The Stability of a Rotating and Heated From Below Horizontal Cylindrical Layer of a Viscous, Incompressible Liquid with Free Boundaries

Abstract

The stability of a rotating and heated from below horizontal cylindrical layer of a viscous, incompressible liquid with free boundaries was theoretically investigated. Neglecting the centrifugal forces, the equations of motion, thermal conductivity and incompressibility of the liquid were written, from which the well-known dispersion equation was derived in the linear approximation. The stability of a rotating cylindrical volume of a liquid with no heating from below was considered, provided that the temperature difference between the horizontal boundaries of the liquid was fixed and equal to zero. It was demonstrated, that with no heating from below the temperature difference between the horizontal boundaries of the rotating liquid was not fixed and not maintained from the outside, the perturbed liquid temperature would increase, but its final value did not exceed the phase transition temperature. The obtained result was used to explain the heating of water in Ranque – Hilsch vortex tubes. It was concluded that the water heating in Ranque -Hilsch tubes should be considered as the inverse Rayleigh problem, in which the temperature gradient can be determined from the known distribution of velocities inside the volume. The stability of a rotating cylindrical volume of a liquid when heated from below was analyzed. It was demonstrated, that the value of the specified temperature difference at cylinder boundaries, as well as the initial rate of its variation, determine the final heating temperature of the liquid. A comparison of the proposed theory and experimental data for water heating shows their good qualitative and quantitative agreement.

Keywords

• theory
• stability
• viscous
• incompressible
• liquid
• horizontal cylindrical layer
• free boundaries