Having trouble writing a coupled differential equation in OpenFOAM
 

Is someone willing to help me write this equation in OpenFOAM?

I want to write this coupled differential equation.
// x = x coordinate.
// Ax = partial derivative of A with ONLY x

(1/x)*Laplacian A - (1/(x*x))*Ax = (B)/x
(1/x)*Laplacian B - (1/(x*x))*Bx = -(A)/x

I have two problems here.
1. How do I write the derivative of A and B only with x and not write it as div(B)?
2. How can I multiply 1/x or 1/x*x to the matrix which is the derivative of A and B only with respect to x?

I have to solve this using a steadystate solver without any time derivative like simpleFoam.

Please Help me.:(:(
Thank you.

I haven't worked in OpenFOAM for a while, but these are my first shot at them. My apologies if my OpenFOAM syntax is a little rusty.

(1) You would have to take the gradient of A and B and then dot with the x-direction unit vector. Something like: fvc::grad(A) & vector(1,0,0)

(2) fvMesh has a volume vector field C that contains the cell centroids. Again, dot with the x-direction unit vector to get the x component.

I would multiply through by 1/x^2 to avoid division by zero. I do this reflexively.

There is a coupled solver in the openfoam water supply. You could investigate how to make that work if performance is absolutely critical, but honestly, you are better off by building a segregated iterative solver. Start by fixing B, and solving the first equation for A. Then fixing that result for A and solve the second equation for B. And then iterate until convergence.

Thank you so much Mr. Prinkey for your reply.
Taking the dot product after computing the gradient is a very good idea.
I will try that. And thanks for letting me know the tutorial case.
I will take a deeper look into it.