Interdisciplinary Applied Mathematics

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4-2.1 Burnett Equations

The governing conservation equations were given in Chapter 2, e.g., equation (2.16). In the transition regime the Burnett equations govern the thermal fluidic transport. The stress tensor for the Burnett equations is given in Section 2.3. To simplify this stress tensor we consider a very long channel of length L and height h, so that

e = h/L ^ 1.

We also neglect any temperature gradients in the gas. Under such conditions the Burnett equations can be simplified considerably, resulting in terms including 0(1) up to 0(e5). Here we present the results of this expansion including the 0(e) terms. In the following analysis, Kn0 and Mrefer to Knudsen and Mach numbers evaluated at the outlet.

The streamwise component of the Burnett momentum equation in a long channel including 0(e) terms is



) (Uy)2

_ Uyy



_ ~ir

+ t)

Uy Uyx

+ 1 Y “    + T ) UxUyv

h»*    P0 ((“l    . U6 TT „ ,

2 MoKnoJJ Y ~ Y + Y)UyVyy+

^MoKno ^ (fuuyxy+ fvuyyy) — Re(PUU)x

Re(pUV )y

Similarly, the cross-flow Burnett momentum equation including 0(e) terms is


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