Springer LINK: The European Physical Journal B - Abstract Volume 4 Issue 4 (1998) pp 519-527

The European Physical Journal B

ISSN: 1434-6028 (printed version)
ISSN: 1434-6036 (electronic version)

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Abstract Volume 4 Issue 4 (1998) pp 519-527

Asymptotic analysis of wall modes in a flexible tube

V. Kumaran (a)

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India

Received: 5 November 1997 / Revised: 10 March 1998 / Accepted: 29 April 1998

Abstract: The stability of wall modes in a flexible tube of radius R surrounded by a viscoelastic material in the region R < r < H R in the high Reynolds number limit is studied using asymptotic techniques. The fluid is a Newtonian fluid, while the wall material is modeled as an incompressible visco-elastic solid. In the limit of high Reynolds number, the vorticity of the wall modes is confined to a region of thickness $O(\epsilon^{1/3})$ in the fluid near the wall of the tube, where the small parameter $\epsilon= Re^{-1}$, and the Reynolds number is $Re = (\rho V R / \eta)$,$\rho$ and $\eta$ are the fluid density and viscosity, and V is the maximum fluid velocity. The regime $\Lambda = \epsilon^{-1/3} (G / \rho V^{2}) \sim 1$ is considered in the asymptotic analysis, where G is the shear modulus of the wall material. In this limit, the ratio of the normal stress and normal displacement in the wall, $(- \Lambda C(k^{\ast}, H))$, is only a function of H and scaled wave number $k^{\ast}= (k R)$. There are multiple solutions for the growth rate which depend on the parameter $\Lambda^* = k^{* 1/3} C(k^*, H) \Lambda$.In the limit $\Lambda^* \ll 1$, which is equivalent to using a zero normal stress boundary condition for the fluid, all the roots have negative real parts, indicating that the wall modes are stable. In the limit $\Lambda^{\ast}\gg 1$, which corresponds to the flow in a rigid tube, the stable roots of previous studies on the flow in a rigid tube are recovered. In addition, there is one root in the limit $\Lambda^{\ast}\gg 1$ which does not reduce to any of the rigid tube solutions determined previously. The decay rate of this solution decreases proportional to $(\Lambda^{\ast})^{-1/2}$ in the limit $\Lambda^{\ast}\gg 1$, and the frequency increases proportional to $\Lambda^{\ast}$.

PACS. 83.50.-v Deformation; material flow - 47.15.Fe Stability of laminar flows - 47.60.+i Flows in ducts, channels, nozzles, and conduits

(a) email: kumaran@chemeng.iisc.ernet.in

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