Finish and plot this separation of variables problem

In summary, the conversation discusses a PDE that has been solved numerically using a gauss-seidel method and the desire to compare it to the analytical solution. Separation of variables has been used to obtain the general solution, but there is uncertainty about how to apply it. The PDE involves a vorticity wave equation for density coupled into the stream field, with an externally applied angular velocity represented by Ω. The functional form of ρ used in the equation is intended to give the stream function and velocity at equilibrium in a rotating system. The conversation also mentions the use of Bessel functions and the need for further clarification on how to handle coefficients.
  • #1
Daniel Sellers
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TL;DR Summary
I have found general solutions to a PDE which I need help applying Dirichlet conditions to so I can plot a final solution.
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 + ρ(c0 + c1ψ) = 0

and applying separation of variables gives (I think):

ψ = (1/ρc1)⋅[AJk(ρs√c1) + BYk(ρs√c1)]⋅[Ccos(kΦ) + Dsin(kΦ)] + (2Ω/ρc1) + (c0/c1)

Where ρ, Ω, c0, c1 are known constants.

I am not sure what to do from here. I want to apply Dirichlet boundary conditions on an inner boundary s0 and outer boundary s1. Should these be series solutions? Like summations over k =1,2 ... and if so then how do I handle the coefficients (A,B,C,D)?

The domain is a polar annulus, so k must be an integer, but that's all I can readily figure out.
 
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  • #3
Fair enough!:
The PDE:

$$ \frac{1}{s}⋅\frac{∂}{∂s}(\frac{s}{ρ}\frac{∂ψ}{∂s}) + \frac{1}{s^2}⋅\frac{∂}{∂Φ}(\frac{1}{ρ}\frac{∂ψ}{∂Φ}) - 2Ω + ρc_0 + ρc_1ψ = 0$$

What I think the general solution is:

$$ ψ = \frac{1}{ρc_1}[AJ_k(ρs\sqrt{c_1}) + BY_k(ρs\sqrt{c_1})]⋅[Ccos(kΦ) + Dsin(kΦ)] + \frac{2Ω}{ρc_1} + \frac{c_0}{c_1}$$

I am inexperienced with using bessel functions, and I admit I have forgotten a lot about solving PDEs, so I could really use some help/advice here.
 
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  • #4
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 order zero for the bessel functions. This gives a solution, but I'm not confident that it is correct, and it requires me to choose A arbitrarily.

Or do I need to solve the Φ dependent part as Fourier series in order to determine A?

Still very confused, any help is appreciated
 
  • #5
Daniel Sellers said:
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 order zero for the bessel functions. This gives a solution, but I'm not confident that it is correct, and it requires me to choose A arbitrarily.

Or do I need to solve the Φ dependent part as Fourier series in order to determine A?

Still very confused, any help is appreciated
I will also advise you to ask this on math.stackexchange, people answer relatively quickly there.
 
  • #6
I find your problem interesting because I see a sort of analog with flux pinning in type II superconductors. For the constant ##\rho## problem I suggest you treat the problem analogous to the wave equation of a circular drum head, for example here: http://ramanujan.math.trinity.edu/rdaileda/teach/s14/m3357/lectures/lecture_4_3_slides.pdf. I treat the problem as being axially symmetric with a constant radial distribution fuction to evaluate the constants in the PDE. I get,$$ \psi(r)= 2 \rho_0 \sum_{n=0}^\infty \frac {J_0 (\lambda_{0m} r)} {\lambda_{0m} J_1 (\lambda_{0m})} +\rho_ 0 (\rho_0 c_0 - 2\Omega)$$ where $$\lambda_{mn}=\frac {\alpha_{mn} \rho_0^2 c_1} {R_{static}}$$ and ##\alpha_{mn}## is the ##n^{th}## positive zero of ##J_m##, ##R_{static}## is the radius where the stream field goes to zero (approximately) and ##\rho_0## is the static constant mass density.

I have some questions about the physical interpretation of the PDE. I rewrite your PDE as $$\frac {1} {\rho} \nabla^2 \psi =2\Omega - \rho c_0 + \vec \nabla \frac {1} {\rho} \cdot \vec \nabla \psi - c_1 \psi =? \omega$$ where ##\omega## is (possibly?) the vorticity pointing along the z axis. Is ##\Omega## an externally applied angular velocity (like from a centrifuge motor or a constant magnetic field penetrating a type II superconductor)? Is this a vorticity wave equation for density coupled into the stream field? What functional form of ##\rho(r,\theta)## are you using? How was this equation derived? I didn't take fluid dynamics in college and I am confused by the physics behind the PDE:-)
 
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  • #7
Fred Wright said:
I find your problem interesting because I see a sort of analog with flux pinning in type II superconductors. For the constant ##\rho## problem I suggest you treat the problem analogous to the wave equation of a circular drum head, for example here: http://ramanujan.math.trinity.edu/rdaileda/teach/s14/m3357/lectures/lecture_4_3_slides.pdf. I treat the problem as being axially symmetric with a constant radial distribution fuction to evaluate the constants in the PDE. I get,$$ \psi(r)= 2 \rho_0 \sum_{n=0}^\infty \frac {J_0 (\lambda_{0m} r)} {\lambda_{0m} J_1 (\lambda_{0m})} +\rho_ 0 (\rho_0 c_0 - 2\Omega)$$ where $$\lambda_{mn}=\frac {\alpha_{mn} \rho_0^2 c_1} {R_{static}}$$ and ##\alpha_{mn}## is the ##n^{th}## positive zero of ##J_m##, ##R_{static}## is the radius where the stream field goes to zero (approximately) and ##\rho_0## is the static constant mass density.

I have some questions about the physical interpretation of the PDE. I rewrite your PDE as $$\frac {1} {\rho} \nabla^2 \psi =2\Omega - \rho c_0 + \vec \nabla \frac {1} {\rho} \cdot \vec \nabla \psi - c_1 \psi =? \omega$$ where ##\omega## is (possibly?) the vorticity pointing along the z axis. Is ##\Omega## an externally applied angular velocity (like from a centrifuge motor or a constant magnetic field penetrating a type II superconductor)? Is this a vorticity wave equation for density coupled into the stream field? What functional form of ##\rho(r,\theta)## are you using? How was this equation derived? I didn't take fluid dynamics in college and I am confused by the physics behind the PDE:-)
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:

$$ξ_z = ρf(ψ) - 2Ω$$

where ##ξ_z## is the system vorticity. We don't have a function for ρ (except that in this case I am calling it constant in space) but in general will be found numerically from a different part of the algorithm.

So... you're saying there are 2 indices to take into account representing double sums over ##m,n## except that we set n = 0 to make it axisymmetric?

Can you explain more about where my boundary conditions come into play? For instance what if I want to apply ψ = 0 at the inner boundary of the annulus and ψ = 1 at the outer boundary (##R_{static}##?)? Is this possible or does the solution you've suggested only work with certain boundary conditions? If that's the case, it may still be useful as a test.

I will study the link you posted and see if I can answer this myself as well

Thanks for all your help so far!
 
  • #8
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 function zeroes have to be determined numerically?

Am I misunderstanding this? Is there a method for applying arbitrary Dirichlet BCs?
 
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  • #9
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 cosine functions in a Fourier series, and apply fouriers trick (or something analogous for bessel functions, since they have the same orthogonality as trig functions) at the boundaries.

Still not sure specifically how to do what I just said, but does this seems like I'm on the right track?
 
  • #10
Daniel Sellers said:
My domain does not include the origin. Should I discard the YkY_k anyway because the function needs to remain physical anyway?
Yes, I think you are on the right track. I had a thought about modeling ##\rho(r',\theta)##, ## r'=\frac {r}{R_{static}}##. The centrifugal potential is$$\phi_{cent}=-\frac {1} {2} \Omega^2 r'^2 sin(\theta)^2$$ and the potential for a "rigid" 2d rotating fluid is$$\phi_{rf}=\frac {1} {2} \rho \Omega^2 r'^2$$I propose (naively) an effective potential,$$\phi_{eff}= -\frac {1} {2}\Omega^2 r'^2 sin(\theta)^2 + \Omega^3 r'^2 r'^{\Omega}$$This potential would tend to increase the density towards ##R_{static}## as ##\Omega## increases, and the density function becomes $$\rho(r,\theta)= \rho_0 -sin(\theta)^2 + \Omega r'^{\Omega} $$Just a thought and probably wrong but I don't know how you can model ##\rho(r,\theta)## without some physics to go by.
 
  • #11
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 of the zeros of each bessel function Jk...
This approach doesn't take the inner boundary condition into account, and I'm not sure I can carry it out properly, but that's a start!
This is a lot of work just to check one very specific case of this equation!
 

Related to Finish and plot this separation of variables problem

1. What is a separation of variables problem?

A separation of variables problem is a mathematical technique used to solve partial differential equations by breaking them down into simpler equations with only one independent variable. This allows for easier integration and solution of the original equation.

2. How do you solve a separation of variables problem?

To solve a separation of variables problem, you must first identify the independent variables in the partial differential equation. Then, you can separate the variables and solve for each one individually. Once you have the solutions for each variable, you can combine them to get the final solution to the original equation.

3. What are the benefits of using separation of variables to solve equations?

Separation of variables is a powerful technique for solving partial differential equations because it allows for the solution to be broken down into simpler equations. This makes the solution process more manageable and can often lead to more elegant and efficient solutions.

4. Are there any limitations to using separation of variables to solve equations?

While separation of variables is a useful technique, it is not applicable to all partial differential equations. It is most effective for linear equations with well-defined boundary conditions. Additionally, it may not work for equations with highly complex or non-linear terms.

5. How can I check my solution to a separation of variables problem?

To check your solution to a separation of variables problem, you can plug it back into the original equation and see if it satisfies all of the conditions. Additionally, you can compare your solution to known solutions or use numerical methods to verify the accuracy of your solution.

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