One-dimensional Moving Boundary Problem Research Articles

Random walks and moving boundaries: Estimating the penetration of diffusants into dense rubbers

For certain materials science scenarios arising in rubber technology, one-dimensional moving boundary problems with kinetic boundary conditions are capable of unveiling the large-time behavior of the diffusants penetration front, giving a direct estimate on the service life of the material. Driven by our interest in estimating how a finite number of diffusant molecules penetrate through a dense rubber, we propose a random walk algorithm to approximate numerically both the concentration profile and the location of the sharp penetration front. The proposed scheme decouples the target evolution system in two steps: (i) the ordinary differential equation corresponding to the evaluation of the speed of the moving boundary is solved via an explicit Euler method, and (ii) the associated diffusion problem is solved by a random walk method. To verify the correctness of our random walk algorithm we compare the resulting approximations to computational results based on a suitable finite element approach with a controlled convergence rate. Our numerical results recover well penetration depth measurements of a controlled experiment designed specifically for this setting.

Solving a one-dimensional moving boundary problem based on wave digital principles

We report on a novel method for solving one-dimensional moving boundary problems based on wave digital principles. Here, we exploit multidimensional wave digital algorithms to derive an efficient and robust algorithm for the solution of the considered problem. Our method lets the wave digital model, on which the algorithm is based, expand according to the size of the solution domain. The expanding model introduces new dynamical elements, which must be properly initialized to obtain a calculable algorithm. To deal with this problem, we make use of linear multistep methods to extrapolate the missing values. Our results show the proposed method to indeed be capable of correctly solving a one-dimensional partial differential equation describing a growing biological axon.

A study on cylindrical moving boundary problem with variable thermal conductivity and convection under the most realistic boundary conditions

The melting of a phase change material is the most applicable process in thermal energy storage system to capture heat transfer phenomena arising in a class of moving boundary problem. Demand of present technology motivates researchers to develop new theories and techniques for thermal management of a material. Experimental work on melting of a material may be difficult and development of robust theoretical formulation in cylindrical geometry with convection is critical. While there is already available study on cylindrical moving boundary problem, but still insufficient modeling of a size-dependent thermal conductivity and convection effect is not addressed properly, which is being considered in this paper and is expected to improve the previous understanding. In this work, a one-dimensional moving boundary problem with size-dependent heat conductivity and convection effect is analyzed in cylindrical geometry. In the mathematical model, we have considered a time-dependent temperature boundary condition which later assumed in periodic form, and a convective boundary condition at the outer surface of the body. The numerical result of the problem is obtained successfully via heat-balance integral method. Our numerical result is compared with a previous work and found in good acceptance. From mathematical framework, it is found that convection delayed melting process. With a size-no independent thermal conductivity, the rate of moving front decreases more in comparison to the fixed thermal conductivity.

A study of solidification on binary eutectic system with moving phase change material

A one-dimensional moving boundary problem describing solidification of a eutectic system under imposed material movement occupying a semi-infinite medium is solved for two different cases of solid fraction distribution within the mushy zone. In the first case it is assumed that the solid fraction distribution has a linear relationship with temperature and in the second case solid fraction distribution is varying linearly with distance within the mushy zone. An exact solution of the problem is obtained with the help of a similarity technique. To demonstrate the current study experimental data of Al-Cu solidification are presented. All the thermal-physical properties of each part are discussed in detail for both models. The temperature profile in each region and moving interfaces are calculated for different Peclet number Pe. In the present study it is shown that the moving interfaces are enhanced, growing relatively faster and assisting in the process of phase-transition when material moves in the direction of freeze but transition is delayed when material moves in the reverse direction. It is also shown that mushy zone becomes thinner when surface temperature is lower than the solidus temperature for different Peclet numbers. In addition, the heat removal Q at the surface ξ=0 is shown with respect to time for different Peclet numbers. To validate our study, we compare our results with a previous published work and they are found to be close.

Wavelet based numerical approach of non-classical moving boundary problem with convection effect and variable latent heat under the most generalized boundary conditions

The major goal of this article is to analysis a mathematical model of a non-classical one-dimensional moving boundary problem in the presence of convection effect when one surface subjected to the most generalized boundary conditions. The control function describes the cooling and heating effect depends on the evolution of heat flux at x=0. The thermo-physical properties are assumed to be constant while latent heat depends on the position of moving interface and its velocity. The parameters affecting the thermal behaviour of the melting process in various enclosures, i.e. the effect of Peclet number, control function, variable latent heat, heat flux and convecting heat transfer coefficient on the moving interface and temperature distribution are comprehensively investigated by the Legendre wavelet Galerkin computational method. In specific cases the Legendre wavelet Galerkin method is used for the numerical solution when surface is subjected to the first kind boundary condition with constant latent heat and thus the results obtained are compared with Turkyilmazoglu (2018) results and found in good compliance. Further when latent heat is variable, the present numerical solution obtained is compared with exact solution and are found in high accuracy and good flexibility.

Convergence rates of moving mesh methods for moving boundary partial integro–differential equations from regime-switching jump–diffusion Asian option pricing

This paper studies the convergence rates of moving mesh methods for a system of moving boundary partial integro-differential equations (PIDEs) which arise in the Asian option pricing under the state-dependent regime-switching jump–diffusion models. The value function of the Asian option under the model is governed by a system of two-dimensional PIDEs. In this paper, the two-dimensional PIDEs are recast into a one-dimensional moving boundary problem of the PIDEs. A moving finite difference method (FDM) is proposed to solve the one-dimensional moving boundary problem and the convergence rates are proved. Numerical examples are provided to confirm the theoretical results.

Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equations with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be stably numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy’s flow problem, the exact analytical solution for the case of Darcy’s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclusion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equations; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pressure gradient increasing for the one-dimensional problem.

A finite difference moving mesh method based on conservation for moving boundary problems

We propose a velocity-based moving mesh method in which we move the nodes so as to preserve local mass fractions. Consequently, the mesh evolves to be finer where the solution presents rapid changes, naturally providing higher accuracy without the need to add nodes. We use an integral approach which avoids altering the structure of the original equations when incorporating the velocity and allows the solution to be recovered algebraically. We apply our method to a range of one-dimensional moving boundary problems: the porous medium equation, Richards’ equation, and the Crank–Gupta problem. We compare our results to exact solutions where possible, or to results obtained from other methods, and find that our approach can be very accurate (1% relative error) with as few as ten or twenty nodes.

Axisymmetric natural convection-driven evaporation of hot water and the Mpemba effect

The Mpemba effect is popularly summarized by the statement that “hot water can freeze faster than cold”, and has been observed experimentally since the time of Aristotle; however, there exist almost no theoretical models that predict the effect. This paper considers experimentally and theoretically the cooling of a circular pool of water. A model is derived that takes into account conduction, natural convection, evaporation and thermal radiation. The governing equations are nondimensionalized and asymptotically reduced to yield a one-dimensional moving boundary problem that is solved numerically. Whilst the model may need some refinement in the future, the preliminary results are encouraging, indicating that it can predict the occurrence of the Mpemba effect.

Moving boundary problems governed by anomalous diffusion

Anomalous diffusion can be characterized by a mean-squared displacement 〈x(2)(t)〉 that is proportional to t(α) where α≠1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion (α<1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.

Solution of one-dimensional moving boundary problem with periodic boundary conditions by variational iteration method

In this paper, the solution of the one dimensional moving boundary problem with periodic boundary conditions is obtained with the help of variational iterational method. By using initial and boundary values, the explicit solutions of the equations have been derived, which accelerate the rapid convergence of the series solution. The method performs extremely well in terms of efficiency and simplicity. The temperature distribution and the position of moving boundary are evaluated and numerical results are presented graphically.

Understanding the Heat Transfer and Solidification of Plasma-Sprayed Yttria-Partially Stabilized Zirconia Coatings

A variety of microstructures have been observed in plasma-sprayed yttria-partially stabilized zirconia (YSZ) thermal barrier coatings. Control of the coating microstructures requires a good understanding of the heat transfer and solidification during the process. This article presents a quantitative analysis of heat transfer and solidification of plasma-sprayed YSZ splats. The analysis is based on a simple heat transfer and solidification model that solves a one-dimensional moving boundary problem with consideration of melt undercooling prior to solidification and nonequilibrium crystalline growth kinetics at the moving interface. The solidification morphology is first assumed to be planar, and the stability of the planar interface is examined against the absolute stability velocity calculated from the linear stability theory. Examining the temperature distribution in a solidifying YSZ zirconia splat indicates that a large positive temperature gradient exists in front of the interface, which leads to a stable planar interface and a segregation-free columnar structure, agreeing well with experimental observation. The model also finds that a low interface velocity results from poor heat transfer, which leads to a formation of cells and, therefore, the segregation of yttria. A steady-state dendrite tip growth model is then employed to calculate the radius of the cell tips and thus the cell spacings, which is then compared with experimental observations.

A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method

In this paper, the enthalpy method based on suitable finite difference approximations has been applied to the one-dimensional moving boundary problem with a Neumann-type boundary condition known as the Stefan problem. The numerical results obtained by the hopscotch technique are compared with the exact solution of the problem. It is shown that all results are found to be in very good agreement with each other, and the numerical solution displays the expected convergence to the exact one as the mesh size is refined.

Solute distribution during rapid solidification into an undercooled melt

Rapid solidification experiments show that a solute-rich core generally exists in a solid rapidly solidified from an undercooled melt. Several simplified models have been proposed to explain and predict this phenomenon. This paper presents a generalized model that includes mass as well as heat diffusion in both solid and liquid phases and considers nonequilibrium solidification kinetics including solute-trapping to treat the local recalescence. For given local temperature gradients and cooling rates, the model leads to a one-dimensional moving boundary problem with a strongly coupled boundary condition at the solid/liquid interface. The solution is obtained by employing an implicit iterative scheme that uses a coordinate transformation. The model predicts successfully a solute-rich core as observed in the experiments. The results show that both melt undercooling and cooling rates strongly affect the solute distribution in the solidified solid. Selected results for dilute aluminum-copper alloys are presented to illustrate the unique features of solute distribution in rapidly solidified alloys.

Microsegregation in cellular solidification

Microsegregation in a binary alloy solidified in the form of deep cells is predicted using a simplified finite difference model. The model accounts for solid state diffusion and for flow of liquid between cells driven by solidification shrinkage. Cell tip undercooling is predicted using the expression originally derived by Bower et al. Cells are assumed to be cylindrical, and solid state diffusion along the cell axis is ignored, simplifying considerably prediction of solid state diffusion and cell shape behind the tip, which are treated as a one-dimensional moving boundary problem. Experiments were conducted on binary Al-4.5 wt pct Cu, solidified in the cellular growth regime using a Bridgman furnace. Microsegregation in the samples was measured and is compared to predictions; good agreement is found, both for cell heights and microsegregation in the fully solidified material. It is found that intercellular fluid flow exerts a small, but discernable, influence on microsegregation and cell shape.

A continuum model of protrusion of pseudopod in leukocytes

The morphology of human leukocytes, the biochemistry of actin polymerization, and the theory of continuum mechanics are used to model the pseudopod protrusion process of leukocytes. In the proposed model, the pseudopod is considered as a porous solid of F-actin network, the pores of which are full of aqueous solution. G-actin is considered as a "solute" transported by convection and diffusion in the fluid phase. The pseudopod grows as actin filaments elongate at their barbed ends at the tip of the pseudopod. The driving force of extension is hypothesized as being provided by the actin polymerization. It is assumed that elongation of actin filaments, powered by chemical energy liberated from the polymerization reaction, does mechanical work against opposing pressure on the membrane. This also gives rise to a pressure drop in the fluid phase at the tip of the pseudopod, which is formulated by an equation relating the work done by actin polymerization to the local state of pressure. The pressure gradient along the pseudopod drives the fluid filtration through the porous pseudopod according to Darcy's Law, which in turn brings more actin monomers to the growing tip. The main cell body serves as a reservoir of G-actin. A modified first-order equation is used to describe the kinetics of polymerization. The rate of pseudopod growth is modulated by regulatory proteins. A one-dimensional moving boundary problem based on the proposed mechanism has been constructed and approximate solutions have been obtained. Comparison of the solutions with experimental data shows that the model is compatible with available observations. The model is also applicable to growth of other cellular systems such as elongation of acrosomal process in sperm cells.

A Front Tracking Method for One-Dimensional Moving Boundary Problems

We introduce a front tracking method for the numerical solution of one-dimensional Stefan problems. It consists in the formulation of the Stefan problem as an ordinary differential initial-value problem for the moving boundary coupled with a parabolic partial differential equation for the distribution of temperatures. We present a variable time step procedure in which the initial-value problem is solved with a predictor-corrector scheme; in the corrector step the function evaluation is done, iteratively, through an implicit time discretization of the parabolic equation. Numerical results for one-dimensional, one-phase Stefan problems with straight and curved moving boundary trajectories are presented. For these cases the front tracking method presented gives greatly improved results.

Moving grid method without interpolations

In their method of solving a one-dimensional moving boundary problem Crank and Gupta suggest a grid system which moves with the interface. The method requires some interpolations to be carried out which they perform by using a cubic spline or an ordinary polynomial. In the present paper these interpolations are avoided by employing a Taylor's expansion in space and time dimensions. A practical diffusion problem is solved and the results are compared with those obtained from other methods.