International Journal of Differential Equations

Volume 2016, Article ID 4063740, 11 pages

http://dx.doi.org/10.1155/2016/4063740

## Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films

^{1}Institute of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USA^{2}Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, USA

Received 7 March 2016; Accepted 5 April 2016

Academic Editor: Jingxue Yin

Copyright © 2016 Matthew Michal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Fluid flow where advective inertial forces are small compared with viscous forces can be described as a Stokes flow. A Stokes flow model is given by where is the pressure, is the velocity of the fluid, is the dynamical viscosity, and is an applied force.

In the special case when viscous fluid is moving slowly through a porous medium, one can simplify the Stokes model by introducing a Hele-Shaw regime. Consider a Hele-Shaw flow, where and directions are parallel to flat plates, and the direction is perpendicular to the plates. Also, there is only a small gap of between the plates that are at . Under these assumptions, the Hele-Show flow is guided by the system of equations: where the pressure , the velocity of the fluid , the dynamical viscosity has a constant value, and is the normal vector to the fluid surface. The Hele-Shaw model describes viscous flow dynamics when a drop of liquid is pressed between two plates. This model is useful both for oil companies to better understand the process of secondary recovery of oil fields [1] and for environmental agencies attempting to analyze the flow of water moving through the soil [2].

It is well known [3] that solutions for the Hele-Shaw model, for some class of initial data, can develop singularities in finite time known as “fingering phenomenon.” An interesting result was obtained by Constantin et al. [4]. It was shown that a thin neck of fluid between two larger droplets will not break in a finite time. Using the lubrication approximation approach, the authors proved that the minimum width of this neck decreases like for large enough time . Likewise, Almgren [5] showed that singularities can develop in flows driven only by surface tension in Hele-Shaw models.

In 2001, Hernández-Machado et al. developed a theory to predict the forced evolution of a liquid-air interface in a Hele-Shall cell [6]. Ruyer-Quil developed a Hele-Shaw model that corrects Darcy’s law for inertial forces [7]. Later in 2002, Lee et al. described the pinch-off in mixtures of fluids and the linear stability of steady states [8]. He also studied the interface dynamics for certain types of fluid mixtures. At about the same time, Crowdy published a method for determining exact solutions in a rotating Hele-Shaw cell when the initial interface is a fluid annulus [9]. Also, at that time, two groups of mathematicians presented two different points of view on applicability of Hele-Shaw models to study instabilities in viscous flow dynamics. Böckmann and Müller showed the example where the slow flow could evolve properly in a two-dimensional Navier-Stokes model exhibiting previously experimentally observed fingering phenomenon but at the same time failed to obey Hele-Shaw model where no finite or infinite time singularity was formed [10]. Martin et al. found the solution for this disagreement in 2002 and showed that the instability actually can be described by a Hele-Shaw model if the equations include a chemical reaction term [11].

In lubrication approximation regime, namely, when the film thickness is much less than the length of the flow (i.e., the small parameter is introduced by ), a coating flow can be modeled by a thin-film equation which, in the general-slip form, is given bywhere is the time-dependent thickness of the thin fluid film and the value of the order of the nonlinearity depends on the slip conditions between the viscous liquid and the solid interface. With the order of the nonlinearity (which corresponds to the no-slip regime), this equation was derived for the lubrication limit of Hele-Shaw flow in [12]. The existence of nonnegative weak solutions, their qualitative behaviour, and regularity for (3) were rigorously studied in [13–15]. The asymptotic analysis results for classical and weak solutions of the thin-film equation were obtained in [16, 17]. One of the most well known and still open questions is the uniqueness of strong nonnegative solutions for (3). Some results in this direction, for particular classes of initial data, can be found in [18, 19].

In this paper, firstly, we will analyze a general-slip model (3) with nonlinearity . Secondly, instead of the linearized curvature term (which was used to obtain (3), the linearized curvature model), we will introduce the full nonlinear curvature expression (*nonlinear curvature model*). This intermediate asymptotic lubrication model allows us to use initial data with large values of the gradient. Recall that the classical thin-film model with linearized curvature term restricts initial data to small gradient values; that is, .

Our paper has the following structure. In Section 2, using a priori energy estimates, we will prove existence of nonnegative weak solutions for the nonlinear curvature model. In Section 3, we will obtain rate of energy dissipation along weak solutions constructed in Section 2. In Section 4, we will prove arc length dissipation property for some class of positive solutions for the linearized curvature model and, finally, in Section 5, we will compare numerical solutions for two models for the special choice of initial data and parameters which lead to formations of finite time singularities.

#### 2. Existence of Weak Solutions in Nonlinear Curvature Model

We consider the thin-film equation with full nonlinear expression for the curvature term; namely,with initial datacoupled with the boundary conditionswhere and , , . Obviously, we have the mass conservation Formally, multiplying (4) by , after integrating over , we obtainwhence we find the arc length dissipation immediately

For numerical simulations of this arc length dissipation property (see Figure 1), obtained above, we use as initial data and apply stable semi-implicit in time pseudospectral method developed in [20].