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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.

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

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

It is well known [

In 2001, Hernández-Machado et al. developed a theory to predict the forced evolution of a liquid-air interface in a Hele-Shall cell [

In lubrication approximation regime, namely, when the film thickness

In this paper, firstly, we will analyze a general-slip model (

Our paper has the following structure. In Section

We consider the thin-film equation with full nonlinear expression for the curvature term; namely,

For numerical simulations of this arc length dissipation property (see Figure

Arc length dissipation for a numerical solution when

Let

Assume that

Given

Let

From (

In particular, from (

Assume that

In fact, Theorem

Using Galerkin method we can observe numerically that the arc length has exponential decay rate.

The exponential dissipation rate is illustrated in Figure

Exponential decay of the arc length for the numerical solution with

We consider a unique positive classical solution

Now, we show that

Next, let

The time evolution behaviour of the arc length for a positive solution of the thin-film equation is bounded from above; namely,

Assume that

Existence of classical solutions which satisfy condition (

If

We rewrite this equality in a more convenient form:

Let us denote

We know from [

To illustrate analytic results of arc length dissipation (see Figures

Arc length dissipation for the numerical solution with

Arc length dissipation for the numerical solution with

Modeling time evolution for different initial data,

Comparison of solutions to (

It was also observed numerically that solutions of linearized and nonlinear curvature models approach zero (touchdown point) with a different speed; namely, the numerical solution for the linearized curvature model loses uniform positivity faster to compare to the numerical solution that corresponds to the nonlinear curvature model. To study the numerical time evolution of solutions near touchdown points, we implemented the numerical method suggested in [

This time, the comparison of solutions to the equations is shown at the time when dashed line solution of (

The authors declare that they have no competing interests.

This work was partially supported by a grant from the Simons Foundation (no. 275088 to Marina Chugunova).