D.E. Pelinovsky, A.R. Giniyatullin, and Y.A. Panfilova

On solutions of a reduced model for the dynamical evolution of contact lines

Transactions of Nizhni Novgorod State Technical University n.a. Alexeev N.4 (94), 45-60 (2012)

Purpose: The goal of this study is to solve the linear advection-diffusion equation with a variable speed on a semiinfinite line. The variable speed is determined by an additional condition at the boundary, which models the dynamics of a contact line of a hydrodynamic flow at 180°contact angle.

Approach: The investigation is carried out by an application of Laplace transform in spatial coordinate. Properties of Green's function for the fourth-order diffusion equation are used in analysis of implicit solutions of the linear advection-diffusion equation.

Findings: We prove local existence of solutions of the initial-value problem associated with the set of overdetermining boundary conditions in the form of the fractional power series in time variable. We also analyze the explicit solutions in the case of a constant speed to show that the inhomogeneous boundary condition induces change of convexity of the flow at the contact line in a finite time.

linear advection-diffusion equation, variable speed, contact line, Laplace transform, Green’s function.