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bhattach
08-31-2008, 02:56 PM
Hi,

I am trying to find an analytical expression for the concentration field phi(r,t) around a cylinder of radius R that is releasing particles into its quiescent surroundings at a given rate. So, the equations governing this problem (after assuming axisymmetry, and no variation along the length of the cylinder, i.e. phi=phi(r,t) ) are:

Evolution eqn: a*[1/r{d/dr(r d phi/d r)}]=d phi/dt
Boundary condition: a*d phi/ dr = J(t) at r=R ; phi(infinity,t)=0
Initial condition: phi(r,0)=0 for all R<r<infinity

where a is the diffusion coeff, d/dr is a partial derivative w.r.t r and d/dt is partial derivative w.r.t time.

A simple separation of variables approach (i.e. putting phi(r,t)=phi_r(r)*phi_r(t)) does not give me a solution that satisfies all the boundary conditions. I am thinking the solution has to be based on some transform method in space and a Green's function approach in time, but am not exactly sure how to go about it.

Any help will be appreciated !

Regards

Amitabh

bhattach
08-31-2008, 08:10 PM
OK, so I managed to solve the problem. Here's a brief outline of the solution anyways, for future reference:

1. write down phi as a summation of (zeroth order) bessel's functions:
phi(r,t)= sum_n A_n(t) J_0(b_n r)
where (b_n R) are the roots of d J_0/dr=0

2. note that \int_{r=R}^{infty} J_0(b_n r) J_0(b_m r) r dr = I_{m} \delta_{mn}

3. We need A_n(t) for solution

4. multiply LHS and RHS of evolution eqn by J_0(b_m r) r and integrate over R to infty

5. After some careful integration by parts, we get something like

R J_0(b_m r) J(t) - a*b_m^2 A_m(t) I_m = I_m d A_m/dt

From here on, it's an ODE, and you can easily proceed to obtain A_n(t) from J(t) [different from J_0(r)] and the initial conditions

Let me know if you are interested in the details of this solution.

Amitabh

bhattach
09-02-2008, 05:44 PM
OK, so I ran into a problem with the approach given above. Turns out that the norm of Bessel's function taken over R to infty is (wonders of wonders !) infinity. So my discrete transform method does not work.

Update: I realized that I can avoid this problem by defining the outer radius of my region as some R_1, where
R_1 is a root of dJ_0/dr. In that case, the transform is a type of discrete Bessel transform.

abcd679
12-05-2008, 09:06 AM
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