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 sufcient to show that the NS equations are recovered [35]; more important, it has to demonstrate that the transport coefcients (such as bulk viscosity , thermal conductivity , and the specic heat ratio ) of the uid can be correctly replicated, because in the solution of the NS equations, these coefcients are specied, 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 specic 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 specications for the solution of the NS equations [711]. An attempt to address this deciency 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

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