Spatiotemporal Stability of Flow through Collapsible, Viscoelastic Tubes

Abstract

The stability of the Hagen‐Poiseuille flow of a Newtonian fluid in an incompressible, collapsible, viscoelastic tube is determined using linear stability analysis. The dependence of the tube’s diameter in the base state on the axial distance is explicitly accounted for in the present formulation. A novel numerical strategy is introduced to study the spatiotemporal stability of the coupled fluid-structure system subjected to infinitesimal axisymmetric or nonaxisymmetric disturbances. Axisymmetric disturbances correspond to the azimuthal wave number n = 0. There, we have identified two convective instability modes, one propagating upstream and the other downstream. For each of the nonaxisymmetric disturbances n = 1‐6, we have found one absolute instability propagating upstream, whereas for n = 1, 2, downstream-propagating convective modes are additionally observed. Two of the standing waves have equal frequencies at their respective cusp points, while a third absolute instability has triple that frequency, in excellent agreement with existing experiments. The n = 1 absolute instability mode is replaced by a convective mode when the Reynolds number exceeds 200, while the other standing waves, at n = 2‐6, persist to high Reynoldsnumber values. Increasing the solid’s viscosity, thickness, or shear modulus causes the absolute instability modes to become convective as well as to ultimately become stable.