Rarefied Gas Dynamics by Felix Sharipov

Author:Felix Sharipov
Language: eng
Format: epub
ISBN: 9783527685530
Publisher: Wiley
Published: 2015-11-16T00:00:00+00:00

12.1.3 Kinetic Equation

As is shown in Section 9.4.4, the axial symmetry allows us to reduce the number of variables and consider , , and instead of , , , and . The relation between these coordinates is given by Eqs. (9.59), (9.62), and Figure 9.2. Similar to the planar Couette flow, the axisymmetrical one is solved by linearizing the BGK model near the global Maxwellian (5.6), that is, Eq. (7.47) is written for , , and without thesource term. The density and temperature deviations for this problem are zero so that the function defined by (7.48) is reduced to

12.13

To eliminate one more variable, namely, , the new perturbation is introduced as

12.14

which obeys the following kinetic equation:

12.15

where the left-hand side of the kinetic equation has been written in the cylindrical coordinates (9.63). Using the same reasonings as those to obtain (11.18) and assuming the diffuse scattering, the boundary conditions for the new perturbation are obtained as

12.16

Analyzing the equation (12.15) and boundary conditions (12.16), we conclude that the perturbation is antisymmetric

12.17

The moments and are obtained from Eqs. (12.1), (12.2), and (12.5) replacing the velocity coordinates and by and according to (9.62)

12.18

where the property (12.17) has been used.

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