Karhunen-Loève Galerkin method with decimated sampling technique for the simulation of complex fluids defined in the phase space

In complex fluids, solute molecules with structural length scales much larger than atomic are dispersed in solvents of simple fluids such as water. The rheological properties of complex fluids are determined by dynamics of solute molecules which can be modeled by the Fokker-Planck equation defined in a six-dimensional phase space. In the present investigation, we devise a method of efficient simulation of complex fluid flows employing the Karhunen-Loève Galerkin (KLG) method. Adopting the decimated sampling of solvent flow fields, a reduced-order model for the Fokker-Planck equation is obtained, which can be employed for the the simulation of complex fluids with a decent computer time. As a specific example, we consider a flow of dilute polymeric liquids over a cylinder, whose constitutive equation is the FENE (finitely extensible nonlinear elastic) model. It is found that the KLG method with the decimated sampling technique yields accurate results at a computational cost less than a hundredth of that for the numerical simulation using the Fokker-Planck equation. The KLG method supplemented by the decimated sampling technique is an efficient method of coarse-graining for equations of complex fluids defined in the phase space.