Interdisciplinary Applied Mathematics

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Lastly, flow induced by a temperature discontinuity has been studied by (Aoki, 2001), who set up a flow in a square enclosure, half of which is at temperature T and the other half at temperature T2 (see Figure 5.4). A flow is induced from the colder to the hotter part along the one wall with the discontinuity at the middle, and this sets up a global circulating flow (see Figure 5.5). The maximum flow speed tends to a constant value at low Kn values and decays for Kn > 0.1.

5.3 Heat Conduction and the Ghost Effect

We examine here the possible breakdown of the heat conduction equation in the limit that Kn ^ 0. This phenomenon was studied by (Sone et al., 1996a) and shows a fundamental inconsistency in the momentum and energy (Navier-Stokes) equations. We consider a gas at rest contained in a tank. According to the continuum description, the gas temperature field is described by the heat conduction equation, i.e., the energy equation with all the convective terms absent. Below, we follow the argument of (Sone, 2002; Sone, 2000), that demonstrates that the heat equation is not always appropriate, e.g., in microscales.

Let us consider the energy equation in the continuum limit and examine only the relevant terms (for a monoatomic gas) as follows:

5    d(RT)    d (,dT

2pUi dxi ~ «‘+    )

where к is the thermal conductivity. The corresponding heat conduction equation for a gas at rest is

d( dT л

dxi V dxi)    ‘

The thermal conductivity (к) of a gas is a function of its mass as well as its temperature. Specifically, к (divided by the density of the gas) is proportional to the mean free path with the proportionality coefficient a general function of temperature, i.e.,

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