One-dimensional model for the unsteady flow of a generalized third-grade viscoelastic fluid

Specific algorithms for non-Newtonian fluids flow depend on the governing constitutive equation. In this work, we present a constitutive equation for a third-grade fluid in which a specific normal stress coefficient depends on the shear rate. This new three-dimensional model is suitable for studies where phenomena like shear-thinning or shear-thickening occur. Using a function of power-law type, we apply the Cosserat theory to fluid dynamics, reducing the exact three-dimensional equations to a one-dimensional system depending only on time and on a single spatial variable. From this reduced system, we solve the unsteady equations for the wall shear stress and mean pressure gradient depending on the volume flow rate, Womersley number, viscoelastic coefficients, and flow index over a finite section of the tube geometry with constant circular cross section. Attention is focused on numerical simulations of unsteady flows regimes by using a Runge–Kutta method.