Abstract
This paper has analyzed the equation of motion in terms of stresses (Navier), as well as its two special cases for an incompressible viscous current. One is the Stokes (Navier-Stokes) equation, and the other was derived with fewer restrictions. It has been shown that the Laplace equation of linear velocity can be represented as a function of two variables ‒ the linear and angular speed of particle rotation. To describe the particle acceleration, all motion equations employed a complete derivative from speed in the Gromeka-Lamb form, which depends on the same variables. Taking into consideration the joint influence of linear and angular velocity allows solving a task of the analytical description of a turbulent current within the average model. A given method of analysis applies the provision of general physics that examines the translational and rotational motion. The third type of mechanical movement, oscillatory (pulsation), was not considered in the current work. A property related to the Stokes equation decomposition has been found; a block diagram composed of equations and conditions has been built. It is shown that all equations for viscous liquid have their own analog in a simpler model of non-viscous fluid. That makes it easier to find solutions to the equations for the viscous flow. The Stokes and Navier equations were used to solve two one-dimensional problems, which found the distribution of speed along the normal to the surface at the flow on a horizontal plate and in a circular pipe. Both solution methods produce the same result. No solution for the distribution of speed along the normal to the surface in a laminar sublayer could be found. A relevant task related to the mathematical part is to solve the problem of closing the equations considered. A comparison of the theoretical and empirical equations has been performed, which has made it possible to justify the assumption that a rarefied gas is the Stokes liquid
Highlights
Underlying the classic method of constructing mathematical models in fluid mechanics is the equation of motion in terms of stresses (Navier), which is a special case of the law of preserving the amount of movement [1,2,3]
Paper [4] analyzes another special case of the Navier equation, which takes into consideration the angular velocity of the particle rotation
Work [11] models a system of equations in which there is a special case of the Stokes equation without viscosity and equation of electromagnetism
Summary
Underlying the classic method of constructing mathematical models in fluid mechanics is the equation of motion in terms of stresses (Navier), which is a special case of the law of preserving the amount of movement [1,2,3]. There is a known Stokes solution for the movement of a ball in a Newtonian fluid, which is consistent with experimental data in experiments involving non-Newtonian fluid (glycerin or castor oil) [1, 2]. These contradictions between the theory and experiment have no satisfactory explanation. Computer programs used in fluid mechanics (Flowvision, Phoenics, etc.) produce good results only in a narrow range of changes in influencing factors while their solutions are often unstable (approximate) This flaw requires an experimental check of numerical calculations, increases the cost and timing of advancements [2, 3]. It is a relevant task to search for the new forms of Stokes equation and exact solutions to them, derived according to the classical scheme in accordance with the provisions of general physics
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