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

happyparticle
Messages
490
Reaction score
24
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?
 
Physics news on Phys.org
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.
 
Last edited:
Thread 'Need help understanding this figure on energy levels'
This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
Thread 'Understanding how to "tack on" the time wiggle factor'
The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
Back
Top