On the solution of Fokker-Planck equations in steady recirculating flows involving short fiber suspensions

Numerical modeling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an advection equation linked to the fluid history. Thus, the numerical modeling of short fiber suspensions flows requires a description of the microstructural evolution (fiber orientation) which affects the flow kinematics and that is itself governed by this kinematics. There are different ways to describe the microstructural state. The use of orientation tensors, whose evolution is governed by advection equations, is widely considered. Other possibility is to describe the fiber orientation state from its probability density whose evolution is described by the Fokker-Planck equation (which does not involve any closure relation). The numerical treatment of advection problems is not a simple matter. Moreover, many industrial flows show often steady recirculating areas which introduce some additional difficulties, because in these flows neither boundary conditions nor initial conditions are known. In some of our former papers, we have proposed accurate techniques to solve the linear and non-linear advection equations governing the evolution of the second-order orientation tensor, in steady recirculating flows. These techniques combine a standard treatment of the non-linearity with a more original treatment of the associated linear problems imposing the periodicity condition. In this paper, we generalize those numerical strategies to compute the steady solution of the Fokker-Planck equation (which governs the evolution of the fiber orientation probability distribution) in steady recirculating flows. (C) 2003 Elsevier B.V. All rights reserved.