The Marangoni-stress-induced dynamics of a two-dimensional inviscid bubble loaded with insoluble surfactant moving in a linear temperature
gradient is studied in the limit of small Reynolds, capillary and thermal P´eclet
numbers. The bubble moves due to the combined effects of thermocapillary
Marangoni stresses and those induced by the advective-diffusive spreading of
an initial concentration of surfactant on the bubble boundary. It is shown that
this nonlinear multiphysics initial value problem is linearizable at any finite
non-zero value of a surface P´eclet number P es governing the size of surface
diffusion of the insoluble surfactant relative to its advective spreading. This is
done by showing that the dynamics can be encoded in the evolution of a function, analytic and single-valued outside the bubble, that satisfies a complex
partial differential equation of Burgers type. It is shown that this equation can
be linearized by a complex generalization of the classical Cole-Hopf transformation. A numerical method is formulated to solve this linear partial differential equation and determine the Marangoni dynamics of a bubble with some
initial surfactant concentration and illustrative calculations are carried out.
Results are shown to be consistent with exact equilibrium solutions available
from the formulation as P es → 0 and P es → ∞ and a perturbative formula
for the bubble migration velocity at large but finite P es.