Cole–Hopf linearization of the thermocapillary Marangoni dynamics of a two-dimensional bubble with insoluble surfactant

Abstract 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 Péclet 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 Péclet number $$Pe_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>P</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math> 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 $$Pe_s \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>P</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> and $$Pe_s \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>P</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> and a perturbative formula for the bubble migration velocity at large but finite $$Pe_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>P</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math> .