Momentum and heat transfer of the Falkner–Skan flow with algebraic decay: An analytical solution

In this paper, an analytical solution of the Falkner–Skan equation with mass transfer and wall movement is obtained for a special case, namely a velocity power index of ?1/3, with an algebraically decaying velocity profile. The solution is given in a closed form. Under different values of the mass transfer parameter, the wall can be fixed, moving in the same direction as the free stream, or opposite to the free stream (reversal flow near the wall). The thermal boundary layer solution is also presented with a closed form for a prescribed power-law wall temperature, which is expressed by the confluent hypergeometric function of the second kind. The temperature profiles and the wall temperature gradients are plotted. Interesting but complicated variation trends for certain combinations of controlling parameters are observed. Under certain parameter combinations, there exist singular points or poles for the wall temperature gradients, namely wall heat flux. With poles, the temperature profiles can cross the zero temperature line and become negative. From the results, it is also found empirically that under certain given values of the Prandtl number (Pr) and flow controlling parameter (b), the number of times for the temperature profiles crossing the zero line is the same as the number of poles when the wall temperature power index varies between zero and a given value n. The current result provides a new analytical solution for the Falkner–Skan equation with algebraic decay and greatly enriches the understanding of this important equation as well as the heat transfer characteristics for this flow.     

DEPENDENCE OF QUALITATIVE BEHAVIOR OF THE NUMERICAL SOLUTIONS ON THE IGNITION TEMPERATURE FOR A COMBUSTION MODEL

We study the dependence of qualitative behavior of the numerical solutions （obtained by a projective and upwind finite difference scheme） on the ignition temperature for a combustion model problem with general initial condition. Convergence to weak solution is proved under the Courant-Friedrichs-Lewy condition. Some condition on the ignition temperature is given to guarantee the solution containing a strong detonation wave or a weak detonation wave. Finally, we give some numerical examples which show that a strong detonation wave can be transformed to a weak detonation wave under some well-chosen ignition temperature.     

A new regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation in a one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under the a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Spectral regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation with one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Heat Transfer in a Region with Non-Uniform Boundary Conditions

Heat transfer in a rectangular region with non-uniform conditions on the walls is considered. The temperature is given on both vertical walls and a part of the upper wall. The remainder of the upper wall and the lower horizontal wall are perfectly insulated. This boundary value problem is reduced to dual Fourier series equations. Those equations are simplified under the assumption that the height of the region is greater than the length or comparable to it. An exact solution of the simplified equations is constructed by using the Schwinger transformation, which has been used successfully in analyzing the electro-dynamics of wave guides. Numerical solutions also are found using a commercial finite element solver and a finite difference solver written in FORTRAN. Results for the average temperature and the temperature distribution in the region for a variety of high temperature boundary locations are in very good agreement among the three solution techniques.     

形变张量的特征值与Boussinesq方程组的正则性估计

讨论了二维及三维满足周期边界条件的Boussinesq方程初边值问题的局部正则解在有限时间内爆破的可能性.在二维情况下,用形变张量的特征值给出温度梯度的L2估计,从中看出若流体微团变形的速率大,则解爆破的可能性就大.在三维情况下,用形变张量的特征值和温度的偏导给出涡量的L2估计,从中发现若流体微团在大部分时间内一般是平面拉伸,且温度的偏导较小时,解爆破的可能性就大;若一般是线性拉伸,温度的偏导又不任意增大时,解爆破的可能性就小.     

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.     

炉温曲线的数学模型及求解

针对2020年全国大学生数学建模竞赛A题——炉温曲线,在一些简化假设下给出了电路板在回焊炉中进行自动焊接时的炉温曲线的数学模型,求出了给定温度设置下的最大过炉速度,讨论了两种最优曲线的温度和过炉速度设置.最后对学生竞赛论文中的一些情况进行了评述.     

A Priori Estimates of the Solution of the Problem of the Unidirectional Thermogravitational Motion of a Viscous Liquid in the Plane Channel

We consider an initial boundary-value problem describing the unidirectional motion of a liquid in the Oberbeck–Boussinesq model in a plane channel with rigid immovable walls on which the temperature distribution is given (or the upper wall is heat-insulated). For this problem, we obtain a priori estimates, find an exact stationary solution, and determine conditions under which the solution converges to its stationary regime.     

State of stress of nonuniformly heated plates loaded along the boundary surfaces : PMM vol. 43. no. 6, 1979, pp. 1065–1072

The general solution of ati elasticity theory problem for a constant thickness plate is constructed under the condition that a force and a nonuniformly heated plate are applied normally to the boundary planes. The solution is obtained as a result of applying the M.E. Vashchenko-Zakharchenko expansion formulas to the infinitely high-order differential equations obtained by A.I. Lur'e by a symbolic method [1,2], by a separate analysis of the symmetric and antisymmetric elasticity theory problems relative to the middle plane: 1) for constant temperature and given forces on the boundary planes; 2) for a given nonuniform heating and no forces. Simple formulas are presented to determine the state of stress in the case of a slowly varying external load and temperature of the unbounded plate. For a bounded plate the general solution for no forces on the boundary planes and heating resulted in the A.I. Lur'e solution [1].     

非Fourier温度场分布的奇摄动解

应用非Fourier热传导定律构建了单层材料中温度场模型，即一类在无界域上带小参数的奇摄动双曲方程，通过奇摄动展开方法，得到了该问题的渐近解.首先应用奇摄动方法得到了该问题的外解和边界层矫正项，通过对内解和外解的最大模估计和关于时间导数的最大模估计以及线性抛物方程理论，得到了内外解的存在唯一性，从而得到了解的形式渐近展开式.通过余项估计，给出了渐近解的L2估计，得到了渐近解的一致有效性，从而得到了无界域上温度场的分布.通过奇摄动分析，给出了非Fourier 温度场与Fourier 温度场的关系，描述了非Fourier温度场的具体形态.     

An Integrodifferential Equation Arising in Thermo-Viscoelasticity

Viscoelastic material at high temperature is subjected to a cooling process. The stresses built up in the body are determined from a system of equations containing a strongly temperature-dependent viscosity η(T), where the temperature T is given by the heat conduction equation. It is shown that for simple geometries such as infinite cylinders and spheres, the basic equations can be reduced to a single Volterra-type integrodifferential equation, which is shown to have a unique solution.     

The modified regularization method for identifying the unknown source on Poisson equation

This paper discusses the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation from one supplementary temperature measurement at an internal point. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the modified regularization method. For the regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is given. Numerical results are presented to illustrate the accuracy and efficiency of this method.     

The applicability of a hereditary model of wear with an exponential kernel in the one-dimensional contact problem taking frictional heat generation into account

An analytic solution is obtained for the contact problem for a stiff thermally insulated plate and an elastic heat-conducting layer, subject to the conditions of wear and frictional heating, when the contacting bodies are not drawn nearer. The evolution of the contact pressure, the temperature and the wear are traced. Conditions for the occurrence of thermoelastic instability are established. The conditions under which the wear model considered is applicable are given.     

ON STABILITY AND REGULARIZATION FOR BACKWARD HEAT EQUATION

Consider a backward heat equation in a bounded domain Ω (?) R2 with the noisy data in the initial time geometry. The aim is to find the temperature for 0 < ε < t < T. For this ill-posed problem, the authors give a continuous dependence estimate of the solution. Moreover, the convergence rate of the approximate solution is also given.     

自然对流换热问题基于混合元法的差分格式及其数值模拟

研究自然对流换热问题,通过对于空间变量采用有限元离散而对于时间变量用差分离散,导出一种基于混合有限元法的最低阶的差分格式,这种格式可以同时求出流体的速度、温度和压力的数值解,并给出了模拟方腔流的自然换热的数值例子。     

The optimum anisotropy in problems of heat conduction in solids

Some problems of optimizing the internal structure of solids, made of a material which is locally orthotropic with respect to the heat-conducting properties, are formulated. The state variable (the inverse temperature) is determined from the solution of the boundary value problem of heat conduction. The orthogonal rotation tensor, which defines the optimum orientation of the orthotropy axes of the material that delivers an extremum to the dissipation functional, is used as the control variable. The necessary conditions for an extremum are derived and some properties of the equations defining the optimal structures are investigated. Examples are given of the solution of problems of the optimum arrangement of the orthotropic material, and the possibility of effectively using the membrane analogy for this purpose is pointed out.     

Problems of determining the temperature and density of heat sources from the initial and final temperatures

Under consideration are the three problems that simulate the process of determining the temperature and density of heat sources from some given initial and final temperatures. In the course of their mathematical formulation, some inverse problems arise for the heat transfer equation in which, together with the solution of the equation, one needs also to find the unknown right-hand side that depends only on the spatial variable. The existence and uniqueness theorems are proved for the solution.     

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the typey″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.     

Numerical methods for coupled systems of quasilinear elliptic equations with nonlinear boundary conditions

This paper is concerned with numerical solutions of a coupled system of arbitrary number of quasilinear elliptic equations under combined Dirichlet and nonlinear boundary conditions. A finite difference system for a transformed system of the quasilinear equations is formulated, and three monotone iterative schemes for the computation of numerical solutions are given using the method of upper and lower solutions. It is shown that each of the three monotone iterations converges to a minimal solution or a maximal solution depending on whether the initial iteration is a lower solution or an upper solution. A comparison result among the three iterative schemes is given. Also shown is the convergence of the minimal and maximal discrete solutions to the corresponding minimal and maximal solutions of the continuous system as the mesh size tends to zero. These results are applied to a heat transfer problem with temperature dependent thermal conductivity and a Lotka-Volterra cooperation system with degenerate diffusion. This degenerate property leads to some interesting distinct property of the system when compared with the non-degenerate semilinear systems. Numerical results are given to the above problems, and in each problem an explicit continuous solution is constructed and is used to compare with the computed solution