# Interdisciplinary Applied Mathematics

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Temperature profiles under different heat flux conditions are shown in Figures 5.8 and 5.9. The temperature jump diminishes if both surfaces of the channel are insulated (see Figure 5.9, top). In this case, the maximum temperature occurs near the walls, where shear stresses are more dominant and the tangential temperature variation becomes positive (see Figure 5.7). This suggests significant changes in the volumetric flowrate of the microchannel, which is the main reason for the differences in the temperature profiles of two cases. Thermal creep in a microchannel can be avoided if the channel is carefully cooled at a cooling rate of q = — |PrEc (see equation (5.11)); this results in constant temperature along the channel wall. The temperature distribution for a microchannel, without thermal creep effects, is also given in Figure 5.9 (bottom). This result suggests that the temperature of the insulated surface will be greater than its counterpart modeled by the continuum theory.

The analytical results presented for pressure-driven and shear-driven channels (Section 5.5) are also verified with numerical simulations. Since the temperature variation is in the form of a fourth-order polynomial, convergence of the numerical scheme to the exact solution can be obtained with very coarse discretizations. Typically, fourth-order polynomial expansion in p,Flow simulation is sufficient to resolve the spatial variations; see Section 14.1.

Remark: The analytical results presented in this section are based on the incompressibility approximation. Therefore, they are valid for small temperature and pressure variations in the microchannels. Small temperature variations can be a good approximation, but it has been shown experimentally that    the    pressure    drop    in microchannels    could    be    large.    In    the

experiments of    (Pfahler    et    al.,    1991), a total    pressure    drop    of    about    one

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