# Interdisciplinary Applied Mathematics

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(5.16)

Kn or (^(777/2) M/Re )

FIGURE 5.12. Variation of tangential temperature gradient (?^r) along the surface of a shear-driven channel as a function of Mach number for different levels of heat fluxes (q). Dashed line: continuum; solid line: rarefied. (Re = 1.0, AT = 1K, and To = 300 K).

0(y) = ^y3 + §y2-(^+B)y + C,

where

RePr^k

A — _ox

1 + 2 Kn7

RePrKn+ 3    (7 — 1) Kn2 Re2Pr /dTs

l + 2Kn    7    Ec у dx

C-P    27 Kn

W — Pq—~~FTrT

2

EcPr

(1 + 2 Kn)2

Y +1 Pr

and

d% =    2

dx RePrQ

q +

EcPr

(l + 2Kn)2

(5.17)

FIGURE 5.13. Variation of temperature profiles in a shear-driven channel flow for continuum and rarefied flows, with specified heat flux at the bottom surface, as a function of Mach number. Y = 0 corresponds to a stationary wall, and Y = 1 corresponds to a moving wall (Re = 1.0 and Pr = 0.7).

A quadratic equation for can be obtained by combining equations (5.14) and (5.17). The solution for for specified heat fluxes is shown in Figure 5.12 as a function of Knudsen number. Equation (2.22) is used to specify the Eckert number variation for both the continuum and the rarefied cases. The Knudsen number variations are also specified by equation (2.21). It is seen that the heat flux required to maintain = 0 for a specified Mach number is smaller in microchannels compared to the continuum case. The viscous heating effects are more dominant in the continuum case compared to microscales; see also (Wendl and Agarwal, 2002). This leads to different results for cooled and heated channels with respect to thermal creep effects.

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