Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this Letter; one is He's variational iteration method (VIM) and the other is the homotopy–perturbation method (HPM). Nonlinear convective–radiative cooling equations are used as examples to illustrate the simple solution procedures. These methods are useful and practical for solving the nonlinear heat diffusion equation, which is associated with variable thermal conductivity condition. Comparison of the results obtained by both methods with exact solutions reveals that both methods are tremendously effective.     

Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method

He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient.     

Numerical simulation of the regularized long wave equation by He's homotopy perturbation method

In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions.     

Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations

In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

Application of He's variational iteration method to nonlinear heat transfer equations

Instead of finding a small parameter for solving nonlinear problems through perturbation method, a new analytical method called He's variational iteration method (VIM) is introduced to be applied to solve nonlinear heat transfer equations in this Letter. In this research, variational iteration method is used to solve an unsteady nonlinear convective-radiative equation and a nonlinear convective-radiative-conduction equation containing two small parameters of ε₁ and ε₂ and evaluate the efficiency of straight fins. VIM can apply to the nonlinear equations with boundary or initial conditions defined in different points just with developing the correction functional using the extra parameters such as Cn, as used in this Letter.     

Solution to the MHD flow over a non-linear stretching sheet by homotopy perturbation method

In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is obtained.The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve.HPM produces analytical expressions for the solution to nonlinear differential equations.The obtained analytic solution is in the form of an infinite power series.In this work,the analytical solution obtained by using only two terms from HPM soluti...     

竖直恒温面自然对流边界层不稳定性研究

在无扰动、随机式扰动以及正弦式扰动下,通过对竖直恒温面处状态Ra为1.328×10^9、Pr为6.24的自然对流进行模拟,探索了热边界层的不稳定性和共振强化自然对流换热。结果表明:(1)竖直自然对流边界层上游位置的随机式扰动对热边界层的影响主要体现在稳定阶段;(2)该状态下的竖直自然对流边界层的特征频率为15 067,且相比于无扰动状态,频率为15 067的正弦式扰动能在竖直恒温面处提高5.15%的换热量;(3)在竖直自然对流边界层上游位置加入特征频率的正弦式扰动,竖直恒温面处的局部努塞尔数Nu均出现明显波动,且波动随着边界层高度的增加而增大。     

Series solutions of stagnation slip flow and heat transfer by the homotopy analysis method

An analytical approximation for the similarity solutions of the two- and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion method, but it is different from the perturbation technique, because it is independent of small physical parameters at all. Instead, it is based on a continuous mapping in topology so that it is applicable for not only weakly but also strongly nonlinear flow phenomena. Convergent [m,m] homotopy Padé approximants are obtained and compared with the numerical results and the asymptotic approximations. It is found that the homotopy Padé approximants agree well with the numerical results. The effects of the slip length ℓ and the thermal slip constant β on the heat transfer characteristics are investigated and discussed. Supported by the National Natural Science Foundation of China (Grant No. 10872129)     

The application of He's variational iteration method to nonlinear equations arising in heat transfer

In this Letter, we purpose to solve nonlinear equations arising in heat transfer through the He's variational iteration method (VIM) and show that it is strongly and simply capable of solving a large class of linear or nonlinear differential equations without the tangible restriction of sensitivity to the degree of the nonlinear term and also is very user friend because it reduces the size of calculations besides, its iterations are direct and straightforward. VIM can apply to the nonlinear equations with boundary or initial conditions defined in different points just with developing the correction functional using the extra parameters such as cn$c_n$, as used in this Letter.     

Heat transfer in boundary layer stagnation-point flow towards a shrinking sheet with non-uniform heat flux

In this paper, the effect of non-uniform heat flux on heat transfer in boundary layer stagnation-point flow over a shrinking sheet is studied. The variable boundary heat fluxes are considered of two types: direct power-law variation with the distance along the sheet and inverse power-law variation with the distance. The governing partial differential equations (PDEs) are transformed into non linear self-similar ordinary differential equations (ODEs) by similarity transformations, and then those are solved using very efficient shooting method. The direct variation and inverse variation of heat flux along the sheet have completely different effects on the temperature distribution. Moreover, the heat transfer characteristics in the presence of non-uniform heat flux for several values of physical parameters are also found to be interesting.     

Endoscopic Effects onPeristaltic Flow of a Nanofluid

In the present investigation we have studied the peristaltic flow of a nanofluid in an endoscope. The flow is investigated in a wave frame of reference moving with velocity of the wavec. Analytical solutions have been calculated using Homotopy perturbation method (HPM) for temperature and nanoparticle equation while exact solutions have been calculated for velocity and pressure gradient. Numerical integration have been used to obtain the graphical results for pressure rise and frictional forces. The effects of various emerging parameters are investigated for five different peristaltic waves.
Streamlines have been plotted at the end of the article.     

The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation

Here, an analytic technique, namely the homotopy analysis method (HAM), is applied to solve a generalized Hirota–Satsuma coupled KdV equation. HAM is a strong and easy-to-use analytic tool for nonlinear problems and dose not need small parameters in the equations. Comparison of the results with those of Adomian's decomposition method (ADM) and homotopy perturbation method (HPM), has led us to significant consequences. The homotopy analysis method contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series.     

Numerical study of the heat transfer and electro-thermo-convective flow patterns in dielectric liquid layer subjected to unipolar injection

In this article we study the electro-thermal convection in a dielectric liquid layer placed between two electrodes and subjected to the simultaneous action of an electric field and a thermal gradient. The full set of equations describing the electro-thermo-convective phenomena is directly solved using a finite volume method. We first heat the liquid from below at time t = 0, wait for the thermal steady state and then inject the electric charges by applying the electric potential. The development of the electro-convective motion is analysed in detail in two cases: 1) strong injection from the lower electrode, 2) strong injection from the upper one. We also study the heat transfer enhancement due to electro-convection. The evolution in time of the Nusselt number Nu for different combinations of the two usual non-dimensional parameters associated to the electro-thermo-convection phenomena (Rayleigh number Ra and the electrical parameter T) is also given and analysed.     

Casson fluid flow and heat transfer over a nonlinearly stretching surface

A boundary layer analysis is presented for non-Newtonian fluid flow and heat transfer over a nonlinearly stretching surface. The Casson fluid model is used to characterize the non-Newtonian fluid behavior. By using suitable transformations, the governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear ordinary differential equations. Numerical solutions of these equations are obtained with the shooting method. The effect of increasing Casson parameter is to suppress the velocity field. However the temperature is enhanced with the increasing Casson parameter.     

Convective heat transport in a fluid layer of infinite Prandtl number: upper bounds for the case of rigid lower boundary and stress-free upper boundary

We present the theory of the multi--solutions of the variational problem for the upper bounds on the convective heat transport in a heated from below horizontal fluid layer with rigid lower boundary and stress-free upper boundary. A sequence of upper bounds on the convective heat transport is obtained. The highest bound is between the bounds for the case of a fluid layer with two rigid boundaries and for the case of a fluid layer with two stress-free boundaries. As an additional result of the presented theory we obtain small corrections of the boundary layer thicknesses of the optimum fields for the case of fluid layer with two rigid boundaries. These corrections lead to systematically lower upper bounds on the convective heat transport in comparison to the bounds obtained in [5]. Received 29 September 1999     

双液滴撞击平面液膜的流动与传热特性

采用耦合的水平集-体积分数法(CLSVOF)对双液滴连续撞击恒定壁温壁面上的热液膜的流动和换热特性进行了数值模拟及分析,得到了双液滴撞击热液膜后形态演变的过程.分析了液滴垂直间距、撞击速度、液膜厚度以及液滴直径对双液滴撞击液膜后的流动与传热特性的影响,结果显示,壁面平均热流密度随液滴撞击速度的增大而增大,液滴垂直间距、液膜厚度和液滴直径对平均热流密度的影响较小,但会对热流密度在撞击区域和交界区的分布产生重要影响.     

Similarity solutions for boundary layer flow and heat transfer of a FENE-P fluid with thermal radiation

This Letter presents a numerical study of the flow and heat transfer of an incompressible FENE-P fluid over a non-isothermal surface. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation. The effects of the thermal radiation are considered in the energy equation, and the variations of dimensionless surface temperature and dimensionless surface temperature gradient, as well as the heat transfer characteristics with various physical parameters are graphed and tabulated. Two cases are studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the sheet with prescribed heat flux (PHF case). Moreover, the mechanical characteristics of the corresponding flow are also presented.     

蒙特卡罗方法在解微分方程边值问题中的应用

介绍了蒙特卡罗方法的基本原理以及随机数的产生方法。基于蒙特卡罗方法的思想,结合有限差分方法,建立了求解微分方程边值问题的随机概率模型,并以第一类边界条件的拉普拉斯方程和一个给定初值及边界条件的非稳态热传导方程为数值算例,研究了蒙特卡罗方法在求解微分方程边值问题中的应用。结果表明:利用蒙特卡罗方法,不仅可以有效解决给定边界条件的微分方程,对于给定初值条件的微分方程,也可以从时域有限差分方程出发,采用蒙特卡罗方法进行求解。数值模拟和对误差的理论分析均表明,增加蒙特卡罗试验中的模拟粒子点数,可以提高计算结果的精度。     

Verification of numerical simulation method for entropy generation of radiation heat transfer in semitransparent medium

The numerical simulation method of radiative entropy generation in participating media presented by Caldas and Semiao [Entropy generation through radiative transfer in participating media: analysis and numerical computation. JQSRT 2005;96:423-37] is extended to analyze the radiative entropy generation in the enclosures filled with semitransparent media. A discrete ordinates method is used to solve radiative transfer equation and radiative entropy generation. Two different examples are employed to verify the numerical simulation method of radiative entropy generation in the enclosure. Numerical results of dimensionless radiative entropy generation of enclosure are identical to that of entire thermodynamics analysis for the enclosure system. This numerical simulation method can be used in the entropy generation analysis of high-temperature systems such as boilers and furnaces, in which radiation is the dominant mode of heat transfer.     

Application of multidimensional scheme and the discrete ordinate method to radiative heat transfer in a two-dimensional enclosure with diffusely emitting and reflecting boundary walls

Radiative heat transfer is the dominant mode of heat transfer in many engineering problems, including combustion chambers, space, greenhouses, rocket plume sensing, among others. The aim of this study is to develop an efficient method capable of eliminating ray effects in complex 2D situations and to use the developed code for other problems including combined conduction and convection in connection with CFD codes. A complete genuinely multidimensional discretization in two-dimensional discrete ordinates method is formulated to solve radiative heat transfer in a rectangular enclosure composed of diffusely emitting and reflecting boundaries and containing homogeneous media that absorbs, emits and scatters radiation. A new genuinely multidimensional differencing scheme is used to solve the radiative transfer equation with S₄, S₆, S₈, T₆, T₇, T₈ and T₉ angular quadrature schemes. Different cases are analyzed and the results are compared when possible with those obtained by others researchers.