Keller-Box Method and Its Application (eBook)

Contents 7
Chapter 0 Introduction 13
References 15
Chapter 1 Basics of the Finite Difference Approximations 17
1.1 Finite difference approximations 17
1.2 The initial value problem for ODEs 23
1.3 Some basic numerical methods 28
1.4 Some basic PDEs 38
1.5 Numerical solution to partial differential equations 44
References 50
Chapter 2 Principles of the Implicit Keller-box Method 53
2.1 Principles of implicit finite difference methods 53
2.2 Finite difference methods 65
2.3 Boundary value problems in ordinary differential equations 83
References 99
Chapter 3 Stability and Convergence of the Implicit Keller-box Method 101
3.1 Convergence of implicit difference methods for parabolic functional differential equations 102
3.1.1 Introduction 102
3.1.2 Discretization of mixed problems 103
3.1.3 Solvability of implicit difference functional problems 106
3.1.4 Approximate solutions of difference functional problems 108
3.1.5 Convergence of implicit difference methods 111
3.1.6 Numerical examples 115
3.2 Rate of convergence of finite difference scheme on uniform/non-uniform grids 117
3.2.1 Introduction 117
3.2.2 Analytical results 118
3.2.3 Numerical results 122
3.3 Stability and convergence of Crank-Nicholson method for fractional advection dispersion equation 124
3.3.1 Introduction 124
3.3.2 Problem formulation 125
3.3.3 Numerical formulation of the Crank-Nicholson method 126
3.3.4 Stability of the Crank-Nicholson method 127
3.3.5 Convergence 128
3.3.6 Radial flow problem 129
3.3.7 Conclusions 130
References 130
Chapter 4 Application of the Keller-box Method to Boundary Layer Problems 133
4.1 Flow of a power-law fluid over a stretching sheet 133
4.1.1 Introduction 133
4.1.2 Formulation of the problem 134
4.1.3 Numerical solution method 136
4.1.4 Results and discussion 137
4.1.5 Concluding remarks 138
4.2 Hydromagnetic flow of a power-law fluid over a stretching sheet 138
4.2.1 Introduction 138
4.2.2 Flow analysis 140
4.2.3 Numerical solution method 142
4.2.4 Results and discussion 142
4.3 MHD Power-law fluid flow and heat transfer over a non-isothermal stretching sheet 147
4.3.1 Introduction 147
4.3.2 Governing equations and similarity analysis 149
4.3.3 Heat transfer 151
4.3.4 Numerical procedure 153
4.3.5 Results and discussion 161
4.4 MHD flow and heat transfer of a Maxwell fluid over a non-isothermal stretching sheet 163
4.4.1 Introduction 163
4.4.2 Mathematical formulation 165
4.4.3 Heat transfer analysis 167
4.4.4 Numerical procedure 170
4.4.5 Results and discussion 171
4.4.6 Conclusions 177
4.5 MHD boundary layer flow of a micropolar fluid past a wedge with constant wall heat flux 178
4.5.1 Introduction 178
4.5.2 Flow analysis 179
4.5.3 Flat plate problem 182
4.5.4 Results and discussion 183
4.5.5 Conclusion 188
References 189
Chapter 5 Application of the Keller-box Method to Fluid Flow and Heat Transfer Problems 195
5.1 Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet 195
5.1.1 Introduction 195
5.1.2 Mathematical formulation 196
5.1.3 Solution of the problem 199
5.1.4 Results and discussion 200
5.1.5 Conclusions 206
5.2 Convection flow and heat transfer of a Maxwell fluid over a non-isothermal surface 206
5.2.1 Introduction 206
5.2.2 Mathematical formulation 208
5.2.3 Skin friction 211
5.2.4 Nusselt number 212
5.2.5 Results and discussion 212
5.2.6 Conclusion 218
5.3 The effects of variable fluid properties on the hydromagnetic flow and heat transfer over a nonlinearly stretching sheet 219
5.3.1 Introduction 219
5.3.2 Mathematical formulation 220
5.3.3 Numerical procedure 224
5.3.4 Results and discussion 225
5.3.5 Conclusions 235
5.4 Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid over a vertical stretching sheet 235
5.4.1 Introduction 235
5.4.2 Mathematical formulation 237
5.4.3 Numerical procedure 241
5.4.4 Results and discussion 241
5.5 The effects of linear/nonlinear convection on the non-Darcian flow and heat transfer along a permeable vertical surface 250
5.5.1 Introduction 250
5.5.2 Mathematical formulation 252
5.5.3 Numerical procedure 255
5.5.4 Results and discussion 265
5.6 Unsteady flow and heat transfer in a thin film of Ostwald-de Waele liquid over a stretching surface 267
5.6.1 Introduction 267
5.6.2 Mathematical formulation 269
5.6.3 Numerical procedure 273
5.6.4 Results and discussion 274
5.6.5 Conclusions 284
References 284
Chapter 6 Application of the Keller-box Method to More Advanced Problems 291
6.1 Heat transfer phenomena in a moving nanofluid over a horizontal surface 291
6.1.1 Introduction 291
6.1.2 Mathematical formulation 293
6.1.3 Similarity equations 295
6.1.4 Numerical procedure 297
6.1.5 Results and discussion 298
6.1.6 Conclusion 309
6.2 Hydromagnetic fluid flow and heat transfer at a stretching sheet with fluid-particle suspension and variable fluid properties 310
6.2.1 Introduction 310
6.2.2 Mathematical formulation 312
6.2.3 Solution for special cases 315
6.2.4 Analytical solution by perturbation 315
6.2.5 Numerical procedure 317
6.2.6 Results and discussion 318
6.2.7 Conclusions 329
6.3 Radiation effects on mixed convection over a wedge embedded in a porous medium filled with a nanofluid 330
6.3.1 Introduction 330
6.3.2 Problem formulation 331
6.3.3 Numerical method and validation 334
6.3.4 Results and discussion 335
6.3.5 Conclusion 349
6.4 MHD mixed convection flow over a permeable non-isothermal wedge 349
6.4.1 Introduction 349
6.4.2 Mathematical formulation 351
6.4.3 Numerical procedure 354
6.4.4 Results and discussion 356
6.4.5 Concluding remarks 366
6.5 Mixed convection boundary layer flow about a solid sphere with Newtonian heating 367
6.5.1 Introduction 367
6.5.2 Mathematical formulation 369
6.5.3 Solution procedure 372
6.5.4 Results and discussion 372
6.5.5 Conclusions 378
6.6 Flow and heat transfer of a viscoelastic fluid over a flat plate with a magnetic field and a pressure gradient 383
6.6.1 Introduction 383
6.6.2 Governing equations 384
6.6.3 Results and discussion 387
6.6.4 Conclusions 394
References 394
Subject Index 403
Author Index 407