Consider the Robin boundary condition for the diffusion/heat equation $\mathrm{u_t=a(t)u_{xx}+f(x,t)}$:
$$\mathrm{-k(t)u_x(0,t)=h(t)u(0,t)}$$
or
$$\mathrm{u_x(0,t)+\frac{h(t)}{k(t)}u(0,t)=0}$$
where $\mathrm{k(t)}$ thermal conductivity and $\mathrm{h(t)}$ heat tranfer coefficient.
My Question: Is it possible that the ratio $\mathrm{h(t)/k(t)}$ to be constant? Could anyone please help me? I have really no idea.
It certainly is possible for the ratio to be constant or very approximately constant. Indeed, it's very much possible for $k$ and $h$ to be constant.
As written, they're only specifically functions of time and could take on just about any form you choose (including a constant value).
One would not commonly encounter $k(t)$, where conductivity is some prescribed function of time (can't think of a way to physically do it off the top of my head). It's more likely to find $k(u(t))$, which is almost always true, but we neglect that if the change is small enough, as the problem becomes non-linear.
Constant $h$ is pretty easy to find, as long as the driving force behind the external flow is constant and any buoyant effects are negligible. Buoyancy will, again make the problem non-linear.
