Network simulation method for solving phase-change heat transfer problems with variable thermal properties

 The transient heat conduction equation in a finite slab undergoing phase change (two-phase problem of melting and solidification), with isothermal, adiabatic or convective boundary conduction is studied by the network simulation method; solid phase conductivity and specific heat are assumed to be dependent on temperature. Ablation, as a particular case, is also analysed. A network model is established for a cell and boundary conditions are added to complete the whole network model. No restrictions exist, as to the kinds of linear and non-linear boundary conditions, Stefan number values or the initial conditions (when hypotheses concern of the Stefan problem, numerical and exact solutions are compared for a large interval of Stefan numbers; simulation values show good agreement). Movement of the solid–liquid boundary and thermal fields are determined in all cases. Received on 10 May 2000 / Published online: 29 November 2001     

Solution of the inverse problem of thermal conductivity for a melting slab with entrainment

This article proposes an approximate solution to the inverse problem of the Stefan type for a finite region with arbitrary boundary and initial conditions. A comparison with exact solutions is made.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 132–137, May–June, 1973.     

Solidification of a binary mixture with arbitrary heat flux and initial conditions

The problem of solidification of a binary mixture in a semi-infinite region with arbitrarily prescribed initial temperature and composition subject to an arbitrary heat flux at its surface is studied. The liquidus and solidus lines of the phase diagram are used to relate the freezing temperature and the composition of the mixture. Solidification, depending on the prescribed data, could occur immediately or at a later time. In the latter case, there is a period of presolidification. Thus the initial condition of the subsequent solidification cannot be assigned arbitrarily; in particular, it cannot be taken as a uniform state. The conditions for occurrence of these cases are studied and specified. The exact solutions for each of these are found. Existence, uniqueness and convergence of the series solutions are also considered and proved.     

Exponential numerical methods for one-dimensional one-phase Stefan problems

Summary Three exponential iterative methods for one-dimensional one-phase Stefan problems based on the transformation of the moving boundary problem into a mixed one, the discretization of the time variable, and the piecewise linearization of the resulting two-point boundary-value problem at each time step are proposed. Two of the methods are based on the strong conservation-law form of the governing equation and analytically solve a piecewise advection–diffusion equation, whereas the third exponential technique accounts for transient, advective, and diffusive effects when determining the solution. These exponential methods provide piecewise-analytical (exponential) solutions, which, by imposing continuity conditions, are globally continuous throughout the domain, and one of them provides globally smooth solutions. The methods have been applied to the classical one-phase Stefan problem and solutions in excellent agreement with the exact ones have been obtained for several Stefan numbers. In addition, it is shown that the method that accounts for transient, advective, and diffusive effects preserves the similarity of the analytical solution to Stefan problems, yields a tridiagonal matrix, and exhibits a spatial accuracy of, at least, fourth order. Application of this method to a forced one-phase Stefan problem indicates that it provides solutions in excellent agreement with those obtained by means of explicit finite difference and nodal integral techniques, and that the melting-front location exhibits some oscillations in the initial stages whose amplitude decreases as the Stefan number is decreased and as time increases, but which increases as the amplitude of the forcing temperature is increased. It is also shown that the temperature profiles in the liquid are affected by the amplitude and frequency of the forcing and the Stefan number.This research was partially financed by Project BFM2001–1902 from the Ministerio de Educación y Ciencia of Spain and Fondos FEDER.The author gratefully acknowledges the comments made by the referees,which have resulted in a clearer presentation.     

A hierarchy of dynamic equations for solid isotropic circular cylinders

This work considers homogeneous isotropic circular cylinders adopting a power series expansion method in the radial coordinate. Equations of motion together with consistent sets of end boundary conditions are derived in a systematic fashion up to arbitrary order using a generalized Hamilton’s principle. Time domain partial differential equations are obtained for longitudinal, torsional, and flexural modes, where these equations are asymptotically correct to all studied orders. Numerical examples are presented for different sorts of problems, using exact theory, the present series expansion theories of different order, and various classical theories. These results cover dispersion curves, eigenfrequencies and the corresponding displacement and stress distributions, as well as fix frequency motion due to prescribed end displacement or lateral distributed forces. The results illustrate that the present approach may render benchmark solutions provided higher order truncations are used, and act as engineering cylinder equations using low order truncation.     

Fourier Law,Phase Transitions and the Stationary Stefan Problem

We study the one-dimensional stationary solutions of the integro-differential equation which, as proved in Giacomin and Lebowitz (J Stat Phys 87:37–61, 1997; SIAM J Appl Math 58:1707–1729, 1998), describes the limit behavior of the Kawasaki dynamics in Ising systems with Kac potentials. We construct stationary solutions with non-zero current and prove the validity of the Fourier law in the thermodynamic limit showing that below the critical temperature the limit equilibrium profile has a discontinuity (which defines the position of the interface) and satisfies a stationary free boundary Stefan problem. Under-cooling and over-heating effects are also studied: we show that if metastable values are imposed at the boundaries then the mesoscopic stationary profile is no longer monotone and therefore the Fourier law is not satisfied. It regains its validity however in the thermodynamic limit where the limit profile is again monotone away from the interface.     

Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient

In this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values. Geometrically, this can be interpreted as finding a minimal Lipschitz extension. Due to the weak convexity of the functional associated to this problem, solutions are generally nonunique. By adopting G. Aronsson's notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation. In fact, we actually prove a weak maximum principle for this partial differential equation, which in some sense is the Euler equation for the minimization problem. This is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity. To overcome this difficulty we use the weak solutions of M. G. Crandall and P.-L. Lions, also known as viscosity solutions, in conjunction with some arguments using integration by parts.     

On solutions of the problem in stresses with the use of Maxwell stress functions

The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses. In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary, special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation method.     

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

. This paper is concerned with the initial‐boundary‐value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing‐viscosity method and finite‐difference schemes (Lax‐Friedrichs‐type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform bounds on the approximate solutions and so dealing with solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois & LeFloch[15]. The local structure of these sets and the well‐posedness of the corresponding initial‐boundary‐value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics. (Accepted July 2, 1998)     

Continuation of the solution of a free boundary problem in cylindrical symmetry

Summary We perform an a priori analysis of the behavior of the solution to a Stefan type free boundary problem in cylindrical symmetry, in arbitrarily large time intervals; a nonlinear flux condition is prescribed on the fixed boundary, for t>0.Next we find some relations between the occurrence of each possible case and the behavior of the initial datum, assuming that the flux is null on the fixed boundary for t>0.

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Sommario Si esegue un'analisi a priori del comportamento, in intervalli di tempo arbitrariamente grandi, della soluzione di un problema a frontiera libera del tipo di Stefan in simmetria cilindrica; sulla frontiera fissa, per t>0, è stabilita una condizione di flusso non lineare.Si trovano quindi relazioni tra i vari casi possibili e il comportamento del dato iniziale, sotto l'ulteriore ipotesi che il flusso sulla frontiera fissa, per t>0, sia nullo.

     

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

Two kinds of second-order non-linear ordinary differential equations (ODEs) appearing in mathematical physics and non-linear mechanics are analyzed in this paper. The one concerns the Kidder equation in porous media and the second the gas pressure diffusion equation. Both these equations are strongly non-linear including quadratic first-order derivatives (damping terms). By a series of admissible functional transformations we reduce the prescribed equations to Abel's equations of the second kind of the normal form that they do not admit exact analytic solutions in terms of known (tabulated) functions. According to a mathematical methodology recently developed concerning the construction of exact analytic solutions of the above class of Abel's equations, we succeed in performing the exact analytic solutions of both Kidder's and gas pressure diffusion equations. The boundary and initial data being used in the above constructions are in accordance with each specific problem under considerations.     

Spatial Analyticity on the Global Attractor for the Kuramoto–Sivashinsky Equation

For the Kuramoto–Sivashinsky equation with L-periodic boundary conditions we show that the radius of space analyticity on the global attractor is lower-semicontinuous function at the stationary solutions, and thereby deduce the existence of a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of the bifurcation parameter L. As an application of the result, we prove that the number of rapid spatial oscillations of functions belonging to this neighborhood is, up to a logarithmic correction, at most linear in L.     

A continuum model for size-dependent deformation of elastic films of nano-scale thickness

Ultra-thin elastic films of nano-scale thickness with an arbitrary geometry and edge boundary conditions are analyzed. An analytical model is proposed to study the size-dependent mechanical response of the film based on continuum surface elasticity. By using the transfer-matrix method along with an asymptotic expansion technique of small parameter, closed-form solutions for the mechanical field in the film is presented in terms of the displacements on the mid-plane. The asymptotic expansion terminates after a few terms and exact solutions are obtained. The mid-plane displacements are governed by three two-dimensional equations, and the associated edge boundary conditions can be prescribed on average. Solving the two-dimensional boundary value problem yields the three-dimensional response of the film. The solution is exact throughout the interior of the film with the exception of a thin boundary layer having an order of thickness as the film in accordance with the Saint-Venant’s principle.     

Initial and boundary value problems of hyperbolic heat conduction

This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux , where e and are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density e is solved by a new kinetic approach that was introduced and evaluated by Dreyer and Kunik in [1], [2] and Pertame [3]. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in [4]. The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields as well as integrals that need knowledge on energy and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by continuity conditions, it turns out that after some very short time the energy density and heat flux are related to the initial data according to the Rankine Hugoniot relation. Received October 6, 1998     

Energy minimization and the formation of microstructure in dynamic anti-plane shear

We investigate the behavior of a continuum model designed to provide insight into the dynamical development of microstructures observed during displacive phase transformations in certain materials. The model is presented within the framework of nonlinear viscoelasticity and is also of interest as an example of a strongly dissipative infinite-dimensional dynamical system whose forward orbits need not lie on a finite-dimensional attracting set, and which can display a subtle dependence on initial conditions quite different from that of classical finite-dimensional chaos.We study the problem of dynamical (two-dimensional) anti-plane shear with linear viscoelastic damping. Within the framework of nonlinear hyperelasticity, we consider both isotropic and anisotropic constitutive laws which can allow different phases and we characterize their ability to deliver minimizers and minimizing sequences of the stored elastic energy (Theorem 2.3). Using a transformation due to Rybka, we recast the problem as a semilinear degenerate parabolic system, thereby allowing the application of semigroup theory to establish existence, uniqueness and regularity of solutions in L ^p spaces (Theorem 3.1). We also discuss the issues of energy minimization and propagation of strain discontinuities. We comment on the difficulties encountered in trying to exploit the geometrical properties of specific constitutive laws. In particular, we are unable to obtain analogues of the absence of minimizers and of the non-propagation of strain discontinuities found by Ball, Holmes, James, Pego & Swart [1991] for a one-dimensional model problem.Several numerical experiments are presented, which prompt the following conclusions. It appears that the absence of an absolute minimizer may prevent energy minimization, thereby providing a dynamical mechanism to limit the fineness of observed microstructure, as has been proved in the one-dimensional case. Similarly, viscoelastic damping appears to prevent the propagation of strain discontinuities. During the extremely slow development of fine structure, solutions are observed to display local refinement in an effort to overcome incompatibility with boundary and initial conditions, with the distribution and shape of the resulting finer scales displaying a subtle dependence on initial conditions.     

Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol –k ²+4|k|–4(1+), where is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for || small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.     

On the vanishing viscosity limit for acoustic phenomena in a bounded region

The problem of the linearized vibrations of a viscous, barotropic fluid in a bounded vessel is studied under various boundary conditions. It is shown that the essential spectrum is formed by one or two points that tend to infinity as the viscosity coefficients tend to zero. The asymptotics of solutions of the initialboundary value problem as the viscosity coefficients tend to zero is established. Some remarks about the physical properties of the associated sound waves are given.The work reported here was done with the partial support of GNAFA-CNR (Italy) and CNRS, L.A. 229 (France). In particular, E. Sanchez-Palencia has been a visiting professor of GNAFA-CNR at the Politecnico di Torino (Italy).     

On the unsteady rotational flow of fractional Oldroyd-B fluid in cylindrical domains

This paper concerned with the unsteady rotational flow of fractional Oldroyd-B fluid, between two infinite coaxial circular cylinders. To solve the problem we used the finite Hankel and Laplace transforms. The motion is produced by the inner cylinder that, at time t=0⁺, is subject to a time-dependent rotational shear. The solutions that have been obtained, presented under series form in terms of the generalized G functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, fractional and ordinary Maxwell, fractional and ordinary second grade, and Newtonian fluids, performing the same motion, are obtained as limiting cases of general solutions.     

On patches of uniform vorticity in a plane of irrotational flow

We derive an evolution equation for the motions of patches of vorticity (vortex). Steady state solutions of this equation that include those of Kirchhoff and Moore & Saffman are established. The m-fold symmetric, m3, hypotrochoid is an exact steady solution of this equation when rotation and strain are present. When strain is absent but rotation is present, the m-fold symmetric, m2, hypotrochoid is a perturbation solution with a dispersion relation extending that of Lamb. The case of m=2 is exact and is the Kirchhoff elliptical vortex.     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.