Chapter
1.2.1.5 Multiple Scales Method
1.2.2 Adomian Decomposition Method
1.2.3 Homotopy Analysis Method
1.2.4 Differential Transformation Method
1.2.5 Variational Iteration Method and Homotopy Perturbation Method
1.3 FRACTAL THEORY AND FRACTIONAL VISCOELASTIC FLUID
1.3.1 The Concept of Fractals
1.3.2 Fractional Order Calculus
1.3.2.1 Definition of Fractional Order Derivatives
1.3.3 Fractional Integral Transformations and Their Properties
1.3.3.1 Fourier Transformation
1.3.3.2 Laplace Transformation
1.3.3.3 Mellin Transformation
1.3.4 Fractional Viscoelastic Fluid
1.5 MODELING AND ANALYSIS FOR MODERN FLUID PROBLEMS
2 - Embedding-Parameters Perturbation Method
2.1 BASICS OF PERTURBATION THEORY
2.1.1 Perturbation Theory
2.1.2 Asymptotic Expansion of Solutions
2.1.3 Regular Perturbation and Singular Perturbation
2.2 EMBEDDING-PARAMETER PERTURBATION
2.2.1 Approximate Solution to Blasius Flow
2.2.2 Approximate Solutions to Sakidias Flow
2.4 MARANGONI CONVECTION IN A POWER LAW NON-NEWTONIAN FLUID
2.4.1 Marangoni Convection Caused by Temperature Gradient
2.4.2 Mathematical Formulation
2.4.3 Embedding-Parameters Perturbation Method Solutions
2.4.4 Results and Discussion
2.5 MARANGONI CONVECTION IN FINITE THICKNESS
2.5.1 Background to the Problem
2.5.2 Mathematical Model for Three Types of Conditions
2.5.3 Embedding-Parameters Perturbation Method Solutions and Discussion
2.5.3.1 Solutions for Cases I and II
3 - Adomian Decomposition Method
3.2 NONLINEAR BOUNDARY LAYER OF POWER LAW FLUID
3.2.1 Physical Background
3.2.2 Mathematical Formulation
3.2.3 Similarity Transformation
3.2.4 Crocco Variable Transformation
3.2.5 Adomian Decomposition Method Solutions
3.2.6 Results and Discussion
3.2.6.1 Analysis of Velocity Field
3.2.6.2 Analysis of Temperature Field
3.3 POWER LAW MAGNETOHYDRODYNAMIC FLUID FLOW OVER A POWER LAW VELOCITY WALL
3.3.1 Physical Background
3.3.2 Basic Governing Equations
3.3.3 Lie Group of Transformation
3.3.4 Generalized Crocco Variables Transformation
3.3.5 Adomian Decomposition Method Solutions
3.3.6 Results and Discussion
3.4 MARANGONI CONVECTION OVER A VAPOR–LIQUID SURFACE
3.4.1 Boundary Layer Governing Equations
3.4.2 Adomian Decomposition Method Solutions
3.4.3 Results and Discussion
4 - Homotopy Analytical Method
4.2 FLOW AND RADIATIVE HEAT TRANSFER OF MAGNETOHYDRODYNAMIC FLUID OVER A STRETCHING SURFACE
4.2.1 Description of the Problem
4.2.2 Mathematical Formulation
4.2.3 Homotopy Analysis Method Solutions
4.2.3.1 Zero-Order Deformation Equations
4.2.3.2 Higher-Order Deformation Equations
4.2.4 Results and Discussion
4.3 FLOW AND HEAT TRANSFER OF NANOFLUIDS OVER A ROTATING DISK
4.3.1 Background of the Problem
4.3.2 Formulation of the Problem
4.3.3 Von Karman's Transformation
4.3.4 Homotopy Analysis Method Solutions
4.3.5 Results and Discussion
4.3.5.1 Effects of Velocity Slip Parameter
4.3.5.2 Effects of Temperature Jump Parameter
4.3.5.3 Effects of Porosity Parameter
4.3.5.4 Effects of Types of Nanoparticle
4.4 MIXED CONVECTION IN POWER LAW FLUIDS OVER MOVING CONVEYOR
4.4.1 Physical Background of the Problem
4.4.2 Mathematical Formulation
4.4.3 Nonlinear Boundary Value Problems
4.4.4 Homotopy Analysis Method Solutions
4.4.5 Results and Discussion
4.4.5.1 Effects of Power Law Exponent n
4.4.5.2 Effects of Incline Angle ϕ
4.4.5.3 Effects of Velocity Ratio Coefficient γu
4.5 MAGNETOHYDRODYNAMIC THERMOSOLUTAL MARANGONI CONVECTION IN POWER LAW FLUID
4.5.1 Background of the Problem
4.5.2 Mathematical Formulation
4.5.3 Homotopy Analysis Method Solutions
4.5.4 Results and Discussion
5 - Differential Transform Method
5.1.1 Ideas of Differential Transform–Padé and Differential Transform–Basic Function
5.1.2 Definition of Differential Transformation Method and Formula
5.1.2.1 Differential Transformation for Function of One Variable
5.1.2.2 Differential Transformation for Functions of Several Variables
5.1.2.3 Differential Transformation Formula
5.1.3 Magnetohydrodynamic Boundary Layer Problem
5.2 MAGNETOHYDRODYNAMICS FALKNER–SKAN BOUNDARY LAYER FLOW OVER PERMEABLE WALL
5.2.1 Mathematical Physical Description
5.2.2 Differential Transformation Method–Padé Solutions
5.2.3 Results and Discussion
5.3 UNSTEADY MAGNETOHYDRODYNAMICS MIXED FLOW AND HEAT TRANSFER ALONG A VERTICAL SHEET
5.3.1 Mathematical Physical Description
5.3.2 Differential Transformation Method–Basic Function Solutions
5.3.3 Results and Discussion
5.4 MAGNETOHYDRODYNAMICS MIXED CONVECTIVE HEAT TRANSFER WITH THERMAL RADIATION AND OHMIC HEATING
5.4.1 Mathematical and Physical Description
5.4.2 Formulation of the Problem
5.4.3 Differential Transformation Method–Basic Function Solutions
5.4.4 Numerical Solutions
5.4.5 Results and Discussion
5.5 MAGNETOHYDRODYNAMIC NANOFLUID RADIATION HEAT TRANSFER WITH VARIABLE HEAT FLUX AND CHEMICAL REACTION
5.5.1 Mathematical and Physical Description
5.5.2 Formulation of the Problem
5.5.3 Differential Transformation Method–Basic Function Solutions
5.5.4 Numerical Solutions
5.5.5 Results and Discussion
6 - Variational Iteration Method and Homotopy Perturbation Method
6.1 REVIEW OF VARIATIONAL ITERATION METHOD
6.2 FRACTIONAL DIFFUSION PROBLEM
6.3 FRACTIONAL ADVECTION-DIFFUSION EQUATION
6.3.1 Formulation of the Problem
6.3.2 Variational Iteration Method Solutions
6.4 REVIEW OF HOMOTOPY PERTURBATION METHOD
6.5 UNSTEADY FLOW AND HEAT TRANSFER OF A POWER LAW FLUID OVER A STRETCHING SURFACE
6.5.1 Boundary Layer Governing Equations
6.5.2 Modified Homotopy Perturbation Method Solutions
6.5.3 Results and Discussion
7 - Exact Analytical Solutions for Fractional Viscoelastic Fluids
7.1.1 The Viscoelastic Non-Newtonian Fluids
7.1.2 The Fractional Calculus
7.2 FRACTIONAL MAXWELL FLUID FLOW DUE TO ACCELERATING PLATE
7.2.1 Governing Equations
7.2.2 Statement of the Problem
7.2.3 Calculation of the Velocity Field
7.2.4 Calculation of the Shear Stress
7.2.6 Analysis and Discussion
7.3 HELICAL FLOWS OF FRACTIONAL OLDROYD-B FLUID IN POROUS MEDIUM
7.3.1 Formulation of the Problem
7.3.2 Helical Flow Between Coaxial Cylinders
7.3.3 Calculation of the Velocity Field
7.3.4 Calculation of the Shear Stress
7.3.5 The Solution of Heat Transfer Equation
7.3.6 Results and Discussion
7.4 MAGNETOHYDRODYNAMIC FLOW AND HEAT TRANSFER OF GENERALIZED BURGERS' FLUID
7.4.1 Governing Equations
7.4.2 Formulation of the Problem
7.4.3 The Solution of Velocity Fields
7.4.4 The Solution of Temperature Fields
7.4.5 Results and Discussion
7.5 SLIP EFFECTS ON MAGNETOHYDRODYNAMIC FLOW OF FRACTIONAL OLDROYD-B FLUID
7.5.1 Governing Equations
7.5.2 Formulation of the Problem
7.5.5 Results and Discussion
7.6 THE 3D FLOW OF GENERALIZED OLDROYD-B FLUID
7.6.2 Formulation of the Problem
7.6.3 Calculation of the Velocity Field
7.6.4 Calculation of the Shear Stress
7.6.6 Results and Discussion
8.1 REVIEW OF NUMERICAL METHODS
8.1.1 Numerical Methods for Linear System of Equations
8.1.2 Numerical Methods for Ordinary/Partial Differential Equations
8.1.2.1 Runge–Kutta Method
8.1.2.3 Control Volume Method
8.1.3 Numerical Methods for Fractional Differential Equations
8.2 HEAT TRANSFER OF POWER LAW FLUID IN A TUBE WITH DIFFERENT FLUX MODELS
8.2.1 Background of the Problem
8.2.2 Formulation of the Problems and Numerical Algorithms
8.2.3 Results and Discussion
8.3 HEAT TRANSFER OF THE POWER LAW FLUID OVER A ROTATING DISK
8.3.1 Background of the Problem
8.3.2 Formulation of the Problem and Governing Equations
8.3.3 Generalized Karman Transformation
8.3.4 Multiple Shooting Method
8.3.5 Results and Discussion
8.4 MAXWELL FLUID WITH MODIFIED FRACTIONAL FOURIER'S LAW AND DARCY'S LAW
8.4.1 Background of the Problem
8.4.2 Mathematical Formulation and Governing Equations
8.4.3 Numerical Algorithms
8.4.4 Results and Discussion
8.5 UNSTEADY NATURAL CONVECTION HEAT TRANSFER OF FRACTIONAL MAXWELL FLUID
8.5.1 Background of the Problem
8.5.2 Mathematical Formulation
8.5.3 Numerical Algorithms
8.5.3.1 Discretization Method
8.5.3.2 Iteration Algorithm
8.5.3.3 Reliability Comparison
8.5.4 Results and Discussion
8.5.4.1 Effects on Temperature Field
8.5.4.2 Effects on Velocity Field
8.5.4.3 Effects on Skin Friction Coefficient
8.5.4.4 Effects on Nusselt Number
8.6 FRACTIONAL CONVECTION DIFFUSION WITH CATTANEO–CHRISTOV FLUX
8.6.1 Fractional Anomalous Diffusion
8.6.2 Mathematical Formulation
8.6.3 Numerical Algorithms
8.6.4 Comparison of Numerical and Analytical Solutions
8.6.5 Results and Discussion
Modeling and Analysis of Modern Fluid Problems
( Mathematics in Science and Engineering )
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