VISCOUS MARANGONI FLOW DRIVEN BY INSOLUBLE SURFACTANT AND THE COMPLEX BURGERS EQUATION

A new mathematical connection is established between a class of two-dimensional viscous Marangoni flows driven by insoluble surfactant and the complex Burgers equation. It is shown that the Marangoni-driven dynamics of a bath of viscous fluid at zero Reynolds and capillary number, and with a linear equation of state, are described by the evolution of a lower-analytic function with nonnegative imaginary part on the real line satisfying the complex Burgers equation. Surface diffusion of surfactant plays the role of viscosity in the more familiar real-valued Burgers equation arising in gas dynamics. Using this mathematical connection it is shown that, at arbitrary surface Peclet number, the Marangoni dynamics is linearizable, and integrable, via a transformation of Cole--Hopf type. A new class of time-evolving exact solutions is identified for the Marangoni-induced fluid motion at any finite surface Peclet number. These are shown to be given by a class of evolving N-pole solutions which differ from, and generalize, known pole dynamics solutions to the real Burgers equation. Analogous meromorphic solutions describing spatially singly periodic Marangoni flows are also reported. For infinite surface Peclet number it is shown how a generalized method of characteristics leads to an implicit form of the general solution. For a special choice of initial condition it is demonstrated that this implicit solution can be made explicit and, from it, the formation at finite time of an instantaneous weak singularity is observed. Together these new solutions afford a mathematical view of the effect of surface diffusion on Marangoni flows via the evolution of complex singularities in a nonphysical region of the complex plane. The observations open up valuable new mathematical connections between viscous Marangoni flows and the theory of caloric functions, Calogero--Moser systems, random matrices, and Dyson diffusion where the complex Burgers equation also plays a role.