The Stefan problem with variable thermophysical properties and phase change temperature

In this paper we formulate a Stefan problem appropriate when the thermophysical properties are distinct in each phase and the phase-change temperature is size or velocity dependent. Thermophysical properties invariably take different values in different material phases but this is often ignored for mathematical simplicity. Size and velocity dependent phase change temperatures are often found at very short length scales, such as nanoparticle melting or dendrite formation; velocity dependence occurs in the solidification of supercooled melts. To illustrate the method we show how the governing equations may be applied to a standard one-dimensional problem and also the melting of a spherically symmetric nanoparticle. Errors which have propagated through the literature are highlighted. By writing the system in non-dimensional form we are able to study the large Stefan number formulation and an energy-conserving one-phase reduction. The results from the various simplifications and assumptions are compared with those from a finite difference numerical scheme. Finally, we briefly discuss the failure of Fourier's law at very small length and time-scales and provide an alternative formulation which takes into account the finite time of travel of heat carriers (phonons) and the mean free distance between collisions.Comment: 39 pages, 5 figure

Mathematical Modelling of Tyndall Star Initiation

The superheating that usually occurs when a solid is melted by volumetric heating can produce irregular solid-liquid interfaces. Such interfaces can be visualised in ice, where they are sometimes known as Tyndall stars. This paper describes some of the experimental observations of Tyndall stars and a mathematical model for the early stages of their evolution. The modelling is complicated by the strong crystalline anisotropy, which results in an anisotropic kinetic undercooling at the interface; it leads to an interesting class of free boundary problems that treat the melt region as infinitesimally thin

Target shape dependence in a simple model of receptor-mediated endocytosis and phagocytosis

Phagocytosis and receptor-mediated endocytosis are vitally important particle uptake mechanisms in many cell types, ranging from single-cell organisms to immune cells. In both processes, engulfment by the cell depends critically on both particle shape and orientation. However, most previous theoretical work has focused only on spherical particles and hence disregards the wide-ranging particle shapes occurring in nature, such as those of bacteria. Here, by implementing a simple model in one and two dimensions, we compare and contrast receptor-mediated endocytosis and phagocytosis for a range of biologically relevant shapes, including spheres, ellipsoids, capped cylinders, and hourglasses. We find a whole range of different engulfment behaviors with some ellipsoids engulfing faster than spheres, and that phagocytosis is able to engulf a greater range of target shapes than other types of endocytosis. Further, the 2D model can explain why some nonspherical particles engulf fastest (not at all) when presented to the membrane tip-first (lying flat). Our work reveals how some bacteria may avoid being internalized simply because of their shape, and suggests shapes for optimal drug delivery.Comment: 18 pages, 5 figure

Ice formation on a smooth or rough cold surface due to the impact of a supercooled water droplet

Ice accretion is considered in the impact of a supercooled water droplet on a smooth or rough solid surface, the roughness accounting for earlier icing. In this theoretical investigation the emphasis and novelty lie in the full nonlinear interplay of the droplet motion and the growth of the ice surface being addressed for relatively small times, over a realistic range of Reynolds numbers, Froude numbers, Weber numbers, Stefan numbers and capillary underheating parameters. The Prandtl number and the kinetic under-heating parameter are taken to be order unity. The ice accretion brings inner layers into play forcibly, affecting the outer flow. (The work includes viscous effects in an isothermal impact without phase change, as a special case, and the differences between impact with and without freezing.) There are four main findings. First, the icing dynamically can accelerate or decelerate the spreading of the droplet whereas roughness on its own tends to decelerate spreading. The interaction between the two and the implications for successive freezings are found to be subtle. Second, a focus on the dominant physical effects reveals a multi-structure within which restricted regions of turbulence are implied. The third main finding is an essentially parabolic shape for a single droplet freezing under certain conditions. Fourth is a connection with a body of experimental and engineering work and with practical findings to the extent that the explicit predictions here for ice-accretion rates are found to agree with the experimental range.

Numerical Modeling of Cooling Water Droplets using a Two-Way Coupling Approach

The present dissertation focuses on the study of the process of cooling and freezing of free falling water droplets. The freezing phenomenon is of extreme relevance in aviation since the impact of drops on lifting surfaces of an aircraft and consequent accretion can lead to the occurrence of incidents and accidents. In order to prevent the formation and accretion of ice, there are several systems to combat this hazard. Critical areas of an aircraft are usually protected by these de-icing systems. However, although these methods can evaporate drops of water or melt the accreted ice, there is still the possibility of downstream ice formation due to new freezing of the ice-water mixture in unprotected areas. Thus, there is a need to study and adapt the existing physical and mathematical models for a better approximation to real-life situations, in order to contribute to a better understanding of this phenomenon and consequently lead to a reduction in the number of incidents and accidents, safety conditions. The objective of this work is to perform a numerical study with the purpose of studying the cooling of free falling water droplets for different diameters and humidity ratios. Ranz-Marshall relations are used, with and without a correction factor, in addition to the Abramzon and Sirignano approach to take into account the effects of convection. A Two-Way Coupling approach is used being the predictions compared with experimental data and numerical predictions in a One-Way Coupling approach.A presente dissertação centra-se no estudo do processo de arrefecimento e congelamento de gotas de água em queda livre. O fenómeno de congelamento é de extrema relevância na área de Engenharia Aeronáutica, uma vez que devido ao impacto de gotas nas superfícies sustentadoras de uma aeronave e consequente acumulação de gelo poderão ocorrer incidentes e acidentes com as aeronaves. Como forma de impedir a formação e acumulação de gelo, existem diversos sistemas para combater este perigo. As áreas críticas de uma aeronave estão normalmente protegidas por sistemas anti gelo. No entanto, embora estes métodos consigam evaporar gotas de água ou derreter o gelo acumulado, existe ainda a possibilidade de formação de gelo a jusante devido a um novo congelamento da mistura gelo-água em áreas desprotegidas. Desta forma, surge uma necessidade de estudar e adaptar os modelos físicos e matemáticos existentes para uma melhor aproximação a situações reais, por forma a contribuir para uma melhor compreensão deste fenómeno e consequentemente levar a uma redução do número de incidentes e acidentes, melhorando as condições de segurança. O objetivo deste trabalho passa pela realização de um estudo numérico com o intuito de estudar o arrefecimento de gotas de água em queda livre para diferentes diâmetros e razões de humidade do ar. As correlações de Ranz-Marshall são utilizadas, com e sem a adição de um fator de correção, para além da abordagem de Abramzon e Sirignano, como forma de considerar os efeitos de convecção. O modelo utilizado é de Two-Way Coupling e as previsões são comparadas com dados experimentais e previsões numéricas cuja abordagem foi One-Way Coupling

Selection Criterion of Stable Dendritic Growth for a Ternary (Multicomponent) Melt with a Forced Convective Flow

A stable growth mode of a single dendritic crystal solidifying in an undercooled ternary (multicomponent) melt is studied with allowance for a forced convective flow. The steady-state temperature, solute concentrations and fluid velocity components are found for two- and three-dimensional problems. The stability criterion and the total undercooling balance are derived accounting for surface tension anisotropy at the solid-melt interface. The theory under consideration is compared with experimental data and phase-field modeling for Ni 98 Zr 1 Al 1 alloy

Precursors of the Spin Glass Transition in Three Dimensions

We study energy landscape and dynamics of the three-dimensional Heisenberg Spin Glass model in the paramagnetic phase, i.e. for temperature $T$ larger than the critical temperature $T_\mathrm{c}$. The landscape is non-trivially related to the equilibrium states even in the high-temperature phase, and reveals an onset of non-trivial behavior at a temperature $T_\mathrm{o}$, which is also seen through the behavior of the thermoremanent magnetization. We also find a power-law growth of the relaxation times far from the spin-glass transition, indicating a dynamical crossover at a temperature $T_\mathrm{d}$, $T_\mathrm{c}<T_\mathrm{d}<T_\mathrm{o}$. The arising picture is reminiscent of the phenomenology of supercooled liquids, and poses questions on which mean-field models can describe qualitatively well the phenomenology in three dimensions. On the technical side, local energy minima are found with the Successive Overrelaxation algorithm, which reveals very efficient for energy minimization in this kind of models.Comment: 16 pages, 6 figure