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Stephen Montgomery-Smith1
science forum Guru

Joined: 01 May 2005
Posts: 487

Posted: Mon Jul 10, 2006 6:59 am    Post subject: Re: taylor couette flow between two cylinders (outer cylinder stationary) solution through eigen value with the help of bessel function

Stephen Montgomery-Smith wrote:

 Quote: Now any other eigenfunction of the Stokes operator for a specific eigenvalue, lambda, has to be a multiple of one of these non-slip eigenfunctions decribed above, and one particular choice of eigenfunction satisfying the boundary conditions. For this choice, I recommend something like (A r^alpha + B r^beta) (-sin(theta),cos(theta)) where the choice of alpha and beta come from solving the problem f''(r) + r^{-1} f'(r) - r^{-2} f(r) = lambda f(r), i.e. the Laplacian in polar coordinates (since the Leray projection isn't going to do anything).

Oops, this looks more like a parameterized Bessel equation of order 1.
So don't look for solutions of the form r^alpha.

But the general approach might still work.
Stephen Montgomery-Smith1
science forum Guru

Joined: 01 May 2005
Posts: 487

Posted: Mon Jul 10, 2006 6:32 am    Post subject: Re: taylor couette flow between two cylinders (outer cylinder stationary) solution through eigen value with the help of bessel function

gateway114@gmail.com wrote:
 Quote: can any one tell be what is completness theorm of non self adjoint matrix

I don't know of any. The matrix needs to be at least normal (i.e. it
commutes with its hermitian transpose).

 Quote: and how it can be apply to cylindrical tayor couette problem

Maybe you could do it directly. Let me check I am solving the right
problem. You are looking at the Stokes operator (which is the Laplacian
followed by the Leray projection) on the space of divergence free vector
fields between two concentric rings, where the boundary conditions on
the rings are that the tangential componant of the field is constant and
the normal componant is zero. (I will do it in 2D as the third
dimension is easy to handle.)

Now you can compute the eigenfunctions/eigenvalues of the Stokes
operator in the non-slip boundary condition (i.e. the field is zero on
the boundary). We know that this is complete in this space by the usual
Sturm-Louiville type theory. (I'm guesing that it might be specific
linear combinations of Bessel functions of the first and second kind,
but then again the Leray projection might do something non-trivial to
mess this up.)

Now any other eigenfunction of the Stokes operator for a specific
eigenvalue, lambda, has to be a multiple of one of these non-slip
eigenfunctions decribed above, and one particular choice of
eigenfunction satisfying the boundary conditions. For this choice, I
recommend something like (A r^alpha + B r^beta) (-sin(theta),cos(theta))
where the choice of alpha and beta come from solving the problem
f''(r) + r^{-1} f'(r) - r^{-2} f(r) = lambda f(r),
i.e. the Laplacian in polar coordinates (since the Leray projection
isn't going to do anything).

Might this work?

Stephen
gateway114@gmail.com
science forum beginner

Joined: 18 Jun 2006
Posts: 2

 Posted: Sun Jul 09, 2006 8:00 am    Post subject: taylor couette flow between two cylinders (outer cylinder stationary) solution through eigen value with the help of bessel function can any one tell be what is completness theorm of non self adjoint matrix and how it can be apply to cylindrical tayor couette problem

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