Numerical Solvability of a Cosserat Model for the Swirling Motion of a Third-grade Fluid in a Constant Radius Straight Circular Tube

Abstract

In this work we consider a thermodynamically compatible three-dimensional third-grade fluid model. With the aim of modelling swirling flow motions, we consider a velocity field approximation provided by the Cosserat theory, where we introduce specific scalar functions associated with the swirling motion effects. Integrating the linear momentum equation over the cross-section of a straight circular tube with constant radius we reduce the threedimensional model into a one-dimensional system, depending only on time and a single spatial variable. From this reduced system, we derive unsteady equations for the wall shear stress and mean pressure gradient depending on the volume flow rate, the Womersley number, the viscoelastic coefficients, the Rossby number and the swirling scalar function. Also, we obtain a partial differential equation for the swirling scalar function. The solvability of the model is demonstrated by presenting some numerical results for unsteady flow regimes over a finite section of the tube geometry.