Interdisciplinary Applied Mathematics

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(5.5)


1    17    9Kn+48Kn2    7 Kn


Nu(Kn) 140    35(1 + 6Kn)2 ^7+1 Pr ’


where the Nusselt number is defined as


Nu



дРн к AT’


Here DH is the hydraulic diameter (DH = 2 x (2h) = 4h for a channel of total width 2h), and AT is the temperature difference between the wall and the bulk of the fluid. Also, Nu(Kn = 0) = 8.235 is the value corresponding to no-slip conditions. The above equation is based on Maxwell’s first-order slip condition and neglects the effect of thermal creep.


Next, we analyze the combined effects of convection and thermal creep. The momentum equation subject to slip boundary conditions with a specified tangential temperature variation (see equation (2.19)) can be solved analytically. The rarefaction effects on momentum transfer can be investigated either by analyzing the volumetric flowrate increase in a pressure-driven channel or by analyzing the change in the skin friction coefficient for a fixed volumetric flowrate, under an appropriately specified pressure gradient. The nondimensional velocity distribution in a channel extending from y = —h to y = h is obtained as


U (y/h)



Re dp 2 dx


3 (7 — 1) Kn2 Re dTs ^27t    7 Ec dx



2



f-(|) +2



2    & v



v



Kn


l + ^Kn



(5.6)


where denotes the tangential temperature variation along the channel surface, and we defined Kn = X/h, with h the half-channel width. Given this parabolic velocity profile we obtain the coefficient of high-order boundary condition from equation (2.39) to be b =X, Correspondingly, the volumetric flowrate Q through the channel, in nondimensional form, becomes

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