Finite element analysis of fibre suspension flows

Abstract

The numerical simulation of the flow of fluids containing short, slender fibres is investigated. The orientations of the fibres are represented in an averaged sense by a second-order orientation tensor A . The governing equations contain, in addition to velocity, pressure and second-order orientation, a fourth-order orientation tensor which is approximated in terms of A through the use of various closure rules. Discretisation is carried out using the standard Galerkin method for the momentum equation, and the discontinuous Galerkin method for the evolution equation. Numerical results focus on two areas. Firstly, the behaviour of the evolution equation is investigated for simple shear flows, and for various closure rules and choices of parameters. Earlier studies by others, in which the use of the linear closure leads to oscillatory behaviour, is confirmed in the present study, though it is shown that a steady state is ultimately achieved. The second study in this work is concerned with the influence of fibres on fluid flow in the benchmark 4:1 contraction problem. The necessity of using upwinding is confirmed, at least for the limiting case of zero rotary diffusivity, in that unrealistic fibre orientations are obtained in the absence of upwinding. Experimental results show an increase in the magnitude of the zone of recirculation with increase in fibre concentration; this behaviour is reproduced in the numerical results presented here.