Stability of the aneurysm-type solution in a membrane tube with localized wall thinning filled with a fluid with a non-constant velocity profile

Abstract

We perform the stability analysis of bulging localized structures on the wall of a fluid-filled axisymmetric membrane elastic tube. The wall of the tube is assumed to be subjected to localized thinning. The problem has no translational invariance anymore, hence the stability of a bulging wave centered in the point of the localization of imperfection is usual, and not orbital stability up to a shift as in the case of translationally invariant governing equations. Motionless wave solutions of the governing equations having the form of a localized bulging are called throughout the paper aneurysm solutions. We assume that the fluid is subjected to the power law for viscous friction of a non-Newtonian fluid, and also that the viscosity of the fluid does not play a significant role and it can be neglected. The velocity profile remains not constant along the cross section of the tube (even in the absence of the viscosity) because no-slip boundary conditions are performed on the tube walls. Stability is established by demonstrating the non-existence of the unstable eigenvalues with a positive real part of the linearized problem. This is achieved by constructing the Evans function depending only on the spectral parameter, analytic in the right half of the complex plane Ω+ and which zeroes in Ω+ coincide with the unstable eigenvalues of the problem. The non-existence of zeroes of the Evans function is performed using the argument principle from the analysis of complex variable. Finally, we discuss the possibility of applying of the results of the present analysis to the aneurysm formation in damaged human vessels under the action of the internal pressure.