Existence and uniqueness of stationary state of elastic tubes conveying power law fluids

Instability of collapsible tubes has been studied in many papers theoretically and experimentally in the context of biological applications. Up to the present day, only Newtonian fluids flowing in collapsible tubes have been studied. However, there are circumstances when blood, gall, and other biological fluids have essentially non-Newtonian properties. When the pressure inside of the tube is substantially lower than external pressure, the tube loses stability that yields the appearance of oscillation. Oscillation can also start due to non-existence of the steady state of the tube conveying fluid. In this paper, existence and uniqueness of axisymmetric states of collapsible tube conveying power law fluids have been theoretically studied. The qualitative analytical investigation of the equations of motion has been carried out for the stationary state of the tube. As a result, it is obtained that the solution of boundary value problem for stationary state of sufficient long tube does not exist. Moreover, the state of tube satisfying the boundary conditions can be non-unique for sufficiently small length of the tube.