Recovery of Transport Coeffcients in Navier-Stokes Equations from Modeled Boltzmann Equation (Full Paper)

The objective of this Letter is to show that the full set of compressible Navier-Stokes (NS) equations (which consist of the continuity, the momentum, the thermal energy, and the gas equation of state) can be derived from the modeled Boltzmann equation (BE) [1] by suitably modifying the collision relaxation time  in the commonly assumed Bhatnagar, Gross, and Krook (BGK) model [2]. The modeled BE thus derived is valid for dense gas only when the mean free path between two successive particle collisions is very small compared with the characteristic spatial scale of the uid system L. It is not sufcient to show that the NS equations are recovered [35]; more important, it has to demonstrate that the transport coefcients (such as bulk viscosity , thermal conductivity , and the specic heat ratio ) of the uid can be correctly replicated, because in the solution of the NS equations, these coefcients are specied, but they are part of the solution of the modeled BE.   RT, the dependence of  on T is not consistent with the Sutherland law [6]. Here,  is uid density and R is the universal gas constant. The derived expression for  is   cp; this leads to an incorrect  dependence on T and a Prandtl number Pr  cp=  1, where cp is the specic heat at constant pressure of the uid. In other words, the Reynolds number Re  UL=; the Mach and Prandtl numbers thus deduced are different from the specications for the solution of the NS equations [711]. An attempt to address this deciency has been made by postulating a relaxation time matrix S to replace  and to solve the modeled BE using the lattice approach [7]. However, the elements of S, except 1, were not derived from physical consideration, but rather empirically. Because the Reynolds, Mach, and Prandtl numbers are part of the solution of the modeled BE, their accuracies are important to a

Back to Featured Studies