Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models

Abstract

We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwell models. This confirms an earlier observation of Tanner. We provide a new interpretation of the temporal instability of the grade-two fluid model for negative coefficients. Our techniques allow a simple proof of the convergence of the steady grade-two model to the Navier–Stokes model as $\alpha \rightarrow 0$ (under suitable conditions) in three dimensions. They also provide a proof of the convergence of the steady Oldroyd models to the Navier–Stokes model as their parameters tend to zero.