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
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.