Laplace equation in cylindrical coordinates

**alexrao** · Nov6-09, 01:59 PM

Can anyone help with the solution of the Laplace equation in cylindrical coordinates
$LaTeX Code: \\frac{\\partial^{2} p}{\\partial r^{2}}$

$LaTeX Code: \\frac{1}{r}$ $LaTeX Code: \\frac{\\partial p}{\\partial r}$

$LaTeX Code: \\frac{\\partial^{2} p}{\\partial z^{2}}$

with Neumann no-flux boundaries:
$LaTeX Code: \\frac{\\partial p}{\\partial r}$ $LaTeX Code: \\left(0,z\\right)$

$LaTeX Code: \\frac{\\partial p}{\\partial r}$ $LaTeX Code: \\left(Rmax,z\\right)$

$LaTeX Code: \\frac{\\partial p}{\\partial z}$ $LaTeX Code: \\left(r,Zmax\\right)$

and a Dirichlet upper boundary:

**arildno** · Nov6-09, 02:38 PM

With a trial solution using separation of variables, p(r,z)=P(r)*Q(z), we get:
$LaTeX Code: \\frac{Psingle-quotesingle-quote(r)+\\frac{1}{r}Psingle-quote(r)}{P(r)}=\\frac{1}{Q(z) }Qsingle-quotesingle-quote(z)$

Thus, we get two ordinary diff.eqs,
$LaTeX Code: Psingle-quotesingle-quote(r)+\\frac{1}{r}Psingle-quote(r)=CP(r) (*)$

LaTeX Code: Qsingle-quotesingle-quote(z)=CQ(z)(**)

You might try to work with these two, (*) clearly being the most problematic one.

**LCKurtz** · Nov6-09, 10:52 PM

Originally Posted by arildno

With a trial solution using separation of variables, p(r,z)=P(r)*Q(z), we get:
$LaTeX Code: \\frac{Psingle-quotesingle-quote(r)+\\frac{1}{r}Psingle-quote(r)}{P(r)}=\\frac{1}{Q(z) }Qsingle-quotesingle-quote(z)$

Thus, we get two ordinary diff.eqs,
$LaTeX Code: Psingle-quotesingle-quote(r)+\\frac{1}{r}Psingle-quote(r)=CP(r) (*)$

You might try to work with these two, (*) clearly being the most problematic one.

But maybe not too problematic. Multiply it through by r² and you have a form of Bessel's equation.

**alexrao** · Nov7-09, 05:33 AM

Thanks for getting me started, and pardon the ignorance of this geochemist who hasn't taken a pde class. I'm trying to follow you, and can't understand how you got that 2nd ode. Shouldn't (**) be

LaTeX Code: Qsingle-quotesingle-quote(z) = -C

**arildno** · Nov7-09, 06:49 AM

You are BOTH right. Mea culpa.

**alexrao** · Nov7-09, 12:44 PM

Moving along here...

So the solution to (**):

$LaTeX Code: Q(z) = Acosh(\\lambda z)+Bsinh(\\lambda z)$

As Arildno predicted, the solution to (*) is more problematic, since this is my first experience with Bessel functions. Here's where I am so far...

$LaTeX Code: P(r) = CJ_{0}(\\lambda r)+DY_{0}(\\lambda r)$

And here I set the second term must go to zero for the solution to be bounded.

So the final solution is of the form:
$LaTeX Code: p(r,z) = [Acosh(\\lambda z)+Bsinh(\\lambda z)]J_{0}(\\lambda r)$

Using the 2nd Neumann BC:
$LaTeX Code: \\frac{\\partial p}{\\partial r}(r=Rmax=9.5) = 0$
This means that
$LaTeX Code: Jsingle-quote_{0}(9.5\\lambda) = \\frac{\\partial J_{0}}{\\partial r}= 0$
But I'm not sure how this helps me right now, so I'll keep it in mind...

Using the 3rd Neumann BC:
$LaTeX Code: \\frac{\\partial p}{\\partial z}(z=Zmax=20) = 0$
This means that
$LaTeX Code: [Acosh(20\\lambda)+Bsinh(20\\lambda)]=0$
Ah, this is more helpful. I learned that A=-B, and I think at the bottom boundary, this whole term drops out.

Using the 1st Neumann BC: (I guess I could have done these in order...)
$LaTeX Code: \\frac{\\partial p}{\\partial r}(r=0) = 0$
This means that
$LaTeX Code: Jsingle-quote_{0}(0\\lambda) = \\frac{\\partial J_{0}}{\\partial r}= 0$
Again I'm not really sure how this helps me right now, so I'll keep it in mind...

So now we have:
$LaTeX Code: p(r,z)=A[cosh(\\lambda z)-sinh(\\lambda z)]J_{0}(\\lambda r)$

with
$LaTeX Code: Jsingle-quote_{0}(0\\lambda, r=0) = Jsingle-quote_{0}(9.5\\lambda, r=9.5) = 0$

Now here's where it gets tricky and I get stuck...

Using the fourth Dirichlet upper (z=0) boundary condition, I get:

$LaTeX Code: p(r,0)=f(r)=\\sum_{\\infty}^{n=1}A_{n}[cosh(0)-sinh(0)]J_{0}(\\lambda _{n}r)=\\sum_{\\infty}^{n=1}A_{n}J_{0}(\\lambda _{n}r)$

Unfortunately, f(r) is not a constant, but rather an ugly empirical function of r.

So I have written:

$LaTeX Code: f(r)=\\sum_{\\infty}^{n=1}A_{n}J_{0}(\\lambda _{n}r)$

And I have no idea where to go from here. I gather this will eventually end up in Matlab or R, but first I need a better understanding of the Bessel function J0 and A. I've read that the Bessel function of the first kind of order n (in my case n=0, right?) can be expressed as:

$LaTeX Code: J_{n}(x) = \\sum_{\\infty}^{k=1}\\frac {(-1)^{k}(x/2)^{n+2k}}{k!\\Gamma (n+k+1)}}$

Again, any help would be greatly appreciated.

**arildno** · Nov7-09, 02:27 PM

Now, assuming that the class of J_0's, with scaled arguments, represents a COMPLETE BASIS for functions on R (i.e, that any function is representable as a linear combination of these J_0's), then the A_n's are simply the required coefficients.

**LCKurtz** · Nov7-09, 03:10 PM

I haven't checked all your details, but your P equation and boundary conditions appear to be a Sturm-Liouville system which answers questions about orthogonality of the eigenfunctions, gives formulas for the eigenfunction expansion coefficients and settles convergence. For example, see:

http://www.efunda.com/math/ode/Sturm_liouville.cfm

**alexrao** · Nov7-09, 05:33 PM

Thanks again. Arildno, you mentioned that the $LaTeX Code: A_{n}$ 's are the required coefficients. But if I want to define a pressure distribution in this cylinder, don't I also need to figure out $LaTeX Code: \\lambda$ ? And is it true that $LaTeX Code: \\lambda$ in P(r) and $LaTeX Code: \\lambda$ in Q(z) are not the same?

**arildno** · Nov7-09, 05:48 PM

No, the $LaTeX Code: \\lambda_{n}$ MUST be the same numbers; otherwise, your diff.eq won't be satisfied. (Remember, it is directly related to your C's!!)