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Stability of the viscous flow of a fluid through a flexible tube

Published online by Cambridge University Press:  26 April 2006

V. Kumaran
V. Kumaran
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
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Abstract

The stability of Hagen-Poiseuille flow of a Newtonian fluid of viscosity η in a tube of radius R surrounded by a viscoelastic medium of elasticity G and viscosity ηs occupying the annulus R < r < HR is determined using a linear stability analysis. The inertia of the fluid and the medium are neglected, and the mass and momentum conservation equations for the fluid and wall are linear. The only coupling between the mean flow and fluctuations enters via an additional term in the boundary condition for the tangential velocity at the interface, due to the discontinuity in the strain rate in the mean flow at the surface. This additional term is responsible for destabilizing the surface when the mean velocity increases beyond a transition value, and the physical mechanism driving the instability is the transfer of energy from the mean flow to the fluctuations due to the work done by the mean flow at the interface.

The transition velocity Γt for the presence of surface instabilities depends on the wavenumber k and three dimensionless parameters: the ratio of the solid and fluid viscosities ηr = (ηs/η), the capillary number λ = (T/GR) and the ratio of radii H, where T is the surface tension of the interface. For ηr = 0 and λ = 0, the transition velocity Γt diverges in the limits k [Lt ] 1 and k [Gt ] 1, and has a minimum for finite k. The qualitative behaviour of the transition velocity is the same for λ < 0 and ηr = 0, though there is an increase in λt in the limit k > 1. When the viscosity of the surface is non-zero (ηr < 0), however, there is a qualitative change in the λtvs. k curves. For ηr < 1, the transition velocity λt is finite only when k is greater than a minimum value kmin, while perturbations with wavenumber k < kmin are stable even for λ → ∞. For ηr > 1, λt is finite only for kmin < k < kmax, while perturbations with wavenumber k < kmin or k > kmax are stable in the limit λ → ∞. As H decreases or ηr increases, the difference kmax − kmin decreases. At a minimum value H = Hmin which is a function of ηr, the difference kmax − kmin = 0, and for H < Hmin, perturbations of all wavenumbers are stable even in the limit λ → ∞. The calculations indicate that Hmin shows a strong divergence proportional to exp (0.0832nr2) for ηr [Gt ] 1.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 294 , 10 July 1995 , pp. 259 - 281
Copyright
© 1995 Cambridge University Press

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References

Benjamin, T. B. 1960 Effect of a flexible boundary layer on hydrodynamic stability. J. Fluid Mech. 9, 513.Google Scholar
Benjamin, T. B. 1963 The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16, 436.Google Scholar
Bertram, C. D. 1986 Unstable equilibrium behavior in collapsible tubes. J. Biomech. 19, 61.Google Scholar
Bertram, C. D. 1987 The effects of wall thickness, axial strain and end proximity on the pressure area relation in collapsible tubes. J. Biomech. 20, 863.Google Scholar
Bertram, C. D., Raymond, C. J. & Pedley, T. J. 1989 Mapping of instabilities during flow through collapsible tubes. J. Fluids Structures 4, 125–154.Google Scholar
Cancelli, C. & Pedley, T. J. 1985 A separated - flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375.Google Scholar
Carpenter, P. W. 1990 Status of transition delay using compliant walls. Prog. Astro. Aero. 123, 79.Google Scholar
Carpenter, P. W. & Gajjar, J. S. G. 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.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.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.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.
Chen, K. P. 1991 Interfacial instability due to the elastic stratification in concentric coextrusion of two viscoelastic fluids. J. Non-Newtonian Fluid Mech. 40, 155.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Green, C. H. & Ellen, C. H. 1972 The stability of a plane Poiseuille flow between flexible walls. J. Fluid Mech. 51, 403.Google Scholar
Gupta, A. S. 1967 Stability of a visco-elastic film flowing down an inclined plane. J. Fluid Mech. 28, 17.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics, Chap. 3. Prentice-Hall.
Harden, J. L., Pleiner, P. & Pincus, P. A. 1991 Hydrodynamic modes on concentrated polymer solutions and gels. J Chem. Phys. 94, 5208.Google Scholar
Hooper, A. & Boyd, W. G. 1983 Shear flow instability at the interface between two viscous fluids. J. Fluid Mech. 128, 507.Google Scholar
Jensen, O. E. & Pedley, T. J. 1989 The existence of steady flow in a collapsed tube. J. Fluid Mech. 206, 339.Google Scholar
Kramer, M. O. 1957 Boundary-layer stabilization by distributed damping. J. Aero. Sci. 24, 459.Google Scholar
Kramer, M. O. 1960 Boundary layer stabilization by distributed damping. J. Am. Soc. Naval Engrs 74, 25.Google Scholar
Krindel, P. & Silberberg, A. 1979 Flow through gel-walled tubes. J. Colloid Interface Sci. 71, 34.Google Scholar
Kumaran, V. 1993 Surface modes on a polymer gel of finite thickness. J. Chem. Phys. 98, 3429.Google Scholar
Kumaran, V., Fredrickson, G. H. & Pincus, P. 1994 Flow induced instability at the interface between a fluid and a gel at low Reynolds number. J. Phys. Paris II 4, 893-904.Google Scholar
Landahl, M. T. 1962 On the stability of a laminar incompressible boundary layer over flexible surface. J. Fluid Mech. 13, 609.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1989 Theory of Elasticity. Pergamon.
Lin, C. C. 1945a On the stability of two-dimensional parallel flows. Part 1.. Q. Appl. Maths 3, 117–142.Google Scholar
Lin, C. C. 1945b On the stability of two-dimensional parallel flows. Part 2.. Q. Appl. Maths 3, 218–234.Google Scholar
Lin, C. C. 1945c On the stability of two-dimensional parallel flows. Part 3.. Q. Appl. Maths 3, 277–301.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Renardy, Y. 1988 Stability of the interface in a two-layer Couette flow of upper convected Maxwell liquids. J. Non-Newtonian Fluid Mech. 28, 99.Google Scholar
Reyn, J. W. 1987 Multiple solutions and flow limitation for steady flow through a collapsible tube held by open ends. J. Fluid Mech. 174, 467.Google Scholar
Riley, J., Gad-El-Hak, M. & Metcalfe, R. W. 1988 Compliant coatings. Ann. Rev. Fluid Mech. 20, 393.Google Scholar
Schlichting, H. 1933 Zur Entstehung der Tubrulenz bei der Plattenstromung. Z. Agnew. Math. Mech. 13, 171.Google Scholar
Shaqfeh, E. S. G., Larson, R. G. & Fredrickson, G. H. 1989 The stability of gravity driven viscoelastic film-flow at low to moderate Reynolds number. J. Non-Newtonian Fluid Mech. 21, 87.Google Scholar
Silberberg, A. 1987 Physico-chemical hydrodynamics in turbulent flows close to an interface. Physico-chem. Hydrodyn. 9, 419.Google Scholar
Su, Y. Y. & Khomami, B. 1992 Interfacial instability in multilayer viscoelastic fluids in slit and converging channel die geometries. J. Rheol. 36, 357.Google Scholar
Tollmien, W. 1929 Uber die Entstehung der Turbulenz. 1. Mitt. Ges. Wiss. Gottingen, Math. Phys. Klasse 21.Google Scholar
Waters, N. D. & Keely, A. M. 1987 The stability of two stratified non-Newtonian liquids in a Couette flow. J. Non-Newtonian Fluid Mech. 24, 161.Google Scholar
Yih, C. S. 1970 Instability due to viscosity stratification. J. Fluid Mech. 27, 337.Google Scholar
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