Register to reply

Reaction diffusion problem concentric spheres

by scg08
Tags: reaction diffusion
Share this thread:
Apr30-09, 07:54 AM
P: 1

I am trying to (numerically) solve the following reaction-diffusion equation for the probability density of the a pair, [tex]\rho (\vec{r}_1,\vec{r}_2)[/tex]:

[tex]\dot{\rho} (\vec{r}_1,\vec{r}_2,t) = D_1 \nabla^2_1 \rho (\vec{r}_1,\vec{r}_2,t) + D_2 \nabla^2_2 \rho (\vec{r}_1,\vec{r}_2,t) - k \left( \left\| \vec{r}_1 - \vec{r}_2 \right\| \right) [/tex],

where the subscripts refer to the first and second particle, respectively. In 2D and polar coordinates, [tex]r_i[/tex] and [tex]\theta_i [/tex]:

[tex] \nabla^2_i = \frac{1}{r_i} \frac{\partial}{\partial r_i} r_i \frac{\partial}{\partial r_i} + \frac{1}{r_i^2} \frac{\partial}{\partial \theta_i} [/tex].

The domain is confined by two concentric spheres: [tex] 0 \leq \left\| \vec{r}_1 \right\| \leq R [/tex] and [tex] \left\| \vec{r}_2 \right\| \geq R [/tex]. The initial condition are spherically symmetric, i.e. only depends on the [tex]r_i[/tex]s. The reaction term is a function of the distance of the two particles, i.e. in 2D [tex] k( \left\| \vec{r}_1 - \vec{r}_2 \right\| ) = k( \sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos ( \theta_1-\theta_2)} ) [/tex]. I hoped to get rid of at least 1 coordinate by a variable transformation and separation of variables. However, so far I just could not come up with a separable problem. Do I really have to retain all 4 variables? Any suggestions of how to reduce this problem to something manageable are highly welcome. Eventually I will be interested in 3D and 4D as well.

Thank you,
Phys.Org News Partner Science news on
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief
May14-09, 10:24 AM
P: 333
Just throwing idea.

If the forces between particles 1 and 2 are conservative, try working in the centre of mass frame. In mechanics we use this frame to solve the central force motion and scattering problem.

Register to reply

Related Discussions
Potential between 2 concentric spheres Advanced Physics Homework 4
Gauss's Law/Energy Problem with Concentric Spheres Introductory Physics Homework 8
Capacitance of a system of 3 concentric spheres Introductory Physics Homework 0
Electric Potentials: concentric spheres Introductory Physics Homework 3
Two concentric spheres Introductory Physics Homework 8