  The overall {\it steady}-state energy balance with two phases in a flow
domain requires that the change in energy of the domain is equal to the
difference between the total energy entering the domain and that leaving the
domain. From the condition, the integral thermal flux across the surface is
studied for a {\it steady} thermocapillary drop migration in a flow field with
uniform temperature gradient at small and large Marangoni (Reynolds) numbers.
The drop is assumed to have only a slight axisymmetric deformation from a
sphere. It is identified that a conservative/nonconservative integral thermal
flux across the surface in the {\it steady} thermocapillary drop migration at
small/large Marangoni (Reynolds) numbers. The conservative flux confirms the
assumption of {\it quasi-steady} state in the thermocapillary drop migration at
small Marangoni (Reynolds) numbers. The nonconservative flux may well result
from the invalid assumption of {\it quasi-steady} state, which indicates that
the thermocapillary drop migration at large Marangoni (Reynolds) numbers cannot
reach {\it steady} state and is thus a {\it unsteady} process.
