We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two-phase materials, where, in …
We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two-phase materials, where, in contrast to the classical Gelfand problem which uses a single temperature approach, the state of the system is described by two different temperatures. We show that similar to the classical Gelfand problem the thermal explosion occurs exclusively owing to the absence of stationary temperature distribution. We also show that the presence of interphase heat exchange delays a thermal explosion. Moreover, we prove that in the limit of infinite heat exchange between phases the problem of thermal explosion in two-phase porous media reduces to the classical Gelfand problem with renormalized constants.
One proves that the moving interface of a two-phase Stefan problem on $\ooo\subset\rr^d$, $d=1,2,3,$ is controllable at the end time $T$ by a Neumann boundary controller $u$. The phase-transition region …
One proves that the moving interface of a two-phase Stefan problem on $\ooo\subset\rr^d$, $d=1,2,3,$ is controllable at the end time $T$ by a Neumann boundary controller $u$. The phase-transition region is a mushy region $\{σ^u_t;\ 0\le t\le T\}$ of a modified Stefan problem and the main result amounts to saying that, for each Lebesque measurable set $\ooo^*$ with positive measure, there is $u\in L^2((0,T)\times\pp\ooo)$ such that $\ooo^*\subsetσ^u_T.$ To this aim, one uses an optimal control approach combined with Carleman's inequality and the Kakutani fixed point theorem.
Journal Article ON A MULTIPHASE STEFAN PROBLEM Get access JEFFREY N. DEWYNNE, JEFFREY N. DEWYNNE Department of Mathematics, The University of WollongongWollongong, N.S.W., Australia Search for other works by this …
Journal Article ON A MULTIPHASE STEFAN PROBLEM Get access JEFFREY N. DEWYNNE, JEFFREY N. DEWYNNE Department of Mathematics, The University of WollongongWollongong, N.S.W., Australia Search for other works by this author on: Oxford Academic Google Scholar JAMES M. HILL JAMES M. HILL Department of Mathematics, The University of WollongongWollongong, N.S.W., Australia Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mechanics and Applied Mathematics, Volume 39, Issue 1, February 1986, Pages 67–84, https://doi.org/10.1093/qjmam/39.1.67 Published: 01 February 1986 Article history Received: 02 January 1985 Revision received: 10 May 1985 Published: 01 February 1986
Static critical phenomena of a three-dimensional Ising model confined in a porous medium made by spinodal decomposition have been studied using large-scale Monte Carlo simulations. We thus examine the influence …
Static critical phenomena of a three-dimensional Ising model confined in a porous medium made by spinodal decomposition have been studied using large-scale Monte Carlo simulations. We thus examine the influence of the geometry of Vycor-like materials on phase transitions. No surface interactions (preference of the Vycor-like material for one phase above the other) are taken into account. We find that the critical temperature depends on the average pore size D as ${\mathit{T}}_{\mathit{c}}$(D)=${\mathit{T}}_{\mathit{c}}$(\ensuremath{\infty})-c/D. The critical exponents are independent of pore size: we find \ensuremath{\nu}=0.8\ifmmode\pm\else\textpm\fi{}0.1, \ensuremath{\gamma}=1.4\ifmmode\pm\else\textpm\fi{}0.1, \ensuremath{\beta}=0.65\ifmmode\pm\else\textpm\fi{}0.13. No divergence is observed for the specific heat, indicating \ensuremath{\alpha}\ensuremath{\le}0. All data for all pore sizes can be collapsed well with the scaling function for the magnetic susceptibility \ensuremath{\chi}${\mathit{L}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}/\ensuremath{\nu}}$=\ensuremath{\chi}\ifmmode \tilde{}\else \~{}\fi{}(${\mathit{tL}}^{1/\ensuremath{\nu}}$), where t=T/${\mathit{T}}_{\mathit{c}}$(D)-1. These critical phenomena are consistent with those computed for the randomly site-diluted Ising model. Experimental realizations of our numerical experiments are discussed. \textcopyright{} 1996 The American Physical Society.
We consider the flow of a viscous incompressible fluid through a porous medium. We allow the permeability of the medium to depend exponentially on the pressure and provide an analysis …
We consider the flow of a viscous incompressible fluid through a porous medium. We allow the permeability of the medium to depend exponentially on the pressure and provide an analysis for this model. We study a splitting formulation where a convection diffusion problem is used to define the permeability, which is then used in a linear Darcy equation. We also study a discretization of this problem, and provide an error analysis for it.
One proves that the moving interface of a two-phase Stefan problem on $\ooo\subset\rr^d$, $d=1,2,3,$ is controllable at the end time $T$ by a Neumann boundary controller $u$. The phase-transition region …
One proves that the moving interface of a two-phase Stefan problem on $\ooo\subset\rr^d$, $d=1,2,3,$ is controllable at the end time $T$ by a Neumann boundary controller $u$. The phase-transition region is a mushy region $\{\sigma^u_t; 0\le t\le T\}$ of a modified Stefan problem and the main result amounts to saying that, for each Lebesque measurable set $\ooo^*$ with positive measure, there is $u\in L^2((0,T)\times\pp\ooo)$ such that $\ooo^*\subset\sigma^u_T.$ To this aim, one uses an optimal control approach combined with Carleman's inequality and the Kakutani fixed point theorem.
A simple one-dimensional model for crystal dissolution and precipitation is presented. The model equations resemble a one-phase Stefan problem and involve nonlinear and multi-valued exchange rates at the free boundary. …
A simple one-dimensional model for crystal dissolution and precipitation is presented. The model equations resemble a one-phase Stefan problem and involve nonlinear and multi-valued exchange rates at the free boundary. The original equations are formulated on a variable domain. By transforming the model to a fixed domain and applying a regularization, we prove the existence and uniqueness of a solution. The paper is concluded by numerical simulations.
We extend the two-scale expansion approach of periodic homogenization to include time scales and thus can tackle the full instationary Navier-Stokes-Cahn-Hilliard model at the pore scale as microscale. Time scale …
We extend the two-scale expansion approach of periodic homogenization to include time scales and thus can tackle the full instationary Navier-Stokes-Cahn-Hilliard model at the pore scale as microscale. Time scale separation allows us to keep microscale dynamics, responsible e.g. for hysteresis, and arrive at a numerically tractable micro-macro model including coupled generalized Darcy's laws.
We perform direct numerical simulations of the flow through a model of a deformable porous medium. Our model is a two-dimensional hexagonal lattice, with defects, of soft elastic cylindrical pillars, …
We perform direct numerical simulations of the flow through a model of a deformable porous medium. Our model is a two-dimensional hexagonal lattice, with defects, of soft elastic cylindrical pillars, with elastic shear modulus $G$, immersed in a liquid. We use a two-phase approach: the liquid phase is a viscous fluid and the solid phase is modeled as an incompressible viscoelastic material, whose complete nonlinear structural response is considered. We observe that the Darcy flux ($q$) is a nonlinear function -- steeper than linear -- of the pressure-difference ($\Delta P$) across the medium. Furthermore, the flux is larger for a softer medium (smaller $G$). We construct a theory of this super-linear behavior by modelling the channels between the solid cylinders as elastic channels whose walls are made of a material with a linear constitutive relation but can undergo large deformation. Our theory further predicts that the flow permeability is a universal function of $\Delta P/G$, which is confirmed by the present simulations.
A multidimensional problem concerning phase transitions of the liquid-liquid type in a moving substance with different phase densities is investigated. Rather special state equations are considered. The motion of the …
A multidimensional problem concerning phase transitions of the liquid-liquid type in a moving substance with different phase densities is investigated. Rather special state equations are considered. The motion of the fluid is assumed to be slow. It is proved that the problem has a unique global solution. Also, some properties of the solution are investigated. It is shown that the mushy region does not increase in time. In the case of the one-phase Stefan problem the velocity of the substance is equal to zero in one of the two phases.
An exact solution is presented for Ising-like transitions in a decorated lattice model of a porous medium. The model is solved by decimation of the spins, leading to a space-filling …
An exact solution is presented for Ising-like transitions in a decorated lattice model of a porous medium. The model is solved by decimation of the spins, leading to a space-filling lattice with renormalized parameters. The critical temperature is found to vary as 1/lnL, where L is the number of sites between intersections of the spin chains. Some of the critical exponents differ from those of the ordinary Ising problem. We have also studied the case of a single, infinitely long pore, using both exact and approximate methods. An exploration of finite-width effects reveals surprisingly small (quantitative) deviations from mean-field theory.
This paper considers estimation of an unknown state function in the heat equation with state-dependent parameter values. The work is motivated by phase transitions in physical media, e.g., thawing of …
This paper considers estimation of an unknown state function in the heat equation with state-dependent parameter values. The work is motivated by phase transitions in physical media, e.g., thawing of water or foodstuff, welding and casting processes. We point out that known solution to standard Stefan problem solutions can be recovered with this formalism, and then propose a simple phase transition estimator that relies only on boundary measurements. Simulations indicate that the estimates converge for noise-free measurements.
This book provides a concise treatment of the theory of nonlinear evolutionary partial differential equations. It provides a rigorous analysis of non-Newtonian fluids, and outlines its results for applications in …
This book provides a concise treatment of the theory of nonlinear evolutionary partial differential equations. It provides a rigorous analysis of non-Newtonian fluids, and outlines its results for applications in physics, biology, and mechanical engineering
Abstract This book gives a general presentation of the mathematical and numerical connections kinetic theory and conservation laws based on several earlier works with P. L. Lions and E. Tadmor, …
Abstract This book gives a general presentation of the mathematical and numerical connections kinetic theory and conservation laws based on several earlier works with P. L. Lions and E. Tadmor, as well as on more recent developments. The kinetic formalism approach allows the reader to consider Partial Differential Equations, such as some nonlinear conservation laws, as linear kinetic (or semi-kinetic) equations acting on a nonlinear quantity. It also aids the reader with using Fourier transform, regularisation, and moments methods to provide new approaches for proving uniqueness, regularizing effects, and a priori bounds. Special care has been given to introduce basic tools, including the classical Boltzmann formalism to derive compressible fluid dynamics, the study of oscillatons through the kinetic defect measure, and an elementary construction of solutions to scalar conservation laws. More advanced material contains regularizing effects through averaging lemmas, existence of global large solutions to isentropic gas dynamics, and a new uniqueness proof for scalar conservation laws. Sections are also devoted to the derivation of numerical approaches, the 'kinetic schemes', and the analysis of their theoretical properties.
Introductory material Auxiliary propositions Linear equations with discontinuous coefficients Linear equations with smooth coefficients Quasi-linear equations with principal part in divergence form Quasi-linear equations of general form Systems of linear …
Introductory material Auxiliary propositions Linear equations with discontinuous coefficients Linear equations with smooth coefficients Quasi-linear equations with principal part in divergence form Quasi-linear equations of general form Systems of linear and quasi-linear equations Bibliography.
Many basic equations in physics take the following form:Ot -(u) O, (1) where x E Rd, t > O, u-(Ul,...,us) represents the local density of the investigated quantities and f= …
Many basic equations in physics take the following form:Ot -(u) O, (1) where x E Rd, t > O, u-(Ul,...,us) represents the local density of the investigated quantities and f= (fl,"',fd) is the flux vector.Such an equation is often used to describe the conservation of the density u in the evolution of physical process.Common examples are conservation of mass, balance of momentum, and balance of energy.Thus, (1) is called a conservation law (equation).In the case s 1, (1) becomes a first order hyperbolic PDE, which we may call a scalar conservation law (equation).After certain preparation, the authors consider the Cauchy problem for the scalar conservation law:---+ V .f(u) 0,in R + xNd, (2) u(0,.u0, in Rd, with f-(fi,'",fd), fj ca(R) and Uo:Rd.It is well-known that, in general, (2) does not admit a classical solution even if both f and u 0 are smooth.Thus, one seeks weak solutions to (2), by which we mean a function u Loc(R + d) satisfying 0 Rd for all p C10(R Rd).To obtain a weak solution of (2), one natural method is the so-called vanishing of viscosity.It introduces the parabolic perturbation to equation (2), which amounts to the introduction of the viscosity.Thus, one considers the