  On the long nuclear time scale of stellar main-sequence evolution, even weak
mixing processes can become relevant for redistributing chemical species in a
star. We investigate a process of "differential heating," which occurs when a
temperature fluctuation propagates by radiative diffusion from the boundary of
a convection zone into the adjacent radiative zone. The resulting perturbation
of the hydrostatic equilibrium causes a flow that extends some distance from
the convection zone. We study a simplified differential-heating problem with a
static temperature fluctuation imposed on a solid boundary. The astrophysically
relevant limit of a high Reynolds number and a low P\'eclet number (high
thermal diffusivity) turns out to be interestingly non-intuitive. We derive a
set of scaling relations for the stationary differential heating flow. A
numerical method adapted to a high dynamic range in flow amplitude needed to
detect weak flows is presented. Our two-dimensional simulations show that the
flow reaches a stationary state and confirm the analytic scaling relations.
These imply that the flow speed drops abruptly to a negligible value at a
finite height above the source of heating. We approximate the mixing rate due
to the differential heating flow in a star by a height-dependent diffusion
coefficient and show that this mixing extends about $4\%$ of the pressure scale
height above the convective core of a $10\,M_\odot$ zero-age main sequence
star.
