# Interdisciplinary Applied Mathematics

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A detailed derivation of thermal creep boundary condition for rarefied flows with X < h is given in (Kennard, 1938; Loeb, 1961). It can also be derived directly from the Boltzmann equation (see Section 15.4.2). In order to accommodate the thermal creep effects, the wall velocity is enhanced with the following term:

(5.1)

3 pR dT

= TY as ’

where Uc is the creep velocity, and ^ is the tangential temperature gradient along the surface. Therefore, the high-order velocity slip boundary condition is modified as

Us = ^[(2-a)Ux+aUw] + Uc.

The velocity profile for a pressure-driven channel flow of thickness h, including the thermal creep effects, is then given by equation (4.5) with (5.1) added on to the right-hand side. Integrating this profile, we obtain the mass flowrate:

M

h3P dP 12 pRT dx

1+6-—— (Kn — Kn2)

7 v

3    ph dT

4    T dx

(5.2)

We conclude that thermal creep can change the mass flowrate in a channel. If the pressure gradient and the temperature gradient along the channel walls act along the same direction, the flowrate is decreased; otherwise, the flowrate is increased.

• Therefore, it is possible to have nonzero flowrate in a microchannel even in the case of zero pressure gradient.

###### 5.1.1 Simulation Results

An interesting aspect of thermal creep is that it causes zero net mass flowrate in channels where thermal creep and pressure gradient balance each other. To demonstrate this we simulated air flow in microchannels of various dimensions connecting two tanks kept at different conditions with

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