Regularized moment equations for binary gas mixtures: Derivation and linear analysis

Abstract

The applicability of the order of magnitude method [H. Struchtrup, “Stable transport equations for rarefied gases at high orders in the Knudsen number,” Phys. Fluids 16, 3921–3934 (2004)] is extended to binary gas mixtures in order to derive various sets of equations—having minimum number of moments at a given order of accuracy in the Knudsen number—for binary mixtures of monatomic-inert-ideal gases interacting with the Maxwell interaction potential. For simplicity, the equations are derived in the linear regime up to third order accuracy in the Knudsen number. At zeroth order, the method produces the Euler equations; at first order, it results into the Fick, Navier–Stokes, and Fourier equations; at second order, it yields a set of 17 moment equations; and at third order, it leads to the regularized 17-moment equations. The transport coefficients in the Fick, Navier–Stokes, and Fourier equations obtained through order of magnitude method are compared with those obtained through the classical Chapman–Enskog expansion method. It is established that the different temperatures of different constituents do not play a role up to second order accurate theories in the Knudsen number, whereas they do contribute to third order accurate theory in the Knudsen number. Furthermore, it is found empirically that the zeroth, first, and second order accurate equations are linearly stable for all binary gas mixtures; however, although the third order accurate regularized 17-moment equations are linearly stable for most of the mixtures, they are linearly unstable for mixtures having extreme difference in molecular masses

Fundamental solutions to moment equations for the simulation of microscale gas flows

“…For example, fundamental solutions can be found to the steady linearised Burnett equations (Chapman & Cowling 1970), and their many variants. Future work can also include finding fundamental solutions to R13-equation extensions for hard sphere molecules ) and diatomic gas mixtures (Gupta, Struchtrup & Torrilhon 2016). Potentially, even, going As an illustration of the utility of such fundamental solutions, in the context of creeping rarefied flow modelling, we have implemented the Gradlet and Thermal Gradlet into a method of fundamental solutions (a linear superposition scheme).…”

Heat flux in binary gas mixtures confined between two parallel plates via moment equations

“…are the ratios directly related to the collisional cross sections; Ω = x 0 α Ω α + x 0 β Ω β ; and the coefficients δ 8 , δ 19 , δ 20 are constants while the other δ i 's depend only on the mass ratios of the constituents given by (24). The explicit values of δ i 's are not given here for brevity but for MM and HS, they can be found in [23].…”

“…[6,11,[15][16][17][18][19][20][21] and references therein. Motivated from the success of moment method for rarefied single-component monatomic gases, the present authors, in the last few years, have extended the Grad moment method to multi-component monatomic gas mixtures interacting with Maxwell [22] and hard-sphere [23] interaction potentials, and derived the regularized moment equations for twocomponent monatomic gas mixtures of Maxwell molecules [24]. Recently, the method has also been extended to single-component rarefied granular gases of hard spheres [25][26][27].…”

Asymptotic modeling of thermal binary monatomic gas flows in plane microchannels—Comparison with DSMC simulations

“…41 We add the simulations obtained with the DS2V code 45 for flows of a two gas mixture in a microchannel. 46 From a theoretical point of view, the regularized 17-moment equations modeling the binary mixtures of monatomic inert ideal gases in the recent paper of Gupta et al 47 should also give interesting results for a flow in a microchannel. In the paper of Rahimi and Struchtrup, 48 a set of regularized partial differential equations is obtained for a binary rarefied polyatomic gas mixture, where two different relaxation times are introduced to take into account the translational and internal energy exchanges.…”