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Stability of fluid flow through deformable tubes and channels: An overview

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Abstract

The aim of this paper is to provide a systematic overview of the study of instabilities in flow past deformable solid surfaces, with particular emphasis on internal flows through tubes and channels. The subject is certainly more than five decades old, and arguably began with Kramer’s pioneering experiments on drag reduction by compliant surfaces. This was immediately followed by the theoretical studies of Benjamin and Landhal in the early 1960s. Most earlier theoretical studies were focused on stability of external flows such as boundary layers, and used relatively simple wall models composed of spring-backed plates. There has been a resurgence in the field since the mid-1980s, and more attention was focused on internal flows through deformable tubes and channels. The wall deformation was described by both phenomenological spring-backed plate models and continuum linear viscoelastic solid models. All these studies predict several types of instabilities in flow past deformable surfaces. This paper will attempt to place the various theoretical results in perspective, and to classify the instabilities predicted by various studies. Recent studies have also emphasized the importance of using a frame-invariant constitutive model, such as the neo-Hookean model, for the solid deformation. Until recently, however, the field has been dominated by theoretical and numerical studies, with very little experimental observations to corroborate the theoretical predictions. Recent experiments in flow through deformable tubes and channels indeed show instability at Reynolds number much lower than their rigid counterparts, and the experimental observations are in qualitative agreement with some of the theoretical predictions. There have also been a few studies on the non-linear aspects of the instability using the weakly non-linear formulation to determine the nature of the bifurcation at the linear instability. A brief discussion on weakly nonlinear analyses is also provided in this paper.

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References

  • Beatty M F and Zhou Z 1991 Universal motion for a class of viscoelastic materials of differential type. Continuum Mech. Thermodyn. 3: 169

    Article  MathSciNet  Google Scholar 

  • Benjamin T B 1960 Effect of a flexible surface on boundary layer stability. J. Fluid Mech. 9: 513–532

    Article  MathSciNet  Google Scholar 

  • Benjamin B 1963 The threefold classification for unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16: 436–450

    Article  Google Scholar 

  • Carpenter P W and Gajjar J S B 1990 A general theory for two and three dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls. Theoret. Comput. Fluid Dyn. 1: 349–378

    Article  Google Scholar 

  • Carpenter P W and Garrad A D 1985 The hydrodynamic stability of flows over Kramer-type compliant surfaces. part 1. Tollmien-schlichting instabilities. J. Fluid Mech. 155: 465–510

    Article  Google Scholar 

  • Carpenter P W and Garrad A D 1986 The hydrodynamic stability of flows over Kramer-type compliant surfaces. part 2. flow induced surface instabilities. J. Fluid Mech. 170: 199–232

    Article  Google Scholar 

  • Carpenter P and Morris P 1990 The effect of anisotropic wall compliance on boundary-layer stability and transition. J. Fluid Mech. 218: 171–223

    Article  Google Scholar 

  • Chandrasekhar S 1981 Hydrodynamic and hydromagnetic stability. New York: Dover

    Google Scholar 

  • Chen K P 1991 Elastic instability of the interface in Couette flow of two viscoelastic liquids. J. Non-Newtonian Fluid Mech. 40: 261–267

    Article  Google Scholar 

  • Chokshi P 2007 Studies in the stability of newtonian and viscoelastic flows past rigid and flexible surfaces. PhD thesis, Indian Institute of Science, Bangalore, India

  • Choskshi P and Kumaran V 2007 Stability of the flow of a viscoelastic fluid past a deformable surface in the low reynolds number limit. Phys. Fluids 19: 104,103

    Article  Google Scholar 

  • Choskshi P and Kumaran V 2008 Weakly nonlinear analysis of viscous instability in flow past a neo-hookean surface. Phys. Rev. E 77: 056,303

    Article  Google Scholar 

  • Davies C 2003 Convective and absolute instabilities of flows over compliant walls. In: Carpenter P W and Pedley T J (eds) IUTAM symposium on flow past highly compliant boundaries and in collapsible tubes, Kluwer Academic, The Netherlands, chap 4, pp 69–94

  • Davies C and Carpenter P W 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352: 205–243

    Article  MathSciNet  Google Scholar 

  • Destarde M and Saccomandi G 2004 Finite–amplitude inhomogeneous waves in mooney–rivlin viscoelastic solids. Wave Motion 40: 251–262

    Article  MathSciNet  Google Scholar 

  • Drazin P and Reid W 1981 Hydrodynamic stability. Cambridge: Cambridge University Press

    Google Scholar 

  • Eggert M D and Kumar S 2004 Observations of instability, hysterisis, and oscillation in low-reynolds number flow past polymer gels. J. Colloid Interface Sci. 278: 234–242

    Article  Google Scholar 

  • Fosdick R L and Yu J H 1996 Thermodynamics, stability and non–linear oscillations of viscoelastic solids – i. differential type solids of second grade. Int. J. Non–Linear Mech. 31: 495

    Article  MathSciNet  Google Scholar 

  • Gajjar J S B and Sibanda S K 1996 The hydrodynamic stability of channel flow with compliant boundaries. Theor. Comput. Fluid Dyn. 8: 105–129

    Article  Google Scholar 

  • Gaurav and Shankar V 2009 Stability of fluid flow through deformable neo-hookean tubes. J. Fluid Mech. 627: 291–322

    Article  MathSciNet  Google Scholar 

  • Gkanis V and Kumar S 2003 Instability of creeping couette flow past a neo-hookean solid. Phys. Fluids 15: 2864–2471

    Article  Google Scholar 

  • Grotberg J B 2011 Respiratory fluid mechanics. Phy. Fluids 23(2): 021301

    Article  Google Scholar 

  • Grotberg J B and Jensen O E 2004 Biofluid mechanics in flexible tubes. Ann. Rev. Fluid Mech. 36: 121–147

    Article  MathSciNet  Google Scholar 

  • Heil M and Jensen O E 2003 Flow in deformable tubes and channels. In: Carpenter P W, Pedley T J (eds) IUTAM symposium on flow past highly compliant boundaries and in collapsible tubes, Kluwer Academic, The Netherlands, chap 2, pp 15–49

  • Holzapfel G A 2000 Nonlinear solid mechanics. Chichester, UK: John Wiley

    Google Scholar 

  • Krindel P and Silberberg A 1979 Flow through gel-walled tubes. J. Colloid Interface Sci. 71: 34–50

    Article  Google Scholar 

  • Ku D N 1997 Blood flow in arteries. Annu. Rev. Fluid Mech. 29: 399–434

    Article  MathSciNet  Google Scholar 

  • Kumar A S and Shankar V 2005 Instability of high-frequency modes in viscoelastic plane couette flow past a deformable wall at low and finite reynolds number. J. Non-Newtonian Fluid Mech. 125: 121–141

    Article  Google Scholar 

  • Kumaran V 1995 Stability of the viscous flow of a fluid through a flexible tube. J. Fluid Mech. 294: 259–281

    Article  MathSciNet  Google Scholar 

  • Kumaran V 1996 Stability of an inviscid flow through a flexible tube. J. Fluid Mech. 320: 1–17

    Article  MathSciNet  Google Scholar 

  • Kumaran V 1998a Stability of fluid flow through a flexible tube at intermediate Reynolds number. J. Fluid Mech. 357: 123–140

    Article  MathSciNet  Google Scholar 

  • Kumaran V 1998b Stability of wall modes in a flexible tube. J. Fluid Mech. 362: 1–15

    Article  MathSciNet  Google Scholar 

  • Kumaran V 2003 Hydrodynamic stability of flow through compliant channels and tubes. In: Carpenter P W and Pedley T J (eds) IUTAM symposium on flow past highly compliant boundaries and in collapsible tubes. Kluwer Academic, The Netherlands, chap 5, pp 95–118

  • Kumaran V and Muralikrishnan R 2000 Spontaneous growth of fluctuations in the viscous flow of a fluid past a soft interface. Phys. Rev. Lett. 84: 3310–3313

    Article  Google Scholar 

  • Kumaran V, Fredrickson G H and Pincus P 1994 Flow induced instability of the interface between a fluid and a gel at low Reynolds number. J. Phys. II France 4: 893–904

    Article  Google Scholar 

  • Lahav J, Eliezer N and Silberberg A 1973 Gel - walled cylindrical channels as models for micro-circulation: Dynamics of flow. Biorheology 10: 595–604

    Google Scholar 

  • Landahl M T 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13: 609

    Article  Google Scholar 

  • Landau L and Lifshitz E 1989 Theory of elasticity. New York: Pergamon

    Google Scholar 

  • LaRose P G and Grotberg J B 1997 Flutter and long-wave instabilities in compliant channels conveying developing flows. J. Fluid Mech. 331: 37

    Article  Google Scholar 

  • Lucey A D and Carpenter P W 1992 A numerical simulation of the interaction of a compliant wall and inviscid flow. J. Fluid Mech. 234: 121–146

    Article  Google Scholar 

  • Lucey A D and Carpenter P W 1993 On the difference between the hydroelastic instability of infinite and very long compliant panels. J. Sound Vibr. 163: 176–181

    Article  Google Scholar 

  • Lucey A D and Peake N 2003 Wave excitation on flexible walls in the presence of a fluid flow. In: Carpenter P W and Pedley T J (eds) IUTAM symposium on flow past highly compliant boundaries and in collapsible tubes, Kluwer Academic, The Netherlands, chap 6, pp 119–146

  • Ma Y and Ng C O 2009 Wave propagation and induced steady streaming in viscous fluid contained in a prestressed viscoelastic tube. Phys. Fluids 21: 051,901

    Article  Google Scholar 

  • Macosko C 1994 Rheology: Principles, measurements and applications. New York: VCH

    Google Scholar 

  • Malvern L E 1969 Introduction to the mechanics of a continuous medium. Englewood Cliffs, NJ: Prentice-Hall

    Google Scholar 

  • McDonald J C and Whitesides G M 2002 Poly(dimethylsiloxane) as a material for fabricating microfluidic devices. Acc. Chem. Res. 35: 491–499

    Article  Google Scholar 

  • Muralikrishnan R and Kumaran V 2002 Experimental study of the instability of viscous flow past a flexible surface. Phys. Fluids 14: 775–780

    Article  Google Scholar 

  • Neelamegam R, Shankar V and Das D 2013 Suppression of purely elastic instabilities in the torsional flow of viscoelastic fluid past a soft solid. Phys. Fluids 25: 124,102

    Article  Google Scholar 

  • Pedley T J 2000 Blood flow in arteries and veins. In: Batchelor G K, Moffat H K and Worster M G (eds) Perspectives in fluid dynamics, Cambridge, chap 3, pp 105–153

  • Renardy Y 1988 Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids. J. Non-Newtonian Fluid Mech. 28: 99–115

    Article  Google Scholar 

  • Schmid P 2007 Nonmodal stability theory. Ann. Rev. Fluid Mech. 39: 129–162

    Article  Google Scholar 

  • Schmid P J and Henningson D S 2001 Stability and transition in shear flows. New York: Springer

    Book  Google Scholar 

  • Sen P K and Arora S K 1987 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13: 609

    Google Scholar 

  • Shankar V and Kumar S 2004 Instability of viscoelastic plane Couette flow past a deformable wall. J. Non-Newtonian Fluid Mech. 116: 371–393

    Article  Google Scholar 

  • Shankar V and Kumaran V 1999 Stability of non-parabolic flow in a flexible tube. J. Fluid Mech. 395: 211–236

    Article  Google Scholar 

  • Shankar V and Kumaran V 2000 Stability of fluid flow in a flexible tube to non-axisymmetric disturbances. J. Fluid Mech. 408: 291–314

    Article  Google Scholar 

  • Shankar V and Kumaran V 2001a Asymptotic analysis of wall modes in a flexible tube revisited. Eur. Phys. J. B 19: 607–622

    Article  Google Scholar 

  • Shankar V and Kumaran V 2001b Weakly nonlinear stability of viscous flow past a flexible surface. J. Fluid Mech. 434: 337–354

    Article  MathSciNet  Google Scholar 

  • Shankar V and Kumaran V 2002 Stability of wall modes in fluid flow past a flexible surface. Phys. Fluids 14: 2324–2338

    Article  MathSciNet  Google Scholar 

  • Squires T M and Quake S R 2005 Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys. 77: 977–1026

    Article  Google Scholar 

  • Stuart J T 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows: Part 1. the basic behaviour in plane poiseuille flow. J. Fluid Mech. 9: 353–370

    Article  MathSciNet  Google Scholar 

  • Verma M K S and Kumaran V 2012 A dynamical instability due to fluid-wall coupling lowers the transition reynolds number in the flow trough a flexible tube. J. Fluid Mech. 705: 322–347

    Article  Google Scholar 

  • Verma M K S and Kumaran V 2013 A multifold reduction in the transition reynolds number, and ultra-fast mixing, in a micro-channel due to a dynamical instability induced by a soft wall. J. Fluid Mech. 727: 407–45

    Article  MathSciNet  Google Scholar 

  • Yeo K 1988 The stability of boundary layer flow over single and multilayer viscoelastic walls. J. Fluid Mech. 196: 359

    Article  MathSciNet  Google Scholar 

  • Yeo K and Dowling A 1987 The stability of inviscid flows over passive compliant walls. J. Fluid Mech. 183: 265–292

    Article  Google Scholar 

  • Yeo K, Khoo B and Chong W 1994 The linear stability of boundary layer flow over compliant walls: The effects of wall mean state induced by flow loading. J. Fluids Struct. 8: 529

    Article  Google Scholar 

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Acknowledgements

The author would like to thank Professor V. Kumaran, Dr. Rochish Thaokar and Dr. Paresh Chokshi for many useful discussions and suggestions, and his collaborator Dr. Gaurav with whom some of the reported work was carried out.

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  1. Department of Chemical Engineering, Indian Institute of Technology, Kanpur, 208016, India

    V SHANKAR

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SHANKAR, V. Stability of fluid flow through deformable tubes and channels: An overview. Sadhana 40, 925–943 (2015). https://doi.org/10.1007/s12046-015-0358-6

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