Exact, approximate and numerical solutions for a variant of Stokes׳ first problem for a new class of non-linear fluids

Abstract

Stress power-law fluids are a special sub-class of fluids defined through implicit constitutive relations, wherein the symmetric part of the velocity gradient depends on a power-law of the stress (see Eq. (2.2)), and were introduced recently to describe the non-Newtonian response of fluid bodies. Such fluids are counterparts to the classical power-law fluids wherein the stress is given in terms of a power-law for the symmetric part of the velocity gradient. Stress power-law fluids can describe phenomena that cannot be described by classical power-law fluids (see [1]). In this paper, first a new exact solution for a variant of Stokes׳ first problem for stress power-law fluids, when the exponent n=0 (Navier–Stokes fluid), is obtained. Such an exact solution for the stress is in terms of a convolution integral, for which we establish bounds. We then compute the convolution integral using Gauss–Kronrod quadrature by ensuring that its value always lies within the bounds. Using the validated quadrature, we can accurately evaluate the exact solution and we the exact solution it to validate the numerical scheme employed in solving the governing equations for stress-power law fluids with arbitrary exponent n. Finally, for stress power-law fluids wherein the exponent n<0 (stress-thickening fluids), we obtain an approximate solution for the stress that agrees well with the numerical solution.