Free Boundary Model Research Articles

Capillary rise in partially saturated rigid porous media

We explore predictions of two models of one-dimensional capillary rise in rigid and partially saturated porous media. One is an existing one from the literature and the second is a free-boundary model based on Richards’ equation with two moving boundaries of the evolving partially saturated region. Both models involve the specification of saturation-dependent functions for local capillary pressure and permeability and connect to classical models for saturated porous media. Existing capillary-rise experiments show two notable regimes: (i) an early-time regime typically well-described by classical capillary-rise theory in a fully saturated porous media, and (ii) a long-time regime that has anomalous dynamics in which the capillary-rise height may scale with a non-classical power law in time or have more complicated dynamics. We demonstrate that the predictions of both models compare well with experimental capillary-rise data over early- and long-time regimes gathered from three independent studies in the literature. The model predictions also shed light on recent scaling laws that relate the capillary pressure and permeability of the partially saturated media to the capillary-rise height. We use these models to probe computationally observed permeability relationships to capillary-rise height. We demonstrate that a recently proposed permeability scaling for the anomalous capillary-rise regime is indeed realized and is particularly apparent in the lower portion of the partially saturated media. For our free-boundary model we also compute capillary pressure measures and show that these reveal the linear relation between the capillary pressure and capillary-rise height expected for a capillarity–gravity balance in the upper portion of the partially saturated porous media.

ASYMPTOTIC STABILITY FOR A FREE BOUNDARY MODEL OF AN ATHEROSCLEROTIC PLAQUE FORMATION IN THE PRESENCE OF REVERSE CHOLESTEROL TRANSPORT WITH HOLLING TYPE-III FUNCTIONAL RESPONSE

Atherosclerosis, as a chronic inflammatory disease, has been a threat to human health. How to diagnose and prevent this disease has long been the focus of medical and biomathematics study. In this paper, we investigate a free boundary model of an atherosclerotic plaque formation in the presence of reverse cholesterol transport, which includes low-density lipoprotein, high-density lipoprotein, endothelial stimulating cytokines, pro-inflammatory and anti-inflammatory macrophages, as well as foam cells. For this model, we use the Holling type-III response function instead of the usual Holling type-II to describe the change of the concentration of each substance. We first introduce the auxiliary function [Formula: see text] to prove the existence and uniqueness of small radially symmetric stationary plaque for appropriate [Formula: see text] and [Formula: see text] by using the contraction mapping principle and maximum principle, and then establish a condition to ensure their asymptotic stability behavior by expanding the corresponding steady-state solution.

Learning parameter values of a fractional model of cancer employing boundary densities of tumor cells

In this paper, a free boundary model of tumor growth with drug application including the heterogeneity or different types of tumor cells (caused by mutations and different values of drug and nutrient concentrations inside the tumor) is studied. Heterogeneity is included in the model by a variable . It is assumed that converting from mutation to mutation happens with probability . A Caputo time fractional‐order hyperbolic equation describes the evolution of tumor cells depending on . It also includes two Caputo time fractional‐order parabolic equations describing the diffusions of nutrients (e.g., oxygen and glucose) and drug concentrations. Instead of integer‐order time derivatives, the fractional ones are considered. In this study, it is aimed to employ the least squares curve fitting method to fit the order of fractional derivatives, coefficient, and rates of the model. For this purpose, using the mathematical model, we have considered the boundary densities of the tumor cells of different types and near‐boundary concentrations of drug and nutrient as the functions of the unknown orders, coefficients and rates (unknown variables). Due to the complexity of the problem, we have obtained the functions numerically. For this, using a change of variable, we have changed the free boundary problem to a problem with fixed domain. Thus, Riemann–Liouville fractional‐order integrals are added to the problem. In the spatial domain, the problem is discretized using the collocation method. In the temporal domain, the fractional derivatives and integrals (with order ) are approximated in mesh points (with step size ) applying a method with error . Then, the unknown variables are obtained by fitting the functions for unknown variables to the data. In order to obtain the variables, a quadratic objective function is considered. The discretized function is substituted into the objective function, then the objective function is minimized using the trust‐region reflective algorithm. Finally, some numerical examples are presented to verify the efficiency of the method. In the examples, we have added noises generated from Gaussian distributions to the data and the effects of noise on the fitted coefficients, and rates are illustrated using some figures and tables. The noisy data are also plotted to have a clear vision of the effects of noises on the data. It is shown that despite the noise, the prediction of the radius of the tumor is acceptable.

Bifurcation for a free boundary model of an atherosclerotic plaque formation in the presence of reverse cholesterol transport with Holling type III functional response

Bifurcation for a free boundary model of an atherosclerotic plaque formation in the presence of reverse cholesterol transport with Holling type III functional response

A Free Boundary Model for Mosquitoes with Conditional Dispersal in a Globally Unfavorable Environment Induced by Climate Warming

A Free Boundary Model for Mosquitoes with Conditional Dispersal in a Globally Unfavorable Environment Induced by Climate Warming

On the reaction–diffusion type modelling of the self-propelled object motion

In this study, we propose a mathematical model of self-propelled objects based on the Allen–Cahn type phase-field equation. We combine it with the equation for the concentration of surfactant used in previous studies to construct a model that can handle self-propelled object motion with shape change. A distinctive feature of our mathematical model is that it can represent both deformable self-propelled objects, such as droplets, and solid objects, such as camphor disks, by controlling a single parameter. Furthermore, we demonstrate that, by taking the singular limit, this phase-field based model can be reduced to a free boundary model, which is equivalent to the L^2-gradient flow model of self-propelled objects derived by the variational principle from the interfacial energy, which gives a physical interpretation to the phase-field model.

Effects of latent heat in additive manufacturing

We analyzed the effects of latent heat in conditions related to the additive manufacturing technology by solving the free boundary problem of a moving heat source. The most important finding of our analysis is a possibility of a genuine (not constitutional) supercooling at the fusion edge of the melt pool. It is measured by the structure forming number of the system, which changes significantly compared to the latent-heat-free system. Other findings of the present treatment are the increase of the longitudinal size and decrease of the maximum temperature of the melt pool due to the latent heat and the role of the surface heat losses in the process. There are several important physical properties, which were not accounted for by the free-boundary model developed in this treatment. They can be accounted for by a phase-field model, which we plan to develop in the future. The latter may also be useful for other technologies, like laser melting, electron (ion)-beam irradiation, and fusion welding.

Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models

<p style='text-indent:20px;'>This paper investigates the effect of environmental heterogeneity on species spreading via numerical simulation of suitable reaction-diffusion models with free boundaries. We focus on the changes of long-time dynamics (establishment or extinction) and spreading speeds of the species as the parameters describing the heterogeneity of the environment are varied. For the single species model in time-periodic environment and in space-periodic environment theoretically treated in [<xref ref-type="bibr" rid="b15">15</xref>,<xref ref-type="bibr" rid="b16">16</xref>], we obtain more detailed properties here. Among other results, our numerical simulation suggests that, in a time-periodic or space-periodic environment, moderate increase of the oscillation scale enhances the chances of establishment as well as the spreading speed of the species. We also numerically examine a related model with two competing species, which was treated in [<xref ref-type="bibr" rid="b34">34</xref>,<xref ref-type="bibr" rid="b28">28</xref>,<xref ref-type="bibr" rid="b24">24</xref>] recently and reduces to the single species free boundary model when one of the species is absent. Our numerical results, obtained by varying the parameters in the time-periodic and space-periodic terms of the model, suggest that heterogeneity of the environment enhances the invasion of the two species (as in the single species model), although there are subtle differences of the influences felt by the two. Some intriguing phenomena revealed in our simulations suggest that heterogeneity of the environment decreases the level of predictability of the competition outcome.</p>

Bifurcation analysis of a free boundary model of vascular tumor growth with a necrotic core and chemotaxis.

A considerable number of research works has been devoted to the study of tumor models. Several biophysical factors, such as cell proliferation, apoptosis, chemotaxis, angiogenesis and necrosis, have been discovered to have an impact on the complicated biological system of tumors. An indicator of the aggressiveness of tumor development is the instability of the shape of the tumor boundary. Complex patterns of tumor morphology have been explored in Lu et al. (J Comput Phys 459:111153, 2022). In this paper, we continue to carry out a bifurcation analysis on such a vascular tumor model with a controlled necrotic core and chemotaxis. This bifurcation analysis, to the parameter of cell proliferation, is built on the explicit formulas of radially symmetric steady-state solutions. By perturbing the tumor free boundary and establishing rigorous estimates of the free boundary system, we prove the existence of the bifurcation branches with Crandall-Rabinowitz theorem. The parameter of chemotaxis is found to influence the monotonicity of the bifurcation point as the mode l increases both theoretically and numerically.

A tumor growth model with autophagy: The reaction-(cross-)diffusion system and its free boundary limit

<p style='text-indent:20px;'>In this paper, we propose a tumor growth model to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we construct a reaction-(cross-)diffusion system describing the evolution of cell densities, where the drift is determined by the negative gradient of the joint pressure, and the reaction terms manifest the unique mechanism of autophagy. Next, in the incompressible limit, such a cell density model naturally connects to a free boundary system, describing the geometric motion of the tumor region. Analyzing the free boundary model in a special case, we show that the ratio of the two phases of cells exponentially converges to a "well-mixed" limit. Within this "well-mixed" limit, we obtain an analytical solution of the free boundary system which indicates the exponential growth of the tumor size in the presence of autophagy in contrast to the linear growth without it. Numerical simulations are also provided to illustrate the analytical properties and to explore more scenarios.</p>

Emergence of traveling waves and their stability in a free boundary model of cell motility

We consider a 2D free boundary model of cell motility, inspired by the 1D contraction-driven cell motility model due to P. Recho, T. Putelat, and L. Truskinovsky [Phys. Rev. Lett. 111 (2013), p. 108102]. The key ingredients of the model are the Darcy law for overdamped motion of the acto-myosin network, coupled with the advection-diffusion equation for myosin density. These equations are supplemented with the Young-Laplace equation for the pressure and no-flux condition for the myosin density on the boundary, while evolution of the boundary is subject to the acto-myosin flow at the edge. The focus of the work is on stability analysis of stationary solutions and translationally moving traveling wave solutions. We study stability of radially symmetric stationary solutions and show that at some critical radius a pitchfork bifurcation occurs, resulting in emergence of a family of traveling wave solutions. We perform linear stability analysis of these latter solutions with small velocities and reveal the type of bifurcation (sub- or supercritical). The main result of this work is an explicit asymptotic formula for the stability determining eigenvalue in the limit of small traveling wave velocities.

The impact factors of the risk index and diffusive dynamics of a SIS free boundary model.

To discuss the impact factors on the spread of infectious diseases, we study a free boundary problem describing a SIS (susceptible-infected-susceptible) model in a heterogeneous environment. Firstly, the existence and uniqueness of the global solution are given. Then the basic reproduction number related to time is defined, and a spreading-vanishing dichotomy of infectious diseases is obtained. The impacts of the diffusion rate of infected individuals, expanding capability, and the scope and scale of initial infection on the spreading and vanishing of infectious disease are analyzed. Numerical simulations are given to show that the large expanding capability is unfavorable to the prevention and control of the disease.

Bifurcation for a free-boundary problem modeling small plaques with reverse cholesterol transport

In this paper, we rigorously analyze a free boundary model of plaque growth under the influence of reverse cholesterol transport (RCT). We will analyze a model with a highly nonlinear and mutually coupled PDE system of plaque growth dynamics, including LDL and HDL cholesterol, M1 inflammatory and M2 anti-inflammatory macrophages, and foam cells. This free boundary problem is associated with two different types of surfaces: a vascular surface with the boundary set to r=1 and the interface Γ(t) between the plaque and blood flow. For this model, the explicit stationary solutions are nonexistent due to the highly nonlinear nature of the problem. We shall first prove the existence and uniqueness of the radially symmetric stationary solution. Then, we shall combine parameters into a single bifurcation parameter μ, and establish the existence of a sequence μn, such that for each μn(n≥2), the symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solutions for sufficiently small plaques.

Asymptotic stability of contraction-driven cell motion.

We study the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two- dimensional free-boundary model that generalizes a previous one-dimensional model [P. Recho, T. Putelat, and L. Truskinovsky, Phys. Rev. Lett. 111, 108102 (2013)10.1103/PhysRevLett.111.108102] by combining a Keller-Segel model, a Hele-Shaw boundary condition, and the Young-Laplace law with a regularizing term which precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We find a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is nonlinear asymptotic stability of traveling solutions that model observable steady cell motion. We derive an explicit asymptotic formula for the stability-determining eigenvalue via asymptotic expansions in small speed. This formula greatly simplifies computation of this eigenvalue and shows that stability is determined by the change in total myosin mass when stationary solutions bifurcate to traveling solutions. Our spectral analysis reveals the physical mechanisms of stability.

Derivation of the Interfacial Surface Tension between Superconducting and Normal States within Ginzburg-Landau Theory

An understanding of the front propagation in a dynamically unstable state requires a theory that treats the superconducting-normal (SN) interfaces as a free boundary problem. The relevance of an interfacial representation based on dynamics of the superconducting order parameter and vector potential. When magnetic flux diffuses into the normal phases the interfacial dynamics can be determined by the free boundary model. This model involves surface tension for the motion of the interface which has been calculated by the exact solution of TDGL equation [8].This analytical result of the surface tension agrees with that of Osborn and Dorsey [9]. The Chittagong Univ. J. Sci. 43(1): 31-40, 2021

Vanishing and spreading conditions for a free-boundary epidemic model with subclinical infections and vaccination.

This paper presents a free-boundary epidemic model with subclinical infections and vaccination.We prove the existence and uniqueness of solutions to the model.Moreover, sufficient conditions for the disease vanishing and spreading are given.The disease will vanish if the basic reproduction number $ R_0 < 1 $, that the corresponding ODE model defines without spatial expansion. However, the disease will spread to the whole area if $ R^F_0(t_0) > 1 $ for some $ t_0 > 0 $ when it is introduced spatial heterogeneity. $ R^F_0(0) < R_0 $ implies that the spillovers from hotspots to areas with no confirmed cases will reduce the outbreak threshold and increase the difficulty of prevention and control in the whole region. Under the condition $ R^F_0(0) < 1 < R_0 $, if the free boundary condition of infectives $ h(t) < \infty $, $ t \to \infty $, then the disease is vanishing, which indicates that $ R^F_0(0) < 1 $ can also control the disease if the scope of hotspots expansion is limited. Furthermore, the numerical simulations illustrate that the routine vaccination would decrease the basic reproduction number and then change the disease from spreading to vanishing.

A transient biological fouling model for constant flux microfiltration.

Microfiltration is a widely used engineering technology for fresh water production and water treatment. The major concern in many applications is the formation of a biological fouling layer leading to increased hydraulic resistance and flux decline during membrane operations. The growth of bacteria constituting such a biological layer implicates the formation of a multispecies biofilm and the consequent increase of operational costs for reactor management and cleaning procedures. To predict the biofouling evolution, a mono-dimensional continuous free boundary model describing biofilm dynamics and EPS production in different operational phases of microfiltration systems has been well studied. The biofouling growth is governed by a system of hyperbolic PDEs. Substrate dynamics are modeled through parabolic equations accounting for diffusive and advective fluxes generated during the filtration process. The free boundary evolution depends on both microbial growth and detachment processes. What is not addressed is the interplay between biofilm dynamics, filtration, and water recovery. In this study, filtration and biofilm growth modeling principles have been coupled for the definition of an original mathematical model able to reproduce biofouling evolution in membrane systems. The model has been solved numerically to simulate biologically relevant conditions, and to investigate the hydraulic behavior of the membrane. It has been calibrated and validated using lab-scale data. Numerical results accurately predicted the pressure drop occurring in the microfiltration system. A calibrated model can give information for optimization protocols as well as fouling prevention strategies.

Two species nonlocal diffusion systems with free boundaries

&lt;p style='text-indent:20px;'&gt;We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [&lt;xref ref-type="bibr" rid="b7"&gt;7&lt;/xref&gt;], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.&lt;/p&gt;

On a free boundary model for three-dimensional MEMS with a hinged top plate: Stationary case

A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter \lambda which is proportional to the square of the applied voltage difference. Existence of a stable stationary solution is established for small values of \lambda . Nonexistence of stationary solutions is obtained when \lambda is large enough.

Invasive behaviour under competition via a free boundary model: a numerical approach.

What will happen when two invasive species are competing and invading the environment at the same time? In this paper, we try to find all the possible scenarios in such a situation based on the diffusive Lotka-Volterra competition system with free boundaries. In a recent work, Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) considered a weak-strong competition case of this model (with spherical symmetry) and theoretically proved the existence of a "chase-and-run coexistence" phenomenon, for certain parameter ranges when the initial functions are chosen properly. Here we use a numerical approach to extend the theoretical research of Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) in several directions. Firstly, we examine how the longtime dynamics of the model changes as the initial functions are varied, and the simulation results suggest that there are four possible longtime profiles of the dynamics, with the chase-and-run coexistence the only possible profile when both species invade successfully. Secondly, we show through numerical experiments that the basic features of the model appear to be retained when the environment is perturbed by periodic variation in time. Thirdly, our numerical analysis suggests that in two space dimensions the population range and the spatial population distribution of the successful invader tend to become more and more circular as time increases no matter what geometrical shape the initial population range possesses. Our numerical simulations cover the one space dimension case, and two space dimension case with or without spherical symmetry. The numerical methods here are based on that of Liu et al. (Mathematics, 6(5):72, 2018, Int J Comput Math, 97(5): 959-979, 2020). In the two space dimension case without radial symmetry, the level set method is used, while the front tracking method is used for the remaining cases. We hope the numerical observations in this paper can provide further insights to the biological invasion problem, and also to future theoretical investigations. More importantly, we hope the numerical analysis may reach more biologically oriented experts and inspire applications of some refined versions of the model tailored to specific real world biological invasion problems.