S. K. Loyalka

Some numerical results for the BGK model: Thermal creep and viscous slip problems with arbitrary accomodation at the surface

Numerically ’’exact’’ results for the thermal creep and viscous slip problems for the BGK model and Maxwellian diffuse specular reflection at the wall are obtained by using equations derived earlier by Cercignani, Loyalka, and Cipolla. Results obtained by the use of both the half range and the ’’full‐range’’ expansions are found to be in complete agreement and provide standards of comparison for various approximate methods.

Thermal Transpiration in a Cylindrical Tube

Thermal transpiration in a cylindrical tube is studied by using the Bhatnagar‐Gross‐Krook model and methods previously used by Ferziger and Cercignani and Sernagiotto in solving the Poiseuille flow problem. It appears that recent results of Sone and Yamamoto are in slight error. Finally, in analogy with the recent work of Mason and co‐workers on the dusty gas model, a relation for ftr, the translational Eucken factor, is developed.

Thermophoretic force on a single particle—I. Numerical solution of the linearized Boltzmann equation

Abstract The force experienced by a small aerosol particle in the presence of a temperature gradient is known as the thermophoretic force. Motion of particles under such a force is known as thermophoresis. Thermophoresis has important technological applications, but uncertainties continue to exist regarding the experimental measurements as well as the theoretical estimations of the thermophoretic force. We describe, in this work, a prescription for calculation of the force on a sphere based on a numerical solution of the linearized Boltzmann equation. Then, we report some explicit results for a rigid sphere gas.

Some exact numerical results for the BGK model: Couette, Poiseuille and thermal creep flow between parallel plates

Numerically “exact” results for the Couette flow, Poiseuille flow and thermal creep flow between parallel plates are given for the BGK model. Equations obtained earlier by Cercignani and Williams, who used the method of elementary solutions, are solved by convergent Liouville-Series and results are compared with works of other authors.ZusammenfassungNumerisch “exakte” Lösungen für die Couette-, Poiseuille- und thermische Kriechströmung zwischen parallelen Platten werden nach dem BGK-Modell angegeben. Die früheren Gleichungen von Cercignani und Williams, die die Methoden der elementaren Lösungen benutzt haben, werden durch konvergierende Liouville-Reihen gelöst. Die Ergebnisse werden mit Arbeiten anderer Verfasser verglichen.

Sound‐wave propagation in a rarefied gas

The problem of sound wave propagation in a rarefied gas is studied by the use of the linearized BGK model and diffuse reflection boundary conditions. The relevant integrodifferental equation and boundary conditions for the half‐space are converted to a system of linear integral equations, which is then solved by a straightforward numerical differencing technique. The present results are in better agreement with the experimental results of Meyer and Sessler and Greenspan than the BGK model results of Sirovich and Thurber and Buckner and Ferziger who, respectively, use analytical continuation and an approximate boundary condition.

Motion of a sphere in a rarefied gas

The problem of the motion of a single spherical particle in a rarefied gas is considered, and results for the velocity profiles and the drag forces for all rarefactions are obtained using the Bhatanagar–Gross–Krook model and diffuse reflection at the surface of the sphere. Results for the drag forces are in good agreement with previously available results of Cercignani, Pagani, and Bassanini. The velocity profile results are also in good agreement with the results of Sone and Aoki who had considered the slip limit.

Velocity profile in the Knudsen layer for the Kramer’s problem

Since the BGK model is based on the assumption of constant collision frequency, and since this model has been found inadequate in describing some experimental data, the numerical study of a variable collision frequency model proposed earlier by Cercignani and Loyalka and Ferziger is described. Specifically, the Kramer’s problem for this model is solved, and it is found that the ’’velocity defect’’ in the Knudsen layer is quite sensitive to the velocity dependence of the collision frequency. In fact, for the hard sphere collision frequency, the present results agree reasonably well with the recent experimental data of Reynolds, Smolderen, and Wendt.

The Kramers problem: velocity slip and defect for a hard sphere gas with arbitrary accommodation

The half-space problem of rarefied gas flow (the Kramers problem) is considered for the linearized Boltzmann equation and arbitrary gas-surface interaction. Accurate numerical results for the velocity slip coefficient and velocity defect are obtained for the rigid sphere interaction and Maxwellian boundary condition.

Temperature jump in a gas mixture

The problem of temperature jump in a multicomponent gas mixture is considered and a variational result valid for the Boltzmann equation and arbitrary gas surface interaction law is given. From this general result, a particularly useful expression for the monatomic binary gas mixtures and Maxwellian diffuse specular reflection is deduced. Explicit results for the He‐Ar mixture are given by using the Lennard‐Jones potential. Generalizations to the polyatomic gases are also discussed.

Temperature‐jump problem with arbitrary accommodation

A concise and accurate result for the temperature‐jump coefficient based on the linearized BGK model and arbitrary accommodation is reported. The jump coefficient is expressed as a power series in (1‐α), and values of the expansion coefficients are given.