# Interdisciplinary Applied Mathematics

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FIGURE 5.8. Variation of temperature profiles in a pressure-driven channel flow for continuum and rarefied flows, with specified heat flux on the bottom surface (Y = —1), as a function of Mach number (Re = 1.0 and Pr = 0.7).

B

C

D

1 + 2

2 — o,

Kn

l + |Kn

+

3 (7 — 1) Kn2 Re dTs Ec dx

2n

Y

dT (1 4 RePr—-    — — В + -EcPr,

dx V3 ) 3    ’

e”-^TT^+IEcPrReP‘^(!B-|

with 00 the reference temperature. The modifications to the coefficients B and D due to Kn shows the thermal creep, velocity slip, and temperature jump effects. The continuum solution is recovered as the rarefaction effects diminish (i.e., Kn ^ 0 ).

A quadratic equation for can be obtained by combining equations (5.7) and (5.11). The solution for for specified heat fluxes is shown in Figure 5.7 as a function of Mach number. Equation (2.22) is used to specify

the Eckert number variation for both the continuum and the rarefied flow cases. The Knudsen number variations are specified by equation (2.21). It is seen that the heat flux required to maintain = 0 is the same for both continuum and rarefied flow curves. The physical significance of this result is    that    for    a specified    Mach    number    there is    only    a single    value    of

the heat flux required to compensate the viscous heating effects (see equation (5.11)). Another significant result is the reduction in the magnitude of in rarefied flows, which implies that microchannels sustain smaller tangential temperature gradients compared to the large-scale channels. Examining equation (5.7) and Figure 5.7, we see that the volumetric flowrate of a heated microchannel increases due to thermal creep effects. However, cooled microchannels allow less volumetric flowrate compared to the continuum case. If the rarefaction effects are increased further, the viscous heating effects will dominate. Under this condition may become positive, which will result in increase of the volumetric flowrate beyond the predictions of continuum theory even for cooled channels.

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