Unsteady non-isothermal flow of a viscoplastic fluid with non-constant material moduli at low Reynolds number

Abstract

In this paper we consider the non-isothermal flow of a Bingham fluid with rheological parameters depending on pressure and temperature. The governing equations are scaled in order to evaluate the relevance of the various kinematical and thermodynamical effects represented by some non-dimensional groups such as the Reynolds, second Froude, Brinkman, Peclet and Bingham numbers. We consider a perturbative approach in which the small parameter is the product of the reference temperature and the thermal expansion coefficient. In a recent paper (Fusi in Meccanica, 2017. doi: 10.1007/s11012-017-0655-8 ) this problem has been investigated under the hypothesis that the Reynolds number is of order one. Here we study the case in which the Reynolds number is small (creeping flow). The differences with Fusi (Meccanica, 2017. doi: 10.1007/s11012-017-0655-8 ) are far from trivial. In particular we show that in the situation studied here we can obtain a non-isothermal Stokes Oberbeck approximation in which the buoyancy and frictional effects are taken into account.