Solution to diffusion equation in 1d spherical polar coordinates

[captainst1985]

Ok,

I have been given the steady state diffusion equation in 1d spherical polar coordinates as;

D.1/(r^2).'partial'd/dr(r^2.'partial'dc/dr)=0

I know that the solution comes in the form c(r) = A+B/r where A and B are some constants. I just don't know how to get from here to there. I have tried doing differentiation by parts on the equation then integrating the result, with no success. I can form a second order differential equation of the form;

r^2.'partial'd2c/dr2 +2r'partial'dc/dr = 0

but again don't know where to go from here. Any help greatly appreciated!

 

[tiny-tim]

D.1/(r^2).'partial'd/dr(r^2.'partial'dc/dr)=0

Hi captainst1985! Welcome to PF!

(btw, if you type alt-d, it prints ∂)

I don't understand what the D is at the beginning of the line.

If I ignore that, the equation says ∂/∂r(r^2.∂c/∂r) = 0;

so r^2.∂c/∂r must be independent of r (-B, say);

so ∂c/∂r = -B/r^2;

so c = A + B/r.

 

[JHZR2]

Hi,

Can you explain why one would say that:

so r^2.∂c/∂r must be independent of r (-B, say)?

Why would you equate r^2.∂c/∂r to be -B? What is the basis for this and/or the technique that does this called? I've gotten a little forgetful on some of these techniques...

The OP also had the multiplier on the front end that is effectively:

D/r^2

How does that come into play when solving?

Thanks!

 

[tiny-tim]

Welcome to PF!

Hi JHZR2! Welcome to PF!

Hi,

Can you explain why one would say that:

so r^2.∂c/∂r must be independent of r (-B, say)?

Because ∂/∂r of that is 0, ie (in words) the derivative of that with respect to r is 0, so changing r doesn't change it, ie it must be independent of r.

The OP also had the multiplier on the front end that is effectively:

D/r^2

How does that come into play when solving?

Because if 1/r² times something is 0, then the something must also be 0, so we can ignore the 1/r² !

(The D probably just indicates that it's the fourth exercise in the homework … A. B. C. D. …)

(btw, typing "alt-d" for "∂" only works on a Mac )

 

[paranormal]

I would first commend you for attempting to solve the diffusion equation in 1d spherical polar coordinates. This is a complex problem and requires a thorough understanding of differential equations and their solutions.

To solve this equation, we need to first understand the physical meaning behind it. The diffusion equation describes the flow of a substance (c) in space (r) and time (t). In this case, we are dealing with a steady state, meaning there is no change in time. This simplifies the equation to D.'partial'd2c/dr2 + 2/r.'partial'dc/dr = 0.

To find the solution, we can use a technique called separation of variables. This involves assuming that the solution can be written as a product of two functions, one that depends only on r and one that depends only on t. In this case, we can write c(r) = R(r)T(t).

Substituting this into the diffusion equation, we get D.'partial'd2(RT)/dr2 + 2/r.'partial'(RT)/dr = 0. We can rearrange this equation to get (1/R).'partial'd2R/dr2 + 2/r.'partial'R/dr = -(1/T).'partial'dT/dt.

The left side of the equation only depends on r, while the right side only depends on t. This means that both sides must be equal to a constant, which we will call -λ. This gives us two separate equations:

(1/R).'partial'd2R/dr2 + 2/r.'partial'R/dr = -λ

and

-(1/T).'partial'dT/dt = λ

The first equation is a second order differential equation in terms of R, which can be solved by standard techniques. The solution will be in the form R(r) = A + B/r.

The second equation is a first order differential equation in terms of T, which can be solved by separation of variables again. This will give us T(t) = Ce^-λt, where C is a constant.

Combining these two solutions, we get the final solution for c(r) = (A + B/r)e^-λt.

I hope this explanation helps you understand the process of solving the diffusion equation in 1d spherical polar coordinates. It is a challenging problem, but with patience and practice, you will be able to solve more complex