Vinay Kumar Gupta

Automated Boltzmann Collision Integrals for Moment Equations

We present a methodology to evaluate the moments of the Boltzmann collision term, in a general automated way, using the computer algebra software Mathematica. Based on Grads distribution function with 26-moments, we compute the non-linear production terms for a simple gas and a granular gas, and the linear production terms for a binary mixture of gases. The results can be shown for general interaction potential, but, in this paper, they are given only for hard-sphere interaction potential.

Higher order moment equations for rarefied gas mixtures

The fully nonlinear Grads N×26-moment (N×G26) equations for a mixture of N monatomic-inert-ideal gases made up of Maxwell molecules are derived. The boundary conditions for these equations are derived by using Maxwells accommodation model for each component in the mixture. The linear stability analysis is performed to show that the 2×G26 equations for a binary gas mixture of Maxwell molecules are linearly stable. The derived equations are used to study the heat flux problem for binary gas mixtures confined between parallel plates having different temperatures.

Comparison of relaxation phenomena in binary gas-mixtures of Maxwell molecules and hard spheres

The strategy for computing the Boltzmann collision integrals for gaseous mixtures is presented and bestowed to compute the fully non-linear Boltzmann collision integrals for hard sphere gas-mixtures. The Boltzmann collision integrals associated with the first 26 moments of each constituent in a gas-mixture are presented. Moreover, the Boltzmann collision integrals are exploited to study the relaxation phenomena of diffusion velocities, stresses and heat fluxes in binary gas-mixtures of Maxwell molecules and hard spheres.

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

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.

Higher-order moment theories for dilute granular gases of smooth hard-spheres

Grads method of moments is employed to develop higher-order Grads moment equations---up to first 26-moments---for granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grads 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haffs law while the other higher-order moments decay on a faster time scale. The constitutive relations for stress and heat flux (the Navier--Stokes and Fourier relations) are obtained by performing a Chapman--Enskog-like expansion on the Grads 26-moment equations and compared with those existing in the literature. The linear stability of the homogeneous cooling state is analyzed through the Grads 26-moment system and various sub-systems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable eigenmode remains unstable for all wavenumbers below a certain coefficient of restitution while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of coefficient of restitution. In particular, the Grads 26-moment theory leads to a smooth profile for the critical wavenumber in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment and existing theories are also in excellent agreement.

Reprint of

The strategy for computing the Boltzmann collision integrals for gaseous mixtures is presented and bestowed to compute the fully non-linear Boltzmann collision integrals for hard sphere gas-mixtures. The Boltzmann collision integrals associated with the first 26 moments of each constituent in a gas-mixture are presented. Moreover, the Boltzmann collision integrals are exploited to study the relaxation phenomena of diffusion velocities, stresses and heat fluxes in binary gas-mixtures of Maxwell molecules and hard spheres.

Coupled constitutive relations: a second law based higher-order closure for hydrodynamics

In the classical framework, the Navier–Stokes–Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier–Stokes–Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier–Stokes–Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.