Mathematical Physics

   

Solutions of Navier-Stokes Equations plus Solutions of Magnetohydrodynamic Equations

Authors: A. A. Frempong

Better news. After nearly 150 years of patience, the Navier-Stokes equations in 3-D for incompressible fluid flow have been analytically solved by two different methods. It is shown that these equations can be solved in 4-dimensions or n-dimensions. The author has 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 is shown that without gravity forces on earth, there would be no incompressible fluid flow as is known (see p. 23). By applying the above law, the hitherto unsolved magnetohydrodynamic equations were also routinely solved. In addition to the usual method of solving these equations, the N-S equations have also been solved by a second method in which the three equations are added to produce a single equation which is then integrated. The solutions by the two methods are identical, except for the constants involved. The Navier-Stokes equation will be linearized, solved, and the solution analyzed. This solution will be followed by the solution of the Euler equation. Following the Euler solution, the Navier-Stokes equation will be 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. The pressure gradient term is also involved in the parabolic motion. The viscosity terms are involved in parabolic, periodic, and decreasingly exponential motion. The variable acceleration term is also involved in the periodic and decreasingly exponential motion. The convective acceleration terms produce square root function behavior and fractional terms containing square root functions with variables in the denominators and consequent turbulence behavior. For a spin-off, the smooth solutions from above are specialized and extended to satisfy the requirements of the CMI Millennium Prize Problems, and prove the existence of smooth solutions of the Navier-Stokes equations.

Comments: 45 Pages. Copyright © A. A. Frempong

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Submission history

[v1] 2014-05-15 22:30:45 (removed)
[v2] 2014-05-16 14:15:22
[v3] 2014-06-20 23:01:10
[v4] 2014-06-22 18:12:52
[v5] 2014-06-24 14:31:33
[v6] 2014-07-04 23:53:22
[v7] 2014-07-16 17:38:41
[v8] 2015-01-18 02:02:49

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