Unsteady bubble migration by nonlinear surfactant spreading

An analytical study of the unsteady nonlinear motion of an inviscid bubble due to surfactant spreading on its surface is presented. Similar modes of migration due to Marangoni effects are common in biological settings where insects such as microvelia strategically release surfactant at an interface to facilitate propulsion along it. This study adapts this phenomenology to study the following initial value problem: assuming a zero Reynolds and capillary number setting, what is the unsteady speed, and ultimate displacement, of a two-dimensional bubble on which an initial distribution of insoluble surfactant is set up that subsequently spreads around its surface causing it to migrate? Despite its multiphysics nature, it is shown that this nonlinear initial value problem can be solved in analytical form by leveraging new insights from a mathematical connection to a complex equation of Burgers type. Indeed it is shown to be linearizable at any finite surface Peclet number by a variant of the classical Cole-Hopf transformation.