Boundary Value Problem In Half-space Research Articles

Linear Moment Models to Approximate Knudsen Layers

We propose a well-posed Maxwell-type boundary condition for the linear moment system in half-space. As a reduction of the Boltzmann equation, the moment equations are available to model Knudsen layers near a solid wall, where proper boundary conditions should be prescribed. Utilizing an orthogonal decomposition, we separate the part with a damping term from the system and then impose a new class of Maxwell-type boundary conditions on it. Due to the new boundary condition, we show that the half-space boundary value problem admits a unique solution with explicit expressions. Instantly, the well-posedness of the linear moment system is achieved. We apply the procedure to classical flow problems with the Shakhov collision term, such as the velocity slip and temperature jump problems. The model can capture Knudsen layers with very high accuracy using only a few moments.

The Classification of Well-Posed Kinetic Boundary Layer for Hard Sphere Gas Mixtures

The kinetic boundary layer for gas mixtures is described by a half space boundary value problem for the two-species steady Boltzmann equation with an incoming distribution. We consider it around a drifting normalized bi-Maxwellian and prove that the boundary layer problem is well-posed when the drifting velocity u exceeds the sound speed , but one (respectively five, six) additional condition must be imposed when (respectively and ).

A general solution of equations of equilibrium in linear elasticity

A general solution of equations of equilibrium in linear elasticity is presented in cylindrical coordinates in terms of three harmonic functions describing an arbitrary displacement field. The structure of this solution is similar to the general solution given by Love (Kelvin’s solution) in spherical coordinates. Galerkin vector representation of our solution leads to an integral connecting the harmonic functions. The connections to Papkovich–Neuber and Muki’s general representations are also provided. Suitable choices of the harmonic functions in our new representation yield general solutions for axisymmetric deformations due to Love, Boussinesq and Michell. Some unbounded deformations induced by singular forces are tabulated in terms of the scalar harmonic functions to justify the simple nature of our representation. Exact solution of the half-space boundary value problem is also provided to demonstrate the power of our approach. The stress components computed via our solution are also listed (see the Appendix).

Temperature jump in degenerate quantum gases with the Bogoliubov excitation energy and in the presence of the Bose-Einstein condensate

We construct a linearized kinetic equation modeling the behavior of degenerate quantum Bose gases with the collision rate dependent on the momentum of elementary excitations. We consider the general case where the energy of elementary excitations depends on the momentum according to the Bogoliubov formula. We analytically solve the half-space boundary value problem of the temperature jump at the boundary of the degenerate Bose gas in the presence of the Bose-Einstein condensate. We obtain an expression for the Kapitza resistance.

Temperature jump in degenerate quantum gases in the presence of a Bose-Einstein condensate

We construct a kinetic equation modeling the behavior of degenerate quantum Bose gases whose collision rate depends on the momentum of elementary excitations. We consider the case where the phonon component is the decisive factor in the elementary excitations. We analytically solve the half-space boundary value problem of the temperature jump at the boundary of the degenerate Bose gas in the presence of a Bose -- Einstein condensate.

Analytic Solution of an Inhomogeneous Kinetic Equation in the Problem of Second-Order Thermal Creep

An analytic solution of the problem of second-order thermal creep is obtained. A method for solving the half-space boundary value problem for an inhomogeneous linearized kinetic BGK equation forms the basis of the solution. The general solution of the input equation is constructed in the form of an expansion of the corresponding characteristic equation in terms of the eigenfunctions. Substitution of the solution in the boundary conditions leads to a Riemann boundary value problem. The unknown thermal creep velocity is found from the condition of solvability of the boundary value problem. The numerical analysis performed confirms the existence of negative thermophoresis (in the direction of the temperature gradient) for high-conductivity aerosol particles at low Knudsen numbers.

Exact solution of the kinetic equation in the problem of nonisothermal flow in the neighborhood of a weakly curved surface

An analytic method for solving the half-space boundary value problem for the inhomogeneous Boltzmann equation with the collision operator in the form of an elliptico-statistical model (the ES-model of the Boltzmann equation) is proposed for the problem of nonisothermal rarefied gas flow in the neighborhood of a curved surface. An exact analytic expression is derived for the thermal slip of a monatomic gas along the surface of a rigid spherical aerosol particle. A numerical value of the gas-kinetic coefficient which takes into account the effect of the curvature of the surface on the thermal slip coefficient is obtained. A comparison with published data is carried out.

Exact solution of a boundary value problem in semiconductor kinetic theory

An explicit solution of the stationary one-dimensional half-space boundary value problem for the linear Boltzmann equation is presented in the presence of an arbitrarily high constant external field. The collision kernel is assumed to be separable, which is also known as “relaxation time approximation;” the relaxation time may depend on the electron velocity. Our method consists in a transformation of the half-space problem into a non-normal singular integral equation, which has an explicit solution.

Half-space problem of the Boltzmann equation for charged particles

For two particular collision kernels, we explicitly solve the one-dimensional stationary half-space boundary value problem of the linear Boltzmann equation including a constant external field via an extension of Case's eigenfunction technique. In the first collision model we reproduce a solution recently obtained by Cercignani; in the second model the solution of the stationary boundary value problem is presented for the first time.

Wave Theory for a Converging Spherical Incident Wave in an Infinite-aperture System

Abstract Weyl's formula describes the decomposition of a monochromatic spherical wave into an angular spectrum of plane waves. In this paper, explicit formulae for the separate contributions of homogeneous and inhomogeneous plane waves are given, and the results used to obtain an exact solution to the half-space boundary-value problem for the focusing of a converging spherical incident wave in an infinite-aperture system.

The boundary value problem in fermion systems

The half-space boundary value problem for fermions near zero temperature in plane geometry is solved for diffuse boundary scattering by numerically constructing the spatial propagator in terms of the eigenfunctions of a generalized eigenvalue problem for the linearized Uehling-Uhlenbeck collision integral. The slip length is calculated for several interparticle scattering laws and compared with a relaxation time ansatz result and the experimental values for normal fluid3He. It is shown that the nonsingular part of the collision operator is relatively compact to the singular part.

A classification of well‐posed kinetic layer problems

AbstractIn the first part of this paper, we study the half space boundary value problem for the Boltzmann equation with an incoming distribution, obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero. We linearize it about a drifting Maxwellian and prove that, as conjectured by Cercignani [4], the problem is well‐posed when the drift velocity u exceeds the sound speed c, but that one (respectively four, five) additional conditions must be imposed when 0 < u < c (respectively −c < u < 0 and u < −c).In the second part, we show that the well‐posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions.

Nonrelativistic electron stream propagation in the solar atmosphere and type III radio bursts

Electron streams with type III burst characteristics are numerically modeled. The electronplasma wave quasilinear interaction is assumed to be the dominant velocity diffusion process. The quasilinear equations with the addition of spontaneous emission, magnetic and collisional effects are numerically solved as an initial value and a half-space boundary value problem with time, distance and velocity as the independent variables for a solar-type background plasma and a type-III-like stream. Background density and temperature coordinate structure, spontaneous emission, magnetic fields, electron-ion collisions, stream reabsorption and wave pileup are shown to affect propagation and are incorporated into a physical description of the stream motion. The calculated electron flux-time profiles at the Earth suggest scatter-free propagation and compare well with type III stream observations.