Kamis, 16 Februari 2017

On plausible methods to solve 3D Navier-Stokes equations

Dear Sergey and Yury

Thanks, yes i aware that Clay prize may be not for mortals like us...but at least it will be fun to find out that many "emperors" are completely naked... ;-)

Thanks also for your prayer for me to go to Jerusalem. Well I know it is highly improbable that Trump will read and respond my email, but I hope that some professors cited in my email can do preparation for second coming to their people...that is my wish.

Btw, now back to possible solution of NS. Terribly sorry, i am not trained in complicated procedure, so I would propose three possible ways to solve NS:

a. Using its analogy with Riccati, then solve the Riccati-NS with Mathematica.

b. using numerical methods to solve PDE. See for instance and old paper by Gabriel Kron, where he used circuit network analog for solving PDE. 
We know that NS is a PDE degree two, so perhaps it can be solved using Kron's analog circuit...

c. Combine procedure a and b. Find the Riccati equivalent of NS then use circuit network analog to solve numerically.

But really i still need time to put these ideas to working paper.. Perhaps Yury has his own ideas too.

How is that?

Humbly yours,


Victor Christianto
*Founder and Technical Director, www.ketindo.com
E-learning and consulting services in renewable energy
**Founder of Second Coming Institute, www.sci4God.com
Twitter: @Christianto2013
***Papers and books can be found at: 
http://nulisbuku.com/books/view_book/9035/sangkakala-sudah-ditiup

On Feb 17, 2017, at 1:16, Ершков Сергей<sergej-ershkov@yandex.ru> wrote:

Dear Victor, I wish every success for your forthcoming science-trip to Jerusalem (Israel) under the guidance of God and with the possible help of Mr. Donald Trump.
 
Btw, regarding your suggestion in the e-mail below: could you kindly check (?) via "Mathematica" or CFD-solver one of partial examples from the class of exact solutions of NSE, which was obtained by me few weeks ago.
 
This is analytical formulae of nonstationary 3D-solution (see presentation of my Report enclosed, in Russian; Yury already has its old version), such the solution should satisfy to NSE at initial conditions (Cartesian coordinates system) in the case of the Cauchy problem in the whole space; appropriate meanings of the components of pressure gradient should be chosen according to Bernoulli invariant.
 
So, algorithm for checking of the solution via "Mathematics" (components of solution) is pointed below:
 
1) solution is the pair (p, u), where components of pressure gradient should be chosen according to Bernoulli invariant Eq. (1.3);
 
2) the velocity field u = u_p + u_w, where components of u_p (potential flow) are given by formulae (1.4.2); as for the appropriate components of u_w (solenoidal flow), you could see it at p.19 and p.21 accordingly.
 
Parameters a(t), b(t) in the aforementioned formulae are pointed at pages 21-22.
 
3) At page 23 you could see the example of presenting of formulae for the Ox-component of flow velocity; Oy-component and Oz-component of flow velocity should be presented via appropriate formulae (see point 2) above).
 
4) At pages 23-24 you could find the plots of the Ox-component of flow velocity, which was calculated analytically by me.
 
For checking of the validity of the components of solution (does it really satisfy to NSE under the chosen initial conditions if we consider the case of the Cauchy problem in the whole space?), we should substitute appropriate terms in NSE with the help of "Mathematica" or CFD-solver.
 
It will be the qualitative basis for our mutual publication in the future (I think, Prof. Smarandache is also interested in solving of NSE).
 
 
Hope, you will find a spare time for implementing of all the calculations via "Mathematica", it should be interesting mathematical experiment!
 
If you have an additional questions or comments, you are welcome!
 
P.S. Millenium prize is not so easy thing to obtain :), we are only at the beginning of our way.
 
 
Yours,
 
Sergey
 


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