Stability of an elastic tube conveying a non-Newtonian fluid and having a locally weakened section

The work is devoted to the stability analysis of the flow of a non-Newtonian power-law fluid in an elastic tube. Integrating the equations of motion over the cross section, we obtain a one-dimensional equation that describes long-wave low-frequency motions of the system in which the rheology of the flowing fluid is taken into account. In the first part of the paper, we find a stability criterion for an infinite uniform tube and an absolute instability criterion. We show that instability under which the axial symmetry of motion of the tube is preserved is possible only for a power-law index of $n<0.611$, and absolute instability is possible only for $n<1/3$; thus, after the loss of stability of a linear viscous medium, the flow cannot preserve the axial symmetry, which agrees with the available results. In the second part of the paper, applying the WKB method, we analyze the stability of a tube whose stiffness varies slowly in space in such a way that there is a “weakened” region of finite length in which the “fluid–tube” system is locally unstable. We prove that the tube is globally unstable if the local instability is absolute; otherwise, the local instability is suppressed by the surrounding locally stable regions. Solving numerically the eigenvalue problem, we demonstrate the high accuracy of the result obtained by the WKB method even for a sufficiently fast variation of stiffness along the tube axis.