Interdisciplinary Applied Mathematics

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in the form of a gradient of some function, and thus it cannot be incorporated in the pressure term, in contrast to the small temperature variation presented earlier. Therefore, the solution ui1 =0 is possible under a special temperature field. The condition for ui1 to be zero is


drо d dxj dxk

дт0 dxi )



which is obtained    by    putting    пц    =0 in    equation    (15.44c).    When    the    dis

tance between isothermal lines or surfaces is constant along such contours, the condition is satisfied. Thus, even when the temperature of the boundary is uniform and the thermal creep flow is absent, a flow may be induced in the    gas.    This flow is    called    nonlinear    thermal    stress    flow,    and it    was

presented in Section 5.1.

15-4-3 Numerical Solutions

Numerical simulations based on the Boltzmann equation are computationally very    expensive,    and    they    have    been    obtained    mostly    for    simple    ge

ometries, such as pipes and channels. In particular, a number of investigators have considered numerical solutions of the linearized BGK and exact Boltzmann equations, valid for flows with small pressure and temperature gradients (Huang et al., 1997; Sone, 1989; Ohwada et al., 1989a; Loyalka and Hamoodi, 1990; Aoki, 1989). These studies used hard sphere and Maxwellian molecular models.

In the following, we summarize some benchmark solutions that can be used for validating numerical simulations and experimental results for microchannels and micropipes.

Rarefied flow in a channel has been studied extensively using both the full Boltzmann equation and different versions of the BGK model. A comparison of different numerical solutions in (Sharipov and Sleznev, 1998), reveals a small difference of about 2% in most of the published solutions for the normalized flowrate. This quantity is defined as

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