Hele-Shaw Equations Research Articles

Derivation and analysis of a nonlocal Hele–Shaw–Cahn–Hilliard system for flow in thin heterogeneous layers

We derive, through the deterministic homogenization theory in thin domains, a new model consisting of Hele–Shaw equation with memory coupled with the convective Cahn–Hilliard equation. The obtained system, which models in particular tumor growth, is then analyzed and we prove its well-posedness in dimension 2. To achieve our goal, we develop and use the new concept of sigma-convergence in thin heterogeneous media, and we prove some regularity results for the upscaled model.

On the Doubly Non-local Hele-Shaw–Cahn–Hilliard System: Derivation and 2D Well-Posedness

Starting from a classic non-local (in space) Cahn–Hilliard–Stokes model for two-phase flow in a thin heterogeneous fluid domain, we rigorously derive by mathematical homogenization a new effective mixture model consisting of a coupling of a non-local (in time) Hele-Shaw equation with a non-local (in space) Cahn–Hilliard equation. We then analyse the resulting model and prove its well-posedness. A key to the analysis is the new concept of sigma-convergence in thin heterogeneous domains allowing to pass to the homogenization limit with respect to the heterogeneities and the domain thickness simultaneously.

A density-constrained model for chemotaxis

We consider a model of congestion dynamics with chemotaxis: the density of cells follows a chemical signal it generates, while subject to an incompressibility constraint. The incompressibility constraint results in the formation of patches, describing regions where the maximal density has been reached. The dynamics of these patches can be described by either Hele-Shaw or Richards equation type flow (depending on whether we consider the model with diffusion or the model with pure advection). Our focus in this paper is on the construction of weak solutions for this problem via a variational discrete time scheme of JKO type. We also establish the uniqueness of these solutions. In addition, we make more rigorous the connection between this incompressible chemotaxis model and the free boundary problems describing the motion of the patches in terms of the density and associated pressure variable. In particular, we obtain new results characterising the pressure variable as the solution of an obstacle problem and prove that in the pure advection case the dynamic preserves patches.

Navier–Stokes–Cahn–Hilliard system of equations

A growing interest in considering the “hybrid systems” of equations describing more complicated physical phenomena was observed throughout the last 10 years. We mean here, in particular, the so-called Navier–Stokes–Cahn–Hilliard equation, the Navier–Stokes–Poison equations, or the Cahn–Hilliard–Hele–Shaw equation. There are specific difficulties connected with considering such systems. Using the semigroup approach, we discuss here the existence-uniqueness of solutions to the Navier–Stokes–Cahn–Hilliard system, explaining, in particular, the limitation of maximal regularity of the local solutions imposed by the chosen boundary conditions.

Locally conservative discontinuous bubble scheme for Darcy flow and its application to Hele-Shaw equation based on structured grids

Locally conservative discontinuous bubble scheme for Darcy flow and its application to Hele-Shaw equation based on structured grids

H1-Solutions for the Hele-Shaw Equation

The Hele-Shaw equation arises in modeling the motion of viscous droplets spreading over a solid surface. It models also the diffusion of dopant in semiconductors. In this paper, we prove the existence of the solutions of the Cauchy problem associated with this equation.

Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation

This article is devoted to the study of the Hele-Shaw equation. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancellations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the equations in a very simple way. By combining these exact identities with convexity inequalities, we prove the existence of hidden Lyapunov functions of different natures. We also deduce from these identities and previous works on the water wave problem a simple proof of the well-posedness of the Cauchy problem. The analysis contains two side results of independent interest. Firstly, we give a principle to derive estimates for the modulus of continuity of a PDE under general assumptions on the flow. Secondly we prove a convexity inequality for the Dirichlet to Neumann operator.

Convexity and the Hele–Shaw Equation

Walter Craig’s seminal works on the water-wave problem established the importance of several exact identities: Zakharov’s hamiltonian formulation, shape derivative formula for the Dirichlet-to-Neumann operator, and normal forms transformations. In this paper, we introduce several identities for the Hele–Shaw equation which are inspired by his nonlinear approach. First, we study convex changes of unknowns and obtain a large class of strong Lyapunov functions; in addition to be non-increasing, these Lyapunov functions are convex functions of time. The analysis relies on a new, simple compact elliptic formulation of the Hele–Shaw equation, which is of independent interest. Then, we study the role of convexity to control the spatial derivatives of the solutions. We consider the evolution equation for the Rayleigh–Taylor coefficient a (this is a positive function proportional to the opposite of the normal derivative of the pressure at the free surface). Inspired by the study of entropies for elliptic or parabolic equations, we consider the special function $$\varphi (x)=x\log x$$ and find that $$\varphi (1/\sqrt{a})$$ is a sub-solution of a well-posed equation.

A remark on memory effects in constrained fluid systems

The goal of this note is to put into perspective the recent results obtained on memory effects in partially congested fluid systems of Euler or Navier-Stokes type with former studies on free boundary obstacle problems and Hele-Shaw equations. In particular, we relate the notion of adhesion potential initially introduced in the context of dense suspension flows with the one of Baiocchi variable used in the analysis of free boundary problems.

Droplet migration in a Hele–Shaw cell: Effect of the lubrication film on the droplet dynamics

Droplet migration in a Hele–Shaw cell is a fundamental multiphase flow problem which is crucial for many microfluidics applications. We focus on the regime at low capillary number and three-dimensional direct numerical simulations are performed to investigate the problem. In order to reduce the computational cost, an adaptive mesh is employed and high mesh resolution is only used near the interface. Parametric studies are performed on the droplet horizontal radius and the capillary number. For droplets with an horizontal radius larger than half the channel height, the droplet overfills the channel and exhibits a pancake shape. A lubrication film is formed between the droplet and the wall and particular attention is paid to the effect of the lubrication film on the droplet velocity. The computed velocity of the pancake droplet is shown to be lower than the average inflow velocity, which is in agreement with experimental measurements. The numerical results show that both the strong shear induced by the lubrication film and the three-dimensional flow structure contribute to the low mobility of the droplet. In this low-migration-velocity scenario, the interfacial flow in the droplet reference frame moves toward the rear on the top and reverses direction moving to the front from the two side edges. The velocity of the pancake droplet and the thickness of the lubrication film are observed to decrease with capillary number. The droplet velocity and its dependence on capillary number cannot be captured by the classic Hele–Shaw equations, since the depth-averaged approximation neglects the effect of the lubrication film.

Traveling wave solution of the Hele–Shaw model of tumor growth with nutrient

Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions. We study a simple free boundary model formed of a Hele–Shaw equation for the cell number density coupled to a diffusion equation for a nutrient. We can prove that a traveling wave solution exists with a healthy region separated from the progressing tumor by a sharp front (the free boundary) while the transition to the necrotic core is smoother. Remarkable is the pressure distribution which vanishes at the boundary of the proliferative rim with a vanishing derivative at the transition point to the necrotic core.

A volume-of-fluid formulation for the study of co-flowing fluids governed by the Hele-Shaw equations

We present a computational framework to address the flow of two immiscible viscous liquids which co-flow into a shallow rectangular container at one side, and flow out into a holding container at the opposite side. Assumptions based on the shallow depth of the domain are used to reduce the governing equations to one of Hele-Shaw type. The distinctive feature of the numerical method is the accurate modeling of the capillary effects. A continuum approach coupled with a volume-of-fluid formulation for computing the interface motion and for modeling the interfacial tension in Hele-Shaw flows is formulated and implemented. The interface is reconstructed with a height-function algorithm. The combination of these algorithms is a novel development for the investigation of Hele-Shaw flows. The order of accuracy and convergence properties of the method are discussed with benchmark simulations. A microfluidic flow of a ribbon of fluid which co-flows with a second liquid is simulated. We show that for small capillary numbers of O(0.01), there is an abrupt change in interface curvature and focusing occurs close to the exit.

Composite waves for a cell population system modelling tumor growth and invasion

The recent biomechanical theory of cancer growth considers solid tumors as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, the latter depending either on the local cell density (contact inhibition), on mechanical stress in the tumor, or both. For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling, we prove there are always traveling waves above a minimal speed and we analyse their shapes. They appear to be complex with composite shapes and discontinuities. Several small parameters allow for analytical solutions; in particular the incompressible cells limit is very singular and related to the Hele-Shaw equation. These singular traveling waves are recovered numerically.

ABLATIVE HELE–SHAW MODEL FOR ICF FLOWS MODELING AND NUMERICAL SIMULATION

We propose a model for the study of ablation fronts encountered in inertial confinement fusion flows. Following the ideas of physicists specially Sanz's ones, we try to justify the modeling of this problem by a Hele–Shaw equation posed in the complement of a bounded domain the boundary of which is evolving with time. This equation is coupled with an advection equation for the vorticity of the fluid. We propose a numerical strategy based on the fat boundary method to discretize the system and we present some numerical results which show that the numerical method is feasible to simulate ablative Raleigh–Taylor instabilities in the context of these plasma flows.

IMAGE ANALYSIS DEDICATED TO POLYMER INJECTION MOLDING

This work follows the general framework of polymer injection moulding simulation whose objectives are the mastering of the injection moulding process. The models of numerical simulation make it possible to predict the propagation of the molten polymer during the filling phase from the positioning of one point of injection or more. The objective of this paper is to propose a particular way to optimize the geometry of mold cavity in accordance with physical laws. A direct correlation is pointed out between geometric parameters issued from skeleton transformation and Hausdorff's distance and results provided by implementation of a classical model based on the Hele-Shaw equations which are currently used in the main computer codes of polymer injection.

Perturbation theorems for Hele-Shaw flows and their applications

In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, so called the Polubarinova-Galin equation. This theorem enables us to explore properties of solutions with initial functions close to but are not polynomial. Applications of this theorem are given in the suction or injection case. In the former case, we show that if the initial domain is close to a disk, most of fluid will be sucked before the strong solution blows up. In the later case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova-Galin equation is given.

Numerical Simulation for a Droplet Fission Process of Electrowetting on Dielectric Device

Electrowetting has been proposed as a technique for manipulating droplets surrounded by air or oil. In this paper, we discuss the modeling and simulation of the droplet fission process between two parallel plates inside an electrowetting on dielectric (EWOD) device. Since the gap between the plates is small, we use the two-phase Hele-Shaw flow as a model. An immersed boundary (IB) method is employed to solve the governing equations and a local level set method is used to track the drop interface. For comparison purposes, the same set of two-phase HeleShaw equations are also solved directly using the ghost-fluid (GF) method. Numerical results are consistent with experimental observations reported in the literature, provided that care is taken to treat the contact angle.

Experimental and numerical study of the validity of Hele–Shaw cell as analogue model for variable-density flow in homogeneous porous media

Several laboratory experiments were conducted to identify the validity domain under which a Hele–Shaw cell may serve as a suitable analogue for variable-density flow in homogeneous porous media. These experiments are concerned with the injection into a Hele–Shaw cell of a salt solute at different concentrations and flow rates. The experimental data analysis highlighted two types of mixing zone shape: with and without ‘fingers’. A semi-empirical criterion based on the ratio between gravitational and injected velocities was used to forecast the change from one shape to another. The experimental data were then analysed using numerical solutions of the classical Hele–Shaw equations by taking into account an anisotropic dispersion tensor whose components depend on fluid density gradients. The good agreement between experimental and numerical results clearly shows that the validity of the concentration-dependent dispersion tensor strongly depends on the local Péclet number variation. For Péclet numbers lower 50, the Hele–Shaw cell can be considered as an analogous model of a homogeneous and isotropic 2D porous medium. It can be successfully used to study, at the laboratory scale, the gravitational instability effects induced by flow and transport phenomena into a porous medium.

Regularity of one-phase Stefan problem near Lipschitz initial data

In this paper we show that if the Lipschitz constant of the initial free boundary is small, then for small positive time the solution is smooth and satisfies the Hele-Shaw equation in the classical sense. A key ingredient in the proof which is of independent interest is an estimate up to order of magnitude of the speed of the free boundary in terms of initial data.

Relevance of dynamic wetting in viscous fingering patterns

We demonstrate that wetting effects at moving contact lines have a strong impact in viscous fingering patterns. Experiments in a rotating Hele-Shaw (HS) cell, dry or prewetted, show consistent morphological differences. When the wetting fluid invades a dry region, contact angle dynamics yield a kinetic contribution to the interface pressure drop that scales with capillary number as Ca(2/3) but is significantly larger than the Park-Homsy kinetic correction. Numerical results are in very good agreement with experiments and show that standard HS equations work best for prewetted cells.