Internal viscoelastic flows for fluids with exponential type pressure-dependent viscosity and relaxation time

Abstract

The isothermal steady-state and pressure-driven flows in a straight channel and a circular tube, of an incompressible viscoelastic fluid which follows the Maxwell constitutive model, are considered. Under the assumption that both the shear viscosity and the single relaxation time of the fluid vary exponentially with pressure, the governing equations are solved analytically using a regular perturbation scheme with small parameter the dimensionless pressure-viscosity coefficient. The solution is found up to sixth order in the small parameter, revealing a two-dimensional (2D) flow field and the dependence of the primary flow variables on the geometrical aspect ratio, the pressure-viscosity coefficient, and the Weissenberg and Reynolds numbers. It is demonstrated that the pressure-dependent viscosity and relaxation time enhance the pressure gradient along the main flow direction, generate another along the wall-normal direction, and cause vertical motion of the fluid. Viscoelastic extra-stresses, which affect significantly the average pressure difference, required to drive the flow and the shear stress at the wall, are also predicted. Moreover, the mean Darcy friction factor shows a substantial deviation from the average pressure difference, as the fluid elasticity increases. For the Newtonian fluid, the effect of the pressure-dependent viscosity on the velocity components is minor, but substantial on the pressure and shear-stress profiles. Most of these features are predicted for the first time, and they are due to the fact that the flow field is fully 2D, indicating the complex nature of fluids with pressure-dependent viscosity and relaxation time.