The Mathematical Theory of Non-Uniform Gases

Abstract

DAB = binary coefficient of mass diffusion in the mixture g = gravity acceleration h = height difference between the interfaces mercury-L; LHg = length of gas capillary and of mercury tube, n = molecular concentration NRe = Reynolds number P ; Ptot. = experimental and total pressure of the gas, respectively AP = pressure drop corresponding to height difference, h q; q H g = volumetric flow rate of the gas and of the mercury, respectively r; rHg = radius of gas capillary and of mercury tube, respectively R = gas law constant So = area of the interface mercury-gas at zero inclination of the viscometer S = area of the interface mercury-gas in the viscometer reservoirs during a run t; t ' = time required for displacing the mercury and the gas during a run, respectively.7 = t ' when P 2 50 atm.

A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion

“…To obtain expressions for Π and q, so as to close the system of equations (2.1a) to (2.1c), a solution is sought by expanding the distribution function and differential operators in Boltzmann's kinetic equation as power series in the Knudsen number, restricted by the assumption that the Knudsen number is much less than unity. To the second order, these expressions are written as (Chapman & Cowling 1970;Woods 1993):…”

“…This method is an asymptotic series expansion in which the unknown molecular distribution function and differential operators appearing therein are expanded in series in terms of a small parameter, taken to be the Knudsen number (Chapman & Cowling 1970;Karlin & Gorban 2002). At zeroth order this expansion method yields Euler's hydrodynamic equations, followed by Navier-Stokes-Fourier's continuum flow model at first order.…”

Information-theoretical derivation of an extended thermodynamical description of radiative systems

“…3, respectively͒, information theory provides a simple derivation for the second-order non-equilibrium correction (2) . This correction is very difficult to find out in the framework of the kinetic theory of gases 17,18 of conductive, purely matter systems. It is worthwhile to mention that information theory is very simple mathematically but has the disadvantage that the question of what constraints should be imposed in maximizing the entropy density is an open one at present.…”

A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

“…The relaxation problems occur quite often in many physical problems, for example in nonequilibrium and extended thermodynamics [8,25], kinetic theory [6,7] and nonlinear waves [45]. A relaxation phenomenon arises when the equilibrium state of a physical system is perturbed.…”

“…Following [12,22] we derive an anti-diffusive Chapman-Enskog distribution for the discrete Boltzmann equation to develop a second order upwind relaxation scheme. It is to be remarked that the Chapman-Enskog method is always associated with nonlinear convection-diffusion equations [6,7,8] and the use of Chapman-Enskog distribution function to reduce the excess numerical diffusion in the first order relaxation scheme is novel. Moreover, our scheme avoids intricate and time consuming solving of Riemann problems and complicated flux splittings.…”