  We study the heat flow from an open, bounded set $D$ in $\R^2$ with a
polygonal boundary $\partial D$. The initial condition is the indicator
function of $D$. A Dirichlet $0$ boundary condition has been imposed on some
but not all of the edges of $\partial D$. We calculate the heat content of $D$
in $\R^2$ at $t$ up to an exponentially small remainder as $t\downarrow 0$.
