Diffusion: Concentration profile for point sink in infinite plane

[anvoice]

Hello,

I have a certain diffusion problem I am trying to solve. Admittedly, I'm further behind on my math than I'd like, and have trouble setting it up properly, and no luck finding an exact analogue in the literature.

I would like to solve for the time-dependent concentration profile given a point sink in an infinite plane. The diffusion problem has the general form:

dC(x,y,z,t)/dt = ∇(D*∇C(x,y,z,t))

where C is the concentration, t is the time, D is the diffusion coefficient, and ∇ is the partial differential operator. Assuming constant D, and that the problem is two-dimensional as well as radially symmetric, we can simplify the equation to:

dC(r,t)/dt = D*∇²C(r,t)

where r is the radius. At this point I'm stuck. I have seen several solutions in the literature to similar problems involving the error function, and that seems to be the right path to take. However those were typically given for diffusion through an infinite rod, etc., so it seems like the solution here should not be the same. At least it seems logical that the concentration profile will look like the right half of a sigmoidal at any given point in time. I would appreciate any help or helpful resources on this.

 

[Chestermiller]

Hello,

I have a certain diffusion problem I am trying to solve. Admittedly, I'm further behind on my math than I'd like, and have trouble setting it up properly, and no luck finding an exact analogue in the literature.

I would like to solve for the time-dependent concentration profile given a point sink in an infinite plane. The diffusion problem has the general form:

dC(x,y,z,t)/dt = ∇(D*∇C(x,y,z,t))

where C is the concentration, t is the time, D is the diffusion coefficient, and ∇ is the partial differential operator. Assuming constant D, and that the problem is two-dimensional as well as radially symmetric, we can simplify the equation to:

dC(r,t)/dt = D*∇²C(r,t)

where r is the radius. At this point I'm stuck. I have seen several solutions in the literature to similar problems involving the error function, and that seems to be the right path to take. However those were typically given for diffusion through an infinite rod, etc., so it seems like the solution here should not be the same. At least it seems logical that the concentration profile will look like the right half of a sigmoidal at any given point in time. I would appreciate any help or helpful resources on this.

Hi Anvoice. Welcome to Physics Forums!

Check out Conduction of Heat in Solids by Carslaw and Jaeger. It should be in there. Of course, transient diffusion is mathematically analogous to transient heat conduction, and D takes the place of the thermal diffusivity.

Chet

 

[anvoice]

Hi Anvoice. Welcome to Physics Forums!

Check out Conduction of Heat in Solids by Carslaw and Jaeger. It should be in there. Of course, transient diffusion is mathematically analogous to transient heat conduction, and D takes the place of the thermal diffusivity.

Chet

Thanks, good to be here. Also thank you for the good reference.

You're right, diffusion and heat transfer are analogous. I did look through that book, but the closest thing I could find there was the time-dependent equation for an instantaneous or continuous point source in 3d space. The continuous source is basically an integration of an instantaneous point source profile over time.

My understanding is that I need the equation of time-dependent concentration profile for the instantaneous point sink, which I do not know, but will probably be similar (constant - point source profile?), and which can then be integrated over time to get the general profile of the continuous sink, which should hopefully resemble the sigmoidal. The fact that I have a 2d geometry instead of 3d should also change the equation, as the flux only has 2 dimensions in which to travel now.

 

[Chestermiller]

Thanks, good to be here. Also thank you for the good reference.

You're right, diffusion and heat transfer are analogous. I did look through that book, but the closest thing I could find there was the time-dependent equation for an instantaneous or continuous point source in 3d space. The continuous source is basically an integration of an instantaneous point source profile over time.

My understanding is that I need the equation of time-dependent concentration profile for the instantaneous point sink, which I do not know, but will probably be similar (constant - point source profile?), and which can then be integrated over time to get the general profile of the continuous sink, which should hopefully resemble the sigmoidal. The fact that I have a 2d geometry instead of 3d should also change the equation, as the flux only has 2 dimensions in which to travel now.

When you say you have a flux in two dimensions, are we talking about spherical geometry or cylindrical geometry? In other words, is it truly a point sink, or is it a line sink?

Regarding point sources and point sinks, a point sink is simply minus a point source.

By the way, before you solve the transient problem, you should first solve the steady state problem, which is what the transient problem will approach at long times.

Chet

 

[anvoice]

When you say you have a flux in two dimensions, are we talking about spherical geometry or cylindrical geometry? In other words, is it truly a point sink, or is it a line sink?

Regarding point sources and point sinks, a point sink is simply minus a point source.

By the way, before you solve the transient problem, you should first solve the steady state problem, which is what the transient problem will approach at long times.

Chet

The problem applies to a thin liquid film, so technically it would be a line. Naturally, it's also not a point: what drains concentration is a growing crystal. However I am trying to first solve the simpler problem so as to get a better idea of how the system evolves over time. So the one I was interested in was indeed a 2d plane, and then a single point sink in there.

So you're saying I can use the negative of the time-dependent equation for the instantaneous point source that I found in Heat Conduction in Solids, once I adapted it to 2 dimensions, and then integrate that to get my continuous sink? I was just not certain that it's valid to take a point sink as a simple negation of a point source.

The steady-state problem seems both fairly straightforward and not very informative. If the plane has an initial non-infinite concentration, the steady-state solution is zero everywhere (because the sink drinks everything), and what I want to know about is the time-dependent concentration profile at the start. Otherwise, in order to keep the area flux constant at different distances from the sink, we need ∇C(r,t) proportional to 1/r, and C(r,t) therefore has a solution of the form C₁ + C₂ ln(r). This does not have an actual solution for a point sink, which makes sense to me since you would need infinite flux going through the point.

 

[Chestermiller]

The problem applies to a thin liquid film, so technically it would be a line. Naturally, it's also not a point: what drains concentration is a growing crystal. However I am trying to first solve the simpler problem so as to get a better idea of how the system evolves over time. So the one I was interested in was indeed a 2d plane, and then a single point sink in there.

So you're saying I can use the negative of the time-dependent equation for the instantaneous point source that I found in Heat Conduction in Solids, once I adapted it to 2 dimensions, and then integrate that to get my continuous sink? I was just not certain that it's valid to take a point sink as a simple negation of a point source.

The steady-state problem seems both fairly straightforward and not very informative. If the plane has an initial non-infinite concentration, the steady-state solution is zero everywhere (because the sink drinks everything), and what I want to know about is the time-dependent concentration profile at the start. Otherwise, in order to keep the area flux constant at different distances from the sink, we need ∇C(r,t) proportional to 1/r, and C(r,t) therefore has a solution of the form C₁ + C₂ ln(r). This does not have an actual solution for a point sink, which makes sense to me since you would need infinite flux going through the point.

So the solution you are really looking for is equivalent to a line sink in a 3D region. You don't need to get the instantaneous point source solution first, and then integrate. The solution you are looking for is directly available. It should be in Carslaw and Jaeger if you look carefully enough. When I worked in groundwater, there was a solution to the same diffusion equation (involving pressure in aquifers) that featured a well (line sink) that was pumping water out of a geological formation. The solution involved exponential integral functions. Look up the transient Theis solution. It is expressed in terms of the formation "transmissivity T" divided by the formation "storativity" S. The ratio T/S is called the hydraulic diffusivity, and replaces the diffusivity D in the diffusion equation. On the other hand, I think you would be better off finding the solution in Carslaw and Jaeger. I'm sure its in there.

Chet