Fully Developed Flow in Microtubes Uniform Surface Flux

10 Fully Developed Flow in Microtubes Uniform Surface Flux 

However, in microchannels, compressibility and rarefaction change this flow pattern, and none of the above conditions hold good. Density changes in microchannel gaseous flows are appreciable, and the flow can no longer be assumed incompressible. The other effect is due to rarefaction. There is an increase in mean free path A due to a decrease in pressure in microchannels. Thus, the Knudsen number increases along a microchannel. As a consequence, the axial velocity varies with axial distance, lateral velocity component does not vanish, streamlines are not parallel, and pressure gradient is not constant. 

For the analysis, we make the following assumptions (1) steady state (2) laminar flow (3) one dimensional (no variation with axial distance x and normal distance z) (4) slip flow regime (0.001 Kn 0.1) (5) ideal gas (6) constant viscosity, conductivity, and specific heats (7) negligible lateral variation of density and pressure (8) negligible gravity and (9) negligible inertia forces. 

The isothermal flow assumption eliminates temperature as a variable in the momentum equation. Thus, density can be expressed in terms of pressure using the ideal gas law. Negligible inertial forces assumption is justified for microchannel flow due to low Reynolds number. 

The dominant viscous force is ) as this term is of the order of and other viscous 

---