# Interdisciplinary Applied Mathematics

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The steady-state pressure distribution along the channel center and in the reservoirs, normalized with atmospheric pressure (initial pressure at both tanks),    is    given    in    Figure    5.1    for    three different    channel    sizes. It is

seen that the pressure change due to thermal creep for high Kn flows is nonnegligible. In Table 5.1, we compare the pressure differences predicted by equation (5.3) with the numerical values obtained by simulations; the agreement is very good.

The above numerical experiment describes an unsteady problem for which the relevant time scales are governed by two different transient processes. The first transient process is due to the fluid starting to creep along the channel surface. As time goes on, the creeping fluid layer starts interacting with the stagnant fluid layers above it, creating a boundary layer similar to the impulsively started wall problem, the so-called Rayleigh problem (Batchelor, 1998). Of course, formation of the boundary layer creates

X

FIGURE 5.1. Normalized pressure variation along the channel center and reservoirs for different rarefaction parameters (P0 = 1 atm). Triangles and circles show the location of calculation points.

shear stresses, which, in turn, activate the velocity slip mechanism (Ken-nard, 1938). This is the initial transient process with time scale

h2

t oc —.

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Considering microchannels with typical height of about one micron, this transient is very fast.

The second time scale of the problem corresponds to the time it takes to get from initial transients to a steady-state solution where the net mass flowrate is zero. This time scale is based on the creep velocity and tank dimensions. In particular, this time scale increases as the tank size is increased. In the limit where the tanks are reservoirs of infinite dimensions, the fluid steadily creeps from the cold to the hot tank, and the pressure at the two reservoirs remains practically the same.

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