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II: Bulk Fluid Flows
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II: Bulk Fluid Flows
by Bastian E. Rapp
Microfluidics: Modeling, Mechanics and Mathematics
Cover image
Title page
Table of Contents
Copyright
Dedication
Preface
Acknowledgement
List of Figures
List of Tables
List of Listings
List of Acronyms
List of Abbreviations
List of Symbols
List of Constants
List of Chemicals
Conversions
I: Fundamentals
Chapter 1: Introduction
1.1 What is Microfluidics?
1.2 A Brief History of Microfluidics
1.3 Commercial Aspects
1.4 About This Book
1.5 Structure of This Book
Chapter 2: Introduction to Maple
2.1 Introduction
2.2 Elementary Maple Commands
2.3 The File Core.txt
2.4 The File Corefunctions.txt
2.5 The Neptunlib
2.6 Summary
Chapter 3: Engineering Mathematics
3.1 Differential Equations
3.2 Important Functions
3.3 Commonly Used Calculus Tricks
3.4 Summary
Chapter 4: Series
4.1 Introduction
4.2 Taylor Series
4.3 Fourier Series
4.4 Fourier-Bessel Series
4.5 Conclusion
Chapter 5: Transforms
5.1 Fourier Transform
5.2 Laplace Transform
5.3 Summary
Chapter 6: Thermodynamics
6.1 Atomic Model
6.2 Weights and Concentrations
6.3 Important Terms and Concepts in Thermodynamics
6.4 Ideal Gases
6.5 Idealized Thermodynamic Processes
6.6 First Law of Thermodynamics
6.7 Second Law of Thermodynamics
6.8 Third Law of Thermodynamics
6.9 Heat and Mass Transfer
6.10 SUMMARY
Chapter 7: Vector Calculus
7.1 Scalars and Vectors
7.2 Important Theorems in Vector Calculus
7.3 Coordinate System Transformation
7.4 Position, Velocity, and Acceleration
7.5 Jacobian Matrix
7.6 Operators Transformed into the different Coordinate Systems
7.7 Summary
Chapter 8: Differential Equations
8.1 Important Differential Equations
8.2 General Solutions to Selected Ordinary Differential Equations
8.3 General Solutions to Selected Partial Differential Equations
II: Bulk Fluid Flows
Chapter 9: Fluids
9.1 Introduction
9.2 Solids, Liquids, and Gases at the Atomic Scale
9.3 Control Volumes
9.4 Fluid Properties
9.5 Momentum Transport
9.6 Heat Transport
9.7 Mass Transport
9.8 Boundary Conditions
9.9 Dimensionless Numbers
9.10 Summary
Chapter 10: Conservation of Mass: The Continuity Equation
10.1 Fluid Flow in the Bulk
10.2 Continuity Equation
10.3 Integral Representation of the Flowrate
10.4 Mass Balance
10.5 Derivation using Gauss’s Theorem
10.6 Combined Convection and Diffusion: The Convection-Diffusion Equation
10.7 Summary
Chapter 11: Conservation of Momentum: The Navier-Stokes Equation
11.1 Introduction
11.2 Momentum Transfer Into and Out of a Control Volume
11.3 Momentum by in- and Outflowing Mass
11.4 Momentum by Shear Forces
11.5 Momentum by Volume Forces
11.6 Balance of Momentum
11.7 Navier-Stokes Equation for Incompressible Newtonian Fluids
11.8 Dimensional Analysis
11.9 Conclusion
Chapter 12: Conservation of Energy: The Energy Equation and the Thermodynamic Equation of State
12.1 Introduction
12.2 Energy Transfer by Convection
12.3 Heat Flow by Conduction
12.4 Work Flow by Boundary Forces
12.5 Heat Flow by Volume Effects
12.6 Work Flow by Volume Forces
12.7 Balance of Contributions
12.8 Thermodynamic Equation of State
12.9 Summary
Chapter 13: Continuity and Navier-Stokes Equations in Different Coordinate Systems
13.1 Cartesian Coordinates
13.2 Cylindrical Coordinates
13.3 Polar Coordinates
13.4 Spherical Coordinates
13.5 Summary
Chapter 14: The Circular Flow Tube
14.1 Introduction
14.2 Conservation of Mass: The Continuity Equation
14.3 Conservation of Momentum: The Navier-Stokes Equation
14.4 Euler Equation
14.5 Bernoulli Equation
14.6 Conservation of Energy
14.7 Deriving the Euler Equation by a Coordinate System Transformation
14.8 Summary
Chapter 15: Analytical Solutions to the Navier-Stokes Equation
15.1 Hydrostatics and Aerostatics
15.2 Shear Force-Driven Flow: Couette Flow
15.3 Gravity-Driven Flow
15.4 Pressure-Driven Flow: Poiseuille Flow
15.5 Summary
Chapter 16: Analytical Solutions to Poiseuille Flow Problems in Different Geometries
16.1 Elliptical and Circular Profiles
16.2 Planar Infinitesimally Extended Channel Cross-Sections
16.3 Flows in Circular Cross-Sections: Hagen-Poiseuille Flow
16.4 Flows in Rectangular Cross-Sections: Solution to Poisson and Laplace Equations
16.5 Summary
Chapter 17: Hydraulic Resistance
17.1 Introduction
17.2 Viscous Dissipation
17.3 Hydraulic Resistance of Important flow Channel Geometries
17.4 Simplification Approaches to Hydraulic Resistances
17.5 Equivalent Circuit Theory
17.6 Summary
Chapter 18: Analytical Solutions to Transient Flow Problems
18.1 Time-Dependent Transient Effects: Acceleration and Deceleration
18.2 Time-Dependent Couette Flow
18.3 Time-Dependent Hagen-Poiseuille Flow
18.4 Time-Dependent Flow in Rectangular Cross-Sections
18.5 Entrance Effects in Hagen-Poiseuille Flow
18.6 Summary
Chapter 19: Taylor-Aris Dispersion
19.1 Introduction
19.2 Dispersion
19.3 Convection-Diffusion Equation for Cylindrical Cross-Sections
19.4 Mass Concentration Function
19.5 Convection-Diffusion Equation
19.6 Solving for P
19.7 Solving for P
19.8 Validity of the Solution
19.9 Example
19.10 SUMMARY
III: Fluid Surface Effects
Chapter 20: Surface Tension
20.1 Fluid Effects at Interfaces
20.2 Contact Angle Measurement
20.3 Surfactants
20.4 Marangoni Effect
20.5 Summary
Chapter 21: Capillarity
21.1 Capillary Pressure
21.2 Capillary Length
21.3 Meniscus Formation
21.4 Summary
Chapter 22: Measuring Surface Tension and Free Surface Energy
22.1 Introduction
22.2 Measuring the Surface Tension of Liquids
22.3 MEASURING THE FREE SURFACE ENERGY
22.4 Summary
Chapter 23: Plateau-Rayleigh Instability
23.1 Introduction
23.2 Stability Considerations
23.3 Fluid Jets
23.4 Instability
23.5 Standing Waves on a Fluid Jet
23.6 Characteristic Breakup Time
23.7 Applicability of the Plateau-Rayleigh Instability
23.8 Summary
Chapter 24: The Shape of Drops
24.1 Introduction
24.2 Derivation
24.3 Bashforth and Adams: Curvature Expressed as Z (X)
24.4 Brien, Ben, and Van den Brule: Curvature Expressed as Function of θ (Sessile Drops)
24.5 Del Río and Neumann: Curvature Expressed as Function of S (Pendant Drop)
24.6 Comparison With Experimental Data
24.7 Drops Inside of a Fluid
24.8 Historical Development of Drop-Shape Analysis
24.9 Summary
IV: Numerics
Chapter 25: Numerical Methods for Linear Systems of Equations
25.1 Introduction
25.2 Solutions to Linear Systems of Equations
25.3 Numerical Solutions to Linear Systems of Equations
25.4 Summary
Chapter 26: Numerical Solutions to Nonlinear Systems: Newton’s Method
26.1 Introduction
26.2 An Example: The Loran System
26.3 Newton’s Method
26.4 A Solver Implemented in Maple
26.5 Summary
Chapter 27: Numerical Methods for Solving Differential Equations
27.1 Introduction
27.2 Numerical Solutions to Ordinary Differential Equations
27.3 Numerical Solutions to Higher-Order Ordinary Differential Equations and Systems of Coupled Ordinary Differential Equations
27.4 Numerical Solutions to Systems of Ordinary Differential Equations with Boundary Conditions
27.5 Summary
Chapter 28: Numerical Solutions to the Navier-Stokes Equation
28.1 Introduction
28.2 Solution to the Poisson Equation
28.3 Solution to the Poisson Equation Using SOR
28.4 Summary
Chapter 29: Computational Fluid Dynamics
29.1 Introduction
29.2 Galerkin Method
29.3 Summary
Chapter 30: Finite Difference Method
30.1 Introduction
30.2 Advantages and Disadvantages
30.3 FDM in Microsoft Excel
30.4 Summary
Chapter 31: Finite Volume Method
31.1 Introduction
31.2 Conservative form Notation
31.3 Integral form of the Conservative Notation
31.4 Discretization
31.5 Function Reconstruction
31.6 Example: One-Dimensional Heat Equation
31.7 Two-Dimensional Problems of First Order in Time and Space
31.8 Two-Dimensional Problems of First Order in Time and Second-Order in Space
31.9 Summary
Chapter 32: Finite Element Method
32.1 Introduction
32.2 Discretization
32.3 Lagrangian Coordinates
32.4 Basis Functions
32.5 One-Dimensional Example: Flow in Infinitesimally Extended Channels
32.6 Two-Dimensional Example: Flow in Rectangular Channels
32.7 Summary
Chapter 33: Numerical Solutions to Transient Flow Problems
33.1 Introduction
33.2 A Numerical Solver for Two-dimensional Time-Dependent Flow Problems
33.3 A Numerical Solver for Two-Dimensional Entrance Flow Problems
33.4 Summary
Chapter 34: Numerical Solutions to Three-Dimensional Flow Problems
34.1 Introduction
34.2 Derivation
34.3 Implementation of a Stationary Flow Numerical Solver
34.4 Usage of the Numerical Solver
34.5 Summary
Bibliography
Index
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Chapter 9: Fluids
II
Bulk Fluid Flows
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