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In the present study, the problem of non-linear model arising in heat transfer
through the porous fin in a natural convection environment is presented and the
homotopy perturbation method is employed to obtain an approximate solution,
which admits a remarkable accuracy.
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We use variational homotopy perturbation method (VHPM) [M. A. Noor and S. T. Mohyud-Din, Math. Probl. Eng. 2008, Article ID 696734 (2008; Zbl 1155.65082)] for solving Fisher’s equation and also compare the approximations obtained by the variational homotopy perturbation method with the approximations obtained by the variational iteration method [M....
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Citations
... Upon comparing similar exponents of p on both sides of Equation (18), we obtain βKNr Sh· Hoshyar et al. 32 and Petroudi et al. 33 βK Nr Sh ...
A unique investigation has been undertaken to analyze the heat transmission by convective and radiative mechanisms in a fully saturated penetrable fin of a longitudinal structure positioned on a leaning surface. This study introduces the fusion of the realms of Homotopy perturbation and Sumudu transform techniques to address a previously unexplored problem involving a moving fin with temperature‐dependent thermal conductivity. In prior research papers, the Homotopy Perturbation Sumudu Transform Method (HPSTM) was utilized to obtain analytical solutions for fins featuring temperature‐dependent thermal conductivity. However, in our current study, we employ the HPSTM to tackle a novel problem involving a moving porous fin. This fin exhibits temperature‐dependent thermal conductivity and is subjected to convection and radiation effects. Through a comparison with numerical results, the present study has validated the dependability of its findings. The dimensionless temperature profile has been investigated by studying its relationship with several parameters. Here we observed that when the Peclet number ( Pe ) $({Pe})$ is augmented by 400%, there is a corresponding 1.11% increase in thermal outline at the fin's extremity. Enhancing the value of radiation parameter Nr ${Nr}$ by 400% declines the temperature of the fin tip by 14.079%. This study encourages the application of the Homotopy perturbation Sumudu transform technique in more complex fin problems.
... Thermal enhancements and augmentations in thermal and electronics have been passively achieved by fins and spines (Fig. 2). The thermal characterizations of the passive devices have been well explored (Kiwan & Nimr, 2001;Kiwan, 2007aKiwan, & 2007bKiwan & Zeitoun, 2008;Abbasbandy et al., 2011;Bhanja & Kundu, 2011;Gorla and Bakier, 2011;Shivanian & Hashim, 2011;Saedodin & Olank, 2011;Petroudi et al. 2012;Kundu et al., 2012;Saedodin & Shahbabaei, 2013;Ghasemi et al., 2014;Hatami et al., 2014;Rostamiyan et al., 2014;Darvishi et al. 2015;Sobamowo, 2016). ...
Passive cooling of electronics systems requires the use of fins and spines. Improved heat transfer enhancements in the systems have been achieved through porosity, magnetic field, functionally graded materials, etc. However, the space and time-dependent analysis of passive device with inherent nonlinearities and explorations of conditions at the tip have not been fully presented. Therefore, this work presents an application of hybrid computation analysis using combined methods of Laplace transformation and Legendre wavelet collocation to the space and time-dependent nonlinear thermal models of a longitudinal conductive-radiative functionally graded extended surface. The spatial-dependent thermal conductivity is considered for three different cases. The present analysis results demonstrated good agreements with numerical results. The significances of the model pertinent parameters on the fin's performance are thoroughly considered and scrutinized by the aid of graphical illustrations. The fin tip thermal response decreases with increase in conductive-convective parameter, but it increases as time progresses. The in-homogeneity index is directly proportional to the FGM fin temperature. However, the convective term is inversely proportional to the fin's thermal distribution. The fin temperature under quadratic-law heat conductivity of the passive device shows an enhanced performance of lower thermal distribution than the linear-and exponential-law heat conductivity. Ultimately, the study provides a very useful in the design of the fin with FGM.
... Fins have been extensively used for heat transfer augmentation in thermal systems [1]. Consequently, there have been different studies on their thermal responses when they are subjected to heat transfer [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The mathematical models of the thermal response of the passive devices are nonlinear which are very hard to be solved explicitly. ...
The present paper develops explicit and non-power series exact solutions to the nonlinear heat transfer models of conductive-radiative-convective moving-porous fin using Laplace transform-Exp-function method, is presented. The developed solutions are employed for investigation of the included parameters on the transient and steady states studies of the moving-porous fin. The results submitted that the fin temperature is augmented with time increase due to increase heat transfer rate as time progresses. However, thermal parameter of the fin reduces from the its base to its tip. As the porosity, moving, convective-conductive-radiative parameters are increased, the fin temperature are decreased due to increased heat transfer rate. The opposite trend is displayed for the conductive-radiative number. It can be stated that present work will be useful in the analysis of the device.
... Such important passive method of heat transfer enhancements has provoked several studies over the past decades. The study of thermal behavior of the passive device has become an area of increasing research interests [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], just to mention a few. In the bid of theoretical investigations of thermal damage problems and heat transfer enhancement by the extended surfaces, the controlling thermal models of the passive devices are always nonlinear. ...
Developments of analytical solutions to nonlinear equations provide very good physical insights
into the nonlinear phenomena. The nonlinear transient thermal responses of extended surfaces have
been widely studied analytically with the aid of different methods based on power series solutions.
Such solutions come with large number of terms which are not convenient for use in practice. In
this work, a non-power series solution is presented for the nonlinear transient thermal analysis of
the passive device. Also, a comparative study of two analytical methods, Laplace transform-
Galerkin weighted residual and differential transformation methods is presented to analyze
transient nonlinear thermal model of radiative-convective fin considering varying thermal
conductivity. The obtained analytical solutions are verified analytically and numerically, and very
good agreements are established. The symbolic results are employed to check the influences of
model parameters on passive device’s thermal performance. It is found that as the conductiveconvective,
and conductive-radiative parameters increase, temperature distribution decreases since
heat transfer becomes augmented and hence, the fin thermal efficiency is improved. The
temperature distribution increases through the fin as thermal conductivity increases. At the
different positions in the fin, the temperature increases with an increase in time. The time histories
of the solution shows that transient solutions converge to a steady state and the fin tip temperature
increases as time progresses. The developed approximate analytical solutions provide very good
platforms for the nonlinear thermal analyses of fins and proper design of the extended surfaces in
thermal systems.
... Most of the phenomena are represented as non-linear equations, and there is difficulty in finding an exact solution to these problems, so researchers resort to finding an approximate solution analytically or numerically. Many efforts have been made to solve nonlinear differential equations by variety of methods such as Homotopy perturbation method (HPM) ISSN: 0067-2904 [1,2], semi-analytical method with Laplace transform [3], Adomian decomposition method (ADM), variation iteration method (VIM) and finite difference method (FDM) [4]. Many numerical and analytical methods have been developed such as finite-difference methods, weighted residuals methods, Adomian decomposition method and variational iteration method. ...
... In this section, HWCM is applied , which is discussed in the previous section, to solve equation (8) with its boundary conditions (9). The highest derivative of equation (8) (27) and (32) in equation (8), we get ∑ ( ) ( ∑ ( ) ) (33) Solving (33) by using numerical calculation that gets from Mathematica Software and choose the value ,that is proposed in [1] we get the Haar coefficients * + Substituting these coefficients in (32), we get the HWCM solution of (8). ...
In this article, the boundary value problem of convection propagation through the permeable fin in a natural convection environment is solved by the Haar wavelet collocation method (HWCM). We also compare the solutions with the application of a semi-analytical method , namely the Temimi and Ansari (TAM), that is characterized by accuracy and efficiency.The proposed method is also characterized by simplicity and efficiency. The possibility of applying the proposed method to many types of linear or nonlinear ordinary and partial differential equations.
... An excursion into the previous published studies confirmed that the controlling models for investigating the thermal performance of fins are nonlinear. Consequently, various methods in mathematical analysis have found applications in obtaining solutions to nonlinear models [9][10][11][12][13][14][15][16][17][18]. ...
In this work, finite element analysis of the transient thermal response of a convective-radiative fin with functionally graded material under the influence of temperature-variant magnetic field is presented. The thermal conductivity of the fin with the functionally graded fin is a linear, quadratic, and exponential variation of space. The comparisons of the exact analytical solution with the numerical solution show good agreement between the two results. The parametric studies reveal that the temperature of the extended surface increases as the in-homogeneity index. The impact of the in-homogeneity index reduces as the value of the thermo-geometric parameter increases. The change in the fin temperature becomes very insignificant for higher values of the thermo-geometric parameter. This same trend is also noticed under increasing values of the magnetic field and radiative parameters. The augmentations of the radiation and magnetic field parameters lead to a decrease in the thermal distribution of the extended surface which consequently increases the rate of heat transfer through the passive device. The thermal distribution increases as time progresses and later converges to a steady state of heat as time evolves. The material under the exponential law function needs a smaller time to reach a steady state than when the material operates under the linear and quadratic law. The results depict that the application of the passive device with functionally graded material reduces the thermal resistance along the length of the fin. Therefore, there is a significantly higher rate of energy saving in an extended surface with functionally graded material than in a heat sink with homogenous material.
... Although, as pointed out in the review of the previous studies, there are various approximate analytical and numerical solutions that gained applications in solving the thermal problems [45][46][47][48][49][50][51][52], most of these solutions involve power series. Indubitably, such power series solutions require rigorous solution procedures with inherent large number of terms which are not convenient for use in practice [15]. ...
The present investigation is concerned with the development of non-power series analytical solutions for the transient nonlinear thermal model of a radiative-convective fin having temperature-variant thermal conductivity using Laplace transform-Galerkin weighted residual method. In the study, it is demonstrated that the analytical solutions do not involve a large number of terms, complex mathematical analysis, high computational cost, and time. The solutions allow effective predictions of the extended surface thermal performance over a large domain and time. The results of the non-power series solutions are verified numerically, and very good agreements are established. Parametric studies are carried out with the aid of the symbolic non-power solutions. It is found that as the conductive-convective and conductive-radiative increase, temperature distribution decreases since the rate of heat transfer becomes augmented and hence, the fin thermal efficiency is improved. Additionally, when the thermal conductivity of the fin increases, the temperature distribution in the passive device increases. The temperature increases with time at the different positions in the fin. Following the time histories of the solution, it is shown that unsteady state solutions converge to a steady state as time progresses. It could therefore be stated the developed non-power series analytical solutions provide a good platform for comparison of the nonlinear thermal analyses of fins in thermal systems.
... Bhanja and Kundu (2011) numerically investigated the behavior of heat transfer in the absence of convection from the porous fin surface and the impacts of the effective thermal conductivity and porosity. Petroudi et al. (2012) investigated the energy dissipation from a porous fin to develop a non-linear model for the heat transport in straight porous fins using a perturbation technique called the HPM. Kundu et al. (2012) used an ADM to calculate maximum heat transfer in porous fins with different profiles, such as rectangular, convex, and exponential forms. ...
In this paper, the mathematical model of an inclined longitudinal porous fin of trapezoidal, rectangular, and dovetail profiles in the presence of convective and radiative environments is considered to study the heat transfer and heat distribution within the fin. The governing equation for the energy transfer in the porous fin is derived by using the Darcy model that simulates the interaction of fluids and solids. The mathematical model has been analyzed so that a common equation can be used to study the trapezoidal, rectangular, and dovetail profiles. Furthermore, to study the temperature distribution in the fin, a supervised machine learning algorithm is developed using Cascade feedforward backpropagated (CFB) neural networks and Levenberg–Marquardt (LM) algorithm. A reference solution of 1001 points for supervised learning of the design scheme is generated by using a numerical solver (RK-4), which is further utilized by the CFB-LM algorithm with the Log-Sigmoid activation function to train, validate and test the data properly. The design algorithm’s outcomes are compared to the results of the homotopy perturbation method, shooting method, and other machine learning algorithms. Extensive graphical and statistical analyses are conducted to study the influence of variations in inclination angle, tip tapering, wet porous parameter, internal heat generation, porosity, progressive natural convective parameter, and dimensionless radiative parameter on the thermal profile and heat transfer rate of the longitudinal porous fin. The dovetail fin profile achieves the maximum heat transfer rate, followed by rectangular and trapezoidal fin profiles, provided that internal heat production is kept to a minimum.
... Also, Rostamiyan et al. (2014) applied variational iterative method (VIM) to provide analytical solution for heat transfer in porous fin. Ghasemi et al. (2014) used differential transformation method (DTM) for heat transfer analysis in porous and solid fin while Petroudi et al. (2012) utilized both HPM and HAM to solve nonlinear equation arising in a natural convection porous fin. Amirkolaei et al. (2014) applied homotopy analysis method and collocation method while Hoshyar et al. (2016) used least square method to predict the temperature distribution in a porous fin which is exposed to uniform magnetic field. ...
In this work, a comparative study of two approximate analytical methods for the thermal behaviour of convective-radiative porous fin subjected to the magnetic field using homotopy perturbation and differential transform methods is presented. Also, parametric studies of the effects of thermal-geometric and thermo-physical fin parameters are investigated. From the study, it is found that an increase in a magnetic field, porosity, convective, radiative, and parameters increase the rate of heat transfer from the fin and consequently improves the efficiency of the fin. There are good agreements between the results of the homotopy perturbation and differential transform method with the results of the numerical method. Also, the results of the two approximate analytical methods agree very well with each other. It is hoped that the present work will serve as the basis of verifications of the other works on the nonlinear thermal analysis of the extended surface.
... In recent decades, much attention has been devoted to the newly-developed methods to construct an analytic solution of equation; such as Perturbation techniques which are too strongly dependent upon the so-called small parameters (SP) [3]. Many other different methods have been introduced to solve nonlinear equations [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] such as the Adomian decomposition method (ADM), homotopy perturbation method (HPM), variational iteration method (VIM), differential transformation method (DTM), homotopy asymptotic method (HAM) and etc. The main purpose of this study is to apply new analytical method to find approximate analytical solutions (A-AS) of the velocity profile on MHD Jeffery-Hamel flow with nanoparticles. ...
The aim of this paper is to analyze the problem of magneto hydrodynamic Jeffrey-Hamel flow (JHF) with nanoparticles. The governing equations for this problem are reduced to an ordinary differential equation and it is solved using new analytical method (NAM) and fourth-order Runge-Kutta Method (RK ∼ 4). The NAM is an iterative method that relies mainly on derivatives with Taylor expansion interference. In addition, the velocity profile has been computed and shown for various values of physical parameters. The objective of the present work is to investigate the effect of the angles between the plates, Reynold number, magnetic number and nanoparticles volume fraction on the velocity profile.
