  Heating a solid sphere at the surface induces mechanical stresses inside the
sphere. If a finite amount of heat is supplied, the stresses gradually
disappear as temperature becomes homogeneous throughout the sphere. We show
that before this happens, there is a temporary lowering of pressure and density
in the interior of the sphere, inducing a transient lowering of the temperature
here. For ordinary solids this effect is small because c_p is almost equal to
c_V. For fluent liquids the effect is negligible because their dynamic shear
modulus vanishes. For a liquid at its glass transition, however, the effect is
generally considerably larger than in solids. This paper presents analytical
solutions of the relevant coupled thermoviscoelastic equations. In general,
there is a difference between the isobaric specific heat, c_p, measured at
constant isotropic pressure and the longitudinal specific heat, c_l, pertaining
to mechanical boundary conditions that confine the associated expansion to be
longitudinal. In the exact treatment of heat propagation the heat diffusion
constant contains c_l rather than c_p. We show that the key parameter
controlling the magnitude of the "cooling-by-heating" effect is the relative
difference between these two specific heats. For a typical glass-forming
liquid, when temperature at the surface is increased by 1 K, a lowering of the
temperature in the sphere center of order 5 mK is expected if the experiment is
performed at the glass transition. The cooling-by-heating effect is confirmed
by measurements on a 19 mm diameter glucose sphere at the glass transition.
