and replace p1 by c2oρ1, we obtain the wave
equation:∂2ρ1∂t2
− c2o∆ρ1 = 0.
Inviscid incompressible flow and Bernoulli’s principle
Bernoulli’s principle expresses a conservation of kinetic energy in the absence
of viscous friction: the variation of kinetic energy is balanced by a variation of
20 Chapter 1. Introduction to interfacial flows
potential energy and the work of pressure forces.
Original Bernoulli’s principle
Bernoulli’s original theorem is applicable if:
• the flow is incompressible div(−→v ) = 0 and hence ∂ρ∂t = 0
• the external force field is conservative: ∃ U,−→fv = −−→∇U , where U is the scalar
potential associated with fv
• the flow is inviscid (µ = 0)
• the flow is steady ∂−→v∂t =
−→0
Then, along a streamline:
p
ρ+v2
2+ U = constant
Demonstration
To demonstrate this theorem, the (steady) momentum conservation of Euler
equation (1.16) can be rewritten under the following form from previous hypotheses
and the relation −→v .∇(−→v ) = −→∇(
v2
2
)
−−→v ×−→∇ × (−→v ):
−→∇(
p
ρ+v2
2+ U
)
= −→v ×−→∇ × (−→v ) (1.20)
Since streamlines are by definition parallel at any point to the velocity field, any
infinitesimal displacement−→dl along a streamline is parallel to −→v . The vector −→v ×−→∇ × (−→v ) is therefore perpendicular to −→v and the dot product of equation (1.20)
with the displacement−→dl gives:
−→∇(
p
ρ+v2
2+ U
)
.−→dl = 0 (1.21)
Thus the function H =(
pρ + v2
2 + U)
is constant.
Unsteady Bernoulli’s principle
The last hypothesis (steady flow) can be released if the flow is irrotational (−→∇×
(−→v ) = 0). In this case, the velocity field is such that ∃ φ,−→v =−→∇φ. Bernouilli’s
principle becomes (at any point of the flow):
∂φ
∂t+p
ρ+v2
2+ U = K(t) (1.22)
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 21
where K is a time-dependent function.
Demonstration
The demonstration is similar to the previous one but this time unsteady Euler
equation can be written under the form:
−→∇(
∂φ
∂t+p
ρ+v2
2+ U
)
= 0 (1.23)
Thus the function K =(
∂φ∂t +
pρ + v2
2 + U)
only experiences times variations.
General considerations
Figure 1.13: Boundary layer appearing on the walls of a projectile flowing at super-
sonic speed. The boundary layer perturbs the projectile wake.
As already mentioned in the historical introduction of this manuscript, these
inviscid incompressible equations suffer d’Alembert’s paradox: the drag force ap-
plied on a body moving with constant velocity relative to the fluid is zero. Indeed,
since the viscosity of the fluid is neglected, the fluid particles slip on the walls and
there is no transmission of tangential momentum. This paradox has been solved
by the discovery and description of boundary layers by Ludwig Prandtl in 1904.
The boundary layer is a thin layer around moving bodies where viscosity cannot be
neglected anymore (see Fig. 1.13). Indeed, the Reynolds Number Re = ρULµ , which
compares convective effects to viscous diffusion is generally computed by taking as
reference length scale L, the average size of the considered object (see definition
in section 1.4.1). If the body moves fast enough this Reynolds number is large.
However if we zoom on a thin layer around this body, the characteristic length
22 Chapter 1. Introduction to interfacial flows
scale becomes smaller and smaller and, at a given point, the Reynolds number is
no more ≫ 1. In the corresponding layer, viscous diffusion cannot be neglected
anymore as compared to convection. This thin layer introduces some friction drag
Figure 1.14: Flow separation behind a wing tilted with a large angle compared to
the direction of the flow
and for bluff bodies results in flow separation (see Fig. 1.14) and low pressure wake
behind the considered object.
Another surprise when we look at inviscid incompressible flow is how much the
flow is regular (see Fig. 1.15) despite the magnitude of the the nonlinear convective
term. Turbulence cannot be obtained in inviscid flows. Indeed, while turbulence
appears at high Reynolds number it requires a hint of viscosity to develop.
Figure 1.15: Inviscid incompressible flow around a cylinder
1.2. Navier-Stokes bulk equations and equilibrium conditions atinterfaces 23
1.2.3 Mass and momentum balance at interfaces
Bulk equations have been established and simplified for different physical cases.
Some boundary conditions are now required for the problem to be well-posed. It is
at this stage that the complexity induced by rigid or deformable interfaces appears.
An interface is a discontinuity of the flow properties (density, velocity, pressure,
viscosity). Such discontinuity can appear between two different phases (a liquid and
a gas, two immiscible liquids, a liquid and a solid, ...) but also in the same phase
for compressible flow. In this last case, they are called shocks. In this section, we
will establish the balance equations at these discontinuities. While these equations
are valid for both interfaces between two phases and shocks, this second case will
not be developed further in this manuscript.
Continuity condition
dS
−→nρ2, ~v2, p2, µ2, µ
′2
ρ1, ~v1, p1, µ1, µ′1
The continuity condition expresses the mass conservation across an interface.
Let’s consider an infinitesimal interface dS between two phases 1 and 2, −→n being
the normal to the surface oriented from fluid 1 to fluid 2. To establish the balance
equations, the mass of fluid crossing the surface during an infinitesimal time dt must
be estimated. As a starting point, we will suppose that the interface is motionless.
The fluid particles move in phase 1 with a velocity −→v 1. If they move tangentially
to the surface, they will never cross the interface. Thus only the normal velocity−→v 1.
−→n contribute to the balance. Now, if the fluid particles are too far away from
the surface, they will never reach it during the infinitesimal time dt. So we must
determine how far the fluid particles can lie from the interface to cross it during
dt. Since they are moving towards the interface with a velocity −→v 1.−→n , this distance
is naturally, −→v 1.−→n dt. Finally, to get the volume of fluid particles crossing the
interface, we have to multiply this distance by the infinitesimal surface dS, and to
get their mass dm1, to multiply it by ρ1: dm1 = ρ1−→v 1.
−→n dtdS. This mass which
comes from phase 1 to cross the interface is equal to the mass of fluid which comes
from the interface and moves into phase 2, that is to say dm2 = ρ2−→v 2.
−→n dtdS. Thus
we get:
ρ1−→v 1.
−→n dtdS = ρ2−→v 2.
−→n dtdS.
If the interface is moving with a velocity −→u , the balance is the same but in the
24 Chapter 1. Introduction to interfacial flows
referential frame of the interface, that is to say (if we simplify dtdS on both sides):
ρ1(−→v 1 −−→u ).−→n = ρ2(
−→v 2 −−→u ).−→n (1.24)
For interfacial flow, this condition expresses phase change, that is to say for a
fluid/gas interface, evaporation or condensation. In the absence of such phase
change, (and in the absence of shock), there is no mass crossing the interface and
the continuity condition becomes:
−→v 1.−→n = −→v 2.
−→n = −→u .−→n (1.25)
Momentum condition
Momentum transfers through an interface can occur in two different ways. First,
the fluid which crosses the interface (phase change) from phase 1 to 2 carries with
it some momentum ρ1−→v 1. Since, the volume of fluid which crosses an infinitesimal
surface dS during an infinitesimal time dt is (−→v 1−−→u ).−→n dtdS, then the momentum
transferred by phase change from phase 1 to 2 is: ρ1−→v 1(
−→v 1 − −→u ).−→n dSdt. Some
momentum is also exchanged because of the stresses⇒
σ1 and⇒
σ2 applied on each side
of the interface. Since the surface normal is directed from fluid 1 to fluid 2, the
stress applied by phase 1 on phase 2 on dS during dt is: −σ1.−→n dSdt. Thus the
momentum balance can be written under the form:
(ρ1−→v 1(
−→v 1 −−→u )− ⇒
σ 1).−→n = (ρ2
−→v 2(−→v 2 −−→u )− ⇒
σ 2).−→n (1.26)
It is important to underline that we suppose here that equilibrium is reached at
the interface and that the interface has no inner mass, momentum or density. This
hypothesis will be discussed further in section 1.3.
No phase change
In absence of phase change, −→v 1.−→n = −→v 2.
−→n = −→u .−→n , and thus the momentum
conservation at the interface reduces to the stress equilibrium condition:
⇒
σ1 .−→n =
⇒
σ2 .−→n
Rigid body
In the case of an interface with a rigid body the conditions at the interface reduce
to a continuity of the velocity field:
−→v 1 =−→v 2
This condition of continuity must be used with care and is inappropriate in specific
cases, e.g. the study of flow in superhydrophobic microchannels, or the study of gas
flow around objects of the same size as the mean free path of molecules.
1.3. Surface tension, Laplace pressure and Marangoni effect 25
1.3 Surface tension, Laplace pressure and Marangoni ef-
fect
1.3.1 Where does surface tension come from ?
Interactions between molecules
Surface tension appears at the interface between two immiscible fluids. To ex-
plain it, one must first understand the nature of molecular interactions in fluids.
A molecule is made of dense positive protons and uncharged neutrons surrounded
by orbits of negatively charged electrons. Positive and negative charges compen-
sate one another resulting in a global neutral charge for the molecule. However
the heterogenous distribution of these charges induces some interactions between
neighboring molecules. Following Pauli’s exclusion principle, two electrons can-
not occupy the same orbit (or more rigorously, the same quantum state). Thus,
when two molecules are brought close to one another, a short range repulsive force
appears since the negatively charged electronic clouds cannot overlap one another.
Moreover, molecules behave as permanent and induced dipoles and some attractive
intermolecular forces can also appear: the Van der Waals forces. They can be
classified into three categories:
• Keesom forces between two permanent dipoles
• Debye forces between a permanent dipole and the corresponding induced
dipole
• London dispersion force between instantaneously induced dipoles.
1 1.5 2 2.5−2
−1
0
1
2
3
4
5
Distance r/σ
Pot
entia
l V(r
/σ)
ε
Figure 1.16: Lennard-Jones potential
26 Chapter 1. Introduction to interfacial flows
The interaction between pair of neutral atoms or molecules is commonly approxi-
mated by the so-called Lennard-Jones potential which takes into account both the
short range Pauli repulsive term and the long range Van der Waals attractive term
(see Fig. 1.16):
VLJ = 4ε
[
(σ
r
)12−(σ
r
)6]
with ε the depth of the potential well, σ the distance at which the inter-particle
interaction is zero and r the distance between the particles. It is important to note
that while a comprehensive view of these interactions can be given through simple
electrostatic analogies, the correct explanations of the existence of Van der Waals
forces lies in quantum mechanics and is well beyond the objective of this course.
Finally, in some common liquids (such as water), so called attractive hydrogen
bonds can appear between hydrogen atoms and electronegative atoms such as
nitrogen, oxygen or fluorine. These bonds are stronger than Van der Waals
interactions but weaker than covalent or ionic bond.
Interface between a condensed and a gas phase:
GAS
Figure 1.17: Interactions of molecules in the bulk (left) and at a liquid-gas interface
(right). The green attractive interactions disappear when an interface is created.
Fluids molecules experience constant thermal agitation, whose temperature is a
measure. When the attractive forces (described in the previous section) dominate
thermal agitation, the molecules condensate and form a liquid phase. Thus, there
are some strong interactions between neighboring molecules in a liquid and an “equi-
librium” is reached between them. In gases however, the molecules are generally too
far away to feel the attractive and repulsive porentials and they only interact when
1.3. Surface tension, Laplace pressure and Marangoni effect 27
they collide. Thus, at the interface between a liquid and a gas, liquid molecules
lose half of their attractive forces and are not in equilibrium. The existence of this
interface “costs” some energy. To reach a minimum energy state, a volume of liquid
surrounded by a gas will deform to reduce its interface. That is why a drop (in
levitation) adopts a spherical shape which minimizes its surface over volume ratio.
The surface tension γ is simply the coefficient which relates the work δW which is
required to stretch a surface to the corresponding increase of this surface dA:
δW = γdA
Interface between two immiscible liquids
For the same reason, a surface tension exists between two immiscible liquids 1
and 2 since the attractive interactions lost by phase 1 because of the presence of the
interface are not compensated by the interactions with phase 2.
1.3.2 Laplace pressure
Origin of Laplace pressure
dR
pp
1
2
R
Figure 1.18: Pressure jump through a spherical interface due to surface tension.
Since the surface tension tends to reduce the interface of a fluid 1 surrounded by
a fluid 2, an excess of pressure appears in fluid 1: p1 > p2. It can be computed by
considering a spherical drop of fluid 1 surrounded by an immiscible fluid 2. Then,
the virtual work which is required to increase its radius by dR is:
δW = −p1dV1 − p2dV2 + γ12dA
with dA = d(4πR2)dR dR = 8πRdR the increase of surface, dV1 = −dV2 = d(4/3πR3)
dR =
4πR2dR the increase of volume, and γ12 is the surface tension between fluid 1 and
28 Chapter 1. Introduction to interfacial flows
2. With the equilibrium condition δW = 0, we obtain the following condition:
∆p = p1 − p2 =2γ12R
= γ12C
with C the curvature of the surface.
Expression for any surface
Intersection lines
R
R’
P
mardi 24 mai 2011
Figure 1.19: Calculation of the curvature of a surface.
The sphere is a specific case since its curvature is constant and its two
principal radii of curvature are equal to the radius of the sphere. In a general
case, the mean curvature of a surface at a given point P can be computed by
plotting the normal vector −→n at this point and then considering the intersection
lines between the considered surface and some planes containing −→n . These
intersection lines all have a radius of curvature in P. The mean curvature is
the sum of the minimum and maximum radius of curvatures R and R′. The
corresponding planes are called planes of principal curvatures. The radius of curva-
ture is positive when the circle (in green) is inside the object and negative otherwise.
1.3. Surface tension, Laplace pressure and Marangoni effect 29
Thus in the general case, Laplace law can be written:
∆P = γC = γ
(
1
R+
1
R′
)
with C the mean curvature.
1.3.3 Marangoni effect
Marangoni effect is the mass transfer along an interface due to surface tension gra-
dients. Such gradients of surface tension can appear because of the dependence
of surface tension on temperature, or on the concentration of chemicals in a fluid
mixture. An originally plane interface will be deformed by these surface tension
gradients.
Figure 1.20: Wine tears at the surface of a glass.
A well known consequence of Marangoni effect is the wine tears. The five basic
steps for wine testing are “see, swirl, sniff, sip and savor”, the famous “five S”. When
the wine is swirled, a thin film of liquid is deposited on the walls. This film rises
progressively and thickens at the top. At a given point, some droplets form and
flow slowly down the inside of the glass. We will not develop the instability process
30 Chapter 1. Introduction to interfacial flows
of drop formation and will focus on the issue of the film thickening at the top
despite gravity effects. Wine is mainly made of alcohol and water. Alcohol has a
smaller surface tension than water and is more volatile. Since the liquid film is thin,
and its surface/volume ratio important, the alcohol evaporates rapidly. Thus the
concentration of alcohol decreases and the surface tension increases. This excess of
surface tension pulls the film to the top and causes more liquid to be drawn up from
the bulk. When the film becomes too thick it becomes unstable and some liquid
drop form and flow down due to gravity.
1.3.4 Momentum equilibrium at interfaces with surface tension
If we take into account Laplace pressure and Marangoni effect in the momentum
condition at the interface (eq. 1.26), we get the following expression:
(ρ1−→v 1(
−→v 1 −−→u )− ⇒
σ 1).−→n = (ρ2
−→v 2(−→v 2 −−→u )− ⇒
σ 2).−→n (1.27)
+ γ12div(−→n )−→n +
(
⇒
I −−→n ⊗−→n)
.−→∇γ12
with −→n the normal vector oriented from fluid 1 to 2. The two additional terms
on the second line correspond respectively to Laplace pressure and Marangoni ef-
fects. Indeed, the curvature C can be expressed as a function of the normal vector
C = div(−→n ). Since, the surface tension introduces a pressure jump, the associated
momentum will be oriented towards −→n . On the opposite, Marangoni effect induces
a tangential force in the interface plane and it is proportional to surface tension
gradient−→∇γ12.
1.4. Useful dimensionless numbers and illustrations 31
1.4 Useful dimensionless numbers and illustrations
1.4.1 The Reynolds number
Reynolds number whose terminology has been introduced by Sommerfeld in 1908
[30] is named after Osborne Reynolds (1842-1912), an Irish scientist working at
the University of Manchester. Reynolds studied the transition from laminar to
turbulent flow in pipes (see setup on Fig. 1.21) and introduced similarity parameters
by dimensional analysis. It is interesting to note that the subcritical transition in
pipes is still a controversial issue [8], while considerable progress has been made in
the understanding of underlying mechanisms. The Reynolds number compares the
Figure 1.21: Sketch of Reynolds experiments studying the transition from laminar
to turbulent flow in pipes. The original setup is still available at the University of
Manchester and it has been reproduced recently in the same condition. Since the
transition is subcritical and depends on the level of perturbation, a much smaller
critical Reynolds number of transition from laminar to turbulent flow has been
measured, ... a consequence of the large increase in ambient noise.
magnitude of convective terms to the one of diffusive terms. If U is the characteristic
velocity of the flow, L the characteristic size of its variations, ρ the density and µ
32 Chapter 1. Introduction to interfacial flows
the dynamic viscosity, we get:
Re =Convection
Viscous diffusion=
[ρ−→v .∇(−→v )][µ∆−→v ] =
ρUL
µ(1.28)
Re = 10 Re = 26
Re = 300 Re = 2000
Re = 10000
Figure 1.22: Flow over a cylinder at different Reynolds number.
A classical example illustrating the role played by the Reynolds number is the
flow around a cylinder. According to its value, the flow is more or less complex
(see Fig. 1.22). At low Reynolds numbers (Re<10), the flow is laminar. When
the Reynolds number increases (Re ∼ 26), some vortex appear in the wake of the
cylinder. At Re ∼ 300, they are shed from each side of the body, forming rows of
1.4. Useful dimensionless numbers and illustrations 33
vortices in its wake, the so-called Karman vortex street. At Re ∼ 2000, there is
flow separation and the wake behind the body becomes turbulent at Re ∼ 10000.
Flow separation is a primary issue in Aerodynamics, since it results in increased
drag because of the difference of pressure between the front and the rear surfaces of
the object.
1.4.2 The Capillary number
The capillary number compares viscous effects to surface tension:
Ca =Viscous diffusion
Surface tension=µU
γ
with U the characteristic speed of the flow, µ the viscosity and γ the surface
tension. This number is widely used in two-phase microfluidics, where the Reynolds
number is small and therefore viscous effects are dominant compared to convective
effects.
The capillary number gives a measure of the ability of a viscous flow to deform
an interface. For example, let’s consider a droplet of a liquid 1 moving in an
immiscible liquid 2. As long as the capillary number is small, it keeps its spherical
shape due to surface energy minimization. However, if the capillary number
becomes large enough, the flow deforms the droplet.
1.4.3 The Weber number
The Weber number compares inertial effects to surface tension:
We =Inertial effects
Surface tension=µU2L
γ
with µ the dynamic viscosity, U the characteristic speed of the flow, L the charac-
teristic size of the interface and γ the surface tension.
This number is widely used for the atomization of jets or bubble breakup in the
inertial regime. It gives a measure of the ability of an inertial flow to deform an
interface. Fig. 1.24 shows the deformation of a pulsed 2D liquid film at different
Weber numbers.
34 Chapter 1. Introduction to interfacial flows
We = 1
We = 10
We = 100
We = 1000
We = 10000
Figure 1.23: Pulsed 2D liquid film at different Weber numbers. The black line
corresponds to the interface between the liquid (beneath) and the air (above). The
colors correspond to the norm of the velocity field. At weak Weber number, the
interface remains unaffected by the pulsed flow while at large Weber number, the
interface is deformed leading to its atomization into small droplet.
1.4.4 The Bond Number
The Bond number (also called Eötvös number) compares the effect of buoyancy
force to surface tension:
Bo =Buoyancy force
Surface tension force=
∆ρgL2
γ
1.4. Useful dimensionless numbers and illustrations 35
with ∆ρ the difference of density between the considered immisible fluids, g the
gravitational acceleration, L the characteristic size of the considered interface, and
γ the surface tension.
samedi 28 mai 2011
Figure 1.24: Sessile droplets of different volumes lying on a solid substrate. When
the drop is larger than the capillary length, it is flattened by gravity.
For example let’s consider a drop lying on the surface. If the Bond number
is small, the drop keeps its hemi-spherical shape due to surface tension. If the
Bond number is larger, the drop is flattened by gravity effects (see Fig. 1.24). The
bond number can also be seen as the ratio between the square of the characteristic
length of the considered surface and the square of the so-called capillary length
Lc = γ/∆ρg:
Bo =L2
L2c
The capillary length is about 2 mm for water surrounded by air, which means that
gravity effects play no role on the dynamics of liquid drops smaller than this length.
Chapter 2
Interfaces and vibrations
Contents
2.1 Bubbles and the Rayleigh-Plesset equation . . . . . . . . . . 38
2.1.1 Why are bubbles outstanding resonators ? . . . . . . . . . . . 39
2.1.2 Static equilibrium of bubbles . . . . . . . . . . . . . . . . . . 42
2.1.3 Modeling of the liquid phase: the Rayleigh-Plesset equation . 45
2.1.4 Modeling of the gas phase . . . . . . . . . . . . . . . . . . . . 53
2.1.5 From bubbles to metamaterials . . . . . . . . . . . . . . . . . 60
2.2 Inertio-capillary “Rayleigh-Lamb” modes of vibration . . . 61
2.2.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 61
2.2.2 Complete solution of the problem . . . . . . . . . . . . . . . . 61
2.2.3 Sessile droplets and possible use of these surface vibrations. . 65
The vibration of interfaces is a wide subject, ranging from Faraday waves
patterns appearing at the surface of a vibrated liquid layer (see Fig. 2.1), to the
multiple scattering of acoustic waves in metamaterials. In this chapter, we will
focus on monopolar vibration of bubbles and the different modes of vibration of an
incompressible liquid droplet.
Figure 2.1: Faraday waves appearing at the surface of a vibrated liquid layer. Source:
http://www.ia.csic.es/Temas.aspx?Lang=EN&Id=8
38 Chapter 2. Interfaces and vibrations
NB: the surface tension will be called σ in this chapter to avoid confusion with the
heat capacity ratio.
2.1 Bubbles and the Rayleigh-Plesset equation
Figure 2.2: Champagne bubbles bursting and associated mist.
The dynamics of bubbles is of primary importance in physics, chemistry, biology
and engineering. Indeed, bubbles create a mist that wafts champagne’s aroma to the
drinker (Fig. 2.2), they are used by pistol shrimps to kill their prey (Fig. 2.3), by
doctors as contrast agents for ultrasonic imaging, by dentists for scaling, by chemists
for sonochemistry. They are also feared by engineers when they design a boat
because of the damages to propellers induced by cavitation (Fig. 2.4). More recently,
bubbles are under consideration to create invisible objects ... toward acoustic waves.
clawbubble
Figure 2.3: Cavitation bubble created by a snapping shrimp [39, 19].
2.1. Bubbles and the Rayleigh-Plesset equation 39
2.1.1 Why are bubbles outstanding resonators ?
Bubbles have this outstanding property that small (but optimized) variations of
pressure or temperature of the hosting liquid can lead to an “explosive” dynamics,
with speeds of its interface larger than the sound speed or temperatures in the
bubble three times larger than the temperature at the surface of the sun. Bubbles
can even emit some light, the so-called sonoluminescence (Fig. 2.5). Thus, even at
small concentrations, they can drastically modify the behavior of the hosting liquid.
For example, the speed of sound in bubbly media decreases at values much smaller
than the one in the corresponding liquid or gas. The response of the suspension
can become highly nonlinear, with the generation of harmonics when an acoustic
wave travels in it. The absorption of ultrasonic waves is also largely increased by
the presence of bubbles.
Figure 2.4: Cavitation induced by the rotation of a propeller (left) and associated
damages (right).
Early studies of the behavior of bubbles have been motivated by some of these
astonishing properties, such as the study of “The damping of sound in Frothy
liquids” by Mallock in 1920 [20] or the study of the “Music of air-bubbles and the
sound of running water” by Minaert in 1933 [23]. Then, more detailed studies of
these phenomena have been performed in connection with the understanding of
underwater explosions [13, 16].
So what is the secret behind bubbles ? While it will be difficult (and disappoint-
ing) to reveal all the secrets of bubble in an introduction, many of its properties
result from the following consideration. Bubbles are resonators with a weak string
(the compressibility of air) and a large mass (the mass of the liquid). Indeed, when
we want to compress a bubble, the resistance to the deformation is due to the gas
pressure change, while the mass that must be moved comes mainly from the liquid.
40 Chapter 2. Interfaces and vibrations
Figure 2.5: Sonoluminescence induced by acoustic excitation. Left: photo of the
experimental setup. Right: zoom on the bubbles (the white dots correspond to
light emitted by the bubbles.)
Minnaert frequency
From this simple analogy, the resonance frequency of bubbles can be estimated.
Indeed the characteristic frequency of a spring-mass oscillator is given by the well
known formula ωo =√
k/m, with k the stiffness of the spring and m the considered
mass. Since we consider a continuum medium, these quantities will be replaced
by their volume counterparts kv and ρ. As mentioned in the previous section, the
density of the gas ρg can be neglected compared to the one of the liquid ρl and thus
ρ = ρl. In the absence of surface tension, the bubble is prevented from collapsing by
the compression of the gas phase. Therefore, simple dimensional analysis shows that
the stiffness per unit volume kv is given by kv = 1/χgR2, with χg =
[
1ρg
∂ρg∂pg
]
sthe
adiabatic compressibility of the considered medium and R the radius of the bubble.
For an ideal gas in adiabatic evolution, the product pgρ−γg is constant, with γ =
Cpg
Cvg
the heat capacity ratio and thus χg = 1/γpg. If we combine all these equations, we
obtain:
ωo ∼1
R
√
γpgρl
(2.1)
The proportionality coefficient in this expression will be determined later in section
2.1.4. From this formula, we can deduce one of the astonishing properties of bubbles.
Usually, an object is seen by a wave (that is to say it will diffuse this wave) when
its characteristic size is of about or larger than the wavelength. This is why milk
(emulsion of cream in water) or clouds (aerosol of droplets in air) are “white”. Here
if we compare the wavelength of acoustic waves in water λ at resonance to the size
of the bubble R, we have:
λ
R=
clωoR
=cl√ρl√
γpg∼ 102 for air bubbles in water
with cl the speed of sound in the liquid phase. Thus resonance appears when the
wavelength is much larger than the size of the bubble.
2.1. Bubbles and the Rayleigh-Plesset equation 41
Speed of sound in bubbly media
Another astonishing property is the speed of sound in bubbly liquids (at low fre-
quency compared to Minnaert’s resonance frequency). To compute it, we can go
back to the definition of the sound speed c:
c2 =
[
∂p
∂ρ
]
s
(2.2)
with p the pressure of the medium, ρ the density of the medium and the subscript
s is used to designate an isentropic evolution. Since the suspension is a mixture of
gas (bubbles) and liquid, we have:
ρ = αρg + (1− α)ρl (2.3)
with α the volume concentration of gas bubbles. If we combine equations (2.2) and
(2.3), we get:1
c2=
[
∂α
∂p
]
s
(ρg − ρl) +α
c2g+
(1− α)
c2l(2.4)
If we assume that there is no dissolution of gas or phase change (evapora-
tion/condensation) during the propagation of the acoustic wave, we have: αρg/(1−α)ρl = K, with K a constant. If we differentiate equation αρg = K(1 − α)ρl with
respect to p, we obtain:
[
∂α
∂p
]
s
ρg +α
c2g= K
([
∂α
∂p
]
s
ρl +(1− α)
c2l
)
=αρg
(1− α)ρl
([
∂α
∂p
]
s
ρl +(1− α)
c2l
)
If we recombine this expression properly, we obtain:
[
∂α
∂p
]
s
= α(1 − α)
(
1
ρlc2l
− 1
ρgc2g
)
(2.5)
Finally replacing equation (2.5) in equation (2.4), gives:
c2 =
[
α
c2g+
(1− α)
c2l+ α(1− α)(ρg − ρl)
(
1
ρlc2l
− 1
ρgc2g
)]
−1
(2.6)
The sound speed calculated from this formula is plotted on Fig. 2.6 for an
air-water suspension. This figure shows the dramatic decrease in the sound speed
(down to c = 20ms−1 at α ∼ 0.5 compared to its value in water (cl ∼ 1500 ms−1
or in air cg = 320 ms−1). The right plot in Fig. 2.6 shows a zoom on small values
of the concentration α. One can see that for concentrations as small as 0.01%, the
sound speed is already smaller than 100 ms−1.... Conclusion: don’t forget removing
bubbles before measuring the sound speed in liquids !
42 Chapter 2. Interfaces and vibrations
0 0.25 0.5 0.75 10
500
1000
1500
Concentration α
Sou
nd s
peed
c (
m s
−1 )
c = 20 ms−1
cl = 1500 ms−1
cg = 320 ms−1
0 0.2 0.4 0.6 0.8 1x 10
−3
0
500
1000
1500
Concentration α
Sou
nd s
peed
c (
m s
−1 )
Figure 2.6: Sound speed in bubbly media at frequencies smaller than Minnaert’s
frequency ωo. Right: zoom on low concentrations.
2.1.2 Static equilibrium of bubbles
Henri’s law and dissolved gas
Most of the liquids that surround us contain some dissolved gas (mostly air). Henri’s
law stipulate that, at constant temperature, the amount of a given gas that dissolves
in a given type and volume of liquid is directly proportional to the partial pressure
of that gas in equilibrium with that liquid, namely:
pg = kHβdg
where pg is the partial pressure of the considered gas, kH Henri’s variable and βdgthe concentration of dissolved gas.
A classical illustration of this law is given by carbonated soft drink. Carbonated
drinks contain some dissolved carbon dioxide in higher concentration that the
value given by Henri’s law at atmospheric pressure. Thus the pressure of the gas
in the bottle above the carbonated drink is higher than atmospheric pressure po.
When the bottle is opened, this pressure drops down and thus the concentration of
gas has to decrease. Thus some bubbles of carbon dioxide form and rise because
of buoyancy force (see Fig. 2.7). If we leave the bottle at atmospheric pressure,
the dissolved gas will reach a new equilibrium and the liquid will become “flat”.
This process takes some time since it is limited by the diffusion of the dissolved
gas through the liquid. Its speed depends on the difference between the actual
concentration of gas βdg(t) and the equilibrium value βe(pa) explaining why it
is quick at the beginning (with the release of bubbles), and becomes slower and
slower. But when the equilibrium value is reached, it does not mean that there
is no more dissolved gas in the liquid. It can be simply demonstrated by putting
“flat” water in a vacuum chamber. In this case, we will see the same phenomenon
as the one previously observed when the bottle was opened.
2.1. Bubbles and the Rayleigh-Plesset equation 43
Figure 2.7: Illustration of Henry’s law. When the bottle is closed (right), the pres-
sure in the bottle is higher than atmospheric pressure and thus there is a large
quantity of dissolved gas. When the bottle is opened, the pressure is now equal to
atmospheric pressure and thus the quantity of dissolved gas decreases through the
formation and expulsion of bubbles (left).
An interesting property of Henry’s constant is that it increases with temper-
ature (see Van’t Hoff equations) and thus the solubility of gases decreases with
temperature. Indeed, when we heat water, air bubbles form long before the boiling
crisis (at temperature smaller than 100C) and the formation of vapor bubbles.
It is important to note that the times associated with this diffusion of gas through
the liquid are “long”, and therefore this mass diffusion process can generally be
neglected if we consider quick oscillations of bubbles. They can nevertheless play a
significant role on a large number of oscillations (the so-called rectified diffusion).
Single bubble in a “flat” liquid
Laplace law and dissolution of bubbles
Now we will consider some “flat” liquid, and create a bubble by injecting some
gas inside it. Some liquid evaporates in the bubble until the partial pressure of
vapor reaches its equilibrium value at the considered temperature pv = pv(T ) given
by Clausius-Clapeyron relation. The gas in the bubble is therefore a mixture of
vapor and added gas and the value of the pressure of the gas phase in the bubble is
the sum of the partial pressures of added gas pa and vapor pv according to Dalton’s
law: pg = pa + pv. The static equilibrium with the liquid phase is given by Laplace
law:
pl(R) = pg(R)−2σ
R= pa(R) + pv(R)−
2σ
R(2.7)
44 Chapter 2. Interfaces and vibrations
Far from the bubbleLiquid
γ
Vapor+gas
R(t)
pg = pa + pvρg = ρa + ρv
Tg
pl(r), vl(r), Tl(r)p∞, v∞ = 0, T∞
Figure 2.8: Sketch of a spherical bubble in an infinite liquid
The vapor pressure pv(R) only depends on the temperature at the bubble
surface TΣ: pv(R) = pv(TΣ). Since the liquid is flat, it contains a quantity of
dissolved gas given by Henri’s law at atmospheric pressure cdg = po/kH . In the
bubble however, surface tension induces an excess of pressure compared to po. This
excess of pressure will inevitably lead to the dissolution of the gas contained in the
bubble according to Henri’s law. Thus, any bubble should dissolve after a sufficient
amount of time. However, experience shows that bubble nuclei remain stable for
times much longer than the time required for mass diffusion. This stability of
bubble nuclei can be explained by the impurities present in the solution. The
impact of these impurities can be seen in a glass of Champagne. A careful look
at the glass shows that bubbles always form at the same locations. In fact, they
appear where the dirts cover the glass. When the sommelier “cleans” the glasses
with a rag before serving Champagne, he does not try to remove the dust but
instead to leave more dust inside the glass so that bubbles can form ...
Blake pressure threshold and cavitation.
In the following, we consider the evolution of bubbles on characteristic times
much shorter than the time associated with gas dissolution. We will see that, even
in this case, the stability of bubbles is not granted. To demonstrate it, let’s consider
that the evolution of the gas in the bubble is isothermal (constant temperature To).
In this case, we have the following relation between the pressure of the added gas
pa and the radius of the bubble R:
paR3 = K = paoR
3o (2.8)
with K a constant and finally Ro and pao the radius and added gas pressure in the
bubble at rest.
If we combine equations (2.7) and (2.8), and assume that the vapor pressure
2.1. Bubbles and the Rayleigh-Plesset equation 45
remains at its equilibrium value pv = pvo (see the discussion about this hypothesis
in section 2.1.4), we obtain:
pl(R) = pao
(
Ro
R
)3
+ pvo −2σ
R(2.9)
Now we can study the stability of the system if a pressure perturbation is applied:
∂R
∂pl=
[
∂pl∂R
]
−1
= R
[
2σ
R− 3pao
(
Ro
R
)3]
−1
If the bubble radius reaches a critical value Rb such that:
RB =
√
3paoR3o
2σ=
√
9mgKgTo8πσ
(2.10)
with mg and Kg the mass of gas in the bubble and the gas constant respectively, the
bubble growth rate becomes “infinite” (explosive growth). The associated pressure
perturbation, called the Blake threshold, is equal to: pB = pL(R = RB). Thus if a
negative pressure variation larger than the Blake threshold is applied to a bubble
nuclei, it cavitates. Indeed, the stiffness of bubbles’ spring comes both from the gas
pressure variations and surface tension. When the bubble radius increases, Laplace
pressure jump decreases, leading to a reduction of the bubble stiffness due to surface
tension. At a given point (Blake’s threshold), the pressure reduction in the bubble
is no more sufficient to prevent the bubble from growing. This principle is used in
ultrasonic cleaners. A more rigorous derivation of this condition will be given in
section 2.1.4.
2.1.3 Modeling of the liquid phase: the Rayleigh-Plesset equation
In this section, we will focus on the dynamical evolution of bubbles.
Inviscid Rayleigh-Plesset equation
To derive Rayleigh-Plesset equation, we will compute the flow around a spherical
bubble whose radius depends on time: R(t). We consider in this first paragraph that
the flow is inviscid, incompressible and keeps a spherical symmetry: pl = pl(r, t),−→vl = vl(r, t)−→er . From the continuity equation, we have:
div(−→vl ) = 0 =⇒ 1
r2∂
∂r(r2vl) = 0 =⇒ vl =
K(t)
r2
with K a time-dependent function. The boundary condition at the surface of the
bubble is (according to equation (1.24) in chapter 1):
ρl(vl(R)− R) = ρg(vg(R)− R)
46 Chapter 2. Interfaces and vibrations
with R the derivative of R with respect to t. Since the density of the gas phase ρgis much smaller than the density of the liquid phase ρl, the condition at the bubble
surface becomes vl(R) = R and thus K(t) = RR2:
vl(r, t) =RR2
r2(2.11)
From this equation, we can deduce the value of the velocity potential φ defined by−→vl =
−→∇φ:∂φ
∂r=RR2
r2=⇒ φ =
−RR2
r(2.12)
Now if we apply unsteady Bernouilli’s principle (equation (1.22) in chapter 1) and
neglect gravity effects, we obtain:
∂φ
∂t+v2l2
+plρl
= C(t) =⇒ − RR2
r− 2
R2R
r+RR4
2r4+pl(r, t)
ρl= C(t) (2.13)
with C(t) a time-dependent function. C(t) can be computed from the boundary
condition at the bubble surface:
C(t) = −RR− 3
2R2R+
pl(R, t)
ρl(2.14)
If we combine equations (2.13) and (2.14), we obtain:
RR
(
1− R
r
)
+RR2
2
(
1 +R4
3r− 4R
3r
)
=pl(R, t)− pl(r, t)
ρl
If we estimate this equation at a distance r ≫ R, the pressure pl(r, t) tends to p∞(t)
(see Fig. 2.8) and R/r → 0:
RR+3
2R2 =
pl(R, t)− p∞(t)
ρl(2.15)
Finally, since the liquid is supposed to be inviscid, Laplace equation can be used to
express the liquid pressure at the interface: pl(R, t) = pg(R, t)−2σ/R and equation
(2.15) becomes:
RR+3
2R2 =
pg(R, t)− p∞(t)
ρl− 2σ
ρlR(2.16)
This is the so-called Rayleigh-Plesset equation. To close it, the pressure in the
bubble pg(R, t) must be expressed as a function of the radius. This issue will be
thoroughly discussed in section 2.1.4. Nevertheless, to get a first insight of the
behavior predicted by Rayleigh-Plesset equations, we can perform some simulations
when the gas phase follows an adiabatic evolution: pg = pgo (Ro/R)3γ . Fig. 2.9 and
2.10 illustrate the bubble response to a driving pulse of about 105Pa. These figures
show the large departure from linear response. First, we see that when the bubble
collapses, the stiffness of the spring increases due to the compression of the gas. This
entails the bubble bouncing with a quick velocity drop from -100 ms−1 to 100 ms−1.
2.1. Bubbles and the Rayleigh-Plesset equation 47
0 2 4 6
x 10−6
−1
0
1
x 105 Driving Pulse
Time [s]
Pre
ssur
e [P
a]
0 2 4 6
x 10−6
0
1
2
3x 10
−6 Bubble Radius
Time [s]
Rad
ius
[m]
0 2 4 6
x 10−6
−100
0
100
Bubble Wall Velocity
Time [s]
Vel
ocity
[m/s
]
Figure 2.9: Adiabatic evolution of the radius (amplitude and speed) of a 1µm bubble
submitted to a driving pulse of ∼ 105 Pa computed from inviscid Rayleigh-Plesset
equation. The blue line corresponds to the driving pulse, red lines to the bubble
response predicted from Rayleigh-Plesset equation and black ones to the bubble
linear response estimated from the linearized version of Rayleigh-Plesset (equation
(2.29)).
48 Chapter 2. Interfaces and vibrations
Moreover nonlinear terms lead to a large increase in the amplitude and velocity of
bubble oscillations (compared to linear response). Finally, the natural frequency of
the drop is shifted to lower values. We can also notice that, at this stage, there is
no dissipation term, and following this equation a bubble would oscillate endlessly.
In the next sections, we will consider the sources of dissipation.
2.2 2.4 2.6 2.8
x 10−6
0
0.5
1
1.5
2
2.5x 10
−6Bubble Radius
Time [s]
Rad
ius
[m]
2 2.2 2.4 2.6
x 10−6
−100
−50
0
50
100
Bubble Wall Velocity
Time [s]
Vel
ocity
[m/s
]
Figure 2.10: Same graph as Fig. 2.9 but zoomed on two oscillation periods.
Viscous dissipation
To take into account viscous dissipation, one must start with Navier incompressible
equation instead of Bernoulli’s principle. Since the continuity condition still holds,
the expression of the velocity field remains unchanged:
vl(r, t) =RR2
r2(2.17)
The momentum equation in spherical coordinates takes the following form:
ρl
(
∂vl∂t
+ vl∂vl∂r
)
= −∂pl∂r
+ µl
(
∂2vl∂r2
+2
r
∂vl∂r
− 2vlr2
)
(2.18)
If we replace equation (2.17) in equation (2.18), we obtain:
ρl
(
2RR2 +R2R
r2− 2
R4R2
r5
)
= −∂pl∂r
+ µlR2R
(
6
r4− 4
r4− 2
r4
)
(2.19)
We can see that the viscous term is zero and thus that the viscosity plays no role
in the bulk momentum equation. Thus equation (2.15) remains valid:
RR+3
2R2 =
pl(R, t)− p∞(t)
ρl(2.20)
2.1. Bubbles and the Rayleigh-Plesset equation 49
Indeed, viscosity only plays a role in the boundary condition. If we neglect the
density and viscosity of vapor compared to their counterparts in the liquid and as-
sume that the surface tension is uniform, the momentum equilibrium at the interface
(equation (1.27)) is:
−pl(R, t) + 2µl
[
∂vl∂r
]
r=R
= −pv(R, t) + 2σ
R
Thus, if we replace vl by its expression given by equation (2.17) in this equation, we
obtain:
pl(R, t) = pv(R, t)− 2σ
R− 4µl
R
R(2.21)
Finally, if by combining equations (2.18) and (2.21) together, we get:
ρl
(
RR+3
2R2
)
+ 4µlR
R= pg(R, t)− p∞(t)− 2σ
R(2.22)
From the comparison of the order of magnitude of the two terms on the left
hand side of this equation, one can estimate the characteristic time τv associated
with viscous attenuation:
τv =ρlR
2o
4µl
This time is about 2.5 × 10−7s for a micrometric bubble oscillating in water. Ef-
fectively, if we solve numerically equation (2.22), we see that bubble oscillations are
completely damped after a time ∼ 10τv (see Fig. 2.11). Thus, to determine whether
viscous dissipation plays a significant role, one must compare the characteristic time
associated with the bubble dynamics, τ , to τv. If τ ≪ τv, viscous damping can be
neglected.
Thermal dissipation
Some thermal dissipation can also occur due to the large temperature variations
experienced by the gas in the bubble. This heat can spread in the liquid leading
to an attenuation of bubble oscillations. Since the liquid phase is assumed to be
incompressible, the Heat equation and Navier equations are uncoupled, and thus
the heat equation must be solved separately to take into account thermal effects.
However, since the thermal conductivity of the liquid is much larger than the one
of air, the heat diffusion process is not limited by the time required for diffusion
through the liquid but by the one required for diffusion through the gas. Thermal
dissipation will therefore be discussed again later in the section dedicated to the
modeling of the gas behaviour.
Radiation damping
On the incompressibility condition
50 Chapter 2. Interfaces and vibrations
0 2 4 6
x 10−6
−1
0
1
x 105 Driving Pulse
Time [s]
Pre
ssur
e [P
a]
0 2 4 6
x 10−6
0
1
2x 10
−6Bubble Radius
Time [s]
Rad
ius
[m]
0 2 4 6
x 10−6
−50
0
50
Bubble Wall Velocity
Time [s]
Vel
ocity
[m/s
]
0 2 4 6 8
x 106
−180
−160
−140
Power Spectra
Frequency [Hz]
Am
plitu
de [d
B]
Figure 2.11: Adiabatic evolution of the radius (amplitude and speed) of a 1µm
bubble submitted to a driving pulse of ∼ 105 Pa computed from Rayleigh Plesset
equation including viscous dissipation. Blue lines correspond to the driving pulse,
red lines to the bubble response predicted from Rayleigh-Plesset equation and black
ones to the bubble linear response estimated from the linearized version of Rayleigh-
Plesset (equation (2.29)).
2.1. Bubbles and the Rayleigh-Plesset equation 51
Rayleigh-Plesset equation have been derived assuming an incompressible liquid
phase. A counter-intuitive result is that Rayleigh-Plesset equation still holds for
the description (at first order) of bubbles oscillations induced by acoustic waves.
It seems surprising since the liquid compressibility is required for acoustic waves to
propagate. To understand it, the linearized mass conservation equation in the liquid
can be written in a dimensionless form:
ωδρl∂ρl
∂t+ρloδvlRo
div(−→v ) = 0 (2.23)
with ω the frequency of the acoustic wave, δρl and δvl the orders of magnitude of
density and velocity variations induced by the acoustic wave, and Ro the bubble
radius at rest. The characteristic length scale is Ro and the characteristic time
scale is 1/ω. For acoustic waves, the relation between the amplitude of density and
velocity variations is: δρl = ρloclδvl, with cl the sound speed in the liquid. Thus
equation (2.23) becomes:Ro
λ
∂ρl
∂t+ ˜div(−→v ) = 0 (2.24)
with λ = ω/cl the acoustic wavelength. As demonstrated in section 2.1.1, when
a bubble is excited around its resonance frequency, the wavelength is much larger
than the radius of the bubble (Roλ ≪ 1). Thus, the first term of the equation
can be neglected compared to the second one and the liquid can be considered as
incompressible close to the bubble.
div(−→v ) ∼ 0
Dissipation by acoustic radiation
However, while the compressibility of the liquid can be neglected over a single
period of oscillation (the characteristic time scale was chosen as the inverse of the
frequency), it can play a significant role on a large number of oscillations. Indeed,
the bubble oscillations induce the emission of acoustic waves, and thus some energy
is radiated inside the liquid. This energy loss entail a damping of the bubble oscil-
lations. This effect has been introduced by Keller and Kolodner in 1956 [16], and
Keller and Miksis in 1980 [17]. We will not demonstrate how the radiation damping
can be accounted for and will just give the form of the modified Rayleigh-Plesset
equation:
RR
(
1− R
cl
)
+3
2R
(
1− R
3cl
)
(2.25)
=
(
1− R
cl
)
pl(R, t)− p∞(t)
ρl− R
ρlcl
d
dt(pl(R, t) + p∞(t))
When the liquid becomes incompressible, cl → ∞ and Rayleigh-Plesset equation is
recovered. Fig. 2.12 shows the evolution of a bubble in the presence of radiation
damping (and in the absence of any other source of dissipation). The acoustic
radiation indeed leads to a damping of the bubble oscillations, although the effect
is small compared to viscous dissipation in the considered regime.
52 Chapter 2. Interfaces and vibrations
0 2 4 6
x 10−6
−1
0
1
x 105 Driving Pulse
Time [s]
Pre
ssur
e [P
a]
0 2 4 6
x 10−6
0
1
2
3x 10
−6Bubble Radius
Time [s]
Rad
ius
[m]
0 2 4 6
x 10−6
−100
0
100
Bubble Wall Velocity
Time [s]
Vel
ocity
[m/s
]
0 2 4 6 8
x 106
−160
−140
−120
Power Spectra
Frequency [Hz]
Am
plitu
de [d
B]
Figure 2.12: Adiabatic evolution of the radius (amplitude and speed) of a 1µm
bubble submitted to a driving pulse of ∼ 105 Pa computed from Rayleigh Plesset
equation including radiation damping. Blue lines correspond to the driving pulse,
red lines to the bubble response predicted from Rayleigh-Plesset equation and black
ones to the bubble linear response estimated from the linearized version of Rayleigh-
Plesset (equation (2.29)).
2.1. Bubbles and the Rayleigh-Plesset equation 53
2.1.4 Modeling of the gas phase
Inertially controlled dynamics
To close Rayleigh-Plesset equations the pressure difference pg(R, t)−p∞(t) must be
expressed as a function of the bubble radius R. If we consider the partial pressures
of added gas pa and vapor pv, this expression becomes:
pg(R, t)− p∞(t) = pa(R, t) + pv(TΣ)− p∞(t)
since the vapor pressure only depends on the temperature at the surface of the
bubble TΣ. Following C.E Brennen textbook [4], this expression can be decomposed
into:
pa(R, t) + [pv(T∞)− p∞(t)] + [pv(TΣ)− pv(T∞)]
The first term simply corresponds to the pressure variation in the added gas, the
second to the driving term (thermal and inertial) and the third one to the varia-
tion of vapor pressure induced by the departure of bubble temperature from the
remote liquid one. In the following, we consider that no variations of temperature
are imposed (T∞(t) = To), and we neglect the effects of phase change (evapora-
tion/condensation at the bubble surface) induced by temperature increase in the
bubble: [pv(TΣ)− pv(T∞)] = 0. This behavior is called inertially controlled dy-
namics, to distinguish from a dynamics controlled by phase change. Obviously, the
inertially controlled dynamics is not appropriate to describe events such as the boil-
ing crisis and more suitable for dynamics induced by acoustic waves. In some cases,
phase change can however play a fundamental role in bubbles dynamics induced by
pressure variations. The validity range of the different approximations is thoroughly
discussed by C.E Brennen in its textbook [4].
Heat exchange and temperature variations in the gas
The heat diffusion equation must be solved in the gas phase to determine its tem-
perature variations:
ρgCpg
dTgdt
=∂pg∂t
+ kg∆Tg (2.26)
with ddt =
∂∂t +
−→vg .−→∇ the material derivative, and ρg, Tg, C
pg , pg and kg the density,
temperature, heat capacity, pressure and heat conductivity of the gaseous phase
respectively.
Heat transfers from the bubble to the liquid are responsible for a damping of
bubbles oscillations. The characteristic time associated with heat transfers is τT ∼ρloR
2o/ρgoκg, with κg the thermal diffusivity. For air bubbles in water, this time if
about 10−4 s. On Fig. 2.13, we can see indeed that thermal dissipation induces a
slower damping of bubble oscillations than viscous effects.
54 Chapter 2. Interfaces and vibrations
0 2 4 6
x 10−6
−1
0
1
x 105 Driving Pulse
Time [s]
Pre
ssur
e [P
a]
0 2 4 6
x 10−6
0
1
2
3x 10
−6Bubble Radius
Time [s]
Rad
ius
[m]
0 2 4 6
x 10−6
−2
0
2x 10
4Bubble Wall Velocity
Time [s]
Vel
ocity
[m/s
]
0 2 4 6 8
x 106
−160
−140
−120
Power Spectra
Frequency [Hz]
Am
plitu
de [d
B]
Figure 2.13: Evolution of the radius (amplitude and speed) of a 1µm bubble sub-
mitted to a driving pulse of ∼ 105 Pa computed from Rayleigh-Plesset equation
including heat transfers. Blue lines correspond to the driving pulse, red lines to
the bubble response predicted from Rayleigh-Plesset equation and black ones to
the bubble linear response estimated from the linearized version of Rayleigh-Plesset
(equation (2.29)).
2.1. Bubbles and the Rayleigh-Plesset equation 55
Simple cases: adiabatic and isothermal behaviors
When can an evolution be considered as adiabatic or isothermal ?
Since we have introduced a characteristic time associated with thermal ex-
changes, we can define properly in which cases bubbles behavior can be considered
as adiabatic or isothermal. The evolution of the gas phase is adiabatic when there
is no heat exchange between the bubble and the liquid. This happens, when the
characteristic time associated with the bubble dynamics τ ≪ τT . In this case, the
bubble dynamics is too fast for heat transfers to occur. On the opposite, when
τ ≫ τT , the temperature in the bubble is at equilibrium with the one in the liquid.
As long as no temperature variations is imposed, the bubble behavior is isothermal
(T∞(t) = To). In these simple cases, the relation between the added gas pressure
and the radius is:
paR3k = K
with K a constant, k = 1 when the evolution is isothermal and k = γ when the
evolution is adiabatic.
Linearized version of Rayleigh-Plesset equation and Minaert’s frequency
If we consider an adiabatic or isothermal evolution and neglect phase changes
pv = pvo, Rayleigh-Plesset equation becomes:
ρl
(
RR+3
2R2
)
+ 4µlR
R− pao
(
Ro
R
)3k
+2σ
R= pvo − p∞(t) (2.27)
with k = 1 if the evolution is isothermal and k = γ if it is adiabatic.
Considering small radius variations around its equilibrium value: R(t) = Ro(1+
ε(t)), with ε≪ 1, Rayleigh-Plesset equation becomes:
ρlR2oε+ 4µlε− pao(1− 3kε) +
2σ
Ro(1− ε) = pvo − p∞(t) (2.28)
After reorganization of the terms of this equation, we finally obtain:
ε+4µlρlR2
o
ε+
(
3kpaoρlR2
o
− 2σ
ρlR3o
)
ε = po − p∞(t) (2.29)
with po = pao+pvo− 2σRo
the equilibrium pressure before the perturbation. This equa-
tion corresponds to the one of a damped linear oscillator as long as the coefficient
in front of ε is positive. From this expression, we can compute the characteristic
frequency of this oscillator, the so called Minnaert’s frequency:
ωo =1
Ro
√
3kpaoρl
− 2σ
ρlRo(2.30)
56 Chapter 2. Interfaces and vibrations
Compared to equation (2.1) previously obtained from dimensional analysis, the
influence of surface tension has been added.
Rigorous derivation of Blake’s threshold
However when the coefficient in front of ε in equation 2.29 becomes negative:
3kpaoρl
<2σ
ρlRo(2.31)
the solution of this equation is no more the sum of harmonic functions but the sum
of two exponential functions, one with a positive coefficient and one with a negative
coefficient. As long as the imposed pressure variation p∞−po is positive, the radius
decreases exponentially and the bubble is stable. Conversely, when the pressure
variation p∞ − po is negative, the radius increases exponentially, that is to say the
bubble nuclei cavitate.
If the added gas behaves as an ideal gas, pao =mgKgTg
4/3πR3o
, with mg the mass of
added gas, Kg the gas constant and Tg the gas temperature. Thus from equation
(2.31), we obtain the expression of Blake’s critical radius:
Rb =
√
9kmgTg8πσ
This expression is similar to the one previously obtained in section 2.1.2 when the
evolution is isothermal (k = 1). Fig. 2.14 shows the difference of behavior when
a bubble is submitted to a negative or a positive driving pulse with an amplitude
higher than Blake’s threshold. We can see how much the pressure threshold is
increased when the imposed pressure variation is negative.
Nonlinear response of a bubble to a sinusoidal excitation
Fig. 2.15 illustrates the response of a bubble to a periodic sinusoidal excitation
at Minnaert’s frequency (∼ 2.78 MHz for a micrometric bubble). The power spectra
shows clearly the appearance of harmonics and subharmonics due to the nonlinear
response of the bubble.
Outstanding characteristics of bubbles oscillations
The evolution of a 1µm bubble submitted to a pulse of ∼ 2 × 105 Pa can be
considered at first order as adiabatic since the characteristic time associated with
the bubble dynamics is much larger than one associated with heat exchange, τT .
Thus the temperature inside the bubble can be estimated simply from the evolution
of the radius: T/To = (Ro/R)3(γ−1). We see on Fig. 2.16 that the radius is divided
by ∼ 7, which means that the temperature is increased by 10 if the added gas is air
(γ = 1.4). Thus if the initial temperature is 20oC (293 K) the final temperature
is about 2930 K, half the temperature at the sun surface (for a pressure
2.1. Bubbles and the Rayleigh-Plesset equation 57
0 0.5 1 1.5
x 10−6
−5
0
5x 10
5 Driving Pulse
Time [s]
Pre
ssur
e [P
a]
0 0.5 1 1.5
x 10−6
0
2
4x 10
−6Bubble Radius
Time [s]
Rad
ius
[m]
0 0.5 1 1.5
x 10−6
−500
0
500
Bubble Wall Velocity
Time [s]
Vel
ocity
[m/s
]
0 2 4 6 8
x 106
−180
−160
−140
−120Power Spectra
Frequency [Hz]
Am
plitu
de [d
B]
Figure 2.14: Evolution of the radius (amplitude and speed) of a 1µm bubble sub-
mitted to a driving pulse either negative (blue) or positive (red) with an ampli-
tude higher than Blake’s threshold (∼ 5× 105 Pa) computed from Rayleigh-Plesset
equation. Black line correspond to the bubble linear response estimated from the
linearized version of Rayleigh-Plesset (equation (2.29)).
58 Chapter 2. Interfaces and vibrations
0 2 4 6 8
x 10−6
−1
0
1x 10
5 Driving Pulse
Time [s]
Pre
ssur
e [P
a]
0 2 4 6 8
x 10−6
0
1
2x 10
−6Bubble Radius
Time [s]
Rad
ius
[m]
0 2 4 6 8
x 10−6
−40
−20
0
20
40Bubble Wall Velocity
Time [s]
Vel
ocity
[m/s
]
0 2 4 6 8
x 106
−180
−160
−140
−120Power Spectra
Frequency [Hz]
Am
plitu
de [d
B]
Figure 2.15: Adiabatic evolution of the radius (amplitude and speed) of a 1µm
bubble submitted to a sinusoidal pressure excitation of ∼ 105 Pa and frequency
2.78 MHz computed from Rayleigh-Plesset equation including viscous dissipation
and radiation damping. Blue lines correspond to the driving wave, red lines to
the bubble response predicted from Rayleigh-Plesset equation and black ones to
the bubble linear response estimated from the linearized version of Rayleigh-Plesset
(equation (2.29)).
2.1. Bubbles and the Rayleigh-Plesset equation 59
wave of only 2 bar). The wall speed also reaches some values close to the
sound speed.
0 2 4 6
x 10−6
−2
0
2
x 105 Driving Pulse
Time [s]
Pre
ssur
e [P
a]
0 2 4 6
x 10−6
0
1
2
3x 10
−6Bubble Radius
Time [s]
Rad
ius
[m]
0 2 4 6
x 10−6
−200
0
200
Bubble Wall Velocity
Time [s]
Vel
ocity
[m/s
]
0 2 4 6 8
x 106
−180
−160
−140
Power Spectra
Frequency [Hz]
Am
plitu
de [d
B]
Figure 2.16: Adiabatic evolution of the radius (amplitude and speed) of a 1µm
bubble submitted to a pulse of ∼ 2× 105 Pa computed from Rayleigh-Plesset equa-
tion including viscous dissipation and radiation damping. Blue lines correspond to
the driving wave, red lines to the bubble response predicted from Rayleigh-Plesset
equation and black ones to the bubble linear response estimated from the linearized
version of Rayleigh-Plesset (equation (2.29)).
Mass diffusion, inertia and asymmetry effects.
For the sake of completeness, we will just mention the hypotheses made in this
lecture and whose validity is questionable in some cases:
1. Mass diffusion induced by phase change or dissolution of gas has been ne-
glected.
2. Bubbles are assumed to remain spherical.
3. The liquid phase is assumed to remain incompressible around the bubble
(which is false if the walls speed reaches the liquid sound speed).
4. There is no chemical reaction or plasma formation in the bubble.
5. The flow in the bubble has not been considered.
Consideration of these phenomena would increase the model complexity.
60 Chapter 2. Interfaces and vibrations
2.1.5 From bubbles to metamaterials
We will conclude this chapter with an exciting contemporary application of bubbles:
metamaterials. Metamaterials are artificial materials engineered to have properties
that cannot be found in nature. These properties are obtained by inserting small
inhomogeneities in a matrix to create effective macroscopic behavior. One of the
primary aims of metamaterial is to investigate materials with negative refractive
index [40] or with gradient index materials. The former would allow the conception
of superlenses, with subwavelength resolution while the latter would allow the design
of “invisibility cloaks” which make object invisible toward a given type of wave
(electromagnetic, acoustic, water surface).
To understand it, we must first define what the expression “invisible object”
means. The detection of an object comes from the reflection or scattering of a given
wave by this object. Thus, if one can design a cloak which allows the wave to follow
the same path as it would in the absence of the object (see Fig. 2.17), the object
becomes invisible. This does not seem a priori so complicated: the wave ray must
be curved to follow the object. However mathematical transformations show that to
enable the curvature of rays in the appropriate way, one must have materials with
large variations of refractive index. In acoustic, the refractive index n is given by:
n2 = ρχ
with ρ the density of the medium and χ the compressibility of the medium. With
Figure 2.17: Trajectories of rays around an object covered by an invisibility cloak
(left: sketch, right: simulation): “any radiation attempting to penetrate the secure
volume is smoothly guided around the cloak to emerge traveling in the same direction
as if it has passed through an empty volume of space.” [27]
conventional materials, it is hard to obtain the adequate properties to distort the
rays around an object. It is even harder to get a negative refractive index, since
in this case the material must have both a negative density and a nega-
tive compressibility ! This latter means that when the pressure is increased, the
material expands. Of course, it cannot be achieved as a static property. But ma-
terials can behave as if they had effective negative density or compressibility in the
2.2. Inertio-capillary “Rayleigh-Lamb” modes of vibration 61
frequency domain. That is where bubbles come into play. If we consider a bubble
cloud surrounded by a soft matrix and make the bubble oscillate in antiresonance
compared to the excitation, the effective medium expands when the pressure in-
creases. Thus the material behaves as if its compressibility were negative [14]. To
obtain a negative density, one can consider some heavy particles in a soft matrix.
When the wave propagates, if the particles move in the opposite way as the sur-
rounded matrix, and their density exceeds the matrix density, the medium behaves
as if its density were negative [21]. If both properties are obtained in the same
frequency range, the metamaterial has a negative refractive index.
2.2 Inertio-capillary “Rayleigh-Lamb” modes of vibra-
tion
In the previous section, we have only considered spherical deformations of bubbles
involving the compressibility of the fluid. Here, we will show that incompressible
modes of deformation can also appear at the surface of droplets or bubbles. These
vibrations come from a competition between inertia and surface tension.
2.2.1 Dimensional analysis
The characteristic frequency ωo of these surface vibrations can be established from
dimensional analysis for a droplet of liquid with a density ρ surrounded by a gas
with a density ρ′ ≪ ρ. In this case, inertial effects correspond to the term ρ∂−→v∂t in
Navier-Stokes equation. Since Ro is the characteristic length associated with the
flow in the droplet, the magnitude of the inertial term is about:
fi = ρ∂−→v∂t
∼ ρRoω2o
This term is homogeneous to a force per volume unit. Since Laplace pressure jump
is equal to 2σ/Ro, the force per unit volume associated with surface tension is:
fs ∼ σ/R2o
If these two terms have the same order of magnitude, the characteristic frequency
scales as:
ωo =
√
σ
ρR3o
2.2.2 Complete solution of the problem
In this second section, we solve the complete problem by considering a droplet
or bubble of density ρ, initially spherical with a radius Ro, and surrounded by an
infinite amount of immiscible fluid of density ρ′ [18] . We will study small vibrations
appearing at the interface between these two fluids. Thus the radius r at any point
of the interface verifies:
r = Ro + ξ with ξ ≪ Ro
62 Chapter 2. Interfaces and vibrations
The flow is assumed to be inviscid, incompressible, and following vorticity equation
irrotational. The surface tension between the two fluids is σ.
Since the flow is incompressible, the velocity in both fluids can be expressed
as a function of a velocity potential: −→v =−→∇φ and −→v ′ =
−→∇φ′. Since it is also
irrotational, these two potentials verify Laplace equation:
∆φ = 0 and ∆φ′ = 0
The solution of Laplace equation in spherical polar coordinates (r, θ, ϕ) are called
solid harmonics. They can be divided into regular solid harmonics φn which vanish
at origin an irregular solid harmonics φ−n−1 which are singular at origin, with n the
degree of the considered harmonic:
φn = rnSn(θ, ϕ) and φ−n−1 = r−n−1Sn(θ, ϕ)
where Sn(θ, ϕ) is a surface harmonic of degree n. The expression of surface harmon-
ics can be calculated from Laplace equation in spherical coordinates by applying the
separation of variables. We will not develop this theory further here and refer to
the book of Lamb [18] or Gumerov and Duraiswami [11] for a complete overview
of spherical harmonics theory. We just mention here how surface harmonics can be
calculated:
Sn = AoTn(ν) +
n∑
s=1
(Ascos(sϕ) +Bssin(sϕ))Psn(ν)
P sn(ν) = (1− ν)s/2
dsTn(ν)
dνs
Tn(µ) =1
2nn!
dn
dνn(ν2 − 1)n
ν = cos(θ)
with As, 2n + 1 coefficients. The functions P sn are called associated Legendre
polynomials.
The small deformations at the sphere surface can be expressed in terms of surface
harmonics:
r = Ro + ξ = Ro + Sn sin(ωt+ ε)
with ω the frequency of the vibration, t the time and ε the vibration phase. Fig.
2.19 shows different spherical harmonics. The colors correspond to the intensity of
the sphere surface deformation.
In this case, the velocity potentials inside and outside the drop take the form:
φ =ωRo
n
rn
Rno
Sn cos(ωt+ ε) and φ′ =ωRo
n+ 1
Rn+1o
rn+1Sn cos(ωt+ ε)
since:
2.2. Inertio-capillary “Rayleigh-Lamb” modes of vibration 63
Figure 2.18: Sketch representing the different modes of deformation of a sphere
(spherical harmonics). The colors correspond to the intensity of the sphere surface
deformation. Source: http://principles.ou.edu/earth_figure_gravity/index.html
1. the potential inside the spherical inclusion cannot be singular at r = 0 and
the potential outside the drop cannot become infinite when r → ∞.
2. the potential must fulfill the boundary condition (continuity of the normal
velocity) at the sphere surface (r = Ro):
∂ξ
∂t=∂φ
∂r=∂φ′
∂r
From unsteady Bernoulli’s principle (equation 1.22 in chapter 1), we can deduce the
expression of internal and external pressures at the interface:
p = po +ρω2Ro
nSn sin(ωt+ ε) and p = p′o −
ρ′ω2Ro
n+ 1Sn sin(ωt+ ε) (2.32)
with po and p′o the pressures inside and outside the spherical inclusion at rest. Since
the flow inside or and outside the sphere are inviscid, the momentum equilibrium
at the interface (r ∼ Ro) is simply Laplace equation:
∆p = p− p′ = σC = σdiv(−→n ) (2.33)
with C = div(−→n ) the interface curvature and −→n the surface normal. The theorem
of solid geometry stipulates that the normal −→n at any point of a surface defined by
F (x, y, z) = 0 is given by:−→n =
−→∇F (x, y, z)
64 Chapter 2. Interfaces and vibrations
Thus, the curvature is given by:
C = div(−→∇F ) = ∆F
Here, the equation of the surface is given by:
F = r −Ro − ξ = 0
Since the deformations of the interface are small, the function ξ = Sn sin(ωt + ε)
can be expanded as a function of ξn = rn/RnoSn sin(ωt+ ε) according to:
ξ = ξn(r)− (r −Ro)∂ξn∂r
(Ro)−(r −Ro)
2
2
∂ξn∂r
(Ro) + ...
= ξn(r)− (r −Ro)n
Roξn(Ro)−
(r −Ro)2
2
n(n− 1)
R2o
ξn(Ro) + ...
If (x, y, z) are the Cartesian coordinates, the radius is equal to r =√
x2 + y2 + z2
and the derivatives of r with respect to x, y and z are given by:
∂r
∂x=x
r,∂r
∂y=y
r, and
∂r
∂z=z
r
Thus, at first order
ξ = ξn(r)
∂ξ
∂x=∂ξn∂x
− ∂r
∂x
n
Roξn(Ro) + ... =
∂ξn∂x
− nx
rRoξn(Ro) + ...
∂2ξ
∂x2=∂2ξn∂x2
−[
1
r− x2
r3
]
n
Roξn(Ro)−
n(n− 1)
R2o
[
Rox2
r3+
(r −Ro)
r
]
ξn(Ro) + ...
The same formula can be obtained for the derivatives with respect to y and z.
∂2ξ
∂y2=∂2ξn∂y2
−[
1
r− y2
r3
]
n
Roξn(Ro)−
n(n− 1)
R2o
[
Roy2
r3+
(r −Ro)
r
]
ξn(Ro) + ...
∂2ξ
∂z2=∂2ξn∂z2
−[
1
r− z2
r3
]
n
Roξn(Ro)−
n(n− 1)
R2o
[
Roz2
r3+
(r −Ro)
r
]
ξn(Ro) + ...
As a consequence,
∂2F
∂x2=∂2r
∂x2−∂
2ξ
∂x2=
[
1
r− x2
r3
]
−∂2ξn∂x2
+
[
1
r− x2
r3
]
n
Roξn(Ro)+
n(n− 1)
R2o
[
Rox2
r3+
(r −Ro)
r
]
ξn(Ro)
Since x2+ y2+ z2 = r2, and ∂2ξn∂x2 + ∂2ξn
∂y2 + ∂2ξn∂z2 = 0 (ξn is a solid harmonic and thus
a solution of Laplace equation), we get:
∂2F
∂x2+∂2F
∂y2+∂2F
∂z2=
[
2
r
]
+[O]+
[
2n
Ror
]
ξn(Ro)+
[
n(n− 1)
(
3(r −Ro)
r+Ro
r
)]
ξn(Ro)
R2o
2.2. Inertio-capillary “Rayleigh-Lamb” modes of vibration 65
Finally, if we estimate this expression in r = Ro and since 2/r ∼ 2/Ro(1 − ξn), we
obtain:
C =
[
∂2F
∂x2+∂2F
∂y2+∂2F
∂z2
]
r=Ro
=2
Ro+
(n− 1)(n + 2)
R2o
Snsin(ωt+ ε) (2.34)
Now if we combine equation (2.32), (2.33) and (2.34), we obtain the expression of
the characteristic frequency of a Rayleigh-Lamb mode of order n:
ωo =
[
(n+ 2)(n+ 1)n(n − 1)σ
[(n+ 1)ρ+ nρ′]R3o
]1/2
(2.35)
When the density ρ of the fluid inside the sphere is much larger than the one of the
surrounding fluid, we obtain:
ωo =
√
(n+ 2)n(n− 1)σ
ρR3o
which is the same as the previous formula obtained from dimensional analysis, but
with the appropriate proportionality coefficient. This expression has been obtained
by Rayleigh in 1879 [29] from energy considerations and Lagrange’s method. Then,
it has been extended by Lamb in 1932 [18] for two inviscid fluids of density (ρ) and
(ρ′), equation (2.34). The viscosity of the fluids has been introduced by Miller and
Scriven (1968) [22] and Prosperetti (1980) [28]. Finally, larger (nonlinear) vibrations
have been considered by Foote (1971) [10], Tsamopoulos and Brown (1983) [38], and
more recently by Smith [33]. An interesting result is that when droplet oscillations
become larger, there is a decrease in the characteristic frequency ωo due to nonlinear
effects.
2.2.3 Sessile droplets and possible use of these surface vibrations.
To observe Rayleigh-Lamb vibrations, one must start with an initially spherical
drop (levitating drop or drop lying on a superhydrophobic surface) and create an
excitation of the drop at the appropriate frequency. For example, a drop of mercury
lying on a Teflon plate (contact angle of ∼ 157o) can be excited through the
vibration of the underlying plate (see Fig. 2.19). Levitating drop can be obtained
acoustically with the use of radiation pressure [32]. In this case, the excitation of
the drop can be achieved by a modulation of the acoustic signal. Levitating drop
can also be obtained through the deposition of a drop on a hot plate (the drop lies
on an air cushion): it is the so-called Leidenfrost effect. However, the size of the
drop is not constant and the characteristic frequency is time-dependent.
In most practical situation, drops do not levitate and lie on substrates with
contact angles different from 0. In this case, Rayleigh-Lamb oscillations are affected
by the value of the contact angle [35] and the pinning or free motion of the contact
line [26]. An interesting application of Rayleigh-Lamb oscillations is their use in
order to unpin the contact line and induce [25] or enhance the motion of droplets.
66 Chapter 2. Interfaces and vibrations
Mode n=2
Mode n=3
Mode n=4
Mode n=5
Figure 2.19: Different modes of vibrations of a mercury drop lying on a Teflon
plate (contact angle ∼ 157o). Source: http://www.youtube.com/watch?v=
MperC7ySjSU by L. Floc’h, A. Hubert, S. Place and S. Remadi
Chapter 3
Dynamics of bubbles or plugs in
confined geometries
The dynamics of bubbles in capillary tubes is a wide subject, ranging from
Marangoni flow to boiling crisis [1]. In this chapter, we will focus on the “pressure-
driven” motion of bubbles in confined geometries.
NB: In this chapter, the surface tension will be called γ.
3.1 Two phase flow at small scales
On the relative importance of surface effects
Figure 3.1: Pygmy shrew picture.
Let’s start this section by drawing a parallel between the physics at small scales
and the “Pygmy Shrew”. The Pygmy Shrew is the smallest mammal in the world.
68 Chapter 3. Dynamics of bubbles or plugs in confined geometries
It has an average weight of 4 grams and its body measures about 50 mm (see Fig.
3.1). A fundamental question is: why is there a critical size for mammals? If we
compare the surface to volume ratio of a body, it scales as ∼ L2/L3 = 1/L. Thus
the smaller the considered length scale is, the stronger the surface effects compared
to volume ones are. To regulate their temperature, mammals must eat. The smaller
a mammal is, the larger its surface to volume ratio and thus heat exchanges with
the surrounding atmosphere are. Thus the Pygmy Shrew has one of the highest
metabolic rates of any animals and it spends most of its time eating to keep its
temperature constant.
Figure 3.2: Picture of an ant trying to drink the water contained in a rain droplet.
Top picture: the ant tries to pierce the drop surface. Bottom picture: the ant is
trapped in a drop by capillary forces. Source: http://www.greenwala.com/channels/
nature/blog/14735-Cool-Nature-Picture-of-the-Day-Ant-Stuck-in-a-Rain-Drop
One must therefore keep in mind that the small scales world is ruled by com-
pletely different laws from our usual environment. A good illustration of the aston-
ishing physics at small scales is the study of an ant fall from the 4th floor of a building
(see http://www.youtube.com/watch?v=RmgIsk19RSM&feature=related). While
3.2. Bretherton law 69
if you try the same experiment (I strongly advise you not doing it), you would have
little chance to survive, the ant is in perfectly good health after such an incredible
jump. Indeed, since the ant is small, gravity (bulk) effects are rapidly counterbal-
anced by drag (surface) effects. Thus, it reaches its maximum velocity after only
a few centimeters of fall. The little impact of gravity on ants is also obvious when
we see them walking on ceilings). On the other hand, surface effects, like surface
tension, can strongly affect it: Fig. 3.2 shows a picture of an ant trapped in a drop
by surface tension.
Singularities
A key idea is therefore that, at small scales, surface forces are “generally” domi-
nant compared to bulk forces. As a consequence, Capillary and Bond numbers are
generally small. So why taking precautions when stating that surface effects are
dominant? We have seen in a previous chapter that viscosity plays a fundamental
role on the dynamics of the triple line, while the characteristic length involved is
small. This comes from the singularity of the flow close to the triple line, which
results in large velocity gradient and thus viscous dissipation. Thus bulk effects can
play a fundamental role close to walls or more generally singularities.
3.2 Bretherton law
3.2.1 Semi infinite bubble
Description of the problem
R
A
C
Liquid
B
y
x
U
U
Liquid
z
y
x
Air
Air
Figure 3.3: Sketch of the motion of a semi-infinite bubble in a cylindrical tube at a
constant velocity U .
In this section, we will demonstrate a surprising result: the harder you blow air
in a capillary tube filled with liquid, the more liquid you leave on the walls. For this
70 Chapter 3. Dynamics of bubbles or plugs in confined geometries
purpose, we will study the flow of a semi-infinite bubble moving at a constant speed
U in a cylindrical capillary tube (see Fig. 3.3) filled with a wetting liquid [36, 5].
The Bond number ρlgR2/γ, the Reynolds number Re = ρlUR/µl and the Capillary
number Ca = µlU/γ are all supposed to be small, with ρl the density of the liquid,
g the gravity constant, R the radius of the tube, µl the viscosity of the liquid and
γ the surface tension between the liquid and the gas. When the liquid-gas interface
moves, it is deformed close to the wall, resulting in the deposition of a thin liquid
film of height H. Indeed, away from the walls, the characteristic length scale of
both viscous effects and surface tension is the radius R. Since the capillary number
is small, viscous stresses are too weak to deform the interface. This corresponds to
the region AB on Fig. 3.3, which will be called the static meniscus. But closer to
the wall (region BC on Fig. 3.3), velocity gradients are high since the velocity of
the fluid is equal to 0 on the walls and U on the moving interface (see triple line
dynamics). Thus, viscous effects are strong enough to deform the interface. This
region will be called the dynamic meniscus. Finally, away from the interface (region
following point C on Fig. 3.3) a thin film of constant thickness H lies on the walls.
Dimensional analysis
From dimensional analysis, one can estimate the evolution of the thickness of the
liquid filmH as a function of the Capillary number [2]. Indeed, in the region BC, the
film thickness is determined by the equilibrium between viscous force and pressure
gradients:
µlU
H2∼ γ
LR(3.1)
with L the (unknown) length of the meniscus. Now the matching between dynamic
and static meniscus (equilibrium of Laplace pressure) gives:
− γ
R− γH
L2∼ −−2γ
R(3.2)
since one of the radius of curvature is equal to R, while the other is equal to[
d2h(x)/dx2]
−1 ∼ L2/H, with h(x) the function defining the shape of the interface
along x-axis. From this equation we get, L ∼√HR, and combined with equation
(3.1):
H/R ∼ Ca2/3 (3.3)
The lubrication approximation
Here we assume that the transition region BC can be considered as plane and not
annular. This approximation is discussed by Bretherton [5] and is valid as long as the
Capillary number is small. The equation is written in the bubble frame of reference,
which is moving at constant speed U . Incompressible steady Navier-Stokes equation
3.2. Bretherton law 71
H
x
y
h(x)v(y)
x
B
C
L
Figure 3.4: Lubrication approximation in region BC.
at small Reynolds, Bond and Capillary numbers can be written under the form:
∂vx∂x
+∂vy∂y
= 0
µl
[
∂2vx∂x2
+∂2vx∂y2
]
=∂p
∂x
µl
[
∂2vy∂x2
+∂2vy∂y2
]
=∂p
∂y
with vx an vy the velocity projections along x-axis and y-axis respectively, p the
pressure and µl the viscosity. If we introduce the characteristic scales of the problem,
L, H, U, V and P, such that:
x = Lx
y = Hy
vx = Uvx
vy = V vy
p = P p
we obtain the following dimensionless equations:
U
L
∂vx∂x
+V
H
∂vy∂y
= 0 (3.4)
µl
[
U
L2
∂2vx∂x2
+U
H2
∂2vx∂y2
]
=P
L
∂p
∂x(3.5)
µl
[
V
L2
∂2vy∂x2
+V
H2
∂2vy∂y2
]
=P
H
∂p
∂y(3.6)
From the first equation we obtain:
U
L∼ V
H
In the second and third equations, the first term can be neglected compared to the
second one since:H2
L2∼ H
R∼ Ca2/3 ≪ 1
72 Chapter 3. Dynamics of bubbles or plugs in confined geometries
according to equation (3.3). Finally, from equation (3.5), we have:
P ∼ µlUL
H2∼ µlV L
2
H3
and thus the second term on left hand side of equation (3.6) can be neglected
compared to the term on the right hand side. The equations (3.4) to (3.6) become:
∂vx∂x
+∂vy∂y
= 0 (3.7)
∂2vx∂y2
=∂p
∂x(3.8)
∂p
∂y= 0 =⇒ p = p(x) (3.9)
Thus the lubrication equation describing the fluid motion in region BC (with dimen-
sion) is:
∂2vx∂y2
(x, y) =1
µl
dp
dx(x) (3.10)
The integration of this equation gives:
vx =1
2µl
dp
dx(x)y2 +K1(x)y +K2(x) (3.11)
Boundary conditions
To determine the different functions appearing in this expression, we must apply
the boundary conditions.
Boundary condition on the wall
On the wall, since the frame of reference is linked to the moving bubble, we have:
vx(y = 0) = −U (3.12)
Boundary condition at the liquid-gas interface
At the interface between the liquid and the gas (y = h(x)) and in the absence
of phase change, the momentum conservation (equation (1.27) in chapter 1) is:
⇒
σ l .−→n =
⇒
σ g .−→n − γ C−→n (3.13)
with⇒
σ l= −p⇒
I +2µl⇒
D and⇒
σ g= −pg⇒
I +2µg⇒
Dg the stress tensors in the liquid
and gas phase respectively, γ the surface tension, and −→n the surface normal oriented
from the liquid phase to the gas phase. The expression of the rate of strain tensor
in the liquid phase is:
⇒
D=
∂vx∂x
12∂vx∂y 0
12∂vx∂y
∂vy∂y 0
0 0 0
3.2. Bretherton law 73
Since∂vy∂x ,
∂vx∂x , and
∂vy∂y ≪ ∂vy
∂x , the rate of stress tensor becomes at first order:
⇒
D=
0 12∂vx∂y 0
12∂vx∂y 0 0
0 0 0
Because of the small interface curvature (H/L ≪ 1), the normal vector −→n and
tangential vector−→t can be approximated by:
−→n ∼ −→y , −→t ∼ −→x
Thus the projection of equation (3.13) onto−→t gives:
(3.13).−→t =⇒
(
⇒
D .−→n)
.−→t =
µgµl
(
⇒
D .−→n)
.−→t ≈ 0 since
µgµl
≪ 1
that is to say:
∂vx∂y
(y = h(x)) = 0 (3.14)
Finally the projection of equation (3.13) onto−→t gives (see demonstration below):
p∗ = p− pg = − γ
R− γ
d2h(x)
dx2(3.15)
with p∗ the difference of pressure between the liquid and gas phases.
Expression of the curvature of a graph defined by y = y(x)
The mathematical definition of the curvature of a graph is:
κ =
∣
∣
∣
∣
∣
d−→t
ds
∣
∣
∣
∣
∣
with−→t the tangential vector and s the arc length defined by ds =
√
dx2 + dy2. The
tangential vector can be written as:
−→t =
1√
1 + y′2
[
1
y′(x)
]
with y′ = dy/dx the derivative of y with respect to x. The curvature can be expressed
as:
κ =
∣
∣
∣
∣
∣
d−→t
ds
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
d−→t
dx
dx
ds
∣
∣
∣
∣
∣
(3.16)
with:d−→t
dx=
1
(1 + y′2)3/2
[ −y′′y′y′′
]
(3.17)
74 Chapter 3. Dynamics of bubbles or plugs in confined geometries
and:ds
dx=√
1 + y′2 (3.18)
Finally, if we replace equation (3.18) and the norm of equation (3.17) in equation
(3.16), we obtain:
κ =
∣
∣y′′2∣
∣
(1 + y′2)3/2(3.19)
Demonstration of equation (3.15)
The curvature C in equation (3.13) is the sum of transverse and axial curvatures
κt and κa. Both curvatures are negative. The transverse curvature is obviously
equal to κt = − 1R at first order (that is to say if we neglect H compared to R). The
axial curvature can be computed from formula (3.19):
κa = − d2h/dx2
(1 + (dh/dx)2)3/2that is to say at first order κa = −d
2h
dx2
Mass conservation
Equations (3.12), (3.14) and (3.15) express momentum conservation on the wall
and at the gas-liquid interface. An additional equation is required to express the
mass conservation in the liquid layer. The conservation of the volumetric flow rate
gives:
φ =
∫ h(x)
ovx(y)dy = −UH (3.20)
since a constant layer of height H is left behind the meniscus.
Complete solution
Equation of the free surface h(x)
From boundary conditions (3.12), (3.14), we can establish the expressions of the
two functions K1(x) and K2(x) appearing in equation (3.11):
K1(x) = − U
K2(x) = − 1
µl
dp∗
dxh(x)
with p∗ = p− pg. Thus the velocity field takes the form:
vx(x, y) =1
2µl
dp∗
dx
(
y2 − 2h(x)y)
− U
From mass conservation (equation (3.20)), we obtain:
φ =
∫ h(x)
ovx(y)dy = − 1
µl
dp∗
dx
h3
3− Uh(x) = −UH
3.2. Bretherton law 75
that is to say:dp∗
dx=
3µlU(H − h)
h3(3.21)
If we differentiate equation (3.15) with respect to x, we obtain a second equation:
dp∗
dx= −γ d
3h(x)
dx3(3.22)
Combining these two equations gives:
d3h
dx3= 3Ca
h(x) −H
h3
with Ca = µlU/γ the Capillary number. Finally, if we introduce the variables:
ψ =h(x)
Hand ξ = 31/3Ca1/3
x
H
and substitute them in previous equation, we get:
ψ3 d3ψ
dξ3= ψ − 1 (3.23)
Asymptotic matching
3Ψζd 3
=Ψ−1Ψ d2
h/H>>1Ψ=
dd3Ψ
ζ3= 0
B
C
L
H
Ψ=h/H~13dΨ
dζ3=Ψ−1
ζ<<−1 ζ>>1ζ=Ο(1)
1 2
h/H~1
3
Ψ=
Figure 3.5: Asymptotic matching in region BC.
As long as 31/3Ca1/3L/H ≫ 1, the dynamic meniscus can be separated into
three asymptotic region (see Fig. 3.5):
76 Chapter 3. Dynamics of bubbles or plugs in confined geometries
1. A first region (ξ ≪ −1), in red on Fig. 3.5, where ψ = h/H ∼ 1 and thus
equation (3.23) can be approximated by:
d3ψ
dξ3= ψ − 1
2. A second region (|ξ| = O(1)), in blue on Fig. 3.5, where no approximation
can be made and the full equation:
ψ3 d3ψ
dξ3= ψ − 1
must be solved
3. A third region (ξ ≫ 1), in green on Fig. 3.5, where ψ = h/H ≫ 1 and thus
equation (3.23) can be approximated by:
d3ψ
dξ3= 0
The solution of the equation in the third region is:
ψ3 =1
2P1ξ
2 +Q1ξ +R1
with P1, Q1 and R1 three constants. If we go back to the definition of ξ and ψ, we
obtain:
h3(x) =1
2P1
(3Ca)2/3
Hx2 +Q1(3Ca)
1/3x+R1H
The matching between the curvature of the dynamic meniscus in this region and
the static meniscus gives:
d2h3(x)
dx2+
1
R=P1
H(3Ca)2/3 +
1
R≈ 2
R
From this equation we find the expression of the thickness of the film H as a function
of the Capillary number:H
R≈ P1 (3Ca)
2/3
Inly the value of P1 has not been established yet. This expression matches the one
previously obtained from dimensional analysis (equation (3.3)).
The solution of the equation in the first region is:
ψ1 = 1 + α1eξ + e−1/2ξ
[
β1 cos(
√3
2ξ) + η1sin(
√3
2ξ)
]
with α1, β1 and η1 three constants. Since in the first region ξ ≪ −1 and the height
ψ1 cannot become infinite when ξ → −∞, thus β1 = η1 = 0. The solution in this
region is therefore:
ψ1 = 1 + α1eξ
3.2. Bretherton law 77
The constant α1 can be made unity by a suitable change of origin of ξ, and thus the
solution in this region is unique:
ψ1 = 1 + eξ
Complete solution of the problem
To get the complete solution of the problem, the values of P1, Q1 and R1 must
be calculated. They can be computed numerically through stepwise integration of
equation (3.23) from the first region where the solution is known to the third region.
Bretherton in his paper obtains:
P1 ≈ 0.643, Q1 = 0 and R1 ≈ 2.79
Thus we have:
h1(x) = H(
1 + exp(31/3Ca1/3x/H))
(3.24)
h3(x) =1
2
x2
R+ 1.79(3Ca)2/3R (3.25)
The final expression of the layer thickness as a function of the capillary number
is:
H
R≈ 0.643 (3Ca)2/3 (3.26)
Drift velocity of the bubble
Because of the liquid layer left on the walls, the speed of the bubble U exceeds
the average speed of the fluid V in the tube by an amount WU such as:
W =U − V
U≈ 2
H
R≈ 1.29(3Ca)2/3 (3.27)
since the flow rate conservation gives:
U × π(R−H)2 = V πR2
Thus, one must be careful when approximating the velocity of a liquid finger by
the velocity of a bubble moving in it.
Pressure drop at the front interface
Finally, because of the liquid film left on the walls, the curvature in region AB
is larger than 2/R. It is equal to 2/(R −H∗), with H∗ the height when the sphere
tangent becomes parallel to the walls (See Fig. 3.6). Since equation (3.24) describes
78 Chapter 3. Dynamics of bubbles or plugs in confined geometries
Air
H H*
Figure 3.6: Definition of H∗.
a continuation of the spherical region AB, we can compute the height H∗ from this
formula:
H∗ = 1.79(3Ca)2/3R
Thus the curvature is:
2
R−H∗≈ 2
R(1 +
H∗
R) ≈ 2
R
(
1 + 1.79(3Ca)2/3)
Therefore, the bubble motion induces a dynamic pressure drop ∆P through the
front meniscus in addition to the static value (2γ/R):
∆P =2γ
R× 1.79(3Ca)2/3 (3.28)
General considerations
In this section, we have solved the problem of a semi-infinite bubble moving at a
constant speed in a capillary tube filled with a wetting liquid. The liquid layer left
on the walls can be seen as a way to remove the singularity at the contact line.
Indeed, in the absence of such a layer, the same singularity as the one appearing at
the meniscus of a triple line advancing on a dry substrate would appear. Now, why
do we leave more liquid when we blow harder in the tube (i.e. U increases) ? Simply
because, if we blow harder, viscous stress increases, and thus the interface can be
deformed more easily. Or, in other words, since viscous stresses in the dynamic
meniscus are of the order of µlU/H2, the higher U is, the higher H must be to
equilibrate surface tension.
3.2.2 Long bubble motion
In this section, we consider a long bubble moving in a cylindrical tube at constant
speed U in a capillary tube filled with a wetting liquid (see Fig. 3.7). This problem
is similar to the previous one but in addition, the solution for the rear interface
must match with the solution for the front interface. As the front meniscus, the
rear meniscus can be decomposed into a first region (region EF on Fig. 3.7) where
3.2. Bretherton law 79
A
BC
Liquid
D
U
y
xE
F Air
Figure 3.7: Sketch of the motion of a bubble in a cylindrical tube at a constant
velocity U .
viscous stress is not sufficient to deform the interface (the static meniscus) and a
second region DE where the lubrication approximation can be used to compute
the deformation (the dynamic meniscus). Again, the dynamic meniscus DE can be
devided into three asymptotic regions (regions 4, 5 and 6 on Fig. 3.8) with the same
equations as in regions 1, 2 and 3.
H
L
6 5 4
E
D
Figure 3.8: Asymptotic matching in region DE.
Thus, in region 6 the solution of the equations is:
ψ6 =1
2P2ξ
2 +Q2ξ +R2
while in region 4, the solution becomes:
ψ4 = 1 + e−1/2ξ
[
β2 cos(
√3
2ξ) + η2sin(
√3
2ξ)
]
The solution in region 4 differs from the one in region 1 since, this time, the film
thickness cannot become infinite when ξ → +∞ instead of −∞. We see that this
solution has an oscillating part, and indeed some “waves” are observed experimen-
tally in this part of the meniscus. We also observe that a simple change of origin is
not sufficient to determine both β2 and η2. This is why the solution in the region
DE must be matched with the solution in the region BC to obtain the complete
solution of the problem.
80 Chapter 3. Dynamics of bubbles or plugs in confined geometries
3.3 Large capillary number and rectangular channels
In this section, we will describe briefly what happens when the geometry is changed,
or the capillary number increased.
3.3.1 Large capillary numbers
B.
FIG. 2. Normalized film thickness as a function of the capillary number for
viscous liquids. The full circles are Taylor’s data, which are compared withFigure 3.9: Comparison of the evolution of the normalized film thickness predicted
by equation (3.31) to experimental data of Taylor (full circle) and Aussilous & Quéré
(open squares).
When the capillary number is increased, viscous stresses induce larger deforma-
tion of the interface and thus more liquid is left on the walls. However above a
certain threshold, this layer thickness reaches a saturation regime. Following Aus-
silous and Quéré [2], this saturation can be explained through dimensional analysis.
When the capillary number is increased, the layer thickness H cannot be neglected
anymore compared to the radius R. Thus this latter must be replaced by R−H in
equations (3.1) and (3.2):µlU
H2∼ γ
L(R−H)(3.29)
with L the (unknown) length of the meniscus given by
− γ
R−H− γH
L2∼ − −2γ
R−H(3.30)
Thus dimensional analysis, gives:
H
R∼ Ca2/3
1 + Ca2/3
3.3. Large capillary number and rectangular channels 81
Indeed, with appropriate coefficients:
H
R=
1.34Ca2/3
1 + 1.34 × 2.5 Ca2/3(3.31)
Aussilous and Quéré obtain a good fit with experimental data (see Fig. 3.9). These
authors also studied the influence of inertia (when the Weber number is not small
anymore). They showed that inertia effects tends to thicken the film.
3.3.2 Rectangular channels
Most of the channels used in microfluidics systems have rectangular cross section,
since they are produced with soft lithography technique. The main difference in
rectangular channels, is that a large amount of liquid is left in the corner of the
channels since the bubble aims at reaching a cylindrical shape to reduce transverse
curvature (see Fig. 3.11). Bubbles motion in rectangular channels has been investi-
Air Air Air
Liquid Liquid Liquid
Ca<<1 Ca<1 Ca>1
Figure 3.10: Section of a semi-infinite bubble moving in a rectangular channel at
different Capillary numbers.
gated both experimentally and numerically [42, 41, 12]. Recently, de Lózar et al. [7]
have shown that the “coating film thickness” in a rectangular channel of aspect ratio
α = Ly/Lz can be inferred from the one in a square channel if the Capillary num-
ber Ca is replaced by a modified one Ca =[
1 + 0.12(α − 1) + 0.018(α − 1)2]
Ca
(see Fig. 3.12). Of course in a rectangular channel, the “film thickness” H is not
appropriate to determine how much liquid is left on the wall since it is not constant
over the whole section. Instead the wet fraction m = (S −Sg)/S is used with S the
surface of the channel section and Sg the surface occupied by air in a section of the
semi-infinite bubble (behind the meniscus), see Fig. 3.11. The data of de Lozár et
al. can be fitted by the following law:
m = 1−[
1 + 2.631Ca2/3
1 + 4.385Ca2/3
]2
which is nothing but the law proposed by Aussilous and Quéré but expressed in
terms of wet fraction (with appropriate coefficients).
82 Chapter 3. Dynamics of bubbles or plugs in confined geometries
y zS = L Lm = (S−S )/Sg
L
L
y
z Sg
Figure 3.11: Section of a semi-infinite bubble moving in a rectangular channel:
definition of the wet fraction
Figure 3.12: Wet fraction m as a function of the modified capillary number Ca
for different aspect ratio α [7]. The divergence of the curves for the low capillary
numbers is due to the appearance of gravity effects.
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