Maxwell boundary condition

The coefficient a is called the accommodation coefficient. This condition is called the Maxwell boundary condition. ... 

The Maxwell boundary condition [Eq. (122)] also gives rise to hydro-dynamic boundary conditions of the form (125) with o>, t, and x roughly prop)ortional to So, as long as a is of order unity (more specifically a IJ/h) the corrections to stick boundary conditions remain small, and only if a becomes of order boundary conditions differing appreciably from stick are obtained.This is an indication why stick boundary conditions are for most purposes a very good approximation in hydrodynamic theory a reflection mechanism that is almost specular is not very likely to occur in nature, due to irregularities in surface structures and thermal motion of the surface molecules. 

A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

The necessary boundary conditions required for E and //to satisfy Maxwell s equations give rise to tire well known wave equation for tire electromagnetic field ... 

The basis for the familiar non-slip boundary condition is a kinetic theory argument originally presented by Maxwell [23]. For a pure gas Maxwell showed that the tangential velocity v and its derivative nornial to a plane solid surface should be related by... 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

Given the boundary condition (A.1.6) it is a straightforward matter to integrate the Navier Scokes equations in a cylindrical tube, and hence to find the molar flux N per unit cross-sectional area. The result, which was also obtained by Maxwell, is... 

The authors of Ref. [12] reconsidered the problem of magnetic field in quark matter taking into account the rotated electromagnetism . They came to the conclusion that magnetic field can exist in superconducting quark matter in any case, although it does not form a quantized vortex lattice, because it obeys sourceless Maxwell equations and there is no Meissner effect. In our opinion this latter result is incorrect, since the equations for gauge fields were not taken into account and the boundary conditions were not posed correctly. 

For the Maxwell-Stefan theory the following set of differential equations with associated boundary conditions have to be solved ... 

Our fundamental task is to construct solutions to the Maxwell equations (3.1)—(3.4), both inside and outside the particle, which satisfy (3.7) at the boundary between particle and surrounding medium. If the incident electromagnetic field is arbitrary, subject to the restriction that it can be Fourier analyzed into a superposition of plane monochromatic waves (Section 2.4), the solution to the problem of interaction of such a field with a particle can be obtained in principle by superposing fundamental solutions. That this is possible is a consequence of the linearity of the Maxwell equations and the boundary conditions. That is, if Ea and Efc are solutions to the field equations,... 

In Chapter 4 a plane wave incident on a sphere was expanded in an infinite series of vector spherical harmonics as were the scattered and internal fields. Such expansions, however, are possible for arbitrary particles and incident fields. It is the scattered field that is of primary interest because from it various observable quantities can be obtained. Linearity of the Maxwell equations and the boundary conditions (3.7) implies that the coefficients of the scattered field are linearly related to those of the incident field. The linear transformation connecting these two sets of coefficients is called the T (for transition) matrix. I f the particle is spherical, then the T matrix is diagonal. 

In the received opinion [5], these are the vacuum Faraday law and Ampere-Maxwell law, respectively. The vacuum charges and currents are missing in the received opinion. Nevertheless, solving Eq. (625) numerically is a useful computational problem with boundary conditions stipulated in the vacuum. The potentials and fields are related as usual by... 

Directly following the development of the optical laser, in 1961 Frankel et al. [10] reported the first observation of optical harmonics. In these experiments, the output from a pulsed ruby laser at 6943 A was passed through crystalline quartz and the second harmonic light at 3472 A was recorded on a spectrographic plate. Interest in surface SHG arose largely from the publication of Bloembergen and Pershan [11] which laid the theoretical foundation for this field. In this publication, Maxwell s equations for a nonlinear dielectric were solved given the boundary conditions of a plane interface between a linear and nonlinear medium. Implications of the nonlinear boundary theory for experimental systems and devices was noted. Ex-... 

It is useful to consider the solution of Maxwell s Equations (5.1) for plane electromagnetic waves in the absence of boundary conditions, which can be written as exp[i(/ 2 — u>t) assuming propagation in z-direction of cartesian coordinates. The quantity / is the complex propagation constant of the medium with dominant real part for dielectrics and dominant imaginary part for metals. The impedance of the medium, Z, defined as ratio of electric to magnetic field is related to / by Z = ojp,0/f3 with /x0 = 1.256 x 10 6 Vs/Am. As it can be derived from Maxwell s equations, the impedance is related to the conductivity/dielectric function by the following expression ... 

The reflecting boundary condition, due to Maxwell, can be written in terms of the momentum accommodation coefficient ac as (4)... 

Solving Maxwell s equations at the metal/dielectric interface at the appropriate boundary conditions yields the surface plasmon dispersion relation, that is, the relation of the angular frequency co and the x-component of the surface plasmon wave vector kSP,... 

A basic waveguide structure, which is sketched in Fig. 1, is composed of a guiding layer surrounded by two semi-infinite media of lower refractive indices. The optical properties of the stmcture are described by the waveguiding layer refractive index Hsf, and thickness t, and by the refractive indices of the two surrounding semi-infinite media, here called (for cover) and (for substrate). Application of Maxwell s equations and boundary conditions leads to the well-known waveguide dispersion equation [6] ... 

This equation is derived by integrating Eq.( 11-29) with boundary condition)/ = 0, T = To at r = 0. Although the model has some elastic character the viscous response dominates at all but short times. For this reason, the element is known as a Maxwell fluid. 

To start with, we consider steady flows of Maxwell-type fluids in a bounded smooth domain f2 of R, iV = 2,3, and with simple boundary conditions, neunely the system... 

For Maxwell models, where Tj = 0, we need to distinguish the sub- and the supercritical ceises. In the subcritical case (i.e., U < yT]j pX)) and in two space dimensions, one can prescribe the diagonal components o-p and 7P, whereas in three space dimensions a correct choice of boundary conditions for tP is not simple. A possible choice of four boundary conditions is a nonlocal one (in terms of the Fourier components of rP—see [29]). An alternative approach leading to first order differential boundary conditions at the inflow boundary is described in [30]. 

For Maxwell models in the supercritical case (i.e., U > iJi]l(pX)), the previous choice of boundary conditions leads to an ill-posed problem (as does the Dirichlet boundary condition for a hyperbolic equation), as shown in [17]. In addition to the normal velocities at both boundaries (inflow and outflow) and to the previous inflow conditions on the stresses, one can prescribe the vorticitj and its normal derivative in two space dimensions, or the second and third components of the vorticity and their normal derivatives in three... 

A discussion of the traction boundary conditions—where the totaJ normal stress is prescribed on the inflow and outflow boundaries—for Jeffreys-type fluids is given in [31], and for Maxwell-type fluids in [32]. 

In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. 

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