Non-Fourier heat conduction in a finite hollow cylinder with periodic surface heat flux

Analytical solution of the non-Fourier axisymmetric temperature field within a finite hollow cylinder exposed to a periodic boundary heat flux is investigated. The problem studied considering the Cattaneo–Vernotte (CV) constitutive heat flux relation. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The standard method of separation of variables is used for solving the problem with time-independent boundary conditions, and the Duhamel integral is used for applying the time dependency. The solution is applied for the special cases of harmonic uniform heat flux and an exponentially pulsed heat flux with Gaussian distribution in outer surface for modeling a laser pulse, and their respective non-Fourier thermal behavior is studied.     

Unsteady-state temperature distribution in a convecting fin of constant area

A solution for the unsteady-state temperature distribution in a fin of constant area dissipating heat only by convection to an environment of constant temperature, is obtained. The partial differential equation is separated into an ordinary differential equation with position as the independent variable, and a partial differential equation with position and time as the independent variables. The problem is solved for either a step function in temperature or a step function in heat flow rate, for zero time, at one boundary while the other boundary is insulated. The initial condition is taken as an arbitrary constant. The unspecified boundary values (temperature or heat flow rate) are presented for both cases by utilizing dimensionless plots. Experimental verification is presented for the case of constant heat flow rate boundary condition.     

Determination of the thermal stress wave propagation in orthotropic hollow cylinder based on classical theory of thermoelasticity

The thermoelasticity problem in a thick-walled orthotropic hollow cylinder is solved analytically using finite Hankel transform and Laplace transform. Time-dependent thermal and mechanical boundary conditions are applied on the inner and the outer surfaces of the cylinder. For solving the energy equation, the temperature itself is considered as boundary condition to be applied on both the inner and the outer surfaces of the orthotropic cylinder. Two different cases are assumed for solving the equation of motion: traction–traction problem (tractions are prescribed on both the inner and the outer surfaces) and traction–displacement (traction is prescribed on the inner surface and displacement is prescribed on the outer surface of the hollow orthotropic cylinder). Due to considering uncoupled theory, after obtaining temperature distribution, the dynamical structural problem is solved and closed-form relations are derived for radial displacement, radial and hoop stress. As a case study, exponentially decaying temperature with respect to time is prescribed on the inner surface of the cylinder and the temperature of the outer surface is considered to be zero. Owing to solving dynamical problem, the stress wave propagation and its reflections were observed after plotting the results in both cases.     

Heat Transfer and Evaporation from a Plane Surface into a Half-Space upon a Sudden Increase in Body Temperature

The one-dimensional time-dependent problem of evaporation from a plane body surface into a half-space filled by a gas (condensed phase vapor) upon a sudden increase in the body surface temperature is studied. The evaporation coefficient is the problem parameter and may take arbitrary values within the limits from zero to unity. The problem is formulated for the kinetic equation and solved by the finite-difference method. It is shown that a deviation of the evaporation coefficient from unity considerably modifies the gas phase flow pattern. However, the evaporation rate divided by the rate of evaporation into a vacuum at the given surface temperature is only weakly dependent on the evaporation coefficient.     

An exact axisymmetric solution for a simply supported piezoelectric cylindrical shell

Summary Three-dimensional axisymmetric solution is presented for a simply supported piezoelectric cylindrical shell. The variables are expanded in Fourier series to satisfy the boundary conditions at the ends. The solution of the governing differential equations with variable coefficients is constructed as a product of an exponential function and a power series. The coefficients of terms of all degrees in the governing equations are set to zero, yielding a characteristic equation for the exponent and recursive relations for the coefficients of the power series. Results are presented illustrating the effect of thickness parameter of the shell. An inverse problem of inferring the applied temperature from the measured potential difference has been solved. Accepted for publication 26 July 1996     

Low speed MHD Couette flow of a slightly rarefied and electrically conducting gas

The effects of an applied magnetic field on the steady, laminar, low speed plane Couette flow of a slightly rarefied and electrically conducting gas are studied. Consideration is given to the slip-flow regime, wherein the gas rarefaction begins to play its important role. The generally accepted method of analysis for slip flows is utilized, i.e. the continuum magnetohydrodynamic equations of motion are used throughout the gas, together with the first and the second order slip velocity and temperature jump boundary conditions. Considerations are further given to (1) the case of zero electric field and (2) the case of a nonconducting channel in which the net current across the channel is zero.     

Thermo-fluid dynamics of the unsteady channel flow

We present an exact analytical representation of the unsteady thermo-fluid dynamic field arising in a two-dimensional channel with parallel walls for a fluid with constant properties. We assume that the axial pressure gradient is an arbitrary function of time that can be expanded in Taylor series; a particular case is the impulsive motion generated by a sudden jump to a constant value; for large time values the flow reaches the well-known steady Poiseuille solution. As boundary conditions for the dynamic field we consider fixed and moving walls (unsteady Couette flow). The assigned temperature on the walls can be an arbitrary function of time. We also consider the coupling of the energy and momentum equations (i.e. Eckert number different from zero). The solution is obtained by series with simple expressions of the coefficients in terms of the error functions. The fundamental physical parameters, such as shear stress, mass flow and heat flux at the wall are obtained in explicit analytical form and discussed by means of their diagrams.     

Axisymmetric non-Fourier temperature field in a hollow sphere

The non-Fourier axisymmetric (2+1)-dimensional temperature field within a hollow sphere is analytically investigated by the solution of the well-known Cattaneo–Vernotte hyperbolic heat conduction equation. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The method of solution is the standard separation of variables method. General linear time-independent boundary conditions are considered. Ultimately, the presented solution is applied to a (1+1)—as well as a (2+1)—dimensional problem, and their respective non-Fourier thermal behavior is studied. The present solution can be reduced to special cases of interest by choosing appropriate boundary conditions parameters. Dedicated to Prof. Gholamali Atefi, with appreciation and admiration on the occasion of his 65th birthday.     

一维半无限压电杆的广义的热冲击问题

采用具有两个热松驰时间的G－L广义热弹性理论，研究了一维无限无限长杆在其端部受到热冲击时的边值问题，借助于拉普拉斯正、反变换技术，在所考虑时间非常短的情况下，对问题进行了求解。得到了位移及温度分布的近似妥析角，发现位移及温度分布中分别存在两上阶跃点，并通过数值计算，把温度的分布规律用图形反映了出来，从温度的分布图上可以看出，当任何x的值大于第二个阶跃点的位置值时，温度值都是零，也即在当前所绘定的时刻，热以波的形式沿压电杆仅传播到第二阶跃点的位置，而在第二个阶跃点之后，压电杆上的温度分布保持初始温度；定不同时刻，热波波前的位置也将相应的在压电杆上移动，也即热波波前在压电杆上的位置随考虑时刻不同而不同，这与经典的热传导是完全不同的，它说明热是以波的形式以有限的速度，而不是以无限的速度在介质中进行传播的。     

The effect of fractional parameter on a perfect conducting elastic half-space in generalized magneto-thermoelasticity

Recently, Sherief et al. (Int. J. Solids Struct. 47:269–275, 2010) proposed a model in generalized thermoelasticity based on the fractional order time derivatives. The propagation of electro-magneto-thermoelastic disturbances in a perfectly conducting elastic half-space is investigated in the context of the above fractional order theory of generalized thermoelasticity. There acts an initial magnetic field parallel to the plane boundary of the half-space. Normal mode analysis together with the eigenvalue approach technique is used to solve the resulting non-dimensional coupled governing equations of the problem. The obtained solution is then applied to two specific problems for the half-space, whose boundary is subjected to (i) thermally isolated surfaces subjected to time-dependent compression and (ii) a time-dependent thermal shock and zero stress. The effects of fractional parameter and magnetic field on the variations of different field quantities inside the half-space are analyzed graphically.     

磁-电-弹性半空间在轴对称热载荷作用下的三维问题研究

考虑力-电-磁-热等多场耦合作用,基于线性理论给出了磁-电-弹性半空间在表面轴对称温度载荷作用下的热-磁-电-弹性分析,并得到了问题的解析解.利用Hankel积分变换法求解了磁-电-弹性材料中的热传导及控制方程,讨论了在磁-电-弹性半空间在边界表面上作用局部热载荷时的混合边值问题,利用积分变换和积分方程技术,通过在边界表面上施加应力自由及磁-电开路条件,推导得到了磁-电-弹性半空间中位移、电势及磁势的积分形式的表达式.获得了磁-电-弹性半空间中温度场的解析表达式并且给出了应力,电位移和磁通量的解析解.数值计算结果表明温度载荷对磁-电-弹性场的分布有显著影响.当温度载荷作用的圆域半径增大时,最大正应力发生的位置会远离半无限大体的边界;反之当温度载荷作用的圆域半径减小时,最大应力发生的位置会靠近半无限大体的边界.电场和磁场在温度载荷作用的圆域内在边界表面附近有明显的强化,而磁-电-弹性场强化区域的强化程度跟温度载荷的大小和作用区域大小相关.本研究的相关结果对智能材料和结构在热载荷作用下的设计和制造具有指导意义.     

THREE-DIMENSIONAL ANALYSIS OF A MAGNETOELECTROELASTIC HALF-SPACE UNDER AXISYMMETRIC TEMPERATURE LOADING 1)

考虑力-电-磁-热等多场耦合作用, 基于线性理论给出了磁-电-弹性半空间在表面轴对称温度载荷作用下的热-磁-电-弹性分析, 并得到了问题的解析解. 利用Hankel 积分变换法求解了磁-电-弹性材料中的热传导及控制方程, 讨论了在磁-电-弹性半空间在边界表面上作用局部热载荷时的混合边值问题, 利用积分变换和积分方程技术, 通过在边界表面上施加应力自由及磁-电开路条件, 推导得到了磁-电-弹性半空间中位移、电势及磁势的积分形式的表达式. 获得了磁-电-弹性半空间中温度场的解析表达式并且给出了应力, 电位移和磁通量的解析解. 数值计算结果表明温度载荷对磁-电-弹性场的分布有显著影响. 当温度载荷作用的圆域半径增大时, 最大正应力发生的位置会远离半无限大体的边界; 反之当温度载荷作用的圆域半径减小时, 最大应力发生的位置会靠近半无限大体的边界. 电场和磁场在温度载荷作用的圆域内在边界表面附近有明显的强化, 而磁-电-弹性场强化区域的强化程度跟温度载荷的大小和作用区域大小相关. 本研究的相关结果对智能材料和结构在热载荷作用下的设计和制造具有指导意义.     

Electromagnetohydrodynamic (EMHD) flow of fractional viscoelastic fluids in a microchannel

This study investigates the electromagnetohydrodynamic (EMHD) flow of fractional viscoelastic fluids through a microchannel under the Navier slip boundary condition. The flow is driven by the pressure gradient and electromagnetic force where the electric field is applied horizontally, and the magnetic field is vertically (upward or downward). When the electric field direction is consistent with the pressure gradient direction, the changes of the steady flow rate and velocity with the Hartmann number Ha are irrelevant to the direction of the magnetic field (upward or downward). The steady flow rate decreases monotonically to zero with the increase in Ha. In contrast, when the direction of the electric field differs from the pressure gradient direction, the flow behavior depends on the direction of the magnetic field, i.e., symmetry breaking occurs. Specifically, when the magnetic field is vertically upward, the steady flow rate increases first and then decreases with Ha. When the magnetic field is reversed, the steady flow rate first reduces to zero as Ha increases from zero. As Ha continues to increase, the steady flow rate (velocity) increases in the opposite direction and then decreases, and finally drops to zero for larger Ha. The increase in the fractional calculus parameter α or Deborah number De makes it take longer for the flow rate (velocity) to reach the steady state. In addition, the increase in the strength of the magnetic field or electric field, or in the pressure gradient tends to accelerate the slip velocity at the walls. On the other hand, the increase in the thickness of the electric double-layer tends to reduce it.

     

Mathematical model of heat transfer in a fluid with account for its relaxation properties

Using the terms that take account for the temporal and spatial nonlocality (time variation of the heat flux and the temperature gradient) in the formula of Fourier’s law for the heat flux a differential equation for a fluid in motion is derived that contains the second time derivative and themixed derivative with respect to the spatial and temporal variables. Numerical solution of the problem of heat transfer in the laminar fluid flow in a plane channel demonstrates that, in view of the lag in the time variation of the heat flux from zero to a certain maximum value, the boundary condition of the first kind (thermal shock) cannot be instantaneously realized. The process of its stabilization on the wall is characterized by a certain time interval, whose duration is determined by the relaxation properties of the fluid. At large values of the dimensionless coefficients of the heat flux relaxation and the temperature gradient the boundary condition of the first kind can be realized only as the steady state is attainted, as Fo→∞. In this case, the flow does not contain temperature jumps and negative temperature values.     

Stability of curvilinear Euler-Bernoulli beams in temperature fields

In this work, stability of thin flexible Bernoulli-Euler beams is investigated taking into account the geometric non-linearity as well as a type and intensity of the temperature field. The applied temperature field T(x,z) is yielded by a solution to the 2D Laplace equation solved for five kinds of thermal boundary conditions, and there are no restrictions put on the temperature distribution along the beam thickness. Action of the temperature field on the beam dynamics is studied with the help of the Duhamel theory, whereas the motion of the beam subjected to the thermal load is yielded employing the variational principles.The heat transfer (Laplace equation) is solved with the use of the finite difference method (FDM) of the third-order accuracy, while the integrals along the beam thickness defining the thermal stress and moments are computed using Simpson's method. Partial differential equations governing the beam motion are reduced to the Cauchy problem by means of application of FDM of the second-order accuracy. The obtained ordinary differential equations are solved with the use of the fourth-order Runge-Kutta method.The problem of numerical results convergence versus a number of beam partitions is investigated. A static solution for a flexible Bernoulli-Euler beam is obtained using the dynamic approach based on employment of the relaxation/set-up method.Novel stability loss phenomena of a beam under the thermal field are reported for different beam geometric parameters, boundary conditions, and the temperature intensity. In particular, it has been shown that stability of the flexible beam during heating the beam surface essentially depends on the beam thickness.     

非均匀复合材料的动态热弹性断裂力学分析

对非均匀复合材料的动态热弹性断裂力学问题进行了研究,假设材料参数沿厚度方向为变化的,沿该方向将复合材料划分为许多单层,取每一单层材料参数为常数,应用Fourier变换法,在Laplace域内推导出了控制问题的奇异积分方程组,给出了热应力强度因子的表达式,然后利用Laplace数值反演,得出了裂纹尖端的动态应力强度因子．本文的方法具有以下特点：（1）多个垂直于厚度方向的裂纹,（2）材料可以为正交各向异性：（3）考虑了惯性效应．作为算例,研究了带有两个裂纹的功能梯度结构,分析了材料参数的变化对应力强度因子的影响．     

Nonlinear problem of an electrical discharge in a flow

The nonlinear intial boundary value problem describing the relaxation of a quasi-neutral discharge in a gas flow with coplanar, heated and nonheated electrodes of finite extension is formulated in the diffusion approximation. Relaxation occurs from an initial breakdown to a steady-state or zero discharge in a weak electric field. A nonlinear transformation is applied to get an equivalent nonlinear problem, where nonlinearity is treated as a small perturbation. An analytic solution is obtained and criterions for existence and sustainment of a steady-state discharge against plasma losses due to convection, diffusion and recombination is discussed. Some numerical results are exhibited.     

半无限长压电杆的瞬态热冲击问题

采用具有一个热松弛时间的L-S广义热弹性理论,研究了一维半无限长压电杆在一端受到热冲击时的边值问题.借助拉普拉斯正、反变换技术,在所考虑时间非常短的情况下,对问题进行了求解,得到了压电杆上的位移、压力及温度分布的近似解析解,发现应力及温度分布中分别存在两个阶跃点,并给出了算例.     

Crack spacing effect for a piezoelectric cylinder under electro-mechanical loading or transient heating

This paper investigates the fracture problem of a piezoelectric cylinder with a periodic array of embedded circular cracks. An electro-mechanical fracture mechanics model is established first. The model is further used to the thermal fracture analysis of a piezoelectric cylinder subjected to a sudden heating on its outer surface. The temperature field and the associated thermal stresses and electric displacements are obtained and are added to the crack surface to form a mixed-mode boundary value problem for the electro-mechanical coupling fracture. The stress and stress intensities are investigated for the effect of crack spacing. Strength evaluation of piezoelectric materials under the transient thermal environment is made and thermal shock resistance of the medium is given.