ASYMPTOTIC ANALYSIS OF WALL MODES IN A FLEXIBLE TUBE

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 Reynol ds number limit is studied using asymptotic techniques. The fluid is a Newtonian fluid, while the wall material is modeled as an incompressi ble visco-elastic solid. In the limit of high Reynolds number, the vor ticity of the wall modes is confined to a region of thickness O(epsilo n(1/3)) in the fluid near the wall of the tube, where the small parame ter epsilon = Re-1, and the Reynolds number is Re = (rho VR/eta), rho and eta are the fluid density and viscosity, and V is the maximum flui d velocity. The regime Lambda = epsilon(-1/3)(G/rho V-2) similar to 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 a nd normal displacement in the wall, (-Lambda C(k, H)), is only a func tion of H and scaled wave number k = (kR). There are multiple solutio ns for the growth rate which depend on the parameter Lambda = k*C-1/3 (k, H)Lambda. In the limit Lambda* much less than 1, which is equival ent to using a zero normal stress boundary condition for the fluid, al l the roots have negative real parts, indicating that the wall modes a re stable. In the limit Lambda much greater than I, which corresponds to the Row in a rigid tube, the stable roots of previous studies on t he flow in a rigid tube are recovered. In addition, there is one root in the limit Lambda much less than 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 Lambd a much greater than 1, and the frequency increases proportional to La mbda.