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.
Highlights
THE STABILITY OF A ROTATING AND HEATED FROM BELOW HORIZONTAL CYLINDRICAL LAYER OF A VISCOUS, INCOMPRESSIBLE LIQUID WITH FREE
It is known that a periodic structure in the form of Benard cells [1] is formed in a horizontal layer of a viscous, incompressible, below-heated liquid
Let us consider a cylindrical volume of radius R0 filled with a viscous, incompressible liquid, the lower and upper boundaries of which co ie nzc,idwehweriethe tzh-ethpelaunneist z 0 vector and z h
Summary
1, Academichna Str., 61108 Kharkiv, Ukraine 2V.N. Karazin Kharkiv National University. The stability of a rotating and heated from below horizontal cylindrical layer of a viscous, incompressible liquid with free boundaries was theoretically investigated. The obtained solutions describe the occurrence of convective rolls in a horizontal liquid layer, on the vertical common boundaries of which the velocity was directed periodically up / down and vice versa These solutions did not describe the experimental fact of the availability of polygonal structures, the number of angles of which varied from four to seven, but with a predominance of six [1]. In this work the onset of convection in a uniformly rotating and below-heated cylindrical tank with a viscous, incompressible liquid with free boundaries was investigated. THE INITIAL EQUATIONS OF CONVECTION IN A ROTATING VISCOUS, INCOMPRESSIBLE LIQUID
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