Roman M. Taranets

Nonnegative Solutions for a Long-Wave Unstable Thin Film Equation with Convection

We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blow-up. Finally, we present numerical simulations of solutions.

The Spatiotemporal Oscillations of Order Parameter for Isothermal Model of the Surface-Directed Spinodal Decomposition in Bounded Binary Mixtures

The asymptotical behavior of order parameter in confined binary mixture is considered in one-dimensional geometry. The interaction between bulk and surface forces in the mixture is investigated. Its established conditions are when the bulk spinodal decomposition may be ignored and when the main role in the process of formation of the oscillating asymptotic periodic spatiotemporal structures plays the surface-directed spinodal decomposition which is modelled by nonlinear dynamical boundary conditions.

Qualitative Analysis of Coating Flows on a Rotating Horizontal Cylinder

We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. We obtain sufficient conditions for finite speed of support propagation and for waiting time phenomena by application of a new extension of Stampacchias lemma for a system of functional equations.

Theoretical aspects of a binary mixture flow

Abstract We study existence and long-time asymptotic behaviour of non-negative weak solutions for the coupled system of nonlinear partial differential equations. The system models dynamics of a binary mixture flow in the lubrication approximation regime. The applications of results include the process of drying of multi-component paint and distribution of swarming bacteria population. We also present analytical estimations of the dry out time.

Blow-up with mass concentration for the long-wave unstable thin-film equation

For a family of long-wave unstable thin-film equations, we prove existence of non-negative weak solutions blowing-up in a finite time. Specifically, building these solutions from initial data with negative energy, we show that their -norms go to infinity as . In addition, using the Bourgain’s type approach, we obtain qualitative information about the blow-up and prove mass concentration phenomenon.

Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films

For the thin-film model of a viscous flow which originates from lubrication approximation and has a full nonlinear curvature term, we prove existence of nonnegative weak solutions. Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipation of the solution arc length that is a well known property for a Hele-Shaw flow. For the classical thin-film model with linearized curvature term, under some restrictions on parameter and gradient values, we also prove analytically the arc length dissipation property for positive solutions. We compare the numerical solutions for both models, with nonlinear and with linearized curvature terms. In regimes when solutions develop finite time singularities, we explain the difference in qualitative behaviour of solutions.

Self-similar magnetic structures during the vortex-glass to vortex-liquid transition of type II superconductors

We examine the response to an external magnetic field by a multi-layer superconductor with an electrical resistance ρff(b)αbσ, where b is the dimensionless magnetic induction and σ is a parameter characterizing the ratio of the pinning activation energy to the energy of thermal fluctuations. When σ > 1 the sample is in the vortex glass phase, when 0 < σ < 1, it is in the vortex liquid phase, and a vortex glass to vortex liquid phase transition takes place at σ = 1. In the vortex glass phase, the magnetic field penetrates into the superconductor in the form of a self-similar wave. At all times it penetrates to a finite depth and its front moves at a finite velocity which depends on the parameters of the problem, such as the rate of pumping by the external magnetic field. In the vortex liquid phase the magnetic field penetrates to an infinite depth. Thus, the magnetic field penetrates to an infinite depth in the superconductor during a transition from the vortex glass phase into the vortex liquid phase.