A review of the BGK approximation to obtain the equations of motion for an incompressible fluid is presented and its drawbacks are revealed. In order to overcome these inherent problems, new models for the particle distribution functions are needed. Using the Finite Difference Lattice Boltzmann Method (FDLBM) due to Fu and So (2009)  and the Thermal Difference Discrete Flux Method (TDDFM) proposed by Fu et al. 2012 , it is shown that the newer distribution functions lead to the mass conservation equation, the equations of motion and the energy balance equation for incompressible fluids in two dimensions, employing the D2Q9 lattice as the model. This derivation is extended to compressible fluids as well. Next, using the D3Q15 lattice as an example, the three dimensional equations of continuum mechanics are derived. Since no restrictions are placed on the constitutive equations, the theoretical development applies to all fluids, whether they be Newtonian, or power law fluids, or viscoelastic and viscoplastic fluids. Finally, some comments are offered regarding the numerical scheme to calculate the particle distribution functions to determine the velocity and temperature fields.
- BGK approximation
- Continuum mechanics
- Lattice Boltzmann equation
- Particle distribution function