Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

Purpose The main purpose of this paper is given below: To present a mathematical model of a two-phase Stefan problem including a moving phase change material and variable thermophysical properties. To find a numerical solution of the problem to discuss the dependence of considered phase change problem on variable thermal conductivity, variable specific heat and Peclet number. Design/methodology/approach In this paper, a numerical solution of the problem is obtained using the front-fixing method in tandem with the explicit finite difference scheme. The authors have also discussed the consistency and stability of proposed numerical scheme. Findings In this study, it is observed that the considered scheme is an efficient tool that provides sufficiently accurate results for exploring the behaviors of moving interface (free boundary) and temperature profile for a nonclassical two-phase free boundary problem. In this study, the authors have observed that the parameters α1 and α2 influence the temperature profiles of the liquid region and the solid region. It is also found that the free boundary propagates faster when the authors increase the parameter α1 or decrease the parameter α2. Originality/value From the literature, it is seen that most of the two-phase problems with free boundary in an infinite domain are considered by the authors with constant thermophysical properties. Because it is possible to establish an analytical solution of two-phase problems with free boundary in case of an infinite domain. Moreover, a two-phase problem in a finite domain involving moving phase change material with the unidirectional speed is not considered. Therefore, the authors have considered a two-phase free boundary problem with variable thermal coefficients.

Determination of one or two unknown thermal coefficients of a semi-infinite material through a two-phase stefan problem

Variation of thermal conductivity and specific heat of a general-purpose unsaturated polyester during cure was investigated experimentally. Measurements were made on the cured, partially cured, and uncured samples, and the results are presented. Both thermal conductivity and specific heat of the cured samples increase nearly linearly with increasing temperature. Effect of degree of cure on the thermal properties is evident during the earlier stages of cure.

We propose new symbolic-numerical algorithms implemented in Maple-Fortran environment for solving the parametric self-adjoint elliptic boundary-value problem (BVP) in a 2D finite domain, using high-accuracy finite element method (FEM) with triangular elements and high-order fully symmetric Gaussian quadratures with positive weights, and no points are outside the triangle (PI type). The algorithms and the programs calculate with the given accuracy the eigenvalues, the surface eigenfunctions and their first derivatives with respect to the parameter of the BVP for parametric self-adjoint elliptic differential equation with the Dirichlet and/or Neumann type boundary conditions on the 2D finite domain, and the potential matrix elements, expressed as integrals of the products of surface eigenfunctions and/or their first derivatives with respect to the parameter. We demonstrated an efficiency of algorithms and program by benchmark calculations of helium atom ground state.

Fuzzy aspects of the boundary element method

In this paper we present a new, direct computational method for calculating the complex thermal transfer impedances between two separate locations of a given physical structure aimed at the implementation into a field-solver based on the SUNRED (SUccessive Node REDuction) algorithm. We tested the method with a simple 2D example containing 125 internal nodes. For testing the proposed new calculation method multiple combinations of Dirichlet and Neumann type boundary conditions were applied. Also, different types of thermal loads such as prescribed unit-step change in dissipation or temperature were assumed (for time domain transient analysis). The test case was also studied with the assumption of sinusoidal dissipation. Results obtained by the proposed new calculation method and results obtained by conventional simulations differ less than the uncertainty of the traditional solution method. The good agreement enables us to use the balanced truncation method to reduce the order of the transfer functions with low computational cost.

Solute dispersion in a porous formation is Mathematically expressed by partial differential equation well known as advection-dispersion equation (ADE). The present study deals with the solute transport governing equation in a semi-infinite homogeneous porous formation under linear sorption. A constant background solute concentration is assumed initially throughout the solute transport domain. Dirichlet and Neumann type boundary conditions are considered to examine the solute concentration distribution profile in the semi-infinite porous medium. The analytical and numerical solutions of the model problem are derived by Laplace transform technique and Crank-Nicolson method, respectively. Solute dispersion behaviour is studied for various form of flow velocities. Solutions obtained by analytical and numerical techniques are illustrated graphically with the help of MATLAB software. Also, the numerical solution is compared with the analytical solution and found great similarity between them.

In this paper we present the development of numerical methods for two problems in chemical engineering: adsorption of carbon dioxide into phenyl glycidyl ether and diffusion and reaction processes in a porous catalyst. Both problems are modelled by systems of two coupled nonlinear ordinary differential equations. Dirichlet and Neumann type boundary conditions are considered. We develop the standard homotopy analysis method and the boundary-domain element method to solve for the unknown steady-state concentrations of reactants. The validity of the proposed methods is checked and demonstrated by numerical examples. Convergence rates are compared and the advantages and drawbacks of the proposed methods are discussed and compared to the Mathematica NDSOLVE solver. We show, that the proposed homotopy analysis method features exponential convergence rate and is highly accurate and efficient. As low as 6 terms are needed to reach error norms of 10 − 5 . Furthermore, the boundary-domain integral approach is also capable of solving the same problem very efficiently, using a very coarse computational mesh (21 nodes).

A two-phase free boundary problem associated with a diffusion–convection equation is considered. The problem is reduced to a system of nonlinear integral equations, which admits a unique solution for small times. The system admits an explicit two-component solution corresponding to a two-component shock wave of the Burgers equation. The stability of such a solution is also discussed.

We provide an overview of some recent results about the regularity of the solution and the free boundary for so-called two-phase free boundary problems driven by uniformly elliptic equations.

Regularity for degenerate two-phase free boundary problems

We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.

This work extends the characteristic-based volume penalization method, originally developed and demonstrated for compressible subsonic viscous flows in (J. Comput. Phys. 262, 2014), to a hyperbolic system of partial differential equations involving complex domains with moving boundaries. The proposed methodology is shown to be Galilean-invariant and can be used to impose either homogeneous or inhomogeneous Dirichlet, Neumann, and Robin type boundary conditions on immersed boundaries. Both integrated and non-integrated variables can be treated in a systematic manner that parallels the prescription of exact boundary conditions with the approximation error rigorously controlled through an a priori penalization parameter. The proposed approach is well suited for use with adaptive mesh refinement, which allows adequate resolution of the geometry without over-resolving flow structures and minimizing the number of grid points inside the solid obstacle. The extended Galilean-invariant characteristic-based volume penalization method, while being generally applicable to both compressible Navier–Stokes and Euler equations across all speed regimes, is demonstrated for a number of supersonic benchmark flows around both stationary and moving obstacles of arbitrary shape.

Expansion formulae for wave structure interaction problems with applications in hydroelasticity

The brick industry is currently facing a shortage of natural resources. Despite this, the demand for construction bricks is progressively increasing. Alternative materials, such as dredged sediments and solid organic waste, have been recently proposed as options to replace natural clay in brick manufacturing. Potential exploitation of dredged sediments in clay bricks is evaluated in this study. The chemical composition of the mixtures and the opto-thermal properties of brick samples, which differed for the dredged sediment content (from 10% to 50% of the clay weight), were investigated. Chemical analyses detected lower concentrations of heavy metals in bricks incorporating dredged sediments (DS). Negligible variations in thermal conductivity, thermal diffusivity, and specific heat were observed by increasing the amount of DS in the mixture. In particular, the thermal conductivity values ranged between 0.45 ± 0.03 W m−1 K−1 (DS-50) and 0.50 ± 0.03 W m−1 K−1 (DS-30). Conversely, the color shift value and spectral reflectance in the infrared field were found directly proportional to the concentration of DS. Using dredged sediments as building material demonstrated to be a solution to the problem of their disposal and the scarcity of raw materials, reducing the global warming score by up to 2.8%.