# Interdisciplinary Applied Mathematics

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nitj = °,

d2T

dxidxj

where    ni    and tj are    components    of    the    unit    vectors    in the    normal    and

tangential directions, respectively. A global flow is then set up circulating between the cylinders along the clockwise direction if the temperature of the outside cylinder T2 is higher. This phenomenon was discovered by (Sone, 1972), who termed it “thermal stress slip flow.”

The nonlinear thermal stress flow is an O(Kn) effect and was discovered by    (Kogan et    al.,    1976)    as a    new    type    of    convection.    When    the

temperature gradient in a gas is so large that its nonlinear effect is not negligible, the thermal stress does not balance by itself and remains in the momentum equation (see momentum equations in Section 15.4.2). This causes a flow when the distance between isothermal lines or surfaces varies along them. In this flow, in contrast to the thermal creep flow and thermal stress slip flow,    a    solid    boundary    plays    only an indirect    role    in    setting    up

the isothermal contours.

(a) Kn = 0.2

FIGURE 5.5. Flow induced in the lower half of the domain for Г21 = 2. The scale /u2 + v2/y/2RTi = 0.02 is indicated next to the plots. The upper plot corresponds to Kn = 0.2 and the lower plot to Kn = 0.02. (Courtesy of K. Aoki.)

The flow induced around the sharp edge of a heated plate cannot be predicted by the asymptotic analysis of the Boltzmann equation as in the previous two cases. This phenomenon was discovered more recently in numerical and experimental work by (Aoki et al., 1995) and (Sone and Yoshimoto, 1997). If a uniformly heated plate is placed in the middle of gas contained in a tank of uniform temperature, flow is induced around the edge of the plate for a wide range of Knudsen number. There is no flow induced by thermal creep, since the temperature is uniform everywhere. The induced velocity scales approximately as Cl (л/Kn).

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