Comment on “A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition” by A. Aziz,Comm. Nonlinear Sci. Numer. Simul. 2009;14:1064–8

In a recent paper published in this Journal the title problem has been investigated numerically. In the present paper the exact solution for the temperature boundary layer is given in terms of the solution of the flow problem (the Blasius problem) in a compact integral form.     

Perturbation analysis of the heat transfer in porous media with small thermal conductivity

An approximate analytical solution for the one-dimensional problem of heat transfer between an inert gas and a porous semi-infinite medium is presented. Perturbation methods based on Laplace transforms have been applied using the solid thermal conductivity as small parameter. The leading order approximation is the solution of Nusselt (or Schumann) problem. Such solution is corrected by means of an outer approximation. The boundary condition at the origin has been taking into account using an inner approximation for a boundary layer. The gas temperature presents a discontinuous front (due to the incompatibility between initial and boundary conditions) which propagates at constant velocity. The solid temperature at the front has been smoothed out using an internal layer asymptotic approximation. The good accuracy of the resulting asymptotic expansion shows its usefulness in several engineering problems such as heat transfer in porous media, in exhausted chemical reactions, mass transfer in packed beds, or in the analysis of capillary electrochromatography techniques.     

The stream function coordinate system for solution of one-dimensional unsteady compressible flow problems

A one-dimensional unsteady compressible isentropic flow problem is solved using a floating grid finite difference method. GENMIX, a computer program for solution of two-dimensional parabolic flow problems, has been adopted for this purpose. The advantages of utilizing the staggered floating grid method are demonstrated through solution of flow in a shocktube. The present method is able to precisely locate the discontinuities in temperature and density profiles.     

Axisymmetric melting of a long cylinder due to an infinite flux

By employing a new embedding technique, a short-time analytical solution for the axisymmetric melting of a long cylinder due to an infinite flux is presented in this paper. The sufficient condition for starting the instantaneous melting of the cylinder has been derived. The melt is removed as soon as it is formed. The method of solution is simple and straightforward and consists of assuming fictitious initial temperature for some fictitious extension of the actual region.     

Numerical study of partial differential equations to estimate thermoregulation in human dermal regions for temperature dependent thermal conductivity

The paper deals with the temperature distribution in multi-layered human skin and subcutaneous tissues (SST). The model suggests the solution of parabolic heat equation together with the boundary conditions for the temperature distribution in SST by assuming the thermal conductivity as a function of temperature.The model formulation is based on singular non-linear boundary value problem and has been solved using finite difference method. The numerical results were found similar to clinical and computational results.     

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.     

Electro-thermo-mechanical behaviors of FGPM spheres using analytical method and ANSYS software

A hollow sphere made from functionally graded piezoelectric material (FGPM) such as PZT_4 has been considered. One-dimensional analytical method for electro-thermo-mechanical response of symmetrical spheres is used. For asymmetric three-dimensional analysis, ANSYS finite element software is employed in this study. Loading is combination of internal and external pressures, a distributed temperature field due to steady state heat conduction and a constant electric potential difference between its inner and outer surfaces for analytical solution. In three-dimensional solutions closed and open spheres with different boundary conditions subjected to an internal pressure and a uniform temperature field are studied. All mechanical, thermal and piezoelectric properties except the Poisson’s ratio are assumed to be power functions of radius. It has been found from analytical solution that the induced radial and circumferential stresses of an imposed electric potential is similar to the residual stresses locked in the homogeneous sphere during the autofrettage process of these vessels. It has been concluded from the three-dimensional analysis that the magnitudes of effective stresses at all node points are higher for the clamped-clamped boundary condition and are lower for the simply-simply supported condition.     

Thermal analysis of hot strip rolling using finite element and upper bound methods

Thermal analysis of hot rolling process has been studied in this work. A finite element method has been coupled with an upper bound solution assuming, triangular velocity field, to predict temperature field during hot strip rolling operation. To do so, an Upwind Petrov–Galerkin scheme together with isoparametric quadrilateral elements has been employed to solve the steady-state heat transfer equation. A comparison has been made between the published and the model predictions and a good agreement was observed showing the accuracy of the proposed model.     

Heat transfer analysis of the steady flow of an Oldroyd 8-constant fluid due to a suddenly moved plate

This paper concentrates on the heat transfer analysis of the steady flow of an Oldroyd 8-constant fluid due to a suddenly moved plate. The heat transfer analysis has been carried out for the prescribed surface temperature. Employing homotopy analysis method, the developed system of equations are solved analytically. The convergence of the obtained series solution is established. The influence of pertinent parameters on temperature profiles and Nusselt number is shown and discussed through several graphs. Further, a comparison between temperature profiles of Newtonian and Oldroyd fluids is also made.     

Microcomputer-based simulation of heat conduction between solid of tapered shape and the fluid

Simulating the heat conduction in between a solid conducting body immersed in fluid at a given temperature is a difficult task, particularly when the body is tapered in shape and the costs have to be kept low. The body in question is cylindrical, symmetrical about z-axis, tapered in shape and has been heated to a high temperature before being immersed into the fluid. The heat conduction equation in cylindrical polar coordinates with all derivative boundary conditions is attempted to be solved in two ways – first analytically making use of Bessel's function and then by numerical modelling with the help of Finite Difference method, and equations thus formed have been solved through ADI explicit and Implicit (Peaceman Rachford) scheme on microcomputer. The paper is an account of work already done on this and includes further possibilities for general solution with analytical methods and a suitable low-cost numerical solution. Also possible analogy with flow of fluids have been explored. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Temperature distribution in a laminar plane wall jet

In the present note the temperature distribution in a laminar plane wall jet has been studied. It is found that a similarity solution of the energy equation exists. The resulting ordinary differential equation is reduced to a hypergeometric equation by a suitable transformation of the similarity variable and the solution, for arbitrary values of the Prandtl number, is obtained. It is concluded that the heat transfer at the wall at a given section and the product of volume and heat-flux through any cross-section of the boundary layer increase with the increase in the value of Prandtl number respectively.     

Finite integral transform-based analytical solutions of dual phase lag bio-heat transfer equation

A finite integral transform (FIT)-based analytical solution to the dual phase lag (DPL) bio-heat transfer equation has been developed. One of the potential applications of this analytical approach is in the field of photo-thermal therapy, wherein the interest lies in determining the thermal response of laser-irradiated biological samples. In order to demonstrate the applicability of the generalized analytical solutions, three problems have been formulated: (1) time independent boundary conditions (constant surface temperature heating), (2) time dependent boundary conditions (medium subjected to sinusoidal surface heating), and (3) biological tissue phantoms subjected to short-pulse laser irradiation. In the context of the case study involving biological tissue phantoms, the FIT-based analytical solutions of Fourier, as well as non-Fourier, heat conduction equations have been coupled with a numerical solution of the transient form of the radiative transfer equation (RTE) to determine the resultant temperature distribution. Performance of the FIT-based approach has been assessed by comparing the results of the present study with those reported in the literature. A comparison of DPL-based analytical solutions with those obtained using the conventional Fourier and hyperbolic heat conduction models has been presented. The relative influence of relaxation times associated with the temperature gradients (τ_T) and heat flux (τ_q) on the resultant thermal profiles has also been discussed. To the best of the knowledge of the authors, the present study is the first successful attempt at developing complete FIT-based analytical solution(s) of non-Fourier heat conduction equation(s), which have subsequently been coupled with numerical solutions of the transient form of the RTE. The work finds its importance in a range of areas such as material processing, photo-thermal therapy, etc.     

Modelling of heat affected zone in cylindrical steel elements surfaced by welding

Computational models of a temperature field in cylindrical steel elements surfaced by the following methods: controlled pitch, spiral welding sequence and spiral welding sequence with swinging motion of the welding head are presented in the paper. The lateral surface of regenerated cylindrical object, subjected to the welding heat source, has been treated as a plane rolled on cylinder and temperature field of repeatedly surfaced plain massive body was solved. Temperature rises, caused by overlaying consecutive welding sequences and self-cooling of areas previously heated, were taken into consideration in the solution. The computations of the temperature field for continuous casting steel machine roll made of 13CrMo4 steel were carried out.     

Semi-exact solution for thermo-mechanical analysis of functionally graded elastic-strain hardening rotating disks

In this paper, distributions of stress and strain components of rotating disks with non-uniform thickness and material properties subjected to thermo-elasto-plastic loading are obtained by semi-exact method of Liao’s homotopy analysis method (HAM) and finite element method (FEM). The materials are assumed to be elastic-linear strain hardening and isotropic. The analysis of rotating disk is based on Von Mises’ yield criterion. A two dimensional plane stress analysis is used. The distribution of temperature is assumed to have power forms with the hotter point located at the outer surface of the disk. A mathematical technique of transformation has been proposed to solve the homotopy equations which are originally hard to be handled. The domain of the solution has been substituted by a new domain through which the unknown variable has been taken out from the argument of the function. This makes the solution much easier. A numerical solution of the governing differential equations is also presented based on the Runge–Kutta’s method. The results of three methods are presented and compared which shows good agreements. This verifies the implementation of the HAM and demonstrates its applicability to provide accurate solution for a very complicated case of strongly high nonlinear differential equations with no exact solution. It is important to notice that compared with other methods, HAM needs significant more computation time and computer hardware requirements which limit its application for those problems that other methods can easily handle them.     

Problème de Cauchy pour une équation parabolique modélisant la relaxation des systèmes stellaires auto-gravitants

In this Note we are concerned with the Cauchy problem for a parabolic equation (with nonlocal terms), which has recently been proposed by Chavanis, Robert and Sommeria as a model for the large-scale dynamics of stellar systems. We show the local existence and the uniqueness of a variational solution, together with the increasing of the entropy and the positiveness of the stellar density and of the temperature when the initial state is radial.     

Calculation of conjugate heat transfer problem with volumetric heat generation using the Galerkin method

A mathematical model of fluid flow across a rod bundle with volumetric heat generation has been built. The rods are heated with volumetric internal heat generation. To construct the model, a volume average technique (VAT) has been applied to momentum and energy transport equations for a fluid and a solid phase to develop a specific form of porous media flow equations. The model equations have been solved with a semi-analytical Galerkin method. The detailed velocity and temperature fields in the fluid flow and the solid structure have been obtained. Using the solution fields, a whole-section drag coefficient C_d and a whole-section Nusselt number Nu have also been calculated. To validate the developed solution procedure, the results have been compared to the results of a finite volume method. The comparison shows an excellent agreement. The present results demonstrate that the selected Galerkin approach is capable of performing calculations of heat transfer in a cross-flow where thermal conductivity and internal heat generation in a solid structure has to be taken into account. Although the Galerkin method has limited applicability in complex geometries, its highly accurate solutions are an important benchmark on which other numerical results can be tested.     

Analytic method for solving a class of three-dimensional thermoelasticity problems for layered transversely isotropic plates with variable thicknesses

This paper examines three-dimensional boundary value problems in the theory of heat conduction and thermoelasticity for layered transversely isotropic rectangular plates with variable thicknesses acted on by a nonuniform temperature field. It is assumed that known temperature and heat flux at the surfaces of the plate or temperature of the surrounding medium allow a representation of the solution in terms of double trigonometric series. An approximate analytic method has been developed for solving this class of problems which makes it possible to reduce the initial boundary value problem for a plate of variable thickness to a recurrence sequence of the corresponding problems for plates with constant thicknesses. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 26–36, 1999.     

On the plane couette flow of a viscous compressible fluid with transpiration cooling

The exact solution for the plane couette flow of a viscous compressible, heat conducting, perfect gas with the same gas injection at the stationary plate and its corresponding removal at the moving plate has been studied. It is found that the gas injection is very helpful in reducing the temperature recovery factor. Effects of injection on the shearing stress at the lower plate, longitudinal velocity profiles and the enthalpy are shown graphically.     

Influence of heat and mass transfer on peristaltic flow of a third order fluid in a diverging tube

In the present investigation we have discussed the heat and mass transfer analysis on peristaltic flow of a third order fluid in a diverging tube. The assumption of low Reynolds number and long wavelength have been used to simplify the complicated problem into relatively simple problem. Two types of analytical solutions named as perturbation solution and solution have been evaluated for velocity, temperature and concentration field. The expression for pressure rise and frictional forces are calculated using numerical integration. In addition, the quantitative effects of pressure rise, frictional forces, temperature and concentration profile are displayed graphically. Trapping phenomena is also discussed at the end of the article.     

Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method

An analysis has been performed to study the natural convection of a non-Newtonian fluid between two infinite parallel vertical flat plates and the effects of the non-Newtonian nature of fluid on the heat transfer are studied. The governing boundary layer and temperature equations for this problem are reduced to an ordinary form and are solved by homotopy analysis method (HAM), and numerical method. Velocity and temperature profiles are shown graphically. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and non-linear equations and so can be found widely applicable in engineering and sciences.