Fractional Modeling of Fluid Flow and Transport Phenomena

Fractional Modeling of Fluid Flow and Transport Phenomena focuses on mathematical and numerical aspects of fractional-order modeling in fluid flow and transport phenomena. The book covers fundamental concepts, advancements, and practical applications, including modeling developments, numerical solutions, and convergence analysis for both time and space fractional order models. Various types of flows are explored, such as single- and multi-phase flows in porous media, involving different fluid types like Newtonian, non-Newtonian, nanofluids, and ferrofluids. This book serves as a comprehensive reference on fractional-order modeling of fluid flow and transport phenomena, offering a single resource that is currently unavailable.

Fractional-order modeling has gained traction in engineering and science, particularly in fluid dynamics and transport phenomena. However, its mathematical and numerical advancements have progressed relatively slowly compared to other aspects. Therefore, this book emphasizes the fractional-order modeling of fluid flow and transport phenomena to bridge this gap. Each chapter in the book delves into a specific topic closely related to the others, ensuring a cohesive and self-contained structure.

Title of Book
Cover image
Title page
Table of Contents
Copyright
Dedication
Biography
About the series editors
Preface
Acknowledgments
Chapter 1: Traditional modeling of fluid flow and transport phenomena
1.1. Introduction
1.2. Basic concepts in fluid mechanics
1.2.1. Fluid properties
1.2.2. Classification of fluids
1.3. Fluid statics
1.3.1. Pressure
1.3.2. Buoyancy
1.4. Equations of fluid motion
1.4.1. Continuity equation
1.4.2. Momentum equation
1.4.3. Energy equation
1.4.4. Solute transport equation
1.5. Dynamics of inviscid flows
1.5.1. Euler's equations
1.5.2. Bernoulli's equation
1.6. Dimensional analysis and similarity
1.7. Boundary-layer theory
1.7.1. Boundary-layer concepts
1.7.2. Laminar and turbulent boundary layers
1.7.3. Boundary-layer separation and control
1.7.4. Non-Newtonian fluids
1.8. Flow in porous media
1.8.1. Darcy's law
1.8.2. Dispersion model
1.8.3. Multiphase flow in porous media
1.8.4. Non-Newtonian fluid flow in porous media
1.9. Suggested readings
1.10. Problems
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Chapter 2: Fundamentals of fractional calculus
2.1. Overview of fractional calculus
2.2. Fractional operators
2.3. Preliminary concepts
2.3.1. Gamma function
2.3.2. Beta function
2.3.3. Mittag-Leffler function
2.4. Different models of fractional derivatives
2.4.1. Riemann–Liouville fractional integral
Example
2.4.2. Riemann–Liouville fractional derivative
Notes
2.4.3. Caputo fractional derivative
Examples
2.4.4. Grünwald–Letnikov fractional derivative
Notes
2.4.5. Caputo–Fabrizio fractional derivative
2.4.6. Atangana–Baleanu fractional derivative
2.5. Constant proportional-Caputo fractional operator
2.6. Fractional derivatives in fluid mechanics
2.7. Laplace transform
2.7.1. Definition and basic idea
2.7.2. Properties of the Laplace transform
2.7.3. Common Laplace transforms
2.7.4. Inverse Laplace transform
2.7.5. Properties of the inverse Laplace transform
2.7.6. Inverse Laplace transform and the Mittag-Leffler function
2.8. Laplace transform of Caputo derivatives
2.9. Laplace transform of Riemann–Liouville derivative
Example 1
Example 2
2.10. Laplace transform of CPC fractional operator
2.11. Hypergeometric function
2.12. Function spaces of fractional operators
Sobolev spaces and their extensions
Smoothness and continuity in fractional contexts
Fractional derivatives in function spaces
2.13. Problems
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Chapter 3: Fundamentals of fractional modeling of fluid flow
3.1. Fractional differential equations
3.2. Derivation of fractional mass equation
3.3. Derivation of fractional momentum conservation equation
Particular cases
3.4. Fractional energy conservation equation
3.4.1. Fractional heat conduction model
3.4.2. Fractional heat convection–conduction model
3.4.3. Fractional transport equation
3.5. Fractional models of flow in porous media
3.5.1. Fractional Darcy's law with time memory
3.5.2. Fractional Darcy's law with space memory
3.5.3. Anomalous diffusion
3.6. Questions for review
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Chapter 4: Analytical solutions of fractional PDEs
4.1. Power-series analytical solution
4.1.1. Gas flow in porous media
4.1.2. Boundary-layer flow
4.2. Adomian decomposition method
Example
4.2.1. Spatial-fractional convection–conduction equation
4.2.2. Time-fractional conduction heat equation
4.2.3. Time-fractional diffusion–reaction equation
Particular case
4.2.4. Time-fractional advection–diffusion equation
4.2.5. Time–space-fractional advection–diffusion equation
4.3. Laplace transform solution of the diffusion equation
4.4. Problems
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Chapter 5: Numerical methods for solving fractional PDEs
5.1. Introduction
5.2. Fractional forward Euler method
5.2.1. Pseudocode for the fractional forward Euler method
5.2.2. Example
5.3. Generalized fractional forward Euler method
5.3.1. Pseudocode for the generalized fractional forward Euler method
5.3.2. Example
5.4. The s-stage explicit fractional-order Runge–Kutta method
5.4.1. Pseudocode for the s-stage EFORK method
5.5. The 2-stage fractional Runge–Kutta method
5.5.1. Pseudocode for the 2-stage FRK method
5.6. The 3-stage EFORK method
5.6.1. Pseudocode for the 3-stage EFORK method
5.7. 2-stage implicit fractional-order Runge–Kutta method
5.7.1. Pseudocode for the 2-stage IFORK method
5.7.2. Example
5.8. Finite difference methods
5.8.1. Explicit finite difference method
5.8.2. Implicit finite difference method
5.9. Finite element methods
5.9.1. Galerkin finite element method
5.9.2. Mixed finite element method
5.9.3. MFEM for fractional diffusion–reaction equations
5.10. Spectral element methods
5.10.1. Function interpolation
Gauss–Legendre interpolants
Gauss–Lobatto–Legendre interpolants
5.10.2. Quadrature
Gauss–Legendre quadrature
Gauss–Jacobi quadrature
5.10.3. Fractional operators in a spectral framework
Riemann–Liouville derivative
Caputo derivative
5.10.4. Spectral methods for time-fractional PDEs
5.10.5. Spectral methods for space-fractional PDEs
5.11. Meshless methods
5.11.1. Radial basis function (RBF)
5.11.2. Radial point interpolation
Time-fractional diffusion equation
5.12. Problems
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Chapter 6: Fractional-order models of fluid flow, heat and mass transfer
6.1. Preliminaries
6.1.1. Conformable fractional derivative
6.2. Fractional-order Navier–Stokes equations
6.2.1. Poiseuille flow
6.2.2. Time fractional Burgers equation
6.3. Semianalytical solution for fractional Navier–Stokes equations
6.4. Fractional modeling of heat transfer
6.4.1. Fractional modeling of conductive heat transfer
6.4.2. Conformable time-fractional convection equation
6.4.3. Time-fractional of convection-conduction
6.4.4. Time-fractional model of fluid flow and heat transfer
6.4.5. Time-fractional of double-diffusive flow
6.4.6. Spatial fractional boundary-layer flow and heat transfer
6.4.7. Fractional modeling of heat transfer in porous media
6.5. Review questions
6.6. Problems
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Chapter 7: Fractional-order models of non-Newtonian fluids
7.1. Introduction
7.2. Prabhakar fractional derivative
7.3. Fractional non-Newtonian fluids
7.3.1. Example
7.4. Fractional power-law fluid model
7.5. Fractional-order cross and Carreau models
7.5.1. Example
7.6. Fractional Herschel–Bulkley model
7.7. Fractional nonlinear rheology model
7.8. Fractional second-grade fluid model
7.9. Fractional viscoelastic materials model
7.10. Fractional Casson fluid model
7.10.1. Example: convective Casson fluid flow
7.10.2. Analytical solutions
7.10.3. Case study: fractional-order modeling in food engineering
7.11. Review questions
7.12. Problems
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Chapter 8: Fractional-order models of nanofluids and ferrofluids
8.1. Introduction
8.2. Nanofluids
8.2.1. Thermal properties of nanofluids
8.2.2. Thermal properties of hybrid nanofluids
8.3. Fractional-order models of nanofluids
8.3.1. Case study: fractional buoyant nanofluids flow
8.4. Nanoparticles transport in porous media
8.4.1. Transport equations
8.4.2. Fractional transport equations
8.5. Nanoparticles transport in two-phase flow
8.5.1. Fractional-order model
8.6. Ferrofluids
8.6.1. Case study
8.6.2. Fractional-order governing equations
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Chapter 9: Fractional models of turbulent flows
9.1. Turbulent flows
9.2. Reynolds-averaged Navier–Stokes equations
9.2.1. Fractional modeling of eddy viscosity
9.2.2. Fractional Kolmogorov energy spectrum
9.2.3. Variable-order fractional model
9.3. Review questions
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Chapter 10: Fractional-order derivatives for multiphase flow in porous media
10.1. Introduction
10.2. Fractional multiphase continuity equation
10.3. Fractional multiphase Darcy's law
10.4. Case studies
10.4.1. Time fractional derivative of countercurrent imbibition
10.4.2. Spatial fractional derivative for imbibition
10.5. Fractional transport in fractured media
10.5.1. Dual-continuum models
10.5.2. Fractional boundary-condition model
10.5.3. Fractional discrete-fracture model
10.6. Fractional groundwater flow model
10.7. Conformable fractional gas equation
10.8. Fractional two-phase flow in fractured media
10.8.1. Fractional discrete-fracture model for two-phase flows
10.8.2. Fractional time derivative for two-phase flow in fractured media
10.9. Review questions
10.10. Problems
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Chapter 11: Fractional models of magneto- and electrohydrodynamics
11.1. Fractional models of electrohydrodynamics
11.2. Fractional models of electroosmosis
11.2.1. Spectral method solution
11.2.2. Lucas polynomials and spectral methods
11.3. Fractional MHD flows in Maxwell fluids
11.4. Problems
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Chapter 12: Fractional models in renewable energy systems
12.1. Fractional models in wind energy systems
12.2. Fractional models of solar energy
12.2.1. Fractional-order solar heating model
12.3. Fractional biochemical reaction model
12.4. Review questions
12.5. Problems
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Appendix A: Cell-centered finite difference method
A.1. Discretization using CCFD
A.1.1. Pressure equation discretization
A.1.2. Darcy's law discretization
Index

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Fractional Modeling of Fluid Flow and Transport Phenomena

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Mohamed F. El-Amin

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