Discrete Symmetry Analysis of Free Convection in a Cylindrical Porous Annular Microchannel at Low-Pressure Level

Abstract

This article studies the natural convection in an annular porous microchannel in case of one wall being heated and another being cooled. For the first time, such a problem was solved using discrete symmetries of the Navier–Stokes equations. Using discrete symmetries, self-similar forms of differential equations were obtained. Solutions of self-similar equations made it possible to obtain velocity and temperature profiles incorporating slip and temperature jump at the channel walls as boundary conditions. The effects of Grashof, Knudsen, Darcy, and Prandtl numbers on the velocity and temperature profiles in the microchannel and Nusselt numbers are demonstrated. At high Grashof numbers, an ascending flow forms near the hot cylinder, while a descending flow develops near the cold cylinder. As the Knudsen number increases, rise of velocity and temperature jump at the walls as well as heat transfer coefficients decrease is observed. An increase in the Darcy number results in higher velocities for both flows. The temperature jump at the heated cylinder increases, remaining unchanged at the cooled cylinder, and the heat transfer coefficient at the heated cylinder drops.