Three-Dimensional Velocity Field Using the Cross-Model Viscosity Function

The study of the three-dimensional fluid model for which the Cauchy stress tensor depends on the cross viscosity function is a challenging and complex model in terms of computational effort. To simplify this computational difficulty presented by the three-dimensional problem, we use an approach based on the Cosserat theory related to fluid dynamics which reduces the three-dimensional problem to a one-dimensional system of ordinary differential equations depending only on time and a single spatial variable. From this new system, we obtain the unsteady equation for the mean pressure gradient depending on the volume flow rate, Womersley number, and viscosity parameters over a finite section of straight, rigid, and impermeable tube with constant circular cross section. In particular, given specific data, we can obtain information about the volume flow rate, and consequently we can illustrate the three-dimensional velocity field.