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# Asymptotic analysis of wall modes in a flexible tube

Kumaran, V (1998) Asymptotic analysis of wall modes in a flexible tube. In: European Physical Journal B, 4 (4). pp. 519-527.

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The stability of wall modes in a flexible tube of radius R surrounded by a viscoelastic material in the region R < r < HR 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($\epsilon^\frac{1}{3}$) 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 A = $\epsilon^\frac{-1}{3}$(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, (- AC($k^\ast$; H)), is only a function of H and scaled wave number $k^\ast$ = (kR). There are multiple solutions for the growth rate which depend on the parameter $A^\ast$ = $k^\ast\frac{1}{3}$C($k^\ast$, H)A. In the limit $A^\ast$ \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 $A^\ast$ \ll 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 $A^\ast$ \ll 1 which does not reduce to any of the rigid tube solutions determined previously. The decay rate of this solution decreases proportional to $(A^\ast)^\frac{-1}{2}$ in the limit $A^\ast$ \ll 1, and the frequency increases proportional to $A^\ast$.