Weighted Residual Methods: Principles, Modifications and Applications

By Mohammad Hatami

Weighted Residual Methods: Principles, Modifications and Applications introduces a range of WRMs, providing examples that show how they can be used to solve complex engineering problems with greater accuracy and computational efficiency. Examples focus on non-linear problems, including the motion of a spherical particle, nanofluid flow and heat transfer, magnetohydrodynamic flow and heat transfer, and micropolar fluid flow and heat transfer. These are important factors in understanding processes, such as filtration, combustion, air and water pollution and micro contamination. In addition to the applications, the reader is provided with full derivations of equations and summaries of important field research.

• Includes the basic code for each method, giving readers a head start in using WRMs for computational modeling
• Provides full derivations of important governing equations in a number of emerging fields of study
• Offers numerous, detailed examples of a range of applications in heat transfer, nanotechnology, medicine, and more

• Language: English
• Publisher: Academic Press
• Release date: Oct 30, 2017
• ISBN: 9780128132197

Mohammad Hatami

Mohammad Hatami is a Mechanical Engineering Associate Professor at Esfarayen University of Technology, Esfarayen, North Khorasan, Iran. He also was an associate professor at Ferdowsi University of Mashhad, and was selected as a young talent associate professor at Xi'an Jiaotong University. He completed his Ph.D. in energy conversion at Babol University of Technology whilst working as a Ph.D. visiting scholar researcher at Eindhoven University of Technology (TU/e) in the Netherlands. Dr. Hatami was previously a post-doctoral researcher at the State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, in China. He has published more than 200 research papers and 10 books/chapter books in the field of experimental, mathematical and numerical modelling of nanofluids, heat recovery and acts as editor-in-chief of the Quarterly Journal of Mechanical Engineering and Innovation in Technology, Associate Editor of Fluid Dynamic & Material Processing, and an Editor for the International Journal of Mechanical Engineering (IJME), American Journal of Modelling and Optimization, and American Journal of Mechanical Engineering.

Book preview

Weighted Residual Methods

Principles, Modifications and Applications

Mohammad Hatami

Table of Contents

Cover image

Title page

Copyright

Dedication

Preface

Chapter 1. Introduction to Analytical Methods

1.1. Introduction

1.2. Adomian Decomposition Method

1.3. Variational Iteration Method

1.4. Differential Transformation Method

1.5. Homotopy Perturbation Method

1.6. Homotopy Analysis Method

1.7. Weighted Residual Methods

1.8. Differential Quadrature Method

1.9. Optimal Homotopy Asymptotic Method

Chapter 2. Weighted Residual Methods Principles and Modifications

2.1. Introduction

2.2. Weighted Residual Methods Principles

2.3. WRMs for Coupled Equations

2.4. Optimal WRMs for Infinite Boundary Conditions

2.5. Combined WRMs With Other Analytical Methods

2.6. Modified WRMs for Combined Boundary Conditions

2.7. Hybrid WRMs for Partial Differential Equations

2.8. Multistep Polynomial WRMs for Fractional Order Differential Equations

2.9. Padé Approximation and Other Analytical Methods

Chapter 3. Weighted Residual Methods in Fluid Mechanic Applications

3.1. Introduction

3.2. Nanofluid Flow in a Porous Channel

3.3. Nanofluid Flow Between Parallel Disks

3.4. Jeffery–Hamel Flow

3.5. Condensation Flow Over Inclined Disks

3.6. Electrohydrodynamic (EHD) Flow

3.7. Magnetohydrodynamic (MHD) Flow in Divergent/Convergent Channels

3.8. Nanofluid Flow in a Microchannel Heat Sink

3.9. Nanofluid Flow in Expanding and Contracting Gaps

Chapter 4. Weighted Residual Methods in Heat Transfer and Energy Conversion Applications

4.1. Introduction

4.2. Heat Transfer of Longitudinal, Convective–Radiative, Porous Fins

4.3. Heat Transfer of Circular, Convective–Radiative, Porous Fins

4.4. Heat Transfer of Convective–Radiative, Semispherical Fins

4.5. Refrigeration of Fully Wet, Circular, Porous Fins

4.6. Refrigeration of Fully Wet, Semispherical, Porous Fins

4.7. Nanofluids Condensation and Heat Transfer

4.8. Nanofluid Heat Transfer in Circular, Concentric Heat Pipes

4.9. Nanofluid Heat Transfer in a Microchannel Heat Sink

Chapter 5. WRMs in Nanoengineering Applications

5.1. Introduction

5.2. Natural Convection of Non-Newtonian Nanofluid

5.3. MHD Jeffery–Hamel Nanofluid Flow

5.4. MHD Nanofluids Over a Cylindrical Tube

5.5. Forced Convection for MHD Nanofluid Flow Over a Porous Plate

5.6. Non-Newtonian Nanofluid in Porous Media Between Two Coaxial Cylinders

5.7. MHD Nanofluid Flow in a Semiporous Channel

5.8. Nanofluid in Microchannel Heat Sink (MCHS) Cooling

5.9. Carbon Nanotube (CNT)-Water Between Rotating Disks

Index

Copyright

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Dedication

Dedicated to my kind and beloved parents, who always supported me during my educational career.

Preface

You, Whose Name is the best at the beginning!

Without Your Name, how can I begin my story?

Layli-o-Majnun, Nizami Ganjavi, Persian Poet (1141–1209)

In recent years, modeling and analyzing physical phenomena has been significant due to a reduction in costs. Numerical and analytical modeling assist researchers in seeing the treatment of an observed event before it occurs in nature. Many problems in electrical engineering, medical engineering, mechanical engineering, chemical engineering, petroleum, energy, industry, engines, etc., can be defined in mathematical language using ordinary differential equations or partially differential equations. These governing equations need powerful numerical and analytical methods to be solved and analyzed. Although there are many numerical methods that can solve these problems, their solution needs much time and professional computers for analysis. Some of these problems are solvable by analytical methods that are simpler and faster than numerical technique. Weighted residual methods (WRMs) are a group of these analytical methods that are presented in this book. Based on my experience in using, improving, and extending these methods in papers, I decided to collect all of them as one source in a complete book. The contents of the current book can benefit the engineers, researchers, and graduate students who want to develop their knowledge in basic phenomena of analytical methods in engineering. In the first chapters (Chapters 1 and 2), WRMs are introduced in simple and complicated versions, including all improvements and developments. In other chapters (Chapters 3–5), the application of WRMs on various examples in engineering is demonstrated, and several examples of recently published papers from high-quality journals are included to illuminate the subject. I am very pleased to receive the readers' comments and amendments on the materials of the book and hope it will be beneficial for them. Finally, I would like to express my sincere thanks to the staff of book publishing at Elsevier for their helpful support.

Mohammad Hatami,     Assistant Professor of Mechanical Engineering Department, Esfarayen University of Technology (EUT), Esfarayen, North Khorasan, Iran

Chapter 1

Introduction to Analytical Methods

Abstract

Most technical problems in fluid mechanics, heat transfer in many physical and engineering applications such as geothermal systems, chemical catalytic reactors, heat exchangers, etc., are inherently nonlinear. These problems and phenomena can be modeled by ordinary or partial nonlinear differential equations to find their behavior in the environment. One of the simple and reliable methods for solving the system of coupled nonlinear differential equations is an analytical solution that recently has been widely used in many problems. Some of the analytical methods need transformation, and some of them deal with the equation without any transformation formula or other restrictive assumption. By this way, analytical methods have some advantages and disadvantages compared to each other. The author has done wide studies in numerical and analytical studies [1–89] in solving the problems.

Keywords

Adomian decomposition; Collocation; Differential; Galerkin; Homotopy; Least square; Rayleigh–Ritz; Variational iteration

1.1. Introduction

Most technical problems in fluid mechanics, heat transfer in many physical and engineering applications such as geothermal systems, chemical catalytic reactors, heat exchangers, etc., are inherently nonlinear. These problems and phenomena can be modeled by ordinary or partial nonlinear differential equations to find their behavior in the environment. One of the simple and reliable methods for solving the system of coupled nonlinear differential equations is an analytical solution that recently has been widely used in many problems. Some of the analytical methods need transformation, and some of them deal with the equation without any transformation formula or other restrictive assumption. By this way, analytical methods have some advantages and disadvantages compared to each other. The author has done wide studies in numerical and analytical studies [1–89] in solving the problems. In this section the most applicable analytical methods are introduced, including the following:

1.2 Adomian Decomposition Method (ADM)

1.3 Variational Iteration Method (VIM)

1.4 Differential Transformation Method (DTM)

1.5 Homotopy Perturbation Method (HPM)

1.6 Homotopy Analysis Method (HAM)

1.7 Weighted Residual Methods (WRMs)

1.8 Differential Quadrature Method (DQM)

1.9 Optimal Homotopy Asymptotic Method (OHAM)

1.2. Adomian Decomposition Method

Consider a general nonlinear equation in this form [2]:

(1.1)

where L is the operator of the highest ordered derivatives with respect to t, and R is the remainder of the linear operator. The nonlinear term is represented by F(u). Thus, we get

(1.2)

The inverse L−¹ is assumed an integral operator given by

(1.3)

By applying the L−¹ operator on both sides of Eq. (1.2), we have

(1.4)

where f0 is the solution of the homogeneous equation.

(1.5)

The integration constants involved in the solution of homogeneous Eq. (1.5) are to be determined by the initial or boundary condition according to the problem as an initial-value problem or boundary-value problem. The ADM assumes that the unknown function u(x,t) can be expressed by an infinite series of the form

(1.6)

And the nonlinear operator F(u) can be decomposed by an infinite series of polynomials given by

(1.7)

where un(x,t) will be determined recurrently, and An are the so-called polynomials of u0,u1,…,un defined by the following:

(1.8)

It is now well known in the literature that these polynomials can be constructed for all classes of nonlinearity according to algorithms set by Adomian.

Application of ADM: In this example :

(1.9)

where a, ρ, and V are second derivatives of the particles' motion in horizontal and vertical directions with respect to time.

To calculate the drag force, the velocities of the spherical particle are considered adequately small, so the Stokes law can be governed:

(1.10)

(1.11)

while μ signifies the viscosity of the fluid.

The rotation and shear portion of the particle's lift force is obtained as shown:

(1.12)

(1.13)

(1.14)

(1.15)

where α is defined as a positive proportionality constant.

An illustration of the spherical particle in plane Couette fluid flow and exerted forces on the particle is shown in Fig. 1.1. The mass of the particle is assumed in the center of the sphere, and the forces caused from the rotation and shear fields and their interactions on drag and lift forces of the particle are illustrated in Fig. 1.1A and B, respectively. By forming the force balance equation of the inertia force to the drag and lift forces, the equations of motion for the particle are given as shown:

(1.16)

Figure 1.1  Schematic view of exerted forces on a spherical particle in Couette fluid flow [2] . (A) Drag interactions. (B) Lift interactions.

Eventually, by substituting Eqs. (1.13) and (1.15) into Eq. (1.16) the system of equations of motion of a spherical particle in plane Couette flow yields the following:

(1.17)

(1.18)

(1.19)

For simplicity the governing equations have been expressed as shown:

(1.20)

where the coefficients A–C are defined as follows:

(1.21)

An appropriate initial condition is required to avoid trapping the procedure in a nontrivial solution:

(1.22)

(1.23)

Since the procedure of solving Eq. (1.19) is autonomous of constants A, B, and C, for generalization and simplification of the problem for future cases with different physical conditions, the constants that represent physical properties are assumed to be these:

(1.24)

(1.25)

Now, ADM is applied for solving Eq. (1.20) with assumptions from Eq. (1.24) and initial conditions from Eq. (1.25):

(1.26)

(1.27)

(1.28)

(1.29)

(1.30)

(1.31)

(1.32)

(1.33)

(1.34)

(1.35)

(1.36)

(1.37)

(1.38)

(1.39)

(1.40)

(1.41)

Eq. (1.26) continued up to n  =  24 to get an exact solution with reasonable computational effort. The two-dimensional profiles of velocity of a particle are depicted in Fig. 1.2. It can be easily deduced in Fig. 1.2 that the results obtained from ADM for n  =  24 are almost the same as the numerical result in the specified time duration. Furthermore, for more understanding of ADM operation the approximant values of ADM results for n  =  12, 17, and 24, as well as the error computed from numerical results, is inserted into Tables 1.1 and 1.2. From these tables, which show the mean square error (MSE) for different series terms involved in ADM, it is found that the velocities of a particle in both the x and y directions for n  =  24 has the least amount of MSE and, hence, cover the moving particle velocities for a longer time.

Figure 1.2  ADM approximant and numerical solutions.

Table 1.1

The results of ADM and fourth-order Runge–Kutta method for νx

Table 1.2

The results of ADM and fourth-order Runge–Kutta method for νy

1.3. Variational Iteration Method

To illustrate the basic concepts of VIM, consider the following differential Eq. (1.2):

(1.42)

where L and N are linear and nonlinear operators, respectively, and g(t) is the source inhomogeneous term. According to VIM, we can write down a correction functional as follows:

(1.43)

where λ is a general Lagrangian multiplier that can be identified optimally via the variational theory. The subscript n indicates the n th .

Application of VIM: The solution of Eq. (1.20) using VIM, assumptions Eq. (1.24), and initial condition Eq. (1.25) are obtained as shown:

(1.44)

(1.45)

(1.46)

(1.47)

(1.48)

(1.49)

(1.50)

(1.51)

(1.52)

(1.53)

(1.54)

(1.55)

(1.56)

(1.57)

(1.58)

(1.59)

In other stage, VIM was applied to obtain an exact solution of Eq. (1.20). The results of Eqs. (1.56) and (1.57) are obtained for four instantaneous counters of n up to n  =  16 which the acceptable results of Eq. (1.20) achieved. The variation of velocities of a particle in both horizontal and vertical motion versus time is illustrated in Fig. 1.3 in which the terminal velocity of a particle approaches the exact value as the number of iterations increases. The numerical solution is referred as the exact value, and the calculated values of νx and νy in various time instances are compared with exact values, and the errors are tabulated in Tables 1.3 and 1.4.

Figure 1.3  VIM approximant and numerical solutions of Eq. (1.20) .

Table 1.3

The results of VIM and fourth-order Runge–Kutta method for νx

Table 1.4

The results of VIM and fourth-order Runge–Kutta method for νy

1.4. Differential Transformation Method

DTM, a powerful analytical method, has large submethods and modifications, all of which are presented in [1]. The main reasons for extending this method are as follows:

1. DTM is independent of any small parameter such as p in HPM. So, DTM can be applied whether governing equations and boundary/initial conditions of a given nonlinear problem contain small or large quantities or not.

through h-curves against HAM.

3. DTM does not need initial guesses and auxiliary linear operator, and it solves the equations directly versus HAM.

4. DTM provides us with great freedom to express solutions of a given nonlinear problem by means of Padé approximant and multistep DTM (Ms-DTM).

For understanding this method's concept, suppose that x(t) is an analytic function in domain D, and t  =  ti represents any point in the domain. The function x(t) is then represented by one power series, the center of which is located at ti. The Taylor series expansion function of x(t) is in the form:

(1.60)

The Maclaurin series of x(t) can be obtained by taking ti  =  0 in Eq. (1.60):

(1.61)

The differential transformation of the function x(t) is defined as follows:

(1.62)

where X(k) represents the transformed function, and x(t) is the original function. The differential spectrum of X(k) is confined within the interval t∈[0,H], where H is a constant value, and it can be assumed unity in common DTM, but for Ms-DTM, it should be considered the length of t steps. When H→∞ the methods no longer works, and Padé approximation should be applied. More information about H is presented in [2]. The differential inverse transform of X(k) is defined as follows:

(1.63)

It is clear that the concept of differential transformation is based upon the Taylor series expansion. The values of function X(k) at values of argument k are referred to as discrete, i.e., X(0) is known as the zero discrete, X(1) as the first discrete, etc. The more discrete available, the more precise it is possible to restore the unknown function. The function x(t) consists of the T-function X(k), and its value is given by the sum of the