Hi all, I'm stuck on the following problem:
We consider the flux of an imcompressible and irrotational fluid around a clynder of radius a, so long that we consider it as infinitely long in the z direction. We assume a stationary regime such that the velocity does not depend on the axial component (I guess they mean the z component), its axial component is null and far from the cylinder the velocity is uniform and constant.
Mathematically, tends asymptotically to when tends to infinity.
From the imcompressibility and the mass conservation, one has that . (1)
Introducing the function "potential current" such that and so that (1) is satisfied and using the irrotationability of the fluid, one reaches Laplace equation for the potential current: for . Furthermore, for .
Using the method of separation of variables in appropriate coordinates, determine and .
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My attempt:
First, I made a sketch of the situation. I realize that . My plan to solve the problem is first to determine (in polar coordinates), then translate it to and then by integration I could get and and therefore .
So I assumed that . I then took the Laplacian in polar coordinates and equated to 0.
It simplified the problem into 2 ODE's, namely:
1)
2)
The solution to 1) is of the form but since it's a physical problem and psi must remain finite inside the disk of radius a, . Thus .
The solution to 2) is .
Therefore and thus .
Now this is where I'm stuck. I'm looking to apply the boundary condition, . I have reached . I don't know how to get the constants and from here and also I'm skeptical on the solution. Because I have an infinite series in which each terms are linearly independent of each other and the whole series is worth 0, hence it makes me think that all coefficients (therefore 's and 's) must equal 0... And I would get the trival solution.
Is there something wrong in what I've done so far? How could I proceed further? Thanks in advance.