How to Solve the Laplace Equation for Potential Flow Around a Sphere?

AI Thread Summary
The discussion centers on solving the Laplace equation for potential flow around a sphere using different coordinate systems. The initial attempt with spherical coordinates using the separable variable method led to equations that were difficult to solve. A shift to cylindrical coordinates resulted in a new equation, which the user is unsure how to interpret, particularly regarding the distinction between S and s. The equation presented is recognized as a Sturm-Liouville problem, indicating that its solutions are Bessel functions. The conversation highlights the complexities involved in finding a solution and the need for clarity in variable representation.
happyparticle
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Homework Statement
Consider the steady flow pattern produced when an impenetrable rigid spherical obstacle is placed in a uniformly flowing, incompressible, inviscid fluid. Find a solution for the potential flow around the sphere.
Relevant Equations
##\nabla^2 \phi = 0##
I tried to find a solution to the Laplace equation using spherical coordinates and the separable variable method. However, I found equations that I simply don't know how to find a solution. Thus, I tried in cylindrical coordinates with an invariance in ##\theta## but now I'm facing this equation.

##\frac{1}{s} \frac{d}{ds}(s \frac{dS}{ds}) = -k^2 S##

Is there a fairly simple solution for it or should I find another way to do this problem?
 
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I am not sure to distinguish S and s in your equation. Let me rewrite it with x for s and y for S
\frac{1}{x} \frac{d}{dx}(x \frac{dy}{dx}) = -k^2 y
Is it the right equation which has constant k with its physical dimension of [1/s] ? It belongs to Sturm -Liouville equation whose solutions are Bessel functions.
 
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