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 \vec u is uniform and constant.
Mathematically, \vec u tends asymptotically to (v,0) when r=\sqrt {x^2+y^2} tends to infinity.
From the imcompressibility and the mass conservation, one has that \nabla \cdot \vec u= \frac{\partial u_1 }{\partial x }+ \frac{\partial u_2 }{\partial y}=0. (1)
Introducing the function "potential current" \psi such that \frac{\partial \psi }{\partial x }=-u_2 and \frac{\partial \psi }{\partial y}=u_1 so that (1) is satisfied and using the irrotationability of the fluid, one reaches Laplace equation for the potential current: \triangle \psi =0 for r>a. Furthermore, \psi =0 for r=a.
Using the method of separation of variables in appropriate coordinates, determine \psi and \vec u.
My attempt:
First, I made a sketch of the situation. I realize that \vec u = (u_1, u_2). My plan to solve the problem is first to determine \psi (r, \theta ) (in polar coordinates), then translate it to \psi (x,y) and then by integration I could get u_1(x,y) and u_2(x,y) and therefore \vec u.
So I assumed that \psi (r, \theta )=R(r)\Theta (\theta ). I then took the Laplacian in polar coordinates and equated to 0.
It simplified the problem into 2 ODE's, namely:
1) \frac{rR'}{R}+r^2\frac{R''}{R}=m^2
2) -\frac{\Theta '' }{\Theta}=m^2
The solution to 1) is of the form R(r)=Ar^m+Br^{-m} but since it's a physical problem and psi must remain finite inside the disk of radius a, B=0. Thus R(r)=Ar^m.
The solution to 2) is \Theta (\theta ) =C \cos (\theta m )+ B \sin (\theta m ).
Therefore \psi _m (r, \theta )=r^m [E \cos ( \theta m) + F \sin ( \theta m )] and thus \psi (r, \theta ) = \sum _{m=0} ^{\infty } r^m [E_m \cos (\theta m )+ F_m \sin (\theta m) ].
Now this is where I'm stuck. I'm looking to apply the boundary condition, \psi (a, \theta )=0. I have reached \sum _{m=0}^{\infty } a^m [E_m \cos (\theta m ) + F_m \sin (\theta m )]=0. I don't know how to get the constants E_m and F_m 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 E_m's and F_m'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.