Micro/Nano Cavity 
Written by Ehsan Roohi Golkhatmi 
Micro/Nano Cavity In the early slip regime, as shown in Fig. 1 (a) and Fig. 1 (d), predicted distribution of temperature obtained by the NS equations are in approximate agreement with the molecular approach. Both methods predict almost similar range of temperature variation for the flow field. In addition, the shapes of temperature contours are quite similar. The predicted direction of heat flux by the DSMC is from hot to cold regions over the entire domain except for a small region near the top right corner. Temperature distribution in the middle slip regime obtained by the NS equations is shown in Fig. 1 (b). Maximum viscous dissipation occurs where the largest velocity gradients exist. Figure 1 (e) shows the temperature contour from the DSMC results at Kn = 0.05. It is seen that the maximum temperature for the DSMC solution is greater than the predicted value by the continuum approach. Moreover, reduction in flow temperature near the left wall shows that the sudden expansion of the rarefied flow in this region dominates the heat transfer mechanism. Surprisingly, heat flux lines are from the colder region of the cavity to the hotter region. The assumed constitutive law of Fourier heat conduction, , incorporated in the NS equations, cannot predict such a direction for transfer of energy.
Figure 1 (e) reveals that in a simple two dimensional flow, even in the middle slip regime the direction of heat flux cannot be predicted by the NS equations. Such an unusual flux of heat in other geometries has been observed under certain flow conditions. Figure 1 (c) shows heat flux lines obtained by the NS solution at Kn = 0.1. Temperature distribution is very similar to the previous case; however, the maximum temperature is decreased. Smaller shear stress at Kn = 0.1 in comparison with Kn = 0.05 makes the viscous dissipation, the dominant heat generation mechanism in the NS equation, decreasing. Consequently, maximum temperature in the domain decreases. Figure 1 (f) shows the heat flux lines in the cavity from the DSMC solution at Kn = 0.1. The heat flux lines and temperature contours are quite similar to the previous test case; however, the minimum temperature is decreased by one degree which is a sign of stronger nonequilibrium effects in the top left corner of the cavity.
FIG. 1. Heat flux lines overlaid on the temperature contour, top row: NS, bottom row: DSMC, (a) and (d) Kn=0.005, (b) and (e) Kn=0.05, (c) and (f) Kn=0.1
The entropy distribution throughout the domain can also provide some information about the heat flux direction. Figure 2 presents the entropy distribution in the cavity flow, which is normalized with the maximum value of entropy in the domain. According to the molecular gas dynamics theory, the entropy is expressed as:
where k_{B} is the Boltzmann constant, ∆c is the width of the velocity bin, N_{h }(c) is the number of particles in the regarding bin and N is the total number of particles in all the velocity bins. It is known that the energy fluxes from the region with higher entropy toward the lower entropy segments. Figure 2 shows that for all the three Knudsen numbers considered here the top left corner of the cavity is where the maximum entropy occurs. Moreover, entropy reduces from the top left corner toward both the left bottom corner and the right wall; therefore the heat flux lines should be originated from the top left corner and directed toward bottom and right.
FIG. 2. Distribution of entropy in the cavity with isothermal walls predicted by the DSMC at (a) Kn=0.005, (b) Kn=0.05, (c) Kn=0.1
