System of generalized nonlinear boundary conditions of a boundary-value problem of heat conduction for a layer under two-sided local high-temperature deposition of coatings

We obtain a system of generalized nonlinear boundary conditions for describing radiant-convective heat exchange of a layer with a working media through local regions of coatings. The corresponding twodimensional problem of heat conduction is formulated.     

A difference scheme for a conjugate-operator model of a heat conduction problem in polar coordinates

In polar coordinates, a discrete analog of the conjugate-operator model of a heat conduction problem is formulated to hold the structure of the original model. The difference scheme converges with second-order accuracy in the case of discontinuous parameters of the medium in the Fourier law and irregular grids. An efficient algorithm for solving the discrete conjugate-operator model when heat conduction tensor is a unit operator is proposed.     

Heat conduction on the ring: Interface problems with periodic boundary conditions

The classical problem of heat conduction in one dimension on a composite ring is examined. The problem is formulated using the heat equation with periodic boundary conditions. We provide an explicit solution of this problem using the Method of Fokas. The location of the interfaces is known, but neither temperature nor heat flux are prescribed there. Instead, the physical assumption of continuity at the interface is imposed.     

Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model

In the present study, we consider a thermoelastic half-space made of a functionally gradient material with an insulated crack, which is subjected to a thermal impact. The memory-dependent heat conduction model is adopted for analysis. By using the Fourier and Laplace transforms, the thermoelastic problem is formulated in terms of singular integral equations which can be solved numerically. Effects of the time delay, kernel function, and nonhomogeneity parameters on the temperature and stress intensity factor are analyzed. Our results are also compared with those based on the Fourier and CV heat conduction models, which can be viewed as two special cases of the present model. In conclusion, the memory-dependent derivative and nonhomogeneity parameters play an essential role in controlling the heat transfer process.     

The estimation of the strength of the heat source in the heat conduction problems

A general method is proposed to determine the strength of the heat source in the Fourier and non-Fourier heat conduction problems. A finite difference method, the concept of the future time and a modified Newton–Raphson method are adopted in the problem. The undetermined heat source at each time step is formulated as an unknown variable in a set of equations from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. Three examples are used to demonstrate the characteristics of the proposed method. The validity of the proposed method is confirmed by the numerical results. The results show that the proposed method is an accurate and stable method to determine the strength of the heat source in the inverse hyperbolic heat conduction problems. Furthermore, the result shows that more future times are needed in the hyperbolic equation than that of parabolic equation. Moreover, the robustness and the accuracy of the estimated results in the non-Fourier problem are not as well as those of the Fourier problem.     

The thermal problem for a submerged stream

The general solution of the thermal problem of convective heat conduction with volume heat dissipation caused by viscous dissipation of kinetic energy of the fluid, whose velocity field is determined by the exact solution /1/ of the Navier-Stokes equations, is considered for a submerged stream. The possible formulation of the heat problem and the characteristic behaviour of the solutions are investigated. The solutions obtained have a special feature, namely the existence, under specified conditions, of two regimes of convective heat exchange.     

Solution of the stationary problem of heat conduction of laminated anisotropic inhomogeneous plates by the method of initial functions

The problem of stationary heat conduction of laminated plates of constant and variable thickness is formulated in the three-dimensional statement. We reduce the three-dimensional problem to a twodimensional one by the method of initial functions. For plates with layers of variable thickness, a system of resolving equations with variable coefficients is obtained. The obtained two-dimensional boundary-value problems are analyzed. For plates with homogeneous layers of constant thickness, we construct a solution in an analytic form. It is shown that this solution coincides with a solution obtained by the method of separation of variables.     

Heat Conduction Problem in a Dilated Pipe: Existence and Uniqueness Result

We study the heat conduction through a pipe filled with incompressible viscous fluid. The goal of this paper is to take into account the effects of the spipe’s dilatation due to the heating. In view of that, we assume that the longitudinal dilatation of the pipe is described by a linear heat expansion law. We prove the existence and uniqueness theorems for the corresponding boundary value problem. The main difficulty comes from the fact that the flow domain changes depending on the solution of the heat equation leading to a nonlinear coupled governing problem.     

一个抛物型方程侧边值问题的正则逼近解在一类Sobolev空间中的最优误差界

逆热传导问题(IHCP)是严重不适定问题,即问题的解(如果存在)不连续依赖于数据．但目前关于逆热传导问题的已有结果主要是针对标准逆热传导问题．文中给出了出现在实际问题中的一个抛物型方程侧边值问题,即一个含有对流项的非标准型逆热传导问题的正则逼近解一类Sobolev空间中的最优误差界．     

基于分数阶热传导方程激光加热瞬态温度场研究

基于分数阶Taylor(泰勒)级数展开原理,建立单相延迟一阶分数阶近似方程,获得分数阶热传导方程.针对短脉冲激光加热问题建立分数阶热传导方程组,并运用Laplace(拉普拉斯)变换方法进行求解,给出非Gauss(高斯)时间分布的激光内热源温度场解析解.针对具体算例数值研究温度波传播特性.结果表明热传播速度与分数阶阶次有关,分数阶阶次增加,热传播速度减小,温度变化幅度增加.分数阶方程可以用于描述介于扩散方程和热波方程间的热传输过程,且对热传播机制与分数阶热传导方程中分数阶项的关系做了深入剖析.     

Partial Decomposition of a Domain Containing Thin Tubes for Solving the Heat Equation

An initial–boundary value problem for the heat equation in a three-dimensional domain containing thin cylindrical tubes is considered. The Neumann condition is set on the lateral boundaries of the tubes. The original three-dimensional problem is reduced to a hybrid-dimensional one in which the heat equation in the tubes is replaced by the one-dimensional heat equation in shorter cylinders (subtubes), and the three- and one-dimensional equations are matched on the bases of the subtubes. The difference between the solutions of the original and hybrid-dimensional problems is estimated using two geometric characteristics: the distance between the bases of the tubes and subtubes and the reciprocals of the minimal positive eigenvalues of the Neumann problem for the Laplace operator in the tube cross sections.     

Mixed boundary-value problems in the theory of elasticity of thin laminated bodies of variable thickness consisting of anisotropic inhomogeneous materials

Starting from the three-dimensional equations of the theory of thermoelasticity, two-dimensional equations for thin laminated bodies are derived in a general formulation and solved by an asymptotic method. The bodies and layers, consisting of anisotropic and inhomogeneous materials (with respect to two longitudinal coordinates), bounded by arbitrary smooth non-intersecting surfaces, also have variable thicknesses. Recursion formulae are derived for determining the components of the stress tensor and the displacement vector when the kinematic or mixed boundary conditions of the static boundary-value problem of the theory of thermoelasticity are specified on the faces of the body, assuming that the corresponding heat conduction problem is solved. An algorithm for constructing of the analytical solutions of the boundary-value problems formulated is developed using modern computational facilities.     

Singularity formation to the 2D Cauchy problem of nonbarotropic magnetohydrodynamic equations without heat conductivity

The question of whether the two-dimensional (2D) nonbarotropic compressible magnetohydrodynamic (MHD) equations with zero heat conduction can develop a finite-time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. Such a problem is interesting in studying global well-posedness of solutions. In this paper, we proved that, for the initial density allowing vacuum states, the strong solution exists globally if the density and the pressure are bounded from above. Our method relies on weighted energy estimates and a Hardy-type inequality.     

Uniqueness of the finite-difference solution of the inverse problem of heat conduction

The article examines the problem of determination of the coefficients of heat conduction and heat capacity from a system of difference equations for the equation of heat conduction with additional information about the solution of the difference problem. Uniqueness of the solution of the inverse problem is proved.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach Matematicheskoi Fiziki, pp. 22–29, 1993.     

Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems

A sequential method is proposed to estimate boundary condition of the two-dimensional hyperbolic heat conduction problems. An inverse solution is deduced from a finite difference method, the concept of the future time and a modified Newton–Raphson method. The undetermined boundary condition at each time step is denoted as an unknown variable in a set of non-linear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. The example problem is used to demonstrate the characteristics of the proposed method. In the example, a well-known problem is used to demonstrate the validity of the proposed direct method and then the inverse solutions are evaluated. In the second example, the larger value of the relaxation time is implemented in the direct solutions and the inverse solutions. The close agreement between the exact values and the estimated results is made to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method to determine the boundary conditions in the two-dimensional inverse hyperbolic heat conduction problems.     

Nonstationary nonaxisymmetric temperature fields of multilayer orthotropic cylinders

A solution of the three-dimensional nonstationary nonaxisymmetric problem of heat conduction for a preliminarily heated multilayer infinite orthotropic cylinder under the action of heat sources and in the presence of convective heat exchange is obtained. In this case, a Green function of the corresponding problem of heat conduction is used. As an example, we consider the heating of the cylinder by a mobile normally distributed heat flow. The temperature field in a two-layer cylinder caused by a heat flow that moves along the generator and a helical curve is investigated.     

Shape-preserving signal forms in heat conduction

Let us consider the heat conduction problem described by a parabolic equation. We study under which conditions is the time-dependence on the boundary preserved inside the solid. The question is how information entering on the boundary penetrates the solid. E.g. consider a heat conducting solid subject to sinusoidally varying boundary condition. After decay of the transients, the temperature at any inner point varies in time sinusoidally with the same circular frequency, with space dependent amplitude and phase delay. So, sinusoidal signals inserted on the boundary are preserved. Information is also preserved in case of linear signals. Farkas and Mudri [H. Farkas, I. Mudri, Shape-preserving time-dependences in heat conduction, Acta Phys. Hung. 55 (1984) 267–273] have formulated this phenomenon, defined the notion of the boundary following solution and the shape-preserving signal forms, determined necessary and heuristic sufficient conditions for the shape-preserving signal forms.

Their work is extended by rigorous proofs of some sufficient conditions in this paper, and the minimum of the phase delay, expected to be attained on the boundary for physical reasons, is examined.     

在模糊随机因素影响下的多层圆筒结构的稳态热传导问题的区间数解法

在多层圆筒结构稳态热传导分析中,根据给定固体壁两侧表面温度总传热量公式,首先推导出当边界温度为随机变量情况下总传热量函数统计参数的均值和方差;然后推导出在导热系数为模糊数,边界温度为随机数下的总传热量的区间表达式．通过比较可以知道由区间数算法得到的区间最大,由概率统计算法得到的区间最小．并给出了两者的相对误差公式．最后引进粗糙集中的上、下近似集,提出用一个参数来统一定义模糊和随机区间进行稳态结构的热传导分析．     

Boundary value problems in the theory of thermoporoelasticity for materials with double porosity

In this paper the linear quasi-static theory of thermoelasticity for solids with double porosity is considered. The system of equations of this theory is based on the equilibrium equations for solids with double porosity, conservation of fluid mass, constitutive equations, Darcy's law for materials with double porosity and Fourier's law for heat conduction. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for classical solutions of the above mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled boundary element method and finite difference method for the heat conduction in laser processing

This paper is presented as a way to model transient heat conduction in a 3-D axisymmetric case where large rates of heat fluxes are applied on the surfaces as done in the case of laser processing. This would result in large temperature gradients in a small area irradiated by the laser on the incident surface that could also reach melting and subsequent vaporization. BEM can handle large fluxes very easily and it also can be formulated if needed to incorporate the moving boundary problem in a unique manner while on the other hand FDM is a fast and efficient method. For these reasons a coupled BEM–FDM method is formulated to simulate the heat conduction process. In the BEM method linear elements for the boundary and quadratic elements for the domain were used. The integrals in BEM were integrated in time using the asymptotic expansion for the modified Bessel functions in the Green’s function. To further improve the accuracy, special techniques were employed in the spatial integration. As for the FDM formulation, a flux conservation scheme with a 4th order formula for the fluxes was used. The FDM and BEM were coupled at the interface by the temperature from the FDM formulation being imposed on the BEM and the flux from the BEM being utilized by the FDM elements near to the interface. To advance in time, the Crank–Nicholson scheme was used on the FDM directly and due to coupling indirectly on the BEM. The relative errors for the simulation of constant and variable flux cases demonstrate the successful nature of the numerical model.