I am having problem in defining boundary conditions for this scenario. Suppose we have a transmitter, which is a point object, transmitting molecules and receiver is an absorbing sphere which has a radius $r_r$ can be illustrated like:
by Fick's second law it is given that:
$$\frac{\partial p(r,t|r_0) }{\partial t}=D\nabla^2 p(r,t|r_0)$$
where $D$ is the diffusion coefficient and $p(r,t|r_0)$ is the concentration function; $r_0$ is the initial distance. We need to model a scenario in which a point source send molecules and they are received by a spherical receiver. The equivalent figure will be something like this:
Laplacian operator given by:
$$\Delta f = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2}.$$
It is written there that since $p(r,t|r_0)$ is spherically symmetric, i.e., depends solely on r so Laplacian will only be with respect to $r$.
To apply boundary condition, we assume coordinate centered at sphere's center and give the initial and boundary conditions as:
$p(r,t → 0|r0) =a δ(r − r_0),$ which says that when $t=0$ concentration will only be at $r_0$.
at $r \rightarrow \infty$ concentration is zero.
- $D\frac{\partial p(r,t|r_0)}{\partial r}= w p(r,t|r_0), r=r_r$ where $r_r$ is the radius of the sphere, $w $ is the reaction rate implying that when the molecules are at the boundary of sphere assuming reaction rate infinite the concentration will be zero.
where $r_0$ is the distance from transmitter to center of the sphere.Now I have to model the same problem with two spheres:
my problem is where I should place the center of the coordinate system so that the third boundary condition can be extended to the second sphere also, i.e., in second sphere i also have an absorbing boundary at which concentration is zero. The issue is if I place center of first sphere(receiver) as the center of coordinate system than how can I put a boundary condition on second sphere which does not have a particular fixed distance from the reference point in first scenario.