On the reduction of stochastic kinetic theory models of complex fluids

Kinetic theory models involving the Fokker-Planck equation are usually solved in the framework of stochastic approaches, which allows us to circumvent the difficulties related to the multidimensional character of that equation. In fact, the Fokker-Planck equation governs the evolution of the distribution function that defines the molecular configuration at each point of the physical space and at each time. As the molecular conformation is usually defined by several coordinates, the resulting distribution function will depend on the physical and configuration coordinates and the time. Although different numerical strategies have recently been proposed for solving that equation with efficiency and accuracy (Ammar et al 2006 J. Non-Newtonian Fluid Mech. 134 136 47, Ammar et al 2006 J. Non-Newtonian Fluid Mech. 139 153-76) the stochastic approach is today the most common for solving general kinetic theory models. This paper presents some preliminary results that provide evidence for the potential applicability of model reduction techniques based on the Karhunen- Loeve decomposition or on separated representations for reducing the computational efforts related to the solution of such models in the Brownian configuration fields framework.