Capturing non-equilibrium effects in cylindrical Couette flow problem using the Extended Onsager-13 moment equations

Abstract

In the present work, cylindrical Couette flow is analyzed using the recently derived third-order accurate 13-moment transport equations by transforming them into cylindrical coordinates. Assuming the Mach number and normalized temperature difference between the cylinders to be relatively small, closed-form expressions for all relevant quantities, velocity, pressure, temperature, heat flux, and stresses, are obtained from the semi-linearized form of the equations. These closed-form expressions from the present study have been validated against the corresponding linearized Grad 13 moment (G13) equation solutions. It is further demonstrated that contrary to the G13 equations, the pressure in the cylinder annulus is not constant, while the temperature depends upon the magnitude of viscous heating apart from non-isothermal boundary conditions. The coupling among velocity, temperature, heat flux, and stress and its effect on the variation of various physical quantities across the annulus has been discussed. The obtained analytical solution shows that the equations correctly capture most known non-equilibrium effects, such as the presence of a Knudsen layer, non-Newtonian stresses, and non-Fourier heat flux for Knudsen numbers falling well into the transition regime through a quantitative agreement with direct simulation Monte Carlo data, G13, and regularized 13-moment equations. The presence of non-zero radial and tangential heat fluxes, even when both the cylinders are at the same temperature, has been observed. The analysis helps us to demonstrate the ability of the recently derived equations in accurately solving complex rarefied flow problems. Moreover, understanding of higher-order rarefaction effects should greatly improve with the availability of closed-form analytical expressions of all physical quantities obtained here.