Suppose I have a 2D channel with bottom boundary $y(x)=0$ and top boundary $y(x)=1+a\sin(x)$ with an initial dye distribution $D(x,y,0)=D(x)$. I have no flux boundary conditions along the walls: $ \frac{\partial D}{\partial n} = 0 $. The dye evolution is described by: $$\frac{\partial D}{\partial t}=k\nabla^2 D$$
I am interested in the longtime form of this solution, in particular I want to know how the effective diffusivity varies from the $a=0$ case. What methods could I use to solve this analytically? As a follow-up, is there any literature that you can refer me to that discusses this (does not need to be analytic)?