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Astron. Astrophys. 359, 788-798 (2000)

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5. The flux vector in moving 3D media

It is obvious from Eqs. (4) and (5) that the direction dependence of w induces a complicated dependence of the flux vector [FORMULA] (I:11) on the directions of the velocity vector and of the temperature gradient. In particular, it appears possible that in case these directions do not coincide there is a flux component perpendicular to the temperature gradient.

While it seems impossible to derive an explicit expression for the flux vector analytically in the general case, this could be done in the limit of small [FORMULA]. However, even in this case the general expressions are quite lengthy and very difficult to interpret.

In order to discuss the essential effects we therefore consider here as example a plane-parallel medium with a temperature gradient in the normal direction (taken to be the z-direction) and with a velocity field ([FORMULA]) which is assumed to vary only with the z coordinate in the neighborhood of the point [FORMULA] of interest, i.e. in cartesian coordinates

[EQUATION]

For the direction [FORMULA] one now obtains from Eq. (2)

[EQUATION]

and the integration over all directions (see Eq. I:12) of Eq. (18) yields

[EQUATION]

where all coefficients are to be taken at [FORMULA].

It is seen that the classical result for static media is fully regained for [FORMULA]. If the velocity vector is in the z direction ([FORMULA]), i.e. if it is parallel to the temperature gradient, the x and y components of the flux vector are still zero, but the z component is modified by the velocities as has been discussed in Sect. 4.1:

[EQUATION]

The coefficients of the [FORMULA] terms are just given by the two angle averages [FORMULA] and [FORMULA], respectively.

If the flow direction and the temperature gradient are not parallel ([FORMULA]), the comparison of Eq. (55) with (54) shows that the flux component in z direction is not changed in first order but it is increased in second order. More significantly, however, we find also non-zero flux components in the x and y directions, i.e. there is now a radiative flux perpendicular to the temperature gradient.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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