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philefou
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Join Date: Mar 2009
Posts: 2

Analytic Navier Stokes Equation

05/20/2009 5:22 PM

∂/∂x(w)+∂/∂y(w)=constante

With the boundary conditions on a simple rectangle (L per H) : w(x=0,y)=0, w(x=L,y)=0 w(x,y=0)=0 and w(x,y=H)=0.

I'm trying to solve this equation with variables separation and Fourier series.. Can someone give me an hint?

Thank you.
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Anonymous Poster
#1

Re: Analytic Navier Stokes Equation

05/21/2009 11:15 PM

first use the gravity force, pressure force, and turbulence force equations to your problem.
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MarkTheHandyman
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Canada - Member - Toronto, Ontario (South Parkdale On The Lakeshore) Engineering Fields - Marine Engineering - Great Lakes School Of Marine Technology (Owen Sound and Port Colbourne) Technical Fields - Architecture - Private Practice 1976-1990 Technical Fields - Education - Toronto Teachers' College 1971 Technical Fields - Marketing/Advertising - Founding Member Hobbies - Hunting - Founding Member Hobbies - Target Shooting - Founding Member

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#2

Re: Analytic Navier Stokes Equation

05/22/2009 4:39 AM

Hi, philefou!

Wescome to CR4! Hopefully, as you continue to participate in various threads, you will come to appreciate the depth and breadth of engineering disciplines represented here. Feel free to chime in on all the ongoing ones that tickle your fancy; or that you think you can cast light on. We're an interesting and fun group.

I'd like to help you out, but it would mean returning to/digging through old skul notes, so I'll just leave it to the fresher minds to refresh us both with the information you seek, and wait right along with you for the answer to show up.

Mark
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nick name
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#4
In reply to #2

Re: Analytic Navier Stokes Equation

05/22/2009 9:03 AM

Apparently there are no "fresher minds" in the broad CR4 team!

I already noticed that as soon as the question requires a quantitative answer the number of active participants drops almost to nil.

I am surprised that for many members engineering is only a qualitative profession, a "let us speak about your question".

Many many times when a comment is done the reader will have the impression that nobody dares use or even is afraid of quantitative, mathematical approaches. Several times the number of enthusiastic was very important for very down level couples of written formula.

When one of the "guests" asked for a solution to a system of differential equations the comment was "yes we can!!" but nobody did anything, neither a hint nor a proposal under the justification that it is a home work and we do not help for home work. I wonder how many tried off line to find a solution? No one suggested to have displayed the steps the guest will make and give suggestions for corrections.

What I mostly consider as damage is the expression of non professional opinions which could bring the used in bad or even dangerous situations, this is the follow up of the wish to give an answer even one has not the slightest idea about what the question means. A simplification of the problem is always necessary and strongly recommended but all has limits a too simplified answer a too short short-cut could be not the right answer.

On the opposite side when it is a "discussion subject" the number of participants grows almost to infinite..........!

I sincerely expect a bit more from an engineering high level forum. Or am I wrong?

I am sure my comment will generate wild reactions against it but I am accustomed since I always say or write what I think.
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MarkTheHandyman
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Canada - Member - Toronto, Ontario (South Parkdale On The Lakeshore) Engineering Fields - Marine Engineering - Great Lakes School Of Marine Technology (Owen Sound and Port Colbourne) Technical Fields - Architecture - Private Practice 1976-1990 Technical Fields - Education - Toronto Teachers' College 1971 Technical Fields - Marketing/Advertising - Founding Member Hobbies - Hunting - Founding Member Hobbies - Target Shooting - Founding Member

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#9
In reply to #4

Re: Analytic Navier Stokes Equation

05/23/2009 5:24 AM

Hi, nick_name!

While my decision to not go back to old theoretical notes was admittedly due to laziness and the hope that somebody in the forum has just gone through the same (or similar) process as I did back then and can be more present with an answer, I have to write that I found your own response really very good (in a humourous way). Nice to know that somebody shares my sense of dry wit.

I got the gist of it, and a good chuckle. Thanks.

Mark
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agua_doc
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#3

Re: Analytic Navier Stokes Equation

05/22/2009 9:00 AM

Havent done that type of problem in the last 25 years, but recently ran across a site that has a math sub-component that might be able to help. Look at: www.wolframalpha.com

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nick name
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#5
In reply to #3

Re: Analytic Navier Stokes Equation

05/22/2009 9:06 AM

As far as I could see the site offers a help for simple linear differential equations but no help for the partial differential equation described above.
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nick name
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#6

Re: Analytic Navier Stokes Equation

05/22/2009 9:17 AM

You ask for a hint:

If you consider w=X*Y and where X=f(x) and Y=f(y) then you have ∂w/∂x= X'*Y and ∂w/∂y= X*Y'. One possible solution is that both partial derivatives are constant since their sum is a constant so that you come to : X'Y+XY'=C

I let you go further on your own.
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philefou
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Join Date: Mar 2009
Posts: 2
#10
In reply to #6

Re: Analytic Navier Stokes Equation

05/23/2009 12:23 PM

thanks for your help. Actually after two days, I finally found the solution. First of all we suppose a solution of the type: sum_m(1,inf) sum_n(1,inf) [ A_ij * sin(m*pi/Lx*x) * sin(n*pi/Ly*y) ] As you can see, the double sum follows the Dirichlet conditions on the boundary. This double sum can now be derivate two times in x, and two times in y and if we put the constante=1 we get. -sum_m(1,inf) sum_n(i_inf) [ A_mn *((m*pi/Lx)^2+(n*pi/Ly)^2) * sin(m*pi/Lx*x) * sin(n*pi/Ly*y) ] = 1 Now the only thing left is to find A_mn. We use a test function to multiply both side of the equation, and integrate on our domain to finally isolate A_mn. (Ritz element method) So our test function is : sin(i*pi/Lx*x)(sin(i*pi/Ly*y) ... I won't write all the furter solution but you'll have to solve 2 differents type of equation (sin*sin and only sin).. I can suggest integration table for those. A_mn = -16/(i*j*pi^4*((i/Lx)^2+(j/Ly)^2)) Feel free to ask if you're running into some problems :) ps: maybe there are some typing error :\ ! Thanks for all your comments.
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nick name
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#11
In reply to #10

Re: Analytic Navier Stokes Equation

05/24/2009 5:58 AM

I think you should change your "name" from philefou to philesage!
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d_m_rosenberg
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#7

Re: Analytic Navier Stokes Equation

05/22/2009 2:06 PM

Here you go:

http://www.grc.nasa.gov/WWW/K-12/airplane/nseqs.html

I guess it is rocket science.

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nick name
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#8
In reply to #7

Re: Analytic Navier Stokes Equation

05/22/2009 2:59 PM

If you look at the presentation ONLY the equations are written NOT how to solve them, and this is what the question was: a hint to come to the solution!

Again a superficial answer. It is a pity.
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agua_doc (1); Anonymous Poster (1); d_m_rosenberg (1); MarkTheHandyman (2); nick name (5); philefou (1)

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