On a variational principle for the Upper Convected Maxwell model

Abstract

A variational principle for the Upper Convected Maxwell model is presented. The stationary value of the appropriate functional is the drag on an immersed object. From the principle, a formula is derived for the derivative of the drag with respect to the Deborah number for an arbitrarily shaped particle in a circular duct under creeping flow conditions. The formalism is compared with the conventional reciprocal theorem. Whereas the reciprocal theorem gives the drag as a volume integral involving the Stokesian stress tensor, the variational principle involves the stress from the adjoint equation. For low Deborah numbers both approaches provide the correction to the Stokes drag as a volume integral involving only the Stokesian rate-of-strain tensor, in line with second-order fluid theory. • A variational principle for the Upper Convected Maxwell model is presented. • At stationarity the functional equals the drag on a partcle in a duct. • For low Deborah numbers the drag correction involves only the Stokesian strain rate. • This is in correspondence with the reciprocal theorem and second-order fluid theory.