Newton's cooling BC

Mohsen

Hi, I am trying to solve a 2-d steady state heat transfer problem with finite difference (successive under-relaxation). I am trying to apply a convective boundary condition on a horizontal surface with a variable convection heat transfer coeffient. i.e. Heat is to convect outside from a horizontal surface with known but variable (in x, of course) convection heat transfer coefficient, h. I expect the temperature on the surface be variable. But it is not. I use this BC:

|K. grad(T)|=h(T-T_infinity)

where || means "magnitude," and K is the conductivity. Then I discretize it: backward in x and forward in y (since the convective surface is a bottom surface) and find T and use it as BC in my equations. It really seemed like an easy problem at first. When I use an scheme O(dx^2, dy^2), backward for x and forward in y, it never converges. But O(dx, dy^2) converges however the temperature is constant on the surface which it should not be.

I really appreciate any help/idea. Thanks, Mohsen