Rarefaction and dissipation effects on transport phenomena associated with an immiscible two-phase flow within a shaft-housing micro configuration

A parametric analytical study is performed to investigate the mechanism of fluid flow and heat transfer associated with an immiscible gas–liquid flow within a shaft-housing micro configuration. The incompressible Navier–Stokes–Fourier (NSF) equations in the cylindrical polar coordinate reference frame are employed, while simultaneous effects of viscous dissipation and rarefaction phenomenon are taken into account. Two different classical thermal boundary conditions, namely, the Uniform Heat Flux (UHF) and the Constant Wall Temperature (CWT) boundary conditions are considered. Throughout a combination of these boundary conditions three thermal case studies are constructed and then treated separately. Solutions for the velocity and temperature distributions for both liquid and gas phases of each thermal case are obtained. Employing these solutions, major quantities such as shear and temperature reductions at the shaft due to the gas layer and the Nusselt number pertained to the gas and liquid phases are evaluated. Effects of parameters such as the Knudsen and Brinkman numbers as well as the aspect ratio of the annuli on the aforementioned major variables are examined. The results indicate that introducing a gas layer into the micro domain always contributes to decreasing the shear stress at the shaft and consequently, reducing the torque required to initiate the flow, which is absolutely desirable. However, depending on the thermal case considered, this may be accompanied by serious thermal drawbacks like higher temperatures at the shaft surface. It is also found that increasing values of the Knudsen and Brinkman numbers may even deteriorate the situation.