Asymptotic analysis of wall modes in a flexible tube

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 \(O({\varepsilon ^{1/3}})\) in the fluid near the wall of the tube, where the small parameter \(\varepsilon = {{\mathop{\rm Re}\nolimits} ^{ - 1}}\), and the Reynolds number is \({\mathop{\rm Re}\nolimits} = (\rho VR/\eta )\),ρ and η are the fluid density and viscosity, and V is the maximum fluid velocity. The regime \(\Lambda = {\varepsilon ^{ - 1/3}}(G/\rho {V^2}) \sim 1\) 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, \(( - \Lambda C(k*,H))\), is only a function of H and scaled wave number \(k* = (kR)\). There are multiple solutions for the growth rate which depend on the parameter \(\Lambda * = k{*^{1/3}}C(k*,H)\Lambda \).In the limit \(A* \ll 1\), 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 \(\Lambda * \gg 1\), 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 \(A* \ll 1\) which does not reduce to any of the rigid tube solutions determined previously. The decay rate of this solution decreases proportional to \({(\Lambda *)^{1/2}}\) in the limit \(A* \ll 1\), and the frequency increases proportional to \(\Lambda *\).