Exact stream function solutions for two-dimensional incompressible couple stress fluid flow

In this work, the separation of variables approach is employed to investigate and assess analytical stream function solutions for two-dimensional incompressible couple stress fluid flow. The suggested method involves transforming the governing partial differential equations (PDEs) of fluid flow into ordinary differential equations (ODEs) by assuming that the stream function solution may be written as an addition of two functions, each based on a single variable. Without imposing any particular boundary requirements on the intended model, this study allows for a wide investigation of flow behaviours. This work explores the effects of several parameters, such as density, pressure coefficient of viscosity and especially the couple stress parameter, on velocity profiles. The derived solutions, expressed as stream functions, are applicable to several two-dimensional flow conditions involving couple stress fluids. By providing crucial insights into the behaviour of incompressible couple stress fluids, these results have practical consequences for biomedical modelling, industrial fluid processes and lubrication systems, among other uses.