Analytical Estimates of Critical Taylor Number for Motion between Rotating Coaxial Cylinders Based on Theory of Stochastic Equations and Equivalence of Measures

Artur V Dmitrenko

doi:10.3390/fluids6090306

Abstract

The purpose of this article was to present the solution for the critical Taylor number in the case of the motion between rotating coaxial cylinders based on the theory of stochastic equations of continuum laws and the equivalence of measures between random and deterministic motions. Analytical solutions are currently of special value, as the solutions obtained by modern numerical methods require verification. At present, in the scientific literature, there are no mathematical relationships connecting the critical Taylor number with the parameters of the initial disturbances in the flow. The result of the solution shows a satisfactory correspondence of the obtained analytical dependence for the critical Taylor number to the experimental data.

Highlights

Analytical solutions are currently of special value, as the solutions obtained by modern numerical methods require verification
Analytical dependences including theoretical estimates are extremely important in the analysis of experimental data, when it is necessary to take into account the effect of substantial quantities, which are random in time and space, instead of only to give average statistical estimates
We present the results of calculations for the conditions of the experiment of Taylor [31]

Summary

Introduction

Analytical solutions are currently of special value, as the solutions obtained by modern numerical methods require verification. As it is known, an advantage of analytical formulas is the visualization of physical relationships between quantities. The development of physical and mathematical theories for complex physical nonlinear processes, which are described by inhomogeneous high-order partial differential equations, is especially significant. Analytical dependences including theoretical estimates are extremely important in the analysis of experimental data, when it is necessary to take into account the effect of substantial quantities, which are random in time and space, instead of only to give average statistical estimates. Different ideas of the theory of turbulence are presented in [1–10]. Mathematical methods for obtaining solutions of the Navier–Stokes equation, the theory of solitons, and the theory of strange attractors are presented in [11–24]

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Results

Conclusion