Consider the solution to the steady state concentration profile ∇2c = 0 using centered finite differences. The domain is discretized into nine nodes with a spacing x and y. The concentration at the top and right sides of the slab is c = 2, and the concentration on the bottom and left sides of the slab is c = 1. Answer the following questions:
(a) If x = y, what is the temperature at the center node?
(b) If x y, how does the temperature at the center node depend on the spacing?
This is equivalent to discretizing a rectangular domain with an equal number of nodes on each side.
(c) If x y and the top side boundary condition is changed to c = 3, how does the temperature at the center node depend on the spacing?
(d) If x = y and the top side boundary condition is replaced with a no-flux condition ∂c/∂y = 0, what is the temperature at each node in the domain?
