Heat flux on a cylinder with two insulators

[chrishans]

I'm used to problems which ask me to find the heat flux for when, for example I have a very long cylinder covered with an insulator, each with their respective conductivity coefficient. I'd use the formula $\frac{\partial Q} {\partial t} =\int -k\vec{\nabla} T \vec {ds}$. But now I have a situation where the cylinder is covered with two insulators, one on the left half of it, and the other one on the right. So I don't know how to use the previous formula here, as k doesn't vary with ρ only, but also with φ. I found this very same problem on a web but, it didn't use that formula. Instead, an electric-like circuit was built, and so on (I'm NOT supposed to solve it this way) Any advice?

Thanks

 

[olivermsun]

I suppose you need to know what are the boundary conditions to know how to solve this. Are the cylinder and the "outside" specified to be held at fixed temperatures or is it some other situation?