Linear stability of Hagen–Poiseuille flow in a chemically reacting fluid

Abstract

In this short paper we study the linearized stability of the flow of a chemically reacting fluid in a cylindrical pipe, under the assumption that the length of the pipe is far greater than its diameter. The fluid models that are considered have relevance to the flow of both polymeric liquids that are capable of undergoing chemical reactions and biological fluids such as the synovial fluid whose viscosity changes due to the concentration of the hyaluronan. The viscosity of the class of fluids that we consider can increase or decrease due to the concentration of the chemical that is being carried by the fluid and it can also shear thin or shear thicken. We non-dimensionalize the equations governing the motion of the fluid and then carry out an approximation wherein we retain terms that are of order unity in the Reynolds number and Péclet number. We further simplify the problem by seeking a special semi-inverse solution, in the same spirit as that which is used in the study of classical Hagen–Poiseuille flow, and look for solutions for the velocity field and the concentration that vary only with the radial coordinate. Under the above mentioned approximation, one can obtain an exact solution for the basic flow which then allows us to analytically consider the stability of the base flow to sufficiently small disturbances. On the basis of earlier studies of such fluids in the modeling of biological fluids, especially the synovial fluid, we consider two types of variation of the viscosity with the concentration. We find that flows in the cylindrical pipe, within the context of our approximation, are stable to sufficiently small disturbances, for both variations of the viscosity that are considered.