Asymptotic analysis of wall modes in a flexible tube revisited

The stability of wall modes in fluid flow through a flexible tube of radius R surrounded by a viscoelastic material in the region R < r < HR is analys ed using a combination of asymptotic and numerical methods. The fluid is Ne wtonian, while the flexible wall is modelled as an incompressible viscoelas tic solid. In the limit of high Reynolds number (Re), the vorticity of the wall modes: is confined to a region of thickness O(Re-1/3) in the fluid nea r the wall of the tube. Previous numerical studies on the stability of Hage n-Poiseuille flow in a flexible tube to axisymmetric disturbances have show n that the flow could be unstable in the limit of high Re, while previous h igh Reynolds number asymptotic analyses have revealed only stable modes. To resolve this discrepancy, the present work re-examines the asymptotic anal ysis of wall modes in a flexible tube using a new set of scaling assumption s. It is shown that wall modes in Hagen-Poiseuille flow in a flexible tube are indeed unstable in the limit of high Re ill the scaling regime Re simil ar to Sigma (3/4). Here Sigma is a nondimensional parameter characterising the elasticity of the wall, and Sigma equivalent to rho GR(2)/eta (2), wher e rho and eta are the density and viscosity of the fluid, and G is the shea r modulus of the wall medium. The results from the present asymptotic analy sis are in excellent agreement with the previous numerical results. Importa ntly, tile present work shows that the different types of unstable modes at high Reynolds number reported in previous numerical studies are qualitativ ely the same: they all belong to the class of unstable wall modes predicted in this paper.