Asymptotic analysis of wall modes in a flexible tube

摘要

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(∈~(1/3)) in the fluid near the wall of the tube, where the small parameter ∈ = Re~(-1), and the Reynolds number is Re = (ρVR/η), ρ and η are the fluid density and viscosity, and V is the maximum fluid velocity. The regime Λ = ∈~(-1/3)(G/ρV~2) ～ 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, (-ΛC(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 Λ~* = k~(*1/3)C(k~*, H)Λ. In the limit Λ~* 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 Λ~* 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 Λ~* 1 which does not reduce to any of the rigid tube solutions determined previously. The decay rate of this solution decreases proportional to (Λ~*)~(-1/2) in the limit Λ~* 1, and the frequency increases proportional to Λ~*.