Static thermoelasticity problems for layered thermosensitive plates with cubic dependence of the coefficients of heat conductivity on temperature

We describe an analytic-numerical method of solution of one-dimensional static thermoelasticity problems for layered plates, heated in different ways. We take into account the cubic dependence of the coefficients of heat conductivity and arbitrary nature of the dependence of other physicomechanical parameters on temperature. Here, using the constructed exact solution of an auxiliary problem, we have reduced the heat conduction problems, irrespective of the number of layers, to the solution of one or a system of two nonlinear algebraic equations. We have also studied the temperature fields and stresses in four-layer plates under conditions of complex heat exchange.     

Solution of nonstationary heat-conduction problems for thermosensitive bodies under convective heat exchange

We propose a technique for solving nonstationary heat-conduction problems for thermosensitive bodies with simple nonlinearity (the coefficients of thermal conductivity and the heat capacity per unit volume depend on temperature, but the coefficient of thermal diffusivity is constant) heated by convective heat exchange from the surrounding medium. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 148–152.     

Temperature distribution in a laminar plane wall jet due to variably heated wall

The heat transfer in a laminar incompressible plane wall jet due to a variably heated wall has been studied. It is assumed that the difference of temperatures between the wall and the issuing jet is inversely proportional to an arbitrary exponent of the distance from the slit. A similar solution of the energy equation is possible. The solutions, for arbitrary values of the Prandtl number and of the exponent are obtained. It is found that in some cases the heat transfer at the wall may become zero or negative.     

Optimization of the two-dimensional temperature fields in inhomogeneous bodies with constraints on the phase coordinates

We state a problem of speed-optimal control of the two-dimensional nonstationary temperature regime in inhomogeneous bodies with constraints on the control (the temperature of the heating medium) and the phase coordinates (the temperature of the body, the temperature range, and the heat flux on the surface of the body), and we propose a method for solving the problem numerically. We give as an example the computation of the optimal control of heating of a hollow infinite cylinder under constraints on the temperature range. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 119–123.     

Global solvability of an inverse problem for an integro-differential equation of electrodynamics

We consider the problem of reconstructing the time history of the electric field from the electrodynamic equation. As additional information, the magnetic field intensity vector at the point x = 0 is given. For this problem, the existence and uniqueness theorem is proved for an arbitrary given interval.     

Heat transfer in granular materials: effects of nonlinear heat conduction and viscous dissipation

In this paper, we study the heat transfer in a one‐dimensional fully developed flow of granular materials down a heated inclined plane. For the heat flux vector, we use a recently derived constitutive equation that reflects the dependence of the heat flux vector on the temperature gradient, the density gradient, and the velocity gradient in an appropriate frame invariant formulation. We use two different boundary conditions at the inclined surface: a constant temperature boundary condition and an adiabatic condition. A parametric study is performed to examine the effects of the material dimensionless parameters. The derived governing equations are coupled nonlinear second‐order ordinary differential equations, which are solved numerically, and the results are shown for the temperature, volume fraction, and velocity profiles. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary control problem for the one-dimensional Klein-Gordon-Fock equation with a variable coefficient. The case of control by displacement at one endpoint with the other endpoint being fixed

We consider the problem of boundary control by displacement at one boundary point x = 0 for a process described by the Klein-Gordon-Fock equation with a variable coefficient on a finite interval 0 ≤ x ≤ l with the Dirichlet condition u(l, t) = 0 at the other boundary point. For the critical time interval T = 2l, we show that there exists a unique boundary function u(0, t) = µ(t) bringing the system from an arbitrary initial state into an arbitrary terminal state.     

Non-Darcy unsteady mixed convection flow near the stagnation point on a heated vertical surface embedded in a porous medium with thermal radiation and variable viscosity

The unsteady mixed convection boundary layer flow near the stagnation point on a heated vertical plate embedded in a fluid saturated porous medium is studied. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase in the surface temperature. Both the buoyancy assisting and the buoyancy opposing flow situations are considered with combined effects of the first and second order resistance of solid matrix of non-Darcy porous medium, variable viscosity and radiation. The problem is reduced to a system of non-dimensional partial differential equations, which is solved numerically using the Keller-box method. The features of the flow and the heat transfer characteristics for different values of the governing parameters are analyzed and discussed. The surface shear stress and the heat transfer of the present study are compared with the available results and a good agreement is found.     

A MAP-modulated fluid flow model with multiple vacations

We consider a MAP-modulated fluid flow queueing model with multiple vacations. As soon as the fluid level reaches zero, the server leaves for repeated vacations of random length V until the server finds any fluid in the system. During the vacation period, fluid arrives from outside according to the MAP (Markovian Arrival Process) and the fluid level increases vertically at the arrival instance. We first derive the vector Laplace–Stieltjes transform (LST) of the fluid level at an arbitrary point of time in steady-state and show that the vector LST is decomposed into two parts, one of which the vector LST of the fluid level at an arbitrary point of time during the idle period. Then we present a recursive moments formula and numerical examples.     

Numerical scheme for augmented Landau–Lifshitz equation in heat assisted recording

The motivation of heat assisted recording is to improve the thermal stability of recorded data bits by increasing the strength of the uniaxial anisotropy. During the recording process the medium is heated by a laser, reducing the coercivity and allowing the head to write data bits. We present a micromagnetic model based on the augmented Landau–Lifshitz equation taking into account a phenomenological power-law describing the dependence of the saturation magnetization on the temperature. A full-discrete numerical scheme is presented and the convergence of approximate solutions to a weak one is shown.     

Analysis of flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of disks contracting, rotation and heat transfer on the viscous fluid between heated contracting rotating disks. By introducing the Von Kármán type similarity transformations through which we reduced the highly nonlinear partial differential equation to a system of ordinary differential equations. This system of differential equations with appropriate boundary conditions is responsible for the flow behavior between large but finite coaxial rotating and heated disks. It is important to note that the lower disk is rotating with angular velocity Ω while the upper one with SΩ, the disks are also contracting and the temperatures of the upper and lower disks are T₁ and T₀, respectively. The agents which driven the flow are the contraction and also the rotation of the disks. On the other hand the velocity components and especially radial component of velocity strongly influence the temperature distribution inside the flow regime. The basic equations which govern the flow are the Navier Stokes equations with well known continuity equation for incompressible flow. The final system of ordinary differential equations is then solved numerically with given boundary conditions. In addition, the effect of physical parameters, the Reynolds number (Re), the wall contraction ratio (γ) and the rotation ratio (S) on the velocity and pressure gradient, as well as, the effect of Prandtl number (Pr) on temperature distribution are also observed.     

Mixed convection of a viscous dissipating fluid about a vertical flat plate

In this study, the effect of the viscous dissipation in steady, laminar mixed convection heat transfer from a heated/cooled vertical flat plate is investigated in both aiding and opposing buoyancy situations. The external flow field is assumed to be uniform. The governing systems of partial differential equations are solved numerically using the finite difference method. A parametric study is performed in order to illustrate the interactive influences of the governing parameters, mainly, the Richardson number, Ri (also known as the mixed convection parameter) and the Eckert number, Ec on the velocity and temperature profiles as well as the friction and heat transfer coefficients. Based on the facts the free stream is either in parallel or reverse to the gravity direction and the plate is heated or cooled, different flow situations are identified. The influence of the viscous dissipation on the heat transfer varied according to the situation. For some limiting cases, the obtained results are validated by comparing with those available from the existing literature. An expression correlating Nu in terms of Pr, Ri and Ec is developed.     

Free convection from a point heat source in a stratified fluid

A non-stationary problem of free convection from a point heat source in a stratified fluid is considered. The system of equations is reduced to a single equation for a special scalar function which determinos the velocity field, and the temperature and salinity distribution. Relations are found connecting the spatial and temporal scales of the phenomenon with the parameters of the medium and the intensity of the heat source. The magnitude of the critical source intensity at which the fluid begins to move in a jet-flow mode is established.The structure of convective flows above the heat sources depends, in the stratified media, essentially on the nature of the stratification /1/ which may be caused by a change in the temperature of the medium /2, 3/ or its salinity /4–7/, and by the form of the heat source. When a temperature gradient exists within the medium, an ascending jet forms above the point source, mushrooming outwards near the horizon of the hydrostatic equilibrium. In the case of a fluid with salinity gradient, the jet is surrounded by a sheet of descending salty fluid, and a regular system of annular convective cells is formed around it /1/.The height of the stationary jet computed in /2, 3/ on the basis of conservative laws agrees with experiment. However, this approach does not enable the temperature and velocity distribution over the whole space to be found and does not enable the problem of determining the flow to be investigated. A stationary solution of the linearized convection equations /8/ does not correspond to detail to the observed flow pattern /1, 5–7/. In this connection the study of the non-linear, non-stationary convection equations is of interest.The purpose of this paper is to construct a non-linear, non-stationary free convection equation above a point heat source, and to analyse the scales of the resulting structure and the critical conditions under which the flow pattern changes.     

Computation of melting with natural convection inside a rectangular enclosure heated by discrete protruding heat sources

This paper presents the results of a numerical investigation of the heat transfer by natural convection during the melting of a phase change material (PCM, n-eicosane with melting point of 36 °C) contained in a rectangular enclosure. This latest is heated by three discrete protruding heat sources (simulating electronic components) placed on one of its vertical walls. The power generated by heat sources is dissipated in PCM. The advantage of using this cooling scheme is that the PCMs are able to absorb high amount of heat generated by the heat sources, without acting the fan during the charging process (melting of the PCM). The thermal behavior and thermal performance of the proposed PCM based-heat sink are numerically investigated by developing a mathematical model based on the mass, momentum and energy conservation equations. The obtained numerical results show the impact of various key parameters on the cooling capacity of the PCM-based heat sink. Correlations encompassing a wide range of parameters were developed in terms of the dimensionless secured operating time (time required by one of the electronic components before reaching its critical temperature, T_cr ∼ 75 °C) and the corresponding liquid fraction, using the asymptotic computational fluid dynamics (ACFD) technique.     

Zone diagrams in Euclidean spaces and in other normed spaces

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain “dominance” map. Asano, Matou?ek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et?al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.     

Temperature distribution in a parallelopiped under the influence of heat source with boundary conditions of second kind

In the present paper, the general expressions for temperature distribution in an orthotropic plate of finite dimensions have been derived under the influence of an arbitrary volume heat source and arbitrary initial temperature distribution. The boundary conditions prescribed on the faces are of second kind. The solution is obtained by constructing an eigen value problem. The expression for unsteady temperature distribution consists of two parts—steady and pseudo-steady. The pseudo-steady parts of the solution have been obtained with the help of finite integral transform technique. The general expressions are further studied under different cases. The whole analysis is presented in the dimensionless form with the help of various dimensionless parameters,viz., F _o , P _o , K _i , etc.     

Optimal Control of a Parabolic Equation with Dynamic Boundary Condition

We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the dynamic boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an L ^p function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.     

The application of the FDEM program package with error estimate to industrial problems

The Finite Difference Element Method (FDEM) program package is a robust and efficient black-box solver that solves arbitrary nonlinear systems of elliptic and parabolic partial differential equations under arbitrary nonlinear boundary conditions on arbitrary domains in 2-D and 3-D. FDEM is an unprecedented generalization of the finite difference method on unstructured finite element meshes. From the difference of formulas of different order, we get an easy access to the discretization error. By the knowledge of this error, the mesh may be refined locally to reduce the error to a prescribed relative tolerance. The error estimate is a unique property for such a general black-box. In addition, the FDEMprogram package is efficiently parallelized on distributed memory parallel computers. In this paper we demonstrate the usefulness of the FDEM program package by its application to several industrial problems. This gives completely new results as up to now people have solved these problems blindly, unaware of the error of their solution. The first problem is the numerical simulation of a microreactor where we have one chemical component entering through the main channel and one chemical component entering through a side channel so that there is a reaction of the components. We want to examine the flow field and the behaviour of the chemical components. The second problem is the heat conduction in a high pressure Diesel injection pump. This problem is based on a preceding fluid-structure interaction problem, and we now compute the temperature distribution in the injection pump additionally. Finally, we simulate the distribution of the temperature in a DC/AC converter module with six power-MOSFETs heated with uniform power. At the bottom of the module air cooling is applied. In contrast to the first two elliptic problems, this is a 3-D parabolic problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gale duality bounds for roots of polynomials with nonnegative coefficients

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.