Numerical Simulation of Inhomogeneous Couette-Poiseuille Flow with Rayleigh Friction

The results of numerical modeling of steady viscous incompressible fluid flow in a flat channel, taking into account linear Rayleigh friction, are presented. The study focuses on a generalized model of hybrid Couette-Poiseuille flow, in which the longitudinal velocity linearly depends on the transverse coordinate, while the coefficients of this expansion vary along the channel height. The mathematical model is based on modified Navier-Stokes equations with an additional dissipative term proportional to velocity, which allows accounting for distributed drag forces arising during flow through porous media or in the presence of dispersed phases. A second-order finite difference method implemented in the MATLAB 2023b environment was used to solve the boundary value problem. Verification of the numerical method was carried out by comparing the results with analytical solutions for limiting cases. The influence of the Rayleigh friction coefficient on the flow structure was investigated for two technical fluids with different rheological properties: polyalphaolefin PAO-40 and Tyfocor LS heat transfer fluid at 40°C. The results demonstrate that an increase in the friction coefficient leads to a significant reduction in flow velocity and the formation of complex spatial structures. The emergence of reverse flow zones (counterflows) at certain parameter values was discovered. Different sensitivities of the fluids to changes in dissipative characteristics were revealed: the less viscous fluid exhibits a stronger dependence of velocity on the friction parameter. It is shown that qualitative changes in the flow structure are determined by the type of differential operator of the system. The obtained results have practical significance for the design of heat exchange systems, microfluidic devices, and technological equipment where additional dissipative mechanisms must be taken into account. The proposed calculation methodology allows optimizing the parameters of such systems considering volumetric drag forces.