Two-fluid Chapman–Enskog theory for binary gas mixtures

Abstract

As a result of the increasing inefficiency in the transfer of energy in collisions between species with a decreasing ratio of molecular masses, the Knudsen number range of validity of the Chapman–Enskog (CE) theory for binary gas mixtures decreases linearly with the molecular mass ratio. To remedy the situation, a two-fluid CE theory uniformly valid in the molecular mass ratio is constructed here. The analysis extends previous two-fluid theories to allow for arbitrary potentials of intermolecular interaction and arbitrary mass ratios. The treatment differs from the CE formulation in that the mean velocities and temperatures of the two gases are not required to be identical to lowest order. To first order, the streaming terms of the Boltzmann equation are thus computed in terms of the derivatives of the two-fluid hydrodynamic quantities, rather than the mean mixture properties as in the CE theory. As a result, associated with the nonconservation of momentum and energy for each species alone, two new ‘‘driving forces’’ appear in the first-order integral equations. The amount of momentum and energy transferred per unit time between the species appear in the theory as free constants, which allow satisfying the constraint that all hydrodynamic information be contained within the lowest-order two-fluid Maxwellians. Simultaneously, this constraint fixes the rate of momentum and energy interchange in terms of the two-fluid hydrodynamic quantities and their gradients. The driving force d12 of the CE theory is directly related to the rate of interspecies momentum transfer, and the corresponding CE functions D1 and D2 appear here unmodified. But the physical interpretation of d12 is very different in the two pictures. On the CE side there is only one momentum equation, while d12 provides constitutive information fixing the diffusion flux (velocity differences) in the mass conservation equation. Here, the similar constitutive information associated to d12 is used to couple two different momentum equations. Although the CE theory captures some of the two-velocity aspects of the problem, no CE analog exists with the functions E1 and E2 associated here with temperature differences, which now require solving new integral equations. Finally, the presence of two velocities and two temperatures leads to four coefficients of viscosity and of thermal conductivity for the two stress tensors and heat flux vectors. Also, two thermal diffusion factors enter now into the expression for d12. Although all these new coefficients arise as portions of the overall CE transport coefficients, their independent optimal determination requires new developments. The corresponding variational formulation is presented here and used to first order to obtain explicit expressions for all two-fluid transport coefficients by means of Sonine polynomials as trial functions.