Instability and transition in ﬂow through deformable … Problem formulation High Reynolds number...

Introduction Problem formulation High Reynolds number limit Pipe Poiseuille flow Plane Poiseuille flow Summary

Instability and transition in flow through deformabletubes and channels

V. Shankar

Department of Chemical Engineering, IIT Kanpur

Pravartana: TEQIP Symposium, IIT Kanpur, October 2013

V. Shankar (ChE, IITK) Flow in deformable tubes and channels TEQIP, Oct 2013 1 / 49

Acknowledgments

Prof. V. Kumaran (IISc Bangalore)

Students at IIT KanpurDr. Gaurav, PhD (2010), currently at IIT Roorkee.R. Neelamegam, PhD (ongoing).A. HemalathaB. SivakumarAashish JainSaurav AnejaJanakiramulu AdepuAkhilesh SahuPavan KumarLalit KumarSameer Kumar

Outline

1 Introduction

2 Problem formulation

3 High Reynolds number limit

4 Pipe Poiseuille flow

5 Plane Poiseuille flow

6 Summary

Outline

1 Introduction

6 Summary

Flow past deformable solids: biological flows

All fluid-conveying vessels within the body are deformable.

Fluid-solid interactions affect vessel’s (dys)function.

Cardiovascular system: transportation of blood from heart to otherorgans.

Flow-induced instability in brachial artery ⇒ Blood-pressuremeasurement (‘Korotkoff sounds’).

Pulmonary flows: air flow in respiratory system.

Flow-induced instabilities: wheezing, snoring and even speech.

Biological flows: a range of Reynolds number (Re ≡ RVρ/µ)O(1)–O(103).

Flow past deformable solids: Microfluidics

Ref: http://blogs.rsc.org/chipsandtips/Microfluidic devices fabricated using a soft elastomer (PDMS).Flow laminar when Re < 1 due to channel/tube dimensions.Mixing of passive scalars purely due to diffusion: Need toenhance mixing.Can we improve mixing by inducing an instability of the laminarflow in channels with soft PDMS walls ?V. Shankar (ChE, IITK) Flow in deformable tubes and channels TEQIP, Oct 2013 6 / 49

Ref: http://blogs.rsc.org/chipsandtips/

Microfluidic devices fabricated using a soft elastomer (PDMS).

Flow laminar when Re < 1 due to channel/tube dimensions.

Mixing of passive scalars purely due to diffusion: Need toenhance mixing.

Can we improve mixing by inducing an instability of the laminarflow in channels with soft PDMS walls ?

Flow past deformable solids: other contexts

Marine and aerospace propulsion: drag reduction by compliantcoatings (transition delay) ?

Do dolphins swim fast because of the compliance of their skin ?

Geophysical phenomena such as volcanic tremors due tointeraction between flow of magma with surrounding rocks.

Laminar-turbulent transition

Osborne Reynolds (1883), “An experimental investigation of thecircumstances which determine whether the motion of water shallbe direct or sinuous..”

(N. Rott,Ann. Rev. Fluid Mech., 1990)Discontinuous transition from laminar to a turbulent flow whenRe ≡ ρVD/µ > 2100.

For rectangular channels, transition at Re ∼ 1200.

Osborne Reynolds (1883), “An experimental investigation of thecircumstances which determine whether the motion of water shallbe direct or sinuous..” (N. Rott, Ann. Rev. Fluid Mech., 1990)Discontinuous transition from laminar to a turbulent flow whenRe ≡ ρVD/µ > 2100.

turbulent

For rectangular channels, transition at Re ∼ 1200.V. Shankar (ChE, IITK) Flow in deformable tubes and channels TEQIP, Oct 2013 8 / 49

Osborne Reynolds (1883), “An experimental investigation of thecircumstances which determine whether the motion of water shallbe direct or sinuous..” (N. Rott, Ann. Rev. Fluid Mech., 1990)

Discontinuous transition from laminar to a turbulent flow whenRe ≡ ρVD/µ > 2100.

For rectangular channels, transition at Re ∼ 1200.

Salient features of flow past soft solids

Rigid walls: Shear modulus (e.g. steel) G ∼ 1010Pa

Soft solids (biological tissues, polymer gels, elastomers):G ∼ 104–106 Pa ∼ O(10−5)Grigid

Soft interfaces easily deformed by fluid stresses.

Elasto-hydrodynamic coupling ⇒ Interfacial waves.

Can affect stability of flow past deformable walls.

Instabilities and transition to turbulence affected by wall elasticity.

A pioneer at IITK: Vijay Garg’s 1977 paper

Transition in gel-coated deformable tubes

Re for transition decreases with wall shear modulus.

Friction factor changes continuously with Re, in contrast to rigidtubes.

Transition in gel-coated deformable tubes

Re for transition decreases with wall shear modulus.

Friction factor changes continuously with Re, in contrast to rigidtubes.

Stability of fluid flows

Laminar flows: always a solution to Navier-Stokes equations.

Landau & Lifshitz“Yet not every solution of the equations of motion, even if it is exact,can actually occur in Nature. The flows that occur in Nature must notonly obey the equations of fluid dynamics, but also be stable”

Need to probe the stability of laminar flows to externaldisturbances.

Objectives of stability analysis:

◮ Determine when the transition takes place (often doable).

◮ Determine the flow after transition (very hard!).

Stability of fluid flows

Laminar flows: always a solution to Navier-Stokes equations.

Landau & Lifshitz“Yet not every solution of the equations of motion, even if it is exact,can actually occur in Nature. The flows that occur in Nature must notonly obey the equations of fluid dynamics, but also be stable”

Need to probe the stability of laminar flows to externaldisturbances.

Objectives of stability analysis:

◮ Determine when the transition takes place (often doable).

◮ Determine the flow after transition (very hard!).

Linear stability: 1-D toy problem

Intuitive picture:

STABLE

STABLEUNSTABLE

NEUTRALLY

Dynamical equation (1-D): dxdt = F (x)

Equilibrium solution: F (x0) = 0

Small perturbations: x(t) = x0 + x ′(t)

Linearized evolution: dx ′

dt = x ′ dFdx |x=x0

Solution: x ′(t) = x ′(t = 0) exp[st ], where s = dFdx |x=x0

Equilibrium solution x0 unstable if Re[s] > 0.

Linear stability: fluid flow

Impose infinitesimal perturbations to the laminar flow:

V v’V = +

v(x, t) = v(x) + v′(x, t)

Fourier normal modes

v′(x, t) = φ(y) exp[ikx ] exp[−ikct ]

Temporal stability: k real wavenumber ; c = cr + ici is complexwavespeed.

Linear stability: fluid flow

Impose infinitesimal perturbations to the laminar flow:

v(x, t) = v(x) + v′(x, t)

MEAN FLOW :

x (y), 0, 0 )(V

Fourier normal modes

= + + ...

v′(x, t) = φ(y) exp[ikx ] exp[−ikct ]Temporal stability: k real wavenumber ; c = cr + ici is complexwavespeed.

Orr-Sommerfeld equation

Navier-Stokes equation:

∂tv + v · ∇v = −∇p +1

Re∇2v

∇ · v = 0

Linearize about a given base state Vx(y): v = V + v′.

∂tv′ + V · ∇v′ + v′·∇V = −∇p′ +1

Re∇2v′

∇ · v′ = 0

(V − c)(d2y − k2)vy − V ′′vy =

(d2y − k2)2vy

Pipe dream: transition in rigid tubes and channels

Both Pipe Poiseuille flow and plane Couette flow asymptoticallystable (t → ∞) at any Re, even as Re → ∞.

For flow in a rectangular channel, theory predicts Re = 5772 forinstability.

Experiments show that flow in a channel becomes unstable atRe = 1200.

Classical temporal analysis fails spectacularly for rigid channelflows.

Observed instability attributed to transient growth at early times &nonlinear effects; linear operators not self-adjoint.

Phenomena poorly understood even today!

Soft walls make the problem less hard

Flow past deformable solids unstable to small disturbances.

Can hope to explain experimental observations with traditionalnormal mode analysis, unlike rigid surfaces.

Soft walls make the problem less hard

Flow past deformable solids unstable to small disturbances.

Can hope to explain experimental observations with traditionalnormal mode analysis, unlike rigid surfaces.

Outline

1 Introduction

6 Summary

Schematic of geometry

soft solid

zero−displacement condition

Plane Couette

Pipe Poiseuillesoft solid

Plane Poiseuille

Solid deformation: Linear viscoelastic solid

Deformation tensor: F = ∇X w

Incompressible solid: det(F) = 1Lagrangian displacement field

u(X) = w(X)− X

Momentum equation: ρ(

∂2u∂t2

X= ∇X · Σ

Constitutive equation: Σ =(

G + ηw∂∂t

(∇Xu +∇XuT )

ρ = solid density = fluid density.Violates material-frame invariance; valid only when thenondimensional strain ≪ 1.

Neo-Hookean solid

Momentum equation: ρ(

∂2w∂t2

X= ∇X · P

First Piola-Kirchoff tensor P = F−1 · σ

Cauchy stress tensor

σ = −psI + G(F · FT ) + σd

Dissipative stress σd = ηgddt (F · FT ).

ps = pw + G, where pw is the pressure in the solid.

Nonlinear stress-strain relationship.

Exhibits nonzero first normal stress difference under simple sheardeformation.

Linear stability analysis

Base state in the fluid: steady unidirectional flow.

Base state in the solid: unidirectional displacement in the flowdirection; Neo-Hookean ⇒ normal stresses.

Impose small fluctuations to the fluid and solid dynamical fields:φ = φ+ φ′

Linearization ⇒ Coupling between base-state quantities andperturbations in both fluid and solid media.

Linearization of interfacial conditions

Continuity of velocities and stresses:

Unperturbed Interface Perturbed

Interface

Perturbed interfacial profile: z = wZ (X ,Z = 0)|X=x

Taylor-expand about unperturbed interface z = 0.

Non-trivial coupling between base flow and fluctuations.

vx = v ′

x |z=0 +dvxdz |z=0wZ + · · · (Linearized).

Stress continuity: t · τ · n = τ ′xz + τ xxn′

Non-dimensional parameters

Re = ρVR/η Reynolds number based on maximum velocity andhalf-width.

Γ = Vη/(GR) Solid deformability parameter; Γ → 0 ⇒ rigid walllimit.

Σ = Re/Γ = ρGR2/η2, a flow independent non-dimensional wallelasticity. Σ → ∞ ⇒ rigid wall limit.

ηr = ηsolid /ηfluid ratio of solid to fluid viscosities.

Non-dimensional interfacial tension γ ≡ γ∗/(GR)

Equal densities: ρsolid = ρfluid

Will use Re vs Γ (or) Re vs Σ to depict results.

Dimensional analysis: Mode classification

Non-dimensional groups: Re = RVρ/µ, Σ = ρGR2/η2. Inaddition, ηr , H etc.

Re ≫ 1, inertial stresses in the fluid (ρV 2) ∼ elastic stresses G inthe solid ⇒ Re ∼ Σ1/2 ⇒ ‘Inviscid modes’

Re ≪ 1, viscous stresses in the fluid (ηV/R) ∼ elastic stresses Gin the solid ⇒ Re ∼ Σ ⇒ ‘Viscous modes’

Re ≫ 1, but viscous stresses in a thin layer near the wall ofO(Re−1/3) R: ηVRe1/3/R ∼ elastic stresses in the solid ⇒Re ∼ Σ3/4 ⇒ ‘Wall modes’

Qualitative picture from scaling: Re vs. Σ

Re ~ Σ3/4

VISCOUS MODES

WALL MODES

INVISCID MODESRe ~ Σ1/2 Log Re

ρ)1/2V ~

V >> (G/ρ)1/2

V << (G/ρ)1/2

GR /2 2η

• Instabilities absent in flow through rigid tubes/channels.

Outline

1 Introduction

6 Summary

Flow in rigid channels: Rayleigh theorem

Rayleigh equation: d2y vy − k2vy −

V ′′vy(V (y)−c) = 0

Multiply by v∗

y and integrate over channel length:

y dyvy )|y=y2y=y1

∫ y1

dy(|dyvy |2 + k2|vy |

∫ y1

dyV ′′|vy |

V − c

Take imaginary part:

y dyvy )|y=y2y=y1

∫ y1

dyV ′′|vy |

(V − cr )2 + c2i

Rayleigh’s inflexion point theorem: Inviscid limitInviscid flow unstable in rigid channels only if it has an inflexion point(U ′′ = 0) in the flow.

Rayleigh theorem for inviscid flow in a deformable tube

Flow in a deformable tube: Laminar velocity profile U(r);G(r) = rU ′(r)/(n2 + k2r2).

Equivalent of Rayleigh’s theorem for a deformable tube

Inviscid flow unstable in deformable tubes only if U dG(r)dr < 0.

Illustration for Hagen-Poiseuille flow:U(r) = (1 − r2) ⇒ G ′(r) = −4rn2/(n2 + k2r2)2.

n = 0: G ′(r) = 0; Flow stable to axisymmetric disturbances.

For n = 0, developing flow and flow in a slightly converging tubecould be unstable.

High Reynolds number inviscid modes

Developing flow in the entrance region, flow in a converging tube.Fully-developed flow subjected to non-axisymmetric perturbations.Re ≫ 1 ⇒ Inviscid fluid governing equations.Fluid inertial stresses ∼ Wall elastic stresses

ρV 2 ∼ G

⇒ V ∼ (G/ρ)1/2 Shear wave speed in a elastic solid.

(U − c)Lvr − (U ′′ − r−1U ′)vr =1

ikReL2vr

L ≡ (d2r + r−1dr − r−2 − k2)

Singular perturbation in the inviscid limit: Viscous effectsimportant in two thin layers even at high Re.

V.S et al., JFM 395, 211 (1999); JFM, 407, 291 (2000)V. Shankar (ChE, IITK) Flow in deformable tubes and channels TEQIP, Oct 2013 30 / 49

Inviscid instability

Inviscid limit: Logarithmic singularity at r = rc where U(rc) = cr

Critical and wall layers:

��

��r = rc

r = 1 O(Re )−1/2 Wall

CriticalO(Re

−1/3) layer

DEFORMABLE WALL

Multiple solutions at leading order for c(0).

Flow unstable in the inviscid limit when ρV 2/G increases.

Inviscid results corroborated by numerical solution at large Re.

Reynolds number Re vs. Nondimensional elasticity Σ

• Flow in entrance region: X = 0.050,H = 2 ; Σ = ρGR2/η2

102 103 104 105 106 107 108 109 1010 1011

k = 0.1k = 1.0k = 2.0k = 6.0k = 10.0

• Instability extends to moderate Reynolds number Re ∼ Σ1/2

• Absent in fully-developed pipe flow subjected to axisymmetricdisturbances and in plane Couette flow.

Wall modes in flow past soft solids: Re ≫ 1

soft solid

O( Re −1/3

) wall layer

inviscid core

Viscous effects in a thin O(Re−1/3) layer near the solid.

Wave speed c∗/V ∝ Re−1/3.

Rigid tubes: Wall modes are stable (Gill, 1965)

Viscous shear stresses in the wall layer ∼ Elastic stresses in thesolid.

ηVRRe−1/3 ∼ G ⇒ Re ∝ Σ3/4 ; V ≫ (G/ρ)1/2

V.S et al., Euro Phys J B 19, 647 (2001); Phys Fluids, 14, 2324 (2002)

Wall modes in flow past deformable solids

Asymptotic analysis: c = c(0) + Re−1/3c(1) + · · ·

Leading order: c(0) ∼ ±(G/ρ)1/2 ⇒ shear waves in the solid.

Multiple solutions for upstream and downstream waves.

First correction: c(1) ∼ Re−1/3(G/ρ)1/2 affected by viscous stressin the wall layer.

Downstream modes destabilized by flow for Γ = Γ0Re−1/3 > Γc .

Instability continues to low Re.

Independent of the details of the velocity profile: a generic featurein flow past deformable solid surfaces at Re ≫ 1.

Wall modes: Comparison with numerics

Σ = ρGR2/η2

102 103 104 105 106 107 108 109

H = 2, k = 1H = 5, k = 1H = 10, k = 1

• Instability extends to moderate Reynolds number Re ∼ Σ3/4

Outline

1 Introduction

6 Summary

Zero-inertia limit: Re = 0

0.001 0.01 0.1 1 10

Wavenumber, k

Γ = 0.5Γ = 5Γ = 10

Couette Flow: H = 5, Re = 0

Γ high-k instability

Pipe Poiseuille flowCreeping flow instability absent at finite k .

High-k instability when surface tension = 0.

Gaurav & V.S, JFM (2009)

0.001 0.01 0.1 1 10

Wavenumber, k

Γ = 0.5Γ = 5Γ = 10

0.001 0.01 0.1 1 10

Wavenumber, k

Γ = 0.5Γ = 5Γ = 10Γ = 20

Poiseuille Flow: H = 5, Re = 0

high-k instability

0.001 0.01 0.1 1 10

Wavenumber, k

Γ = 0.5Γ = 5Γ = 10

0.001 0.01 0.1 1 10

Wavenumber, k

Γ = 0.5Γ = 5Γ = 10Γ = 20

Poiseuille Flow: H = 5, Re = 0

high-k instability

Re ≪ 1: a new low-k instability

Re ≪ 1, k = k∗R ∝ Re1/2, Γ ∼ O(1).

Shear waves in the solid: frequency ω∗ ∼ (G/ρR2)1/2,c∗ ∼ ω∗/k∗ ∼ (G/ρ)1/2 1

k∗R ∼ (G/ρ)1/2Re−1/2

Solid inertia important even at Re ≪ 1.