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FFlluuiidd MMeecchhaanniiccss
CChhaapptteerr--11 IInnttrroodduuccttiioonn
PPrreeppaarreedd BByy
BBrriijj BBhhoooosshhaann
AAsssstt.. PPrrooffeessssoorr
BB.. SS.. AA.. CCoolllleeggee ooff EEnngggg.. AAnndd TTeecchhnnoollooggyy
MMaatthhuurraa,, UUttttaarr PPrraaddeesshh,, ((IInnddiiaa))
SSuuppppoorrtteedd BByy::
PPuurrvvii BBhhoooosshhaann
In This Chapter We Cover the Following Topics
S. No. Contents Page No.
1.1 Viscosity 3
1.2 Non-Newtonian Fluids 5
1.3 Flow Patterns 7
1.4 What Is Surface 9
1.5 Surface Tension
Surface Energy
Surface Tension on Liquid Droplet
Surface Tension on Hollow Bubble
Surface Tension on A Liquid Jet
9
10
10
11
11
1.6 Capillarity 11
1.7 Properties Of Fluids 13
References:
1. Andersion J. D. Jr., Computational Fluid Dynamics “The Basics with applications”,
1st Ed., McGraw Hill, New York, 1995.
2. Frank M. White, Fluid Mechanics, 6th Ed., McGraw Hill, New York, 2008.
3. Frank M. White, Viscous Fluid Flow, 2nd Ed., McGraw Hill, New York, 1991.
4. Fox and McDonald’s, Introduction to Fluid Mechanics, 6th Ed., John Wiley & Sons,
Inc., New York, 2004.
5. Welty James R., Wicks Charles E., Wilson Robert E. and, Rorrer Gregory L.,
Fundamentals of Momentum, Heat, and Mass Transfer, 5th Ed. John Wiley & Sons,
Inc., New York, 2008.
6. Mohanty A. K., Fluid Mechanics, 2nd Ed, Prentice Hall Publications, New Delhi,
2005.
7. Gupta Vijay, and, Gupta S. K., Fluid Mechanics & its Applications, 1st Ed., New Age
International (P) Limited, Publishers, New Delhi 2005.
8. Cengel & Cimbala, Fluid Mechanics Fundamentals and Applications, 1st Ed.,
McGraw Hill, New York, 2006.
9. Modi & Seth, Hydraulics and Fluid Mechanics, Standard Publications.
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2 Chapter 1: Introduction
Please welcome for any correction or misprint in the entire manuscript and your
valuable suggestions kindly mail us [email protected].
The study of Fluid Mechanics is directed towards the behaviour of a fluid at rest or in
motion. Some of the notable examples of application are: consideration of fluid statics for
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3 Fluid Mechanics By Brij Bhooshan
the design of a dam; flow of water through pipes and distribution to domestic service
lines; production of mechanical power in hydraulic, steam or gas turbines; working of a
hydraulic pump or of a compressor; load carrying capacity of a hydrodynamic bearing;
motion of an aircraft or a missile in the atmosphere and the like. The study also
encompasses methods and devices for the measurement of various parameters, e.g., the
pressure and velocity in a fluid at rest or in motion. Indeed the scope of the study of
fluid mechanics is quite vast and is of considerable interest to mechanical, aeronautical,
civil, chemical and mining engineers. Apart from engineers, the mathematicians and
physicists have made significant contributions to our knowledge of fluid mechanics.
A fluid is a substance that continues to deform so long as a shear force is acted upon it.
The term 'fluid' encompasses both liquids and gases. The study of Fluid Mechanics is
concerned with the macroscopic behaviour of fluids at rest or in motion.
1.1 VISCOSITY
We have defined a fluid as a substance that continues to deform under the action of a
shear stress. In the absence of a shear stress, there will be no-deformation.
Consider the behavior of a fluid element between the two infinite plates shown in
Diagram 1.1(a). The rectangular fluid element is initially at rest at time t. Let us now
suppose a constant rightward force δFx is applied to the upper plate so that it is dragged
across the fluid at constant velocity δu. The relative shearing action of the infinite plates
produces a shear stress, τyx, which acts on the fluid element and is given by
𝜏𝑦𝑥 = lim𝛿𝐴𝑦→0
𝛿𝐹𝑥𝛿𝐴𝑦
=𝑑𝐹𝑥𝑑𝐴𝑦
where δAy is the area of contact of the fluid element with the plate and δFx is the force
exerted by the plate on that element.
Diagram 1.1
During time interval δt, the fluid element from position MNOP to MNOP. The rate of
deformation is given by
deformation rate = lim𝛿𝑡→0
𝛿𝛼
𝛿𝑡=
𝑑𝛼
𝑑𝑡
We want to express d/dt in terms of readily measurable quantities. This can be done
easily. The distance, δl, between the points M and M is given by
δl = δuδt
Equating these two expressions for δl gives
𝛿𝑙
𝛿𝑦= tan 𝛿𝛼 =
𝛿𝑢 ∙ 𝛿𝑡
𝛿𝑢
where δ is very small.
(b) newtonian shear distribution
in a shear layer near a wall. (a) Fluid element at time t
Force Fx
Velocity u
𝜏 ∝𝑑𝛼
𝑑𝑡
Fluid element
at time t + dt
Fluid element
at time t y
x O N
P P l
M M Velocity
profile
No slip at wall x
y
dy
du
u(y)
𝜏 = 𝜇𝑑𝑢
𝑑𝑦
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4 Chapter 1: Introduction
Taking the limits of both sides of the equality, we obtain
𝑑𝛼
𝑑𝑡=
𝑑𝑢
𝑑𝑦
Thus, the fluid element, when subjected to shear stress τ, experiences a rate of
deformation (shear rate) given by du/dy. We have established that any fluid that
experiences a shear stress will flow (it will have a shear rate). What is the relation
between shear stress and shear rate? Fluids in which shear stress is directly
proportional to rate of deformation are Newtonian fluids. The term non-Newtonian is
used to classify all fluids in which shear stress is not directly proportional to shear rate.
𝜏 ∝𝑑𝛼
𝑑𝑡=
𝑑𝑢
𝑑𝑦
𝜏 = 𝜇𝑑𝛼
𝑑𝑡= 𝜇
𝑑𝑢
𝑑𝑦 [1.1]
The constant of proportionality in Eq. 1.1 is the absolute (or dynamic) viscosity, μ, and
the term du/dy is velocity gradient.
All gases and water obeys Newton’s law of shear stress. Substance like Tar, mlasses,
which are commonly named as visco-elastic materials, and physical mixture of fluids
and solids such as a slurries, exhibit no-linear stress-strain behaviour is termed as Non-
Newtonian fluid. When a real fluid has no-relative velocity with respect to the solid
surface in contact. This is the condition of no-slip. Because of the no-slip condition, the
fluid layer on the lower surface remains stationary where as the top layer moves with
the same speed as the upper plate.
The term kinematic viscosity is defined as
= /.
The shear strain in a fluid is proportional to the time rate of deformation, in contrast to
only deformation in a solid.
The shear stress is proportional to the slope of the velocity profile and is greatest at the
wall. Further, at the wall, the velocity u is zero relative to the wall: This is called the no-
slip condition and is characteristic of all viscous-fluid flows.
Flow between Plates
A classic problem is the flow induced between a fixed lower plate and an upper plate
moving steadily at velocity V, as shown in Diagram 1.2. The clearance between plates is
h, and the fluid is newtonian and does not slip at either plate. If the plates are large,
this steady shearing motion will set up a velocity distribution u(y), as shown, with v = w
= 0. The fluid acceleration is zero everywhere.
Diagram 1.2 Viscous flow induced by relative motion between two parallel plates.
With zero acceleration and assuming no pressure variation in the flow direction, you
should show that a force balance on a small fluid element leads to the result that the
shear stress is constant throughout the fluid. Then Eq. (1.1) becomes
Fixed plate u = 0
Viscous
fluid
x
y
h u (y)
V
Moving plate u = V
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5 Fluid Mechanics By Brij Bhooshan
𝑑𝑢
𝑑𝑦=
𝜏
𝜇
which we can integrate to obtain
u = a + by
The velocity distribution is linear, as shown in Diagram 1.2, and the constants a and b
can be evaluated from the no-slip condition at the upper and lower walls:
u = 0 = a + b(0) at y = 0
u = V = a + b(h) at y = h
Hence a = 0 and b = V/h. Then the velocity profile between the plates is given by
𝑢 = 𝑉 ∙𝑦
ℎ [1.2]
1.2 NON-NEWTONIAN FLUIDS
Fluids which do not follow the linear law of Eq. (1.1) are called nonnewtonian and are
treated in books on rheology. Diagram 1.3a compares four examples with a Newtonian
fluid. Numerous empirical equations have been proposed to model the observed relations
between τ and du/dy for time-independent fluids. They may be adequately represented
for many engineering applications by the power law model, which for one-dimensional
flow becomes
𝜏 = 𝑘 𝑑𝑢
𝑑𝑦 𝑛
[1.3]
where the exponent, n, is called the flow behavior index and the coefficient, k, the
consistency index. This equation reduces to Newton’s law of viscosity for n = 1 with k =
μ.
To ensure that τ has the same sign as du/dy, Eq. 1.3 is rewritten in the form
𝜏 = 𝑘 𝑑𝑢
𝑑𝑦 𝑛−1
.𝑑𝑢
𝑑𝑦= 𝜂 ∙
𝑑𝑢
𝑑𝑦 [1.4]
The term η = kdu/dyn–1 is referred to as the apparent viscosity. The idea behind Eq.
1.4 is that we end up with a viscosity η that is used in a formula that is the same form
as Eq. 1.1, in which the Newtonian viscosity μ is used. The big difference is that while μ
is constant (except for temperature effects), η depends on the shear rate. Most non-
Newtonian fluids have apparent viscosities that are relatively high compared with the
viscosity of water.
An ideal fluid (since = 0) or a real fluid at rest (u = 0) cannot develop shear stress. The
ideal fluid, therefore, offers no resistance to deformation and motion.
A dilatant, or shear-thickening, fluid increases resistance with increasing applied stress.
If the apparent viscosity increases with increasing deformation rate (n > 1) the fluid is
termed dilatant (or shear thickening). Suspensions of starch and of sand are examples of
dilatant fluids.
𝜏 = 𝜇 𝑑𝑢
𝑑𝑦 𝑛
(𝑛 > 1) [1.4𝑎]
Fluids in which the apparent viscosity decreases with increasing deformation rate (n <
1) are called pseudoplastic (or shear thinning) fluids. Most non-Newtonian fluids fall
into this group; examples include polymer solutions, colloidal suspensions, and paper
pulp in water.
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6 Chapter 1: Introduction
Diagram 1.3 Rheological behavior of various viscous materials
A pseudoplastic, or shear-thinning, fluid decreases resistance with increasing stress. If
the thinning effect is very strong, as with the dashed-line curve, the fluid is termed
plastic.
𝜏 = 𝜇 𝑑𝑢
𝑑𝑦 𝑛
(𝑛 < 1) [1.4𝑏]
A “fluid” that behaves as a solid until a minimum yield stress, τ0, is exceeded and
subsequently exhibits a linear relation between stress and rate of deformation is
referred to as an ideal or Bingham plastic. The corresponding shear stress model is
𝜏 = 𝜏0 + 𝜇 𝑑𝑢
𝑑𝑦 𝑛
(𝑛 < 1) [1.4𝑐]
Clay suspensions, drilling muds, and toothpaste are examples of substances exhibiting
this behavior. Alternately, the limiting case of a plastic substance is one which requires
a finite yield stress before it begins to flow. The linear-flow Bingham plastic idealization
is shown, but the flow behavior after yield may also be nonlinear. An example of a
yielding fluid is toothpaste, which will not flow out of the tube until a finite stress is
applied by squeezing.
In non-Newtonian fluids also some substances which can not be classified as entire
fluids or solids, but intermediate behaviour. These are termed as visco-elastic fluid.
A visco-elastic fluid is
𝜏 = 𝛼𝐸 + 𝜇 𝑑𝑢
𝑑𝑦 [1.4𝑑]
where E is Young modulus, and < 1.
The study of non-Newtonian fluids is further complicated by the fact that the apparent
viscosity may be time-dependent. Thixotropic fluids show a decrease in η with time
under a constant applied shear stress; many paints are thixotropic. Rheopectic fluids
show an increase in η with time. After deformation some fluids partially return to their
original shape when the applied stress is released; such fluids are called viscoelastic.
A further complication of nonnewtonian behavior is the transient effect shown in
Diagram 1.3(c). Some fluids require a gradually increasing shear stress to maintain a
constant strain rate and are called rheopectic. The opposite case of a fluid which thins
out with time and requires decreasing stress is termed thixotropic.
1.3 FLOW PATTERNS
(c) effect of time on
applied stress.
1- Ideal fluid; 2- Pseudoplastic; 3- ideal Bingham plastic;
4- Ideal Solid; 5- Plastic; 6- non-Newtonian; 7- Dilatants
(b) apparent viscosity,
η, as a function of
deformation rate
𝑑𝑢
𝑑𝑦=
𝑑𝑥
𝑑𝑡
η
Const.
Strain
rate
Common
fluid
Thixotropic
Rheopectic
t
Pseudoplastic
Dilatants
Newtonian
1
(a) stress versus
strain rate
𝑑𝜃
𝑑𝑡=
𝑑𝑢
𝑑𝑦
Y
ield
str
ess
7 6
5
Ideal plastic
4
Newtonian
3
2
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7 Fluid Mechanics By Brij Bhooshan
Fluid mechanics is a highly visual subject. The patterns of flow can be visualized in a
dozen different ways, and you can view these sketches or photographs and learn a great
deal qualitatively and often quantitatively about the flow. Four basic types of line
patterns are used to visualize flows:
1. A streamline is a line everywhere tangent to the velocity vector at a given
instant.
2. A pathline is the actual path traversed by a given fluid particle.
3. A streakline is the locus of particles which have earlier passed through a
prescribed point.
4. A timeline is a set of fluid particles that form a line at a given instant.
The streamline is convenient to calculate mathematically, while the other three are
easier to generate experimentally. Note that a streamline and a timeline are
instantaneous lines, while the pathline and the streakline are generated by the passage
of time. The velocity profile shown in Diagram 1.4 is really a timeline generated earlier
by a single discharge of bubbles from the wire. A pathline can be found by a time
exposure of a single marked particle moving through the flow. Streamlines are difficult
to generate experimentally in unsteady flow unless one marks a great many particles
and notes their direction of motion during a very short time interval. In steady flow the
situation simplifies greatly:
Streamlines, pathlines, and streaklines are identical in steady flow.
Diagram 1.4 The most common method of flow-pattern presentation:
In fluid mechanics the most common mathematical result for visualization purposes is
the streamline pattern. Diagram 1.4a shows a typical set of streamlines, and Diagram
1.4b shows a closed pattern called a streamtube. By definition the fluid within a
streamtube is confined there because it cannot cross the streamlines; thus the
streamtube walls need not be solid but may be fluid surfaces.
Diagram 1.5 shows an arbitrary velocity vector. If the elemental arc length dr of a
streamline is to be parallel to V, their respective components must be in proportion:
Stream lines
𝑑𝑥
𝑢=
𝑑𝑦
𝑣=
𝑑𝑧
𝑤=
𝑑𝑟
𝑉 [1.5]
If the velocities (u, v, w) are known functions of position and time, Eq. (1.5) can be
integrated to find the streamline passing through the initial point (x0, y0, z0, t0). The
method is straightforward for steady flows but may be laborious for unsteady flow.
The pathline, or displacement of a particle, is defined by integration of the velocity
components, as:
Pathline: x = u dt, y = v dt, z = w dt [1.6]
Given (u, v, w) as known functions of position and time, the integration is begun at a
specified initial position (x0, y0, z0, t0). Again the integration may be laborious.
(a) Streamlines are everywhere
tangent to the local velocity vector
V
(b) a streamtube is formed by a
closed collection of streamlines.
No flow across stream
tube wall
Individual stream lines
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8 Chapter 1: Introduction
Streaklines, easily generated experimentally with smoke, dye, or bubble releases, are
very difficult to compute analytically.
Diagram 1.5 Geometric relations for defining a streamline.
Application 1.1: Given the steady two-dimensional velocity distribution
u = Kx, v = –Ky w = 0 [1.7]
where K is a positive constant, compute and plot the streamlines of the flow, including
directions, and give some possible interpretations of the pattern.
Solution: Since time does not appear explicitly in Eq. (1.7), the motion is steady, so that
streamlines, pathlines, and streaklines will coincide. Since w = 0 everywhere, the
motion is two dimensional, in the xy plane. The streamlines can be computed by
substituting the expressions for u and v into Eq. (1.5):
u = Kx, v = –Ky
𝑑𝑥
𝐾𝑥= −
𝑑𝑦
𝐾𝑦
𝑑𝑥
𝑥= −
𝑑𝑦
𝑦
Integrating, we obtain ln x = –ln y + ln C, or
xy = C [1.8]
Diagram 1.6 Streamlines for the velocity distribution given by Eq. (1.7), for K > 0.
This is the general expression for the streamlines, which are hyperbolas. The complete
pattern is plotted in Diagram 1.6 by assigning various values to the constant C. The
arrowheads can be determined only by returning to Eqs. (1.7) to ascertain the velocity
component directions, assuming K is positive. For example, in the upper right quadrant
(x > 0, y > 0), u is positive and v is negative; hence the flow moves down and to the right,
establishing the arrowheads as shown.
Note that the streamline pattern is entirely independent of constant K. It could
represent the impingement of two opposing streams, or the upper half could simulate
C = 3
2
1
y
x
3 2
1
–1 –2
C = –3
–3
O
–2 –1
x
z
y
w
v
u
dx
dy
dz dr
V V
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9 Fluid Mechanics By Brij Bhooshan
the flow of a single downward stream against a flat wall. Taken in isolation, the upper
right quadrant is similar to the flow in a 90° corner.
Finally note the peculiarity that the two streamlines (C = 0) have opposite directions
and intersect.
This is possible only at a point where u = v = w = 0, which occurs at the origin in this
case. Such a point of zero velocity is called a stagnation point.
1.4 WHAT IS SURFACE
Consider a liquid body which is contact with air as shown in Diagram 1.7(a), the upper
most layer of the liquid with the thickness of molecules diameter is consider to be liquid
air surface. Difference between the state of molecules present on the surface and in the
bulk.
Diagram 1.7
Surface tension is a property of surface in which the surface tries to minimize its area.
Why does a surface try to minimize its area?
The molecule present inside the bulk experience uniform distance of force according to
the net force acting is equal to zero. The molecule present on the surface is pulled
inverts potential energy of the molecule on the surface is moved then the potential
energy of the bulk as shown in Diagram 1.7(b). Every liquid system tried to minimum
its surface area lesser the surface area of the liquid system less will be its potential
energy and greater its stability.
1.5 SURFACE TENSION
The intermolecular forces within a given fluid are balanced on the average. The
magnitude of intermolecular force differs from fluid to fluid. When we consider the
interface of two different fluids, say air and water, the unequal intermolecular forces of
the two give rise to an apparent unbalance force at the interface. This interface force is
named as surface tension and said to act along the line of contact of two fluids.
Consider an imaginary line of length l to have been drawn on the surface as shown in
Diagram 1.8. The molecules is present on this line remain in equilibrium because they
experience equal and opposite force on both side.
Surface tension is force per unit length acting on this line.
S = F/l [1.9]
Diagram 1.8
Surface F
F B
A
Bulk
Liquid
Surface
Air
(b) (a)
Air
U2 > U1 –U2
–U1
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10 Chapter 1: Introduction
The behaviour of the liquid surface is similar to the behaviour of a stretch a rubber
sheet. A stretched vapour sheet always increases to surface area. Work done has to the
external agent that increase the surface the angry is stored in the form of elastic
potential energy.
Properties on which the value of surface tension depends:
1. temperature of surrounding,
2. density and viscosity of the liquid,
3. Magnitude of cohesive and chessive force acting on the molecules present of
layer.
Surface property is the second medium.
Surface Energy
In the Diagram 1.9 shown two surfaces, the upper and lower surface on the both side, it
is connected air and atmosphere. If we applied a force slider so that the slider move
slowly then the x distance the external work done is Fx, this is the increase surface
energy of the shoap film.
Fx = 2Slx = S(2lx)
W = SA [1.10]
where A is the total increase the surface area (upper and lower).
Diagram 1.9
If the slider is move and accelerate then we have to consider the change in the kinetic
energy of the system.
Surface Tension on Liquid Droplet
Consider a small spherical droplet of a liquid of radius r. On the entire surface of the
droplet, the tensile force due to surface tension will be acting.
Diagram 1.10 Forces on droplet
Let = Surface tension of the liquid, p = Pressure intensity inside the droplet (in excess
of the outside pressure intensity), d = Dia. of droplet.
Let the droplet is cut into two halves. The forces acting on one half (say left half) will be
tensile force due to surface tension acting around the circumference of the cut portion as
shown in Diagram 1.10(b) and this is equal to
= Circumference
= d
Pressure force on the area d2/4 and p d2/4 as shown in Diagram 1.10(c). These two
forces will be equal and opposite under equilibrium conditions, i.e.,
(a) Droplet (b) Surface Tension (c) Pressure Force
p
x
Soap film
F 2Sl
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11 Fluid Mechanics By Brij Bhooshan
p d2/4 = d
or p = 4/d [1.11]
Equation (1.11) shows that with the decrease of diameter of the droplet, pressure
intensity inside the droplet increases.
Surface Tension on Hollow Bubble
A hollow bubble like a soap bubble in air has two surfaces in contact with air, one inside
and other outside. Thus two surfaces are subjected to surface tension. Then,
p d2/4 = 2 d
or p = 8/d [1.12]
Surface Tension on a Liquid Jet
Consider a liquid jet of diameter "d" and length 'L' shown in Diagram 1.11.
Diagram 1.11 Forces on liquid jet
Let p = Pressure intensity inside the liquid jet above the outside pressure
= Surface tension of the liquid.
Consider the equilibrium of the semi jet, we have
Force due to pressure = p area of semi jet
= p L d
Force due to surface tension = 2L
Equating the forces, we have
p L d = 2L
𝑝 =𝜍 × 2𝐿
𝐿 × 𝑑 [1.13]
1.6 CAPILLARITY
Capillarity is defined as a phenomenon of rise or fall of a liquid surface in a small tube
relative to the adjacent general level of liquid when the tube is held vertically in the
liquid. The rise of liquid surface is known as capillary rise while the fall of the liquid
surface is known as capillary depression. It is expressed in terms of cm or mm of liquid.
Its value depends upon the specific weight of the liquid, diameter of the tube and surface
tension of the liquid.
Capillarity Action Due to Surface Tension
Basically capillarity action due to surface tension is wetting or non-wetting liquid.
In case of wetting liquids the force of cohesion which acts between the molecules of the
liquid is lesser then a force of cohesive adhesive which acts between the molecules liquid
and molecules of surface on which are presents due to this liquid.
These force depends upon the intermolecular distance and their magnitude is directly
proportional to 1/rn when n lies between 6 to 8. To spread itself to the maximum extent
on the surface. The liquid wets the surface and it is termed as wetting liquid. Examples
are glass, and, water from a combination of wetting liquid.
L
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12 Chapter 1: Introduction
In case of non-wetting liquids the force of cohesion is stronger than the force of adhesion.
The liquid molecules tries to decrease there are of the container with the surface.
Examples are Marking, and, anything.
Diagram 1.12
Let us suppose angle of contact ia acute for wetting liquid, and, angle of contact ia
obtuse for non-wetting liquid. Now, at the contact surface we have to obtain force
balance
𝜍𝑠𝑙 cos 𝜃 + 𝜍𝑠𝑎 = 𝜍𝑠𝑙
If 𝜍𝑠𝑙 > 𝜍𝑠𝑎 , physically it means that If 𝜍𝑠𝑙 > 𝜍𝑠𝑎 , therefore it is difference created solid
and liquid surface. In case of mercury drop, we know that drop tries to minimize this
area.
Expression for Capillary Rise
Consider a glass tube of small diameter d opened at both ends and is inserted in a
liquid, say water. The liquid will rise in the tube above the level of the liquid.
Let h = height of the liquid in the tube. Under a state of equilibrium, the weight of liquid
of height h is balanced by the force at the surface of the liquid in the tube. But the force
at the surface of the liquid in the tube is due to surface tension.
Diagram 1.13 Capillary rise.
Let = Surface tension of liquid
= Angle of contact between liquid and glass tube.
The weight of liquid of height h in the tube = (Area of tube h) g
=𝜋
4𝑑2 × ℎ × 𝜌 × 𝑔
where = Density of liquid
Vertical component of the surface tensile force
= ( Circumference) x cos = d cos
For equilibrium equating above two equations, we get 𝜋
4𝑑2 × ℎ × 𝜌 × 𝑔 = 𝜍 × 𝜋𝑑 × cos 𝜃
ℎ =4𝜍 cos 𝜃
𝜌 × 𝑔 × 𝑑 [1.14]
The value of between water and clean glass tube is approximately equal to zero and
hence cos is equal to unity. Then rise of water is given by
ℎ =4𝜍
𝜌 × 𝑔 × 𝑑 [1.15]
h
Liquid
𝜍𝑠𝑙 𝜍𝑙𝑎
𝜍𝑠𝑎
Liquid
Liquid
Water
Drop
Hg
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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)
Copyright by Brij Bhooshan @ 2013 Page 13
13 Fluid Mechanics By Brij Bhooshan
Expression for Capillary Fall
If the glass tube is dipped in mercury, the level of mercury in the tube will be lower than
the general level of the outside liquid as shown in Diagram 1.14.
Let h = Height of depression in tube.
Diagram 1.14
Then in equilibrium, two forces are acting on the mercury inside the tube. First one is
due to surface tension acting in the downward direction and is equal to d cos .
Second force is due to hydrostatic force acting upward and is equal to intensity of
pressure at a depth 'h' Area
= 𝑝𝜋
4𝑑2 = 𝜌 × ℎ × 𝑔 ×
𝜋
4𝑑2
Equating the two, we get
𝜌 × ℎ × 𝑔 ×𝜋
4𝑑2 = 𝜍 × 𝜋𝑑 × cos 𝜃
ℎ =4𝜍 cos 𝜃
𝜌 × 𝑔 × 𝑑 [1.16]
Value of for mercury and glass tube is 128.
1.8 PROPERTIES OF FLUIDS
Density or Mass Density
Density or mass density of a fluid is defined as the ratio of the mass of a fluid to its
volume. Thus mass per unit volume of a fluid is called density. The unit of mass density
in SI unit is kg per cubic metre, i.e., kg/m3. The density of liquids may be considered as
constant while that of gases changes with the variation of pressure and temperature.
Mathematically, mass density is written as
𝜌 =Mass of fluid
Volume of fluid
The value of density of water is 1 gm/cm3, or 1000kg/m3.
Specific Weight or Weight Density
Specific weight or weight density of a fluid is the ratio between the weight of a fluid to
its volume. Thus weight per unit volume of a fluid is called weight density.
Mathematically,
𝑤 =Weight of fluid
Volume of fluid=
Mass of fluid × Acceleration due to gravity
Volume of fluid
w = g
The value of specific weight or weight density (w) for water is 9.81 1000 Newton/m3 in
SI units.
Specific Volume
Mercury
h
p
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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)
Copyright by Brij Bhooshan @ 2013 Page 14
14 Chapter 1: Introduction
Specific volume of a fluid is defined as the volume of a fluid occupied by a unit mass or
volume per unit mass of a fluid is called specific volume. Mathematically, it is expressed
as
Specific Volume =Volume of fluid
Mass of fluid=
1
𝜌
Thus specific volume is the reciprocal of mass density. It is expressed as m3/kg. It is
commonly applied gases.
Specific Gravity
Specific gravity is defined as the ratio of the weight density (or density) of a fluid to the
weight density (or density) of a standard fluid. For liquids, the standard fluid is taken
water and for gases, the standard fluid is taken air. Specific gravity is also called
relative density. It is dimensionless quantity and is denoted by the symbol S.
𝑆 for liquids =Weight density liquid of liquid
Weight density liquid of water
𝑆 for gases =Weight density liquid of gas
Weight density liquid of air
Thus weight density of a liquid = S Weight density of water
= S 1000 9.81 N/m3
The density of a liquid = S Density of water
= S 1000 kg/m3
If the specific gravity of a fluid is known, then the density of the fluid will be equal to
specific gravity of fluid multiplied by the density of water. For example the specific
gravity of mercury is 13.6, hence density of mercury = 13.6 1000= 13600 kg/m3.
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