A vector valued Stefan problem from aluminium industry

Dissolution of stoichiometric multi-component particles in ternary alloys is an important process occurring during the heat treatment of as-cast aluminium alloys prior to hot-extrusion. A mathematical model is proposed to describe such a process. In this model an equation is given to determine the position of the particle interface in time, using two diffusion equations which are coupled by nonlinear boundary conditions at the interface. Moreover the well-posedness of the moving boundary problem is investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an analytical asymptotic approximation based on self-similarity is derived. This asymptotic approximation gives insight into the well-posedness of the problem

A level-set method for the evolution of cells and tissue during curvature-controlled growth

Most biological tissues grow by the synthesis of new material close to the tissue's interface, where spatial interactions can exert strong geometric influences on the local rate of growth. These geometric influences may be mechanistic, or cell behavioural in nature. The control of geometry on tissue growth has been evidenced in many in-vivo and in-vitro experiments, including bone remodelling, wound healing, and tissue engineering scaffolds. In this paper, we propose a generalisation of a mathematical model that captures the mechanistic influence of curvature on the joint evolution of cell density and tissue shape during tissue growth. This generalisation allows us to simulate abrupt topological changes such as tissue fragmentation and tissue fusion, as well as three dimensional cases, through a level-set-based method. The level-set method developed introduces another Eulerian field than the level-set function. This additional field represents the surface density of tissue synthesising cells, anticipated at future locations of the interface. Numerical tests performed with this level-set-based method show that numerical conservation of cells is a good indicator of simulation accuracy, particularly when cusps develop in the tissue's interface. We apply this new model to several situations of curvature-controlled tissue evolutions that include fragmentation and fusion.Comment: 15 pages, 10 figures, 3 supplementary figure

On Similarity Solutions and Interface Reactions for a Vector-Valued Stefan Problem

In this paper first it is shown for several geometries that classical similarity solutions for particle growth exist if and only if the Stefan problem is well-posed in the sense of being mass conserving. The extension of the similarity solutions to multicomponent alloys, which makes the problem nonlinear, is illustrated by the application to a hypothetic alloy with realistic input values. The similarity solutions are based on the assumption of local equilibrium at the interface. In the second part, the assumption of local equilibrium is relaxed using a first-order interface reaction. The influence of the interface reaction on the movement of the interface and on the interface concentrations is evaluated using Finite Difference calculations. A Newton scheme is used to solve the nonlinear problem

A mathematical model for the dissolution of particles in multi-component alloys

AbstractDissolution of stoichiometric multi-component particles is an important process occurring during the heat treatment of as-cast aluminum alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model equations are given to determine the position of the particle interface in time, using a number of diffusion equations which are coupled by nonlinear boundary conditions at the interface. This problem is known as a vector valued Stefan problem. A necessary condition for existence of a solution of the moving boundary problem is proposed and investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an asymptotic approximation based on self-similarity is derived. The asymptotic approximation is used to gain insight into the influence of all components on the dissolution. Subsequently, a numerical treatment of the vector valued Stefan problem is described. The numerical solution is compared with solutions obtained by the analytical methods. Finally, an example is shown

Biomedical implications from a morphoelastic continuum model for the simulation of contracture formation in skin grafts that cover excised burns

<p>A continuum hypothesis-based model is developed for the simulation of the (long term) contraction of skin grafts that cover excised burns in order to obtain suggestions regarding the ideal length of splinting therapy and when to start with this therapy such that the therapy is effective optimally. Tissue is modeled as an isotropic, heterogeneous, morphoelastic solid. With respect to the constituents of the tissue, we selected the following constituents as primary model components: fibroblasts, myofibroblasts, collagen molecules, and a generic signaling molecule. Good agreement is demonstrated with respect to the evolution over time of the surface area of unmeshed skin grafts that cover excised burns between outcomes of computer simulations obtained in this study and scar assessment data gathered previously in a clinical study. Based on the simulation results, we suggest that the optimal point in time to start with splinting therapy is directly after placement of the skin graft on its recipient bed. Furthermore, we suggest that it is desirable to continue with splinting therapy until the concentration of the signaling molecules in the grafted area has become negligible such that the formation of contractures can be prevented. We conclude this study with a presentation of some alternative ideas on how to diminish the degree of contracture formation that are not based on a mechanical intervention, and a discussion about how the presented model can be adjusted.</p

A mathematical model for the dissolution of particles in multi-component alloys

Dissolution of stoichiometric multi-component particles is an important process ocurring during the heat treatment of as-cast aluminium alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model equations are given to determine the position of the particle interface in time, using a number of diffusion equations which are coupled by nonlinear boundary conditions at the interface. This problem is known as a vector valued Stefan problem. Moreover the well-posedness of the moving boundary problem is investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an analytical asymptotic approximation based on self-similarity is derived. Moreover, this self-similar solution and the asymptotic approximation are extended to the vector valued Stefan problem. The approaches are compared to each other and the asymptotic approximation is used to gain insight into the influence of all components on the dissolution. Subsequently a numerical treatment of the vector valued Stefan problem is described. The numerical method is compared with solutions by analytical methods. Finally, an example is shown

Crisis management: The response of a small Dutch hospitality company during the COVID-19 pandemic

The purpose of this study was to explore how a small Dutch hospitality company responded to the COVID-19 pandemic and hence create an understanding of how hospitality businesses can potentially use this knowledge when facing similar crises in the future. This study is based on exploratory research and used interviews to collect primary data. Five themes were found: initial crisis response, operational expenses, health scare, marketing, and crisis impact. It is seen that crisis management was implemented during the COVID-19 pandemic, where reactive strategies were key for survival. Immediate actions were taken and implementing change was seen as easier due to the small size of the company. Further, operational expenses were adjusted to the changed demand and a favourable reputation helped to rebuild customers’ trust, where marketing initiatives were seen as important to reach customers. Finally, the impact of COVID-19 can strengthen organisational efficiency when handled well. Research about the impact of COVID-19 on the hospitality industry is limited, therefore further research is recommended on the long-term crisis response and the crisis consequences as well as the attitude of owners and employees of the hospitality industry during the COVID-19 pandemic. Keywords: COVID-19, crisis management, crisis response, hospitality managemen