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

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 essential, and not orbital stability up to a shift as in the case of translationally invariant governing equations. Localized bulging motionless wave solutions of the governing equations are called aneurysm solutions. We assume that the fluid is subjected to the power law for viscous friction of a non-Newtonian fluid, though the viscosity of the fluid does not play a significant role and 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 of the linearized problem with a positive real part. This is achieved by constructing the Evans function depending only on the spectral parameter, analytic in the right half of the complex plane $\Omega^+$ and which zeroes in $\Omega^+$ coincide with the unstable eigenvalues of the problem. The non-existence of the zeroes of the Evans function is performed using the argument principle from the analysis of complex variables. Finally, we discuss the possibility of applying the results of the present analysis to the aneurysm formation in damaged human vessels under the action of internal pressure.