Three Dimensional Dynamics of Polar Fluids: A Lattice Boltzmann Approach

In this paper we propose a Lattice-Boltzmann-based mesoscopic model for the three-dimensional flow of an incompressible polar fluid.The mesoscopic model is equivalent to the usual incompressible three-dimensional Navier-Stokes equation coupled with the angular momentum equation, which describes the evolution of the angular velocity vector of the fluid particles.The proposed model is applied to investigate the effects of the fluid polar structure on three steady flows: the steady Couette and Poiseuille flows in a square channel and the three-dimensional lid-driven cavity flow at $Re=100,$ $Re=400.$The effects of the fluid polar structure on the above mentioned flows are investigated by varying the relevant dimensionless parameters: the coupling parameter $N$ and the geometric parameter $L.$Results are consistent with the predictions of the theory of polar fluids. In particular, it is shown for the three-dimensional lid-driven cavity flow that the effect of the coupling parameter $N$ is to lower the effective Reynolds number and thus to increase the viscosity.