I need to solve
∂2Φ/∂s2 + (1/s)*∂Φ/ds - C = 0
Where s is a radial coordinate and C is a constant.
I know this is fairly simple but I haven't had to solve a problem like this in a long time. Can someone advise me on how to begin working towards a general solution?
Is the method of...
As I understand it the density is going to be calculated based on the enthalpy once psi is known for each iteration of a larger algorithm.
So setting m = 0 and B = 0, and choosing ψ (s1) = ψouter = 0 as a boundary condition then I can state the R(r) solution as a sum, with λ0,n defined in terms...
Ok after studying that link I might be a little closer to making sense of this.
My domain does not include the origin. Should I discard the ##Y_k## anyway because the function needs to remain physical anyway?
In any case I should treat the bessel functions the same way I would treat sine or...
As I said I am not practiced with using bessel functions in any capacity (besides a few homework problems over a year ago) but I am getting the impression that they are difficult to work with unless you can state a boundary R(a) = 0. This had something to do with the fact that the bessel...
There is a lot going into this derivation but it is intended to give the stream function and therefore velocity at equilibrium of a rotating system. Ω is simply frame rotation (taken to be known for the purposes of this question). Indeed the more compact form of this PDE is:
$$ξ_z = ρf(ψ) -...
I will also elaborate on my recent attempts to solve this:
If I want the solution to be axisymmetric (which physically, I think I do), then it seems I need to require k = 0, then I can solve for A and B using inner and outer boundary conditions ##(ψ(s_0) = ψ_{inner} , ψ(s_1) = ψ_{outer})## with...
I have a PDE which I have solved numerically using a guass-seidel method, but I want to compare it to the analytical solution. I have used separation of variables to get the general solution, but I need help applying it.
The PDE is
(1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω...
A couple of things I'd like to clarify as I'm having trouble with this. Shouldn't b = ρ2c1 per the first substitution?
For the second set of substitutions, if we replace Ψ in the eq by Ψ' I can't get a0 to vanish algebraically as you have.
Sorry if this is obvious, I've only worked with...
As an update if anyone is still interested, I constructed a guass-seidel algorithm which converges for most of the examples I've tried.
One of my next steps will be to test the solution for a constant density using the analytical solution, so thanks for helping with that!
This is a numerical problem solving exercise. You're suggesting solve the two second order ODEs for R(r) and Θ(θ) by discretizing them and generating a tridiagonal matrix?
I'm not sure I can justify defining ρ as constant in the first step though. We are assuming equilibrium (dψ/dt) = 0 but the...
Sorry I did neglect to identify my variables in that long post.
ρ is mass density, assumed to be a known function of s and Φ.
The work I've done so far trying to solve this numerically has been in python.
I am attempting to solve the following PDE for Ψ representing a stream function on a 2D annulus grid:
(1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω + ρ(c0 + c1ψ) = 0
I have made a vertex centered discretization:
(1/sj)⋅(1/Δs2)⋅[(sj+1/2/ρj+1/2,l){ψj+1,l - ψj,l} -...
No, the density depends on s and Φ, the entire equation is:
(1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω + ρ(c0 + c1ψ) = 0
Where Ω is frame rotation rate and c0, c1 are arbitrary constants (from the first and second term of a taylor expansion).
Oh, I should probably mention that I am only giving the terms which I need help discretizing. The rest of the equation is straightforward to discretize and is mostly constant terms.