On the impact of nonlinear continuum mechanics on particle distribution functions for Newtonian and non-Newtonian fluids: Applications to the flows of Bingham fluids

Abstract

From the lattice Boltzmann equation, it is possible to derive the continuity equation and Cauchy’s equations of motion for a compressible medium, when one uses the Bhatnagar-Gross-Krook (BGK) - Welander approximation. From this, one can obtain the equations relevant to incompressible fluids. However, these require that the pressure be proportional to the density and the viscosity be dependent on the collision relaxation time. Clearly, these restrictions on the pressure and the viscosity are unacceptable in modelling the flows of Newtonian or non-Newtonian, incompressible fluids. In order to overcome these inherent problems, new models for the particle distribution functions are needed which are as general as possible. The motivation for the development of these models is driven by the history of nonlinear continuum mechanics, which shows that this subject evolved at the level of utmost generality, whether the specific topic was finite deformations in isotropic elasticity or the flows of viscoelastic fluids; or, the formulations of constitutive equations; and, the kinematics of flows. In order to maintain this generality in deriving the continuity equation and the equations of motion in Cartesian, cylindrical and spherical coordinates for all fluids using particle distribution functions, their evolution equations are written in a divergence form applicable in three dimensions. From this set, it is shown that the equations of continuum mechanics for Newtonian and non- Newtonian fluids can be derived in the three coordinate systems, when additional source terms are added to the equations of evolution in the latter two coordinate systems. If the body forces are present, a new set of source functions is required in each coordinate system and these are described as well. Next, the energy equation is derived by using a separate set of particle distribution functions. Modifications of the relevant equations to be applicable to incompressible fluids are described. The incorporation of boundary conditions and the description of the numerical scheme for the simulation of the flows employing the new approach is given. Validation results obtained through the modelling of a mixed convection flow of a Bingham fluid in a lid-driven square cavity, and the steady flow of a Bingham fluid in a pipe of square cross-section are presented. Finally, some comments on the theoretical differences between the present approach and the existing formulations regarding Lattice Boltzmann Equations are offered.