Stokes' first problem for some non-Newtonian fluids: Results and mistakes

TL;DR

This paper revisits Stokes' first problem for non-Newtonian fluids, providing corrected solutions, identifying common errors in prior work, and validating results with numerical schemes.

Contribution

It offers accurate analytical solutions for second-grade and Oldroyd-B fluids and highlights and corrects mistakes in previous literature.

Findings

Correct analytical solutions for non-Newtonian fluids

Identification of errors in previous studies

Numerical schemes confirming analytical results

Abstract

The well-known problem of unidirectional plane flow of a fluid in a half-space due to the impulsive motion of the plate it rests upon is discussed in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The governing equations are derived from the conservation laws of mass and momentum and three correct known representations of their exact solutions given. Common mistakes made in the literature are identified. Simple numerical schemes that corroborate the analytical solutions are constructed.