On a Cahn--Hilliard--Darcy system for tumour growth with solution dependent source terms

We study the existence of weak solutions to a mixture model for tumour growth that consists of a Cahn--Hilliard--Darcy system coupled with an elliptic reaction-diffusion equation. The Darcy law gives rise to an elliptic equation for the pressure that is coupled to the convective Cahn--Hilliard equation through convective and source terms. Both Dirichlet and Robin boundary conditions are considered for the pressure variable, which allows for the source terms to be dependent on the solution variables.Comment: 18 pages, changed proof from fixed point argument to Galerkin approximatio

A multiphase Cahn--Hilliard--Darcy model for tumour growth with necrosis

We derive a Cahn–Hilliard–Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. A multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, are included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth are investigated numerically. In particular, for certain modelling choices, we obtain some form of focal and patchy necrotic growth that have been observed in experiments

Numerical computations of facetted pattern formation in snow crystal growth

Facetted growth of snow crystals leads to a rich diversity of forms, and exhibits a remarkable sixfold symmetry. Snow crystal structures result from diffusion limited crystal growth in the presence of anisotropic surface energy and anisotropic attachment kinetics. It is by now well understood that the morphological stability of ice crystals strongly depends on supersaturation, crystal size and temperature. Until very recently it was very difficult to perform numerical simulations of this highly anisotropic crystal growth. In particular, obtaining facet growth in combination with dendritic branching is a challenging task. We present numerical simulations of snow crystal growth in two and three space dimensions using a new computational method recently introduced by the authors. We present both qualitative and quantitative computations. In particular, a linear relationship between tip velocity and supersaturation is observed. The computations also suggest that surface energy effects, although small, have a larger effect on crystal growth than previously expected. We compute solid plates, solid prisms, hollow columns, needles, dendrites, capped columns and scrolls on plates. Although all these forms appear in nature, most of these forms are computed here for the first time in numerical simulations for a continuum model.Comment: 12 pages, 28 figure

Towards a quantitative phase-field model of two-phase solidification

We construct a diffuse-interface model of two-phase solidification that quantitatively reproduces the classic free boundary problem on solid-liquid interfaces in the thin-interface limit. Convergence tests and comparisons with boundary integral simulations of eutectic growth show good accuracy for steady-state lamellae, but the results for limit cycles depend on the interface thickness through the trijunction behavior. This raises the fundamental issue of diffuse multiple-junction dynamics.Comment: 4 pages, 2 figures. Better final discussion. 1 reference adde

A variational formulation of anisotropic geometric evolution equations in higher dimensions

Accepted versio

Theoretical and methodological approaches to the determination of the "capital of enterprise" economic essence

Розглянуто основні підходи до обґрунтування сутності поняття "капітал підприємства". Сформовано власне визначення категорії "капітал" підприємства як матеріальні, грошові та нематеріальні ресурси, що авансовано у формування активів підприємства, необхідних для здійснення його господарської діяльності в довгостроковій перспективі, з метою отримання доходу та прибутку. Визначено склад взаємопов'язаних і взаємообумовлених внутрішніх і зовнішніх факторів, що впливають на структуру капіталу підприємства та визначають можливості управління ним.The main approaches to substantiating the essence of the concept of "capital of an enterprise" are considered. The actual definition of the category of "capital" of the enterprise as material, monetary and intangible resources, which was advanced in forming the assets of an enterprise necessary for its economic activity in the long run, was formed for the purpose of obtaining income and profits. The composition of interconnected and mutually determined internal and external factors influencing the structure of the enterprise capital and determine the possibilities of management of it are determined. The internal factors determining the peculiarities of the formation and composition of the capital of enterprises are: the organizational and legal form of the enterprise's activity, the existing capital structure, the level of profitability of the operating acti vity, the size of the enterprise and the stage of its life cycle, the degree of financial stability, the priorities of owners and management in choosing a method of financial provision, etc. External factors are the following: the state of the legislative process, the level of administrative influence on the economy of enterprises, the stability of the commodity market, the financial market situation, the tax burden on the enterprise, the ratio of creditors and investors to a particular enterprise, the degree of credit risk and the level of potential of the banking system, tendencies of development of other branches of economy

A pilot study of a phenomenological model of adipogenesis in maturing adipocytes using Cahn–Hilliard theory

We consider the accumulation and formation of lipid droplets in an adipocyte cell. The process incorporates adipose nucleation (adipogenesis) and growth. At later stages, there will be merging of droplets and growth of larger droplets at the expense of the smaller droplets, which will essentially undergo lipolysis. The process is modeled by the use of the Cahn–Hilliard equation, which is mass-conserving and allows the formation of secondary phases in the context of spinodal decomposition. The volume of fluid (VOF) method is used to determine the total area that is occupied by the lipids in a given cross section. Further, we present an algorithm, applicable to all kinds of grids (structured or unstructured) in two spatial dimensions, to count the number of lipid droplets and the portion of the domain of computation that is occupied by the lipid droplets as a function of time during the process. The results are preliminary and are validated from a qualitative point using experiments carried out on cell cultures. It turns out that the Cahn–Hilliard theory can model many of the features during adipogenesis qualitatively

Well posedness of an isothermal diffusive model for binary mixtures of incompressible fluids

We consider a model describing the behavior of a mixture of two incompressible fluids with the same density in isothermal conditions. The model consists of three balance equations: continuity equation, Navier-Stokes equation for the mean velocity of the mixture, and diffusion equation (Cahn-Hilliard equation). We assume that the chemical potential depends upon the velocity of the mixture in such a way that an increase of the velocity improves the miscibility of the mixture. We examine the thermodynamic consistence of the model which leads to the introduction of an additional constitutive force in the motion equation. Then, we prove existence and uniqueness of the solution of the resulting differential problem

Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels, Garcke, and Gr\"un for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system. The density of the mixture depends on an order parameter.Comment: 33 page

A phase-field model for phase transformations in glass-forming alloys

A phase-field model is proposed for phase transformations in glass-forming alloys. The glass transition is introduced as a structural relaxation, and the competition between the glass and crystalline phases is investigated. The simulations are performed for Cu-Zr alloys, employing thermodynamic and kinetic parameters derived from reported thermodynamic modeling and molecular dynamics simulation results,[1–3] respectively. Four distinct phase fields are treated with a multi-phase-field approach, representing the liquid/glass, Cu10Zr7, CuZr, and CuZr2 phases. In addition, a continuum-field method is applied to the liquid to accommodate the liquid–glass transformation. The combined phase-field approach is used to investigate the glass formation tendency, and critical cooling rates are estimated and compared with the reported experimental values