Determination of Unknown Boundary in the Composite Materials with Stefan-Boltzmann Conditions

Abstract The authors consider one specific kind of heat transfer problems in a threedimensional layered domain, with nonlinear Stefan-Boltzmann conditions on the boundaries as well as on the interfaces. To determine the unknown part of the boundary （or corrosion） by the Cauchy data on the reachable part is an important inverse problem in engineering. The mathematical model of this problem is introduced, the well-posedness of the forward problems and the uniqueness of the inverse problems are obtained.     

Theoretical analysis of an optimal control problem of conductive–convective–radiative heat transfer

The problem of optimal heat removal from a three-dimensional domain is considered. The specific of the study consist in accounting for the radiative heat transfer. The so-called P₁ approximation of the radiative heat transfer equation is used, which reduces the model to a nonlinear elliptic system. A problem of optimal boundary control of this system is considered. The solvability of the control problem is proved, and necessary optimality conditions of first order are derived. Examples of non-singularity of these conditions are given.     

A numerical method for identifying heat transfer coefficient

In this paper, we consider an inverse problem of heat equation with Robin boundary condition for identifying heat transfer coefficient. Combining the method of fundamental solutions with discrepancy principle for the choice of the locations for source points, we give a method for solving the reconstruction problem. Since the resultant matrix is severe ill-conditioning, Tikhonov regularization with L-curve method is employed. Some numerical examples are given for verifying the efficiency and accuracy of the presented method.     

Unsteady MHD stagnation-point flow with heat and mass transfer in a micropolar fluid in the presence of thermophoresis and suction/injection

The effect of suction or injection on unsteady MHD flow with heat and mass transfer in a micropolar fluid near the forward stagnation point flow with thermophoresis has been investigated. The problem is reduced to a system of non-dimensional partial differential equations, which are solved numerically using the implicit finite-difference scheme. Profiles for velocity, microrotation, temperature and concentration as well as the skin friction, the rate of heat and mass transfer are determined and presented graphically for physical parameters. The results show that the suction increases the skin friction, the rate of heat and mass transfer while opposite trend is observed for the case of injection. It is also found that the effect of thermophoresis is decrease the concentration boundary layer thickness.     

The direct and inverse problem for an inclusion within a heat-conducting layered medium

This paper is concerned with the problem of heat conduction from an inclusion in a heat transfer layered medium. Making use of the boundary integral equation method, the well-posedness of the forward problem is established by the Fredholm theory. Then an inverse boundary value problem, i.e. identifying the inclusion from the measurements of the temperature and heat flux on the accessible exterior boundary of the medium is considered in the framework of the linear sampling method. Based on a careful analysis of the Dirichlet-to-Neumann map, the mathematical fundamentals of the linear sampling method for reconstructing the inclusion are proved rigorously.     

非常量导热系数几例微分方程的推立

在本文里,曾先后假设物体的导热系数是依直线和指数函数空间地起改变,就这样来建立了六个二阶热传导微分方程;又对于变密度、变比热、变导热系数这样的更一般的情况也推立了六个二阶热传导的微分方程.     

A nonlinear model and viscosity solution for the temperature field of an oil-immersed self-cooled three-phrase transformer

In this paper, on the basis of the characteristics of an oil-immersed self-cooled three-phrase transformer, we establish a mathematical model of the three-dimensional temperature field. But because the specific heat, density, heat sources and coefficient of heat transfer are discontinuous and non-differentiable, the problem has no analytical solution. We decompose the problem into seven subproblems, and prove the existence and uniqueness of a viscosity solution for every subproblem, by combining Perron’s method with the technique of coupled solutions.     

Identification of nonlinear heat transfer laws from boundary observations

We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperature-dependent heat transfer coefficient in a parabolic equation from boundary observations. As a practical example, this model applies to the heat transfer coefficient that describes the intensity of heat exchange between a hot wire and the cooling water in which it is placed. We reformulate the inverse problem as a variational one which aims to minimize a misfit functional and prove that it has a solution. We provide a gradient formula for the misfit functional and then use some iterative methods for solving the variational problem. Thorough investigations are made with respect to several initial guesses and amounts of noise in the input data. Numerical results show that the methods are robust, stable and accurate.     

Stationary problem of complex heat transfer in a system of semitransparent bodies with boundary conditions of diffuse reflection and refraction of radiation

A boundary value problem describing complex (radiation-conductive) heat transfer in a system of semitransparent bodies is considered. Complex heat transfer is described by a system consisting of a stationary heat equation and an equation of radiative transfer with the boundary conditions of diffuse reflection and diffuse refraction of radiation. The dependence of the radiation intensity and optical properties of bodies on the frequency of radiation is taken into account. The unique existence of the weak solution to this problem is established. The comparison theorem is proven. Estimates of the weak solution are derived, and its regularity is established.     

MHD-mixed convection from a vertical slender cylinder

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical slender cylinder is studied numerically. A uniform magnetic field is applied perpendicular to the cylinder. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. The velocity and temperature profiles as well as the local skin friction and the local heat transfer parameters are determined for different values of the governing parameters, mainly the transverse curvature parameter, the magnetic parameter, the electric field parameter and the Richardson number. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase, increasing the Richardson number, Ri (i.e. the mixed convection parameter), electric field parameter E₁ and magnetic parameter Mn.     

Steady one-dimensional heat flow in a longitudinal triangular and parabolic fin

We consider a heat transfer problem of a longitudinal fin with triangular and parabolic profiles. Both thermal conductivity and heat transfer coefficient are assumed to be temperature-dependent, and given by power laws. We construct exact solution when the problem is linearizable. In the other case, classical Lie symmetry techniques are employed to analyze the problem. The obtained exact solutions satisfy the realistic boundary conditions. The effects of the physical applicable parameters such as thermo-geometric fin parameter and the fin efficiency are analyzed.     

A look at a problem in heat transfer

Consideration and careful investigation of a simple physical problem in heat transfer shows that even such a problem may not always have solutions and meaningful solutions exist only under restricted conditions. In other words, not all physical problems are well posed. The analysis given here shows that in many practical problems in fluid mechanics and heat transfer the question ‘how well posed is it?’ is non-trivial.     

Numerical study of a mathematical model of a catalytic fuel processor with cocurrent oxidation and conversion flows

The results are presented of the numerical study of a mathematical model in the form of a nonlinear boundary value problem describing the stationary regimes in a catalytic fuel processor. We study a two-dimensional model for the endoblock, with the longitudinal heat and mass transfer by the gas and the transversal heat conductivity along the catalyst in the two-temperature approximation. For the exochannel, a model is considered with the longitudinal heat and mass transfer by the gas flow and the longitudinal heat transfer along the catalytic wall. These two blocks are related to each other through the equality of the temperature and heat flux on the boundary. The results obtained are in good agreement with experimental data.     

On Analytical Results for the Optimal Control of the quasi-stationary Radiative Heat Transfer System

We state an optimal control problem of the coupled quasi-stationary radiative heat equations consisting of the radiative transfer equation and the instationary heat transfer equation that model radiative-conductive heat transfer. We give an existence and uniqueness result for the state equations and the adjoint equations of the quasi-stationary radiative heat transfer system. For the optimal control problem the existence of a minimizer is proven. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, f_w. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer.     

Analytical description of the radiative-conductive heat transfer in a gray medium contained between two diffuse parallel plates

Explicit analytical solutions for the temperature and heat flux in a gray medium contained between two diffuse parallel plates are derived for both pure thermal radiation and coupled conduction-radiation heat transfer. This is achieved by combining the integral equations for the heat flux and temperature predicted by the radiative transfer equation with the corresponding predictions of the discrete ordinates method. The algebraic formulation of this well-known method is used to derive analytical results that agree with their corresponding numerical ones with an accuracy greater than 99.9%, for a large interval of optical thicknesses and conduction-to-radiation factors. The explicit and original solutions, for both pure radiation and radiative-conductive heat transfer, therefore solve the problem of one dimensional steady-state heat transfer in gray cavities.     

Noniterative Reconstruction Method for an Inverse Potential Problem Modeled by a Modified Helmholtz Equation

This paper deals with an inverse potential problem posed in two dimensional space whose forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial measurements of the associated potential. Since the inverse problem, we are dealing with, is written in the form of an ill-posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional is defined to measure the misfit of the solution obtained from the model and the data taken from the partial measurements. This shape functional is minimized with respect to a set of ball-shaped anomalies using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to noisy data and independent of any initial guess. Finally, some numerical experiments are presented to show the effectiveness of the proposed reconstruction algorithm.     

Optimization technique in solving laminar boundary layer problems with temperature dependent viscosity

In this paper, a novel numerical method is proposed to solve specific third order ODE on semi-infinite interval. These kinds of problems often occur in laminar boundary layer with temperature dependent viscosity. Runge-Kutta method incorporating with optimization techniques is used to solve the problem. First, the semi-infinite interval is transformed into a finite interval. Second, by converting the boundary value problem, with some initial and distributed unknowns, into an optimization problem, solving the original problem is limited to solving a multiobjective optimization problem. Third, we use shooting-Newton’s method for solving this optimization problem. It is shown that the Falkner-Skan problem with constant surface temperature, that arise during the solution for the laminar forced convection heat transfer from wedges to flow, can be solved accurately and simultaneously by this strategy. Numerical results for different values of wedge angle and Prandtl number are presented, which are in good agreement with some of the successful provided solutions in the literature.     

Numerical study of viscous dissipation effect on free convection heat and mass transfer of MHD non-Newtonian fluid flow through a porous medium

The problem of free convection heat with mass transfer for MHD non-Newtonian Eyring–Powell flow through a porous medium, over an infinite vertical plate is studied. Taking into account the effects of both viscous dissipation and heat source. The temperature and concentration are of periodic variation. The governing non-linear partial differential equations of this phenomenon are transformed into non-linear algebraic system utilizing finite difference method. Numerical results for the velocity, temperature and concentration distributions as well as the skin friction, heat and mass transfer are obtained and reported in tabular form and graphically for different values of physical parameters of the problem. Also, the stability condition is studied.     

Global existence result for thermoviscoelastic problems with hysteresis

We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a three-dimensional setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouët. Finally a global existence result is derived.