Interdisciplinary Applied Mathematics

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1    3 (7-l)Kn2RedTs

2    2ty 7 Ec dx

It is seen that thermal creep effects result in change of the flowrate of the channel. The ratio of friction coefficients of the shear-driven slip flow to a continuum flow is given by


FIGURE 5.11. Temperature profile across the channel as a function of the Knud-sen and    the    Mach    numbers.    In the    figure,    the    Knudsen    number    increases    going

from left to the right, while the Mach number increases going from the bottom to the top. The dimension of the simulated channel was L/W = 20. The Knudsen and Mach numbers (Kn, M) used in the simulations were: Plot F1: (0.001, 0.01), Plot F2: (0.001,0.02), Plot F3: (0.001,0.05), Plot F4: (0.20,0.01), Plot F5: (0.20, 0.05), Plot F6: (0.20, 0.12), Plot F7: (0.45, 0.01), Plot F8: (0.45, 0.04), Plot F9: (0.45, 0.17), respectively. (Courtesy of I. Karlin and S. Ansumali.)

Cf    1

Gfo l + 2^Kn

The above equation is obtained for constant mass flowrate in the channel. If thermal creep effects are considered, the driving velocity Uo of the channel must be modified to keep the volumetric flowrate constant. Therefore, the

thermal creep effects are not included in the derivation of equation (5.15).

Heat convection analysis for a steady and thermally fully developed shear-driven microchannel is obtained by decomposing the temperature profile into two parts, as given in equation (5.10). The channel is assumed to have an insulated top surface and a bottom surface with a specified heat flux (q). With this decomposition, the temperature variation across the channel becomes a third-order polynomial given by

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