Mathematical Physics

   

Navier-Stokes Equations Solutions Completed

Authors: A. A. Frempong

Over nearly a year and half ago, the Navier-Stokes (N-S) equations in 3-D for incompressible fluid flow were analytically solved by the author. However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author proposed and applied a new law, the law of definite ratio for incompressible fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. It was mathematically shown that without gravity forces on earth, there would be no incompressible fluid flow on earth as is known, and also, there would be no magnetohydrodynamics. In addition to the usual method of solving these equations, the N-S equations were also solved by a second method in which the three equations in the system were added to produce a single equation which was then integrated. The solutions by the two methods were identical, except for the constants involved. Ratios were used to split-up the equations; and the resulting sub-equations were readily integrable; and even, the nonlinear sub-equations were readily integrated. The examples in the preliminaries show everyday examples on using ratios to divide a quantity into parts, as well as possible applications of the solution method in mathematics, science, engineering, business, economics, finance, investment and personnel management decisions. The x-direction Navier-Stokes equation was linearized, solved, and the solution analyzed. This solution was followed by the solution of the Euler equation of fluid flow. The Euler equation represents the nonlinear part of the Navier-Stokes equation. Following the Euler solution, the Navier-Stokes equation was solved essentially by combining the solutions of the linearized equation and the Euler solution. For the Navier-Stokes equation, the linear part of the relation obtained from the integration of the linear part of the equation satisfied the linear part of the equation; and the relation from the integration of the non-linear part satisfied the non-linear part of the equation. The solutions and relations revealed the role of each term of the Navier-Stokes equations in fluid flow. The gravity term is the indispensable term in fluid flow, and it is involved in the parabolic and forward motion of fluids. The pressure gradient term is also involved in the parabolic motion. The viscosity terms are involved in the parabolic, periodic and decreasingly exponential motion. Periodicity increases with viscosity. The variable acceleration term is also involved in the periodic and decreasingly exponential motion. The fluid flow in the Navier-Stokes solution may be characterized as follows. The x-direction solution consists of linear, parabolic, and hyperbolic terms. The first three terms characterize parabolas. If one assumes that in laminar flow, the axis of symmetry of each parabola for horizontal velocity flow profile is in the direction of fluid flow, then in turbulent flow, the axes of symmetry of some of the parabolas would be at right angles to that for laminar flow. The characteristic curve for the integral of the x-nonlinear term is such a parabola whose axis of symmetry is at right angles that of laminar flow. The integral of the y-nonlinear term is similar, parabolically, to that of the x-nonlinear term. The characteristic curve for the integral of the z-nonlinear term is a combination of two similar parabolas and a hyperbola. If the above x-direction flow is repeated simultaneously in the y-and z-directions, the flow is chaotic and consequently turbulent. For a spin-off, the smooth solutions from above are specialized and extended to satisfy the requirements of the CMI Millennium Prize Problems, and thus prove the existence of smooth solutions of the Navier-Stokes equations.

Comments: 47 Pages. Copyright © by A. A. Frempong

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Submission history

[v1] 2015-12-16 00:15:11
[v2] 2015-12-16 22:28:55
[v3] 2016-02-22 03:06:28
[v4] 2016-02-26 01:37:13
[v5] 2017-07-03 01:48:23

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