Theory of Boundary Conditions for the Boltzmann Equation

1977 ◽  
Vol 32 (6) ◽  
pp. 521-531 ◽  
Author(s):  
L. Waldmann
Keyword(s):  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Temperature Jump ◽  
Functional Relation ◽  
Linearized Boltzmann Equation ◽  
Kinetic Boundary ◽  
The Difference ◽  
Molecular Distribution Function ◽  
Transport Relaxation ◽  
Slip Coefficients

Abstract In preceding papers, Refs. 1,2, boundary conditions were developed for transport-relaxation equations by aid of a general reciprocity postulate for the interface. The same method is now used for the linearized Boltzmann equation. A new scheme emerges: the kinetic boundary conditions consist in a linear functional relation between interfacial "forces and fluxes" - in the sense of non-equilibrium thermodynamics - which are, broadly speaking, given by the sum and the difference of the molecular distribution function and its time-reversed, at the wall. The general properties of the kernels occurring in this atomistic boundary law are studied. The phenomenological surface coefficients of (generalized) linear thermo-hydrodynamics, as e. g. temperature jump, slip coefficients etc., can in a simple way be expressed by the kernel of the atomistic boundary law. This kernel is explicitly worked out for completely thermalizing wall collisions.

Heat Transfer Through a Gas Between Parallel Plates

1977 ◽  
Vol 32 (9) ◽  
pp. 914-926 ◽  
Author(s):  
L. Waldmann
Keyword(s):  
Heat Transfer ◽  
Particle Distribution ◽  
Temperature Jump ◽  
Exact Formula ◽  
Parallel Plates ◽  
Total Heat Transfer ◽  
Linearized Boltzmann Equation ◽  
Scattering Kernel ◽  
Interfacial Scattering ◽  
Transport Relaxation

Within the framework of boundary conditions recently developed for the linearized Boltzmann equation 1 the problem of heat transfer between parallel plates can be solved in terms of "transport- relaxation eigenfunctions". The particle distribution function and the total heat transfer in the Knudsen case are exactly expressed by integrals over the interfacial scattering kernel occurring in the new scheme. A detailed discussion of the general case gives an exact formula and sign statement for the temperature jump at parallel plates. An approximation, which encompasses v. Smoluchowski's approach, lies at hand. This approximation is also readily confirmed by a moment method.

The Slip Problems for a Simple Gas

1971 ◽  
Vol 26 (6) ◽  
pp. 964-972 ◽  
Author(s):  
S.K. Loyalka
Keyword(s):  
Boltzmann Equation ◽  
Temperature Jump ◽  
Velocity Slip ◽  
Polyatomic Gas ◽  
Extra Effort ◽  
Linearized Boltzmann Equation ◽  
Multicomponent Gas ◽  
Interaction Law ◽  
Temperature Jump Coefficient ◽  
Polyatomic Gas Mixtures

Abstract Simple and accurate expressions for the velocity slip coefficient, the slip in the thermal creep, and the temperature jump coefficient are obtained by applying a variational technique to the linearized Boltzmann equation for a simple gas. Completely general forms of the boundary conditions are used, and the final results are presented in a form such that the results for any particular intermolecular force law or the gas-surface interaction law can easily be calculated. Further, it is shown that, with little extra effort, the present results can be easily extended to include the case of a polyatomic gas. It is felt that the present work, together with a recent paper in which the author has considered the solutions of the linearized Boltzmann equation for a monatomic multicomponent gas mixture, provide the desired basis for the consideration of the various slip problems associated with the polyatomic gas mixtures.

scholarly journals Higher order slip according to the linearized Boltzmann equation with general boundary conditions

2011 ◽  
Vol 369 (1944) ◽  
pp. 2228-2236 ◽  
Author(s):  
Silvia Lorenzani
Keyword(s):  
Boltzmann Equation ◽  
Collision Operator ◽  
Velocity Slip ◽  
Lattice Boltzmann Methods ◽  
Linearized Boltzmann Equation ◽  
Potential Applications ◽  
Microscale Flows ◽  
Scattering Kernel ◽  
Linearized Collision Operator ◽  
Slip Coefficients

In the present paper, we provide an analytical expression for the first- and second-order velocity slip coefficients by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator and the Cercignani–Lampis scattering kernel of the gas–surface interaction. The polynomial form of the Knudsen number obtained for the Poiseuille mass flow rate and the values of the velocity slip coefficients are analysed in the frame of potential applications of the lattice Boltzmann methods in simulations of microscale flows.

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