Interdisciplinary Applied Mathematics

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5.2 Other Temperature-Induced Flows


In addition to the flow induced by thermal creep, which is an O(Kn) effect, there are other possibilities for setting up a flow in rarefied flows in the absence of any external forces, e.g., gravity. Here we list a few representative cases and provide a short explanation:


   thermal stress slip flow,


   nonlinear thermal stress flow,


   flow induced around the edge of a heated plate, and


   flow induced by a temperature discontinuity.


Most of these cases were discovered by Sone, Aoki, and their collaborators by analyzing the Boltzmann equation (see Section 15.4.2). They are most probably present in microsystem flows, but they are difficult to isolate and be detected as individual effects. Here we give a brief summary for each following the work in (Sone, 2002; Sone, 2000).


x2.


У//////////у


1


L/2


1


у T, 1


!


-L/2 0


-L/2


L/2



У//////////У

FIGURE 5.4. Flow domain showing the temperature discontinuities at the upper and lower walls. (Courtesy of K. Aoki.)


The thermal stress slip flow is an O(Kn2) effect, and it is derived following a consistent second-order expansion in Kn of the Boltzmann equation. In that case, in addition to the thermal creep that is present in the first-order expansion of the boundary terms, there is a correction term for the slip velocity proportional to the normal gradient of the temperature. A nonuniform normal gradient can cause a slip velocity even if the boundary is isothermal. To explain this in more detail, we follow the argument of Sone (see Section 15.4.2 and (Sone, 2002)). Let us consider a gas between two eccentric circular cylinders with different uniform temperatures T and T2. No thermal creep flow is induced; however, there is, in general, a nonuniform normal temperature gradient on each cylinder, that is,

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