Interdisciplinary Applied Mathematics

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-=г(£Л V2RTRX.

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Therefore, in the energy equation the conduction term divided by the density is 0(A). For a gas flow of the order of the mean free path (A), which is neglected in classical gas dynamics, its convection term is of the same order as the conduction term and cannot be neglected. More specifically, the order of magnitude for the convection term is

Convection ж 0(pVRT/L),

and the order of magnitude for the conduction term is

Conduction ж 0(p(RT)3/2A/L2),

where V and L are, respectively, the characteristic flow speed of the gas and the length scale of variation of the temperature of the gas. Thus, the two terms are comparable when

V oc 0(VRTX/L) oc 0(Kn).

The question then is whether there are flows of such magnitude; however, we have already seen in this section that both the thermal creep flow and the nonlinear thermal stress flow are possible candidates. These flows are

always present, even at extremely small Knudsen numbers, unless all surfaces in the device have uniform temperature and all temperature isocontours are parallel. These two conditions are difficult to satisfy simultaneously except    in the    trivial    case    of    two infinite    plates    at    exactly    uniform

temperatures with no end effects! Therefore, according to (Sone, 2002) the heat-conduction equation is inappropriate for describing the temperature field of a stationary gas in the continuum limit.

Sone termed this the “ghost effect” in the sense that something that does not exist in the world of a gas in the continuum limit has a finite effect in the molecular world. In (Sone, 2000) an example of the ghost effect is given for a channel with small periodic temperature variation at the two walls. No experiments in microscales have reported the ghost phenomenon, and it is difficult to assess its practical significance; nevertheless, it is a significant result from the fundamental principles point of view.

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