When two obects $A$ and $B$ exchange heat irreversibly ($T_A\neq T_B$), the entropy of $A+B$ increases. My question is where: in $A$, in $B$, on the boundary? Is there some entropy exchange between $A$ and $B$? For the moment, my question is not really clear. I'm going to try to give a better example.
Consider a one-dimensional metal rod with temperature $T(x)$ along the rod. The total entropy is:
$$S = \int s(x)dx$$
with $s(x) = \rho C\log(T(x))$ and $\rho$ (mass per unit of length) and $C$ (specifc heat capacity) are constants.
The heat diffuses according to the heat equation. The entropy $s(x)$ will change at each point. Is it possible to separate, at each point $x$, the entropy being created there from the entropy being "exchanged" with the neighborhood? Can you write something like:
$$ds(x)=\delta s_e(x) + \delta s_c(x)$$
where $\delta s_e(x)$ would be the entropy being "exchanged with the neighbourhood" and $\delta s_c(x)$ would be the entropy being "created" at $x$? If so, what would they be?