Shear-driven flows are encountered in micromotors, microcomb mechanisms and microbearings. In the simplest form, the linear Couette flow can be used as a prototype flow to model such flows driven by a moving plate. An analytical solution is easy to obtain given the simplicity of the geometry consisting of two plates separated by a distance h. The flow is driven by moving the upper plate at a constant velocity U_o; the upper plate temperature is at T_o while the bottom plate is assumed to be adiabatic. Also for simplification, viscosity and thermal conductivity are assumed to vary linearly with temperature, and the Prandtl number is fixed (for air Pr=0.72). In this case it is possible to obtain the friction coefficient (C_f) analytically.
In the simulations the Mach number M_o is specified by varying the driving velocity of the top plate U_o. Correspondingly, rarefaction effects are specified through the Knudsen number, since Kn_o = 1.14 M_o / Re.
The variation of friction coefficient as a function of Mach number and corresponding Knudsen number is shown below. It is seen that the friction coefficient of no-slip compressible flow increases quadratically in agreement with analytic results, well above the constant value of the corresponding incompressible flow. The no-slip compressible flow simulations match the theoretical results exactly. For rarefied flows, slip effects change the friction coefficient significantly. Compressible slip-flow results are denoted by open circles in the figure. It is seen that the compressible slip-flow results correspond to small deviations from the incompressible slip-flow results obtained analytically.
In linear Couette flow the pressure is constant and therefore compressibility effects are due to temperature changes only. As M_o increases the temperature difference between the two plates gets larger (due to the viscous heating effects). Thus, compressibility effects become significant. It is seen in the figure that significant deviations from incompressible flows (slip/no-slip) are obtained for M_o > 0.3. In particular, we investigated a case where the bottom plate is kept at T_w=350 K while the top plate is kept at T_o=300 K. The friction coefficient of this case is also given in the above figure. The results are shown by solid and open triangles for the no-slip and the slip cases, respectively. The trend is different than the adiabatic bottom plate case. No-slip results show small variation of C_f as a function of M_o, while for slip flows C_f is reduced significantly as Kn is increased.
Flow in a micromotor or a microbearing is more complicated than the linear Couette flow. We therefore consider a two dimensional shear-driven grooved channel flow (see figure below) in order to model the geometric complexity of these microdevices. The presence of grooves complicates the geometry and an analytic solution for this flow does not exist. The flow separates and starts to recirculate in the grooves even for small Reynolds number flows. In our numerical model we have assumed that the top wall is moving with a speed U_o, and both surfaces are kept at T=300 K. We also assumed that the geometry repeats itself along the flow direction. Therefore the flow is periodic, and only a section of the channel shown below is simulated. In our simulations the Reynolds number is fixed to be Re=5.0, and the Knudsen number is increased by decreasing the channel gap. Therefore the top wall speed U_o is increased to keep the Reynolds number constant, resulting in an increase of flow Mach number according to Kn_o = 1.14 M_o / Re. The temperature contours for no-slip (left) and slip (right Kn=0.086) flows is shown below. The increase in temperature in the middle of the channel is due to viscous heating resulted from large shear stresses in this low Reynolds number flow. The viscous heating effect for slip flow is less than that of no-slip flow due to the reduction in shear stresses. The temperature of the gas near the walls is not the same with the prescribed wall temperature. Since the temperature of the fluid is higher in the middle of the channel, the channel looses heat and due to the temperature jump effects, the gas temperature is higher than the surface temperature. This may create a problem for micro-gas-flow temperature measurements. Although, the change in the temperature due to the viscous heating effects seems to be small in magnitude, the gradients in temperature (as seen by the contour density in the figure) can be quite large due to the small length scales associated with microflows. For 1K temperature difference within a micron, the temperature gradient is 1.0 E6 K/m. In thermal creep section we discuss effects of tangential temperature gradients in rarefied flows.
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