pdf-file - Institut für Theoretische Physik
pdf-file - Institut für Theoretische Physik
pdf-file - Institut für Theoretische Physik
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4 Boundary Conditions<br />
A solution of the hydrodynamic equations is possible only if in addition to the<br />
equations we also have the appropriate initial and boundary conditions. The<br />
latter are obtained from the bulk equations themselves. Therefore, number<br />
and type of the boundary conditions depend on the two systems comprising<br />
the interface. The boundary conditions are best derived in the rest frame of the<br />
interface, (or more generally in the rest frame of the portion of the interface<br />
under consideration).<br />
4.1 The Dielectric-Dielectric Interface, without Phase Transition<br />
We shall first consider the simpler case in which no mass current may cross<br />
the interface, ie in the absence of the possibility of a phase transition, such as<br />
given at an interface made from two different substances.<br />
The first boundary condition is the continuity of the normal component of the<br />
energy flux,<br />
∆(Qn + Q D n )=0, (78)<br />
where the subscript n denotes the component normal to the interface. This<br />
condition is obtained by integrating the energy conservation,<br />
˙ɛ tot + ∇·(Q + Q D )=0,<br />
over an infinitesimally thick slab around the stationary interface. Physically, it<br />
simply implies that the same amount of energy enters and leaves the interface.<br />
The notation, here and below, is given as<br />
∆A ≡ A(r − rsf →−0)<br />
−A(r − rsf → +0) ≡ A1 − A2 , (79)<br />
A ≡〈A〉 ≡ 1<br />
2 (A1 + A2) , (80)<br />
where rsf is a given point on the surface. The normal vector n hence points<br />
into region 2. Also,<br />
∆(AB) ≡ A1 B1 − A2 B2 = A ∆B + B ∆A. (81)<br />
All vectors are divided into an tangential and a normal component, say<br />
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