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Consistent formulations for stability of fluid flow through deformable channels and tubes

Published online by Cambridge University Press:  18 August 2017

Ramkarn Patne ,
D. Giribabu  and
V. Shankar Open the ORCID record for V. Shankar [Opens in a new window]
Ramkarn Patne
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur, 208016, India
D. Giribabu
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur, 208016, India
V. Shankar*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur, 208016, India
*
†Email address for correspondence: [email protected]
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Abstract

In the formulation of stability of fluid flow through channels and tubes with deformable walls, while the fluid is naturally treated in an Eulerian framework, the solid can be treated either in a Lagrangian or Eulerian framework. A consistent formulation, then, should yield results that are independent of the chosen framework. Previous studies have demonstrated this consistency for the stability of plane Couette flow past a deformable solid layer modelled as a neo-Hookean solid, in the creeping-flow limit. However, a similar exercise carried out in the creeping-flow limit for the stability of pressure-driven flow in a neo-Hookean tube shows that while the flow is stable in the Lagrangian formulation, it is unstable in the existing Eulerian formulation. The present work resolves this discrepancy by presenting consistent Lagrangian and Eulerian frameworks for performing stability analyses in flow through deformable tubes and channels. The resolution is achieved by making important modifications to the Lagrangian formulation to make it fundamentally consistent, as well as by proposing a proper formulation for the neo-Hookean constitutive relation in the Eulerian framework. In the neo-Hookean model, the Cauchy stress tensor in the solid is proportional to the Finger tensor. We demonstrate that the neo-Hookean constitutive model within the Eulerian formulation used in the previous studies is a special case of the Mooney–Rivlin solid, with the Cauchy stress tensor being proportional to the inverse of the Finger tensor unlike in a true neo-Hookean solid. Remarkably, for plane Couette flow subjected to two-dimensional perturbations, there is perfect agreement between the results obtained using earlier Eulerian and Lagrangian formulations despite the crucial difference in the constitutive relation owing to the rather simple kinematics of the base state. However, the consequences are drastic for pressure-driven flow in a tube even for axisymmetric disturbances. We propose a consistent neo-Hookean constitutive relation in the Eulerian framework, which yields results that are in perfect agreement with the results from the Lagrangian formulation for both plane Couette and tube flows at arbitrary Reynolds number. The present study thus provides an unambiguous formulation for carrying out stability analyses in flow through deformable channels and tubes. We further show that unlike plane Couette flow and Hagen–Poiseuille flow in rigid-walled conduits where there is a remarkable similarity in the linear stability characteristics between these two flows, the stability behaviour for these two flows is very different when the walls are deformable. The instability of plane Couette flow past a deformable wall is very robust and is not sensitive to the constitutive nature of the solid, but the stability of pressure-driven flow in a deformable tube is rather sensitive to the constitutive nature of the deformable solid, especially at low Reynolds number.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 827 , 25 September 2017 , pp. 31 - 66
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Copyright
© 2017 Cambridge University Press 

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References

Benjamin, T. B. 1960 Effect of a flexible surface on boundary layer stability. J. Fluid Mech. 9, 513–532.Google Scholar
Benjamin, T. B. 1963 The threefold classification for unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16, 436–450.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1977 Dynamics of Polymeric Liquids, Vol. 1 Fluid Mechanics. John Wiley.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Carpenter, P. W. & Morris, P. J. 1990 The effect of anisotropic wall compliance on boundary-layer stability and transition. J. Fluid Mech. 218, 171–223.Google Scholar
Carpenter, P. W. & 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. Theor. Comput. Fluid Dyn. 1, 349–378.Google Scholar
Carpenter, P. W. & 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.Google Scholar
Carpenter, P. W. & 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.Google Scholar
Chokshi, P. & Kumaran, V. 2007 Stability of the flow of a viscoelastic fluid past a deformable surface in the low Reynolds number limit. Phys. Fluids 19, 104103.Google Scholar
Chokshi, P. & Kumaran, V. 2008a Weakly nonlinear analysis of viscous instability in flow past a neo-Hookean surface. Phys. Rev. E 77, 056303.Google Scholar
Chokshi, P. & Kumaran, V. 2008b Weakly nonlinear stability analysis of a flow past a neo-Hookean solid at arbitrary Reynolds numbers. Phys. Fluids 20, 094109.Google Scholar
Davies, C. & Carpenter, P. W. 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.Google Scholar
Destarde, M. & Saccomandi, G. 2004 Finite–amplitude inhomogeneous waves in Mooney–Rivlin viscoelastic solids. Wave Motion 40, 251–262.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Eggert, M. D. & Kumar, S. 2004 Observations of instability, hysterisis, and oscillation in low-Reynolds number flow past polymer gels. J. Colloid Interface Sci. 278, 234–242.CrossRefGoogle Scholar
Garg, V. K. 1977 Effect of tube elasticity on the stability of Poiseuille flow. J. Fluid Mech. 81, 625–640.Google Scholar
Gaurav & Shankar, V. 2009 Stability of fluid flow through deformable neo-Hookean tubes. J. Fluid Mech. 627, 291–322.Google Scholar
Gaurav & Shankar, V. 2010 Stability of pressure-driven flow in a deformable neo-Hookean channel. J. Fluid Mech. 659, 318–350.Google Scholar
Giribabu, D. & Shankar, V. 2016 Consistent formulation of solid dissipative effects in stability analysis of flow past a deformable solid. Phys. Rev. Fluids 1, 033602.Google Scholar
Gkanis, V. & Kumar, S. 2003 Instability of creeping Couette flow past a neo-Hookean solid. Phys. Fluids 15, 2864–2871.Google Scholar
Gkanis, V. & Kumar, S. 2005 Stabilty of pressure-driven creeping flows in channels lined with a nonlinear elastic solid. J. Fluid Mech. 524, 357–375.Google Scholar
Grotberg, J. B. 2011 Respiratory fluid mechanics. Phys. Fluids 23 (2), 021301.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Holzapfel, G. A. 2000 Nonlinear Solid Mechanics. John Wiley.Google Scholar
Krindel, P. & Silberberg, A. 1979 Flow through gel-walled tubes. J. Colloid Interface Sci. 71, 34–50.Google Scholar
Kumaran, V. 1995a Stability of the flow of a fluid through a flexible tube at high Reynolds number. J. Fluid Mech. 302, 117–139.Google Scholar
Kumaran, V. 1995b Stability of the viscous flow of a fluid through a flexible tube. J. Fluid Mech. 294, 259–281.Google Scholar
Kumaran, V. 1998 Stability of fluid flow through a flexible tube at intermediate Reynolds number. J. Fluid Mech. 357, 123–140.Google Scholar
Kumaran, V. 2003 Hydrodynamic stability of flow through compliant channels and tubes. In IUTAM symposium on flow past highly compliant boundaries and in collapsible tubes (ed. Carpenter, P. W. & Pedley, T. J.), chap. 5, pp. 95–118. Kluwer.Google Scholar
Kumaran, V. 2015 Experimental studies on the flow through soft tubes and channels. Sadhana 40, 911–923.Google Scholar
Kumaran, V. & Bandaru, P. 2016 Ultra-fast microfluidic mixing by soft-wall turbulence. Chem. Engng Sci. 149, 156–168.Google Scholar
Kumaran, V., Fredrickson, G. H. & 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.Google Scholar
Kumaran, V. & Muralikrishnan, R. 2000 Spontaneous growth of fluctuations in the viscous flow of a fluid past a soft interface. Phys. Rev. Lett. 84, 3310–3313.Google Scholar
Landahl, M. T. 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13, 609.Google Scholar
Larson, R. G. 1988 Constitutive Equations for Polymer Melts and Solutions. Butterworths.Google Scholar
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.Google Scholar
Lee, S. H., Maki, K. L., Flath, D., Weinstein, S. J., Kealey, C., Li, W., Talbot, C. & Kumar, S. 2014 Gravity-driven instability of a thin liquid film underneath a soft solid. Phys. Rev. E 90, 053009.Google Scholar
Ma, Y. & Ng, C.-O. 2009 Wave propagation and induced steady streaming in viscous fluid contained in a prestressed viscoelastic tube. Phys. Fluids 21, 051901.Google Scholar
Macosko, C. W. 1994 Rheology: Principles, Measurements, and Applications. VCH.Google Scholar
Malvern, L. E. 1969 Introduction to the Mechanics of a Continuous Medium. Prentice-Hall.Google Scholar
Mcdonald, J. C. & Whitesides, G. M. 2002 Poly(dimethylsiloxane) as a material for fabricating microfluidic devices. Acc. Chem. Res. 35, 491–499.Google Scholar
Neelamegam, R. & Shankar, V. 2015 Experimental study of the instability of laminar flow in a tube with deformable walls. Phys. Fluids 27, 024102.Google Scholar
Norris, A. N. 2007 Small-on-large theory with applications to granular materials and fluid/solid systems. In Waves in Nonlinear Pre-Stressed Materials (ed. Destrade, M. & Saccomandi, G.), pp. 27–62. Springer.Google Scholar
Norris, A. N. 2008 Eulerian conjugate stress and strain. J. Mech. Mater. Struct. 3, 243–260.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Shankar, V. 2015 Stability of fluid flow through deformable tubes and channels: an overview. Sadhana 40, 925–943.CrossRefGoogle Scholar
Shankar, V. & Kumaran, V. 1999 Stability of non-parabolic flow in a flexible tube. J. Fluid Mech. 395, 211–236.Google Scholar
Shankar, V. & Kumaran, V. 2000 Stability of fluid flow in a flexible tube to non-axisymmetric disturbances. J. Fluid Mech. 408, 291–314.Google Scholar
Shankar, V. & Kumaran, V. 2001 Weakly nonlinear stability of viscous flow past a flexible surface. J. Fluid Mech. 434, 337–354.Google Scholar
Shankar, V. & Kumaran, V. 2002 Stability of wall modes in fluid flow past a flexible surface. Phys. Fluids 14, 2324–2338.Google Scholar
Shrivastava, A., Cussler, E. L. & Kumar, S. 2008 Mass transfer enhancement due to a soft elastic boundary. Chem. Engng Sci. 63, 4302–4305.Google Scholar
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977–1026.Google Scholar
Srinivas, S. S. & Kumaran, V. 2015 After transition in a soft-walled microchannel. J. Fluid Mech. 780, 649–686.Google Scholar
Srinivas, S. S. & Kumaran, V. 2017 Effect of viscoelasticity on the soft-wall transition and turbulence in a microchannel. J. Fluid Mech. 812, 1076–1118.Google Scholar
Toupin, R. A. & Bernstein, B. 1961 Sound waves in deformed perfectly elastic materials: acoustoelastic effect. J. Acoust. Soc. Am. 33, 216–225.Google Scholar
Verma, M. K. S. & Kumaran, V. 2012 A dynamical instability due to fluid-wall coupling lowers the transition Reynolds number in the flow through a flexible tube. J. Fluid Mech. 705, 322–347.Google Scholar
Verma, M. K. S. & 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, 407–455.Google Scholar
Verma, M. K. S. & Kumaran, V. 2015 Stability of flow in a soft tube deformed due to applied pressure gradient. Phys. Rev. E 91, 043001.Google Scholar