Contents
Introduction v
1 Non Newtonian fluids and their mathematical model 1
1.1 Non Newtonian fluids and their characteristics 1
1.2 Incompressible and isothermal bipolar non Newtonian fluids models 3
1.3 lsothermal compressible viscous fluids models 8
1.4 0ther related models 9
2 Global solutions to the equations of non Newtonian fluids 15
2.1 Global solutions to the periodic initial value problems for the incompressible non Newtonian fluids 15
2.1.1 Existence and uniqueness of global solutions to the incompressible bipolar fluids 15
2.1.2 Existence ofweak solutions to the incompressible monopolar fluids 23
2.2 Global solutions to the compressible non Newtonian fluids Existence and uniqueness of weak solution to the bipolar compressible non Newtonian fluids 24
2.3 Time periodic solutions to the incompressible bipolar fluids 29
2.3.1 Time periodic weak solutions to the incompressible bipolar fluids 29
2.3.2 Existence and uniqueness of strong time periodic solutions to the incompressible bipolar fluids 33
2.4 Existence and uniqueness and stability of global solutions to the initial boundary value problems for the incompressible bipolar viscous fluids 43
2.4.1 Existence 44
2.4.2 Regularity 50
2.4.3 Uniqueness 53
2.4.4 Stability 53
2.5 The periodic initial value problem and initialvalue problem for the modified Boussinesq approximation 55
2.6 Periodic initial value problem and initial value problem for the non Newtonian Boussinesq approximation 67
3 Global attractors ofincompressible non Newtonian fluids 89
3.1 Global attractors ofincompressible non Newtonian fluids on bounded domain 93
3.1.1 Existence ofAbsorbing Sets 93
3.1.2 Consistently differentiability for the solution semigroup 98
3.1.3 For Pi > 0, the upper bounded estimates of dH(Api) and dF (Aui) of attractor Alui 105
3.2 Global attractors ofincompressible non Newtonian fluids on unbounded domain 107
3.3 Exponential attractors ofincompressible non Newtonian fluids 119
3.3.1 Estimates for the nonlinear terms 120
3.3.2 Compressibility on L2(Q) 128
4 Global attractors of modified Boussinesq approximation 133
5 Inertial manifolds ofincompressible non Newtonian fluids 159
5.1 Inertial manifolds of incom pressible bipolar non Newtonian fluids 159
5.1.1 Lipschitz property 160
5.1.2 The squeezing property 163
5.1.3 Fixed point theorem 167
5.1.4 Inertial manifolds 177
5.2 Approximated inertial manifolds ofincompressible bipolar non Newtonian fluids 179
5.2.1 The analyticity in time and behavior of higher order modes 180
5.2.2 Approximated inertial manifolds 185
6 The regularity of solutions and related problems 191
6.1 Stationary solutions ofthe incompressible bipolar non Newtonian fluids 191
6.2 Decay estimates of one kind ofincompressible monopolar non Newtonian fluid 195
6.3 Partial regularity of one kind ofincompressible monopolar non Newtonian fluid 202
6.4 The convergence of solution and attractors between one kind of incompressible non Newtonian fluid and the Newtonian fluids 211
6.5 0ther decay estimates ofincompressible non Newtonian fluids 215
7 Globalattractors and time spatial chaos 227
7.1 Global attractor of low regularity 227
7.2 Attractors and their spatial complexity of reaction diffusion equations on bounded domain 244
8 Non Newtonian generalized fluid and their applications 293
8.1 An inverse problem ofa heated generalized second grade fluid 293
8.1.1 Formulation 293
8.1.2 0utline of the optimization method 294
8.1.3 Illustrative examples 298
8.2 A numerical study of a generalized Maxwell fluid through a porous medium 304
8.2.1 Mathematical model 305
8.2.2 HPM solutions 308
8.2.3 Numerical results and discussion 309
8.2.4 Conclusions 313
8.3 Viscoelastic fluid with fractional derivative models 313
8.3.1 Preliminaries 314
8.3.2 Eigenfunction expansion of the solution and properties of the time dependent components 316
8.3.3 Finite difference approximation 321
8.3.4 Duhamel type representation of the solution 322
8.3.5 Numerical experiments 324
8.3.6 Two dimensional problem 330
8.3.7 Conclusion 332
Bibliography 333
Index 339