Transcendental representations with applications to solids and fluids

879 Pages·2012·17.449 MB·English

by  Campos, Luis Manuel Braga da Costa

Mathematics and Physics for Science and Technology
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Transcendental Representations with ihn Transcendental Representations
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Building on the author’s previous book in the series, Complex Analysis Fan
with Applications to Flows and Fields (CRC Press, 2010), Transcendental ltt
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Representations with Applications to Solids and Fluids focuses on four
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infinite representations: series expansions, series of fractions for meromorphic dn
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functions, infinite products for functions with infinitely many zeros, and
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continued fractions as alternative representations. This book also continues e

the application of complex functions to more classes of fields, including tp
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incompressible rotational flows, compressible irrotational flows, unsteady flows, r
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rotating flows, surface tension and capillarity, deflection of membranes under
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load, torsion of rods by torques, plane elasticity, and plane viscous flows. The
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two books together offer a complete treatment of complex analysis, showing
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how the elementary transcendental functions and other complex functions are
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applied to fluid and solid media and force fields mainly in two dimensions. a
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The mathematical developments appear in odd-numbered chapters while o
the physical and engineering applications can be found in even-numbered
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chapters. The last chapter presents a set of detailed examples. Each chapter
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begins with an introduction and concludes with related topics.
Written by one of the foremost authorities in aeronautical/aerospace engineering,
this self-contained book gives the necessary mathematical background and
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physical principles to build models for technological and scientific purposes.
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It shows how to formulate problems, justify the solutions, and interpret the m
results. p
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L.M.B.C. Campos
K11546
K11546_Cover.indd 1 3/8/12 4:52 PM
Transcendental Representations
with Applications to
Solids and Fluids
Mathematics and Physics for
Science and Technology
Series Editor
L.M.B.C. Campos
Director of the Center for Aeronautical
and Space Science and Technology
Lisbon Technical University
Complex Analysis with Applications to Flows and Fields
L.M.B.C. Campos
Transcendental Representations with Applications to Solids and Fluids
L.M.B.C. Campos
Mathematics and Physics for Science and Technology
Transcendental Representations
with Applications to
Solids and Fluids
L.M.B.C. Campos
Director of the Center for Aeronautical
and Space Science and Technology
Lisbon Technical University
CRC Press
Taylor & Francis Group
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© 2012 by Taylor & Francis Group, LLC
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Version Date: 20120301
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to the memory of António Gouvea Portela
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Contents
List of Tables, Notes, Diagrams, Classifications, Lists, and Examples ................................xxi
Series Preface .............................................................................................................................xxix
Preface .......................................................................................................................................xxxiii
Acknowledgments ...................................................................................................................xxxv
Author ......................................................................................................................................xxxvii
Mathematical Symbols ...........................................................................................................xxxix
Physical Quantities ...................................................................................................................xlvii
Introduction .....................................................................................................................................li
1. Sequences of Fractions or Products ....................................................................................1
1.1 Power Series, Singularities, and Functions ...............................................................1
1.1.1 Taylor Series for Analytic Functions .............................................................2
1.1.2 Laurent Series near an Isolated Singularity .................................................2
1.1.3 Classification of Singularities from the Principal Part ...............................3
1.1.4 Coefficients of the Principal Part at a Pole ...................................................4
1.1.5 Pole or Essential Singularity at Infinity ........................................................5
1.1.6 Classification of Functions from Their Singularities ..................................6
1.1.7 Integral Functions as Extensions of Polynomials .......................................6
1.1.8 Exponential and Sine/Cosine as Integral Functions ..................................6
1.1.9 Rational–Integral and Polymorphic Functions ...........................................8
1.1.10 Meromorphic Functions as Extensions of the Rational Functions ...........8
1.1.11 Tangent/Cotangent and Secant/Cosecant as Meromorphic Functions .........9
1.2 Series of Fractions for Meromorphic Functions (Mittag-Leffler 1876, 1884) .......10
1.3 Meromorphic Function as a Ratio of Two Integral Functions ..............................12
1.3.1 Two Cases of Ratios of Integral Functions .................................................12
1.3.2 Series of Fractions for the Circular Secant .................................................13
1.3.3 Series of Fractions for the Circular Cotangent ..........................................14
1.3.4 Extended Theorem on Series of Fractions ..................................................15
1.4 Factorization with Infinite Number of Zeros ..........................................................16
1.4.1 Removal of Poles and Relation with Zeros ................................................16
1.4.2 Theorem on Infinite Products ......................................................................16
1.4.3 Extended Theorem on Infinite Products ....................................................17
1.5 Infinite Products for Circular Functions .................................................................18
1.5.1 Infinite Product for the Circular Cosine .....................................................18
1.5.2 Infinite Product for the Circular Sine .........................................................18
1.5.3 Series of Fractions for Circular Tangent .....................................................19
1.5.4 Series of Fractions for the Circular Cosecant .............................................20
1.5.5 Zeros and Poles of the Circular Tangent ....................................................21
1.6 Recurrence Formulas and Continued Fractions (Wallis 1656; Euler 1737) .........22
1.6.1 Multiple Backward Linear Recurrence Formulas .....................................22
1.6.2 Continuants, Termination, Periodicity, and Convergence .......................23
1.6.3 Recurrence Formula for Numerators and Denominators of
Convergents (Wallis 1656; Euler 1737) ........................................................24
vii
viii Contents
1.6.4 Differences of Continuants and Odd/Even Commutators ......................25
1.6.5 Continued Fraction for the Algebraic Inverse ...........................................26
1.7 Optimal and Doubly Bounded Sharpening Approximations ..............................27
1.7.1 Error of Truncation for a Continued Fraction of the First Kind ..............27
1.7.2 Convergence/Oscillation of a Continued Fraction of the First Kind .....28
1.7.3 Optimal Rational Approximation for a Simple Continued Fraction ......28
1.7.4 Unique Simple Terminating (Nonterminating) Continued Fraction
for a Rational (Irrational) Number ..............................................................29
1.8 Transformation of Series and Products into Fractions (Euler 1785) ....................30
1.8.1 Continued Fraction for a Series (Euler 1748) ..............................................30
1.8.2 Continued Fraction with Given Continuants ............................................31
1.8.3 Continued Fraction for a Product ................................................................32
1.8.4 Continued Fraction for the Exponential .....................................................32
1.8.5 Calculation of the Napier Irrational Number e .........................................33
1.8.6 Continued Fraction (Euler 1739) for the Sine .............................................35
1.8.7 Quadrature of the Circle and the Irrational Number π ............................36
1.9 Continued Fraction for the Ratio of Two Series (Lambert 1770) ..........................37
1.9.1 Convergence of the Hypogeometric Series ................................................38
1.9.2 Integral (Meromorphic) Function of the Variable (Parameter) ...............39
1.9.3 Contiguity Relations and Continued Fractions .........................................40
1.9.4 Terminating and Nonterminating Continued Fractions .........................42
1.9.5 Continued Fraction for the Circular Tangent ............................................43
1.10 Conclusion ....................................................................................................................48
2. Compressible and Rotational Flows .................................................................................49
2.1 Source, Sink, and Vortex in a Compressible Flow ..................................................49
2.1.1 Momentum Equation in an Inviscid Fluid (Euler 1755, 1759) .................50
2.1.2 Potential Flow: Irrotational and Incompressible (Bernoulli 1693) ..........51
2.1.3 Irrotational Flow of a Barotropic Fluid .......................................................52
2.1.4 Bernoulli Equation for a Homentropic Flow .............................................53
2.1.5 Sound Speed as a Flow Variable ..................................................................53
2.1.6 Critical Flow at the Sonic Condition ...........................................................54
2.1.7 Local, Reference, and Critical Mach Numbers ..........................................55
2.1.8 Minimum/Critical Radii for a Vortex in a Homentropic Flow ...............56
2.1.9 Conservation of the Mass Rate for a Source/Sink ....................................58
2.1.10 Combination in a Compressible Spiral Monopole ....................................60
2.2 Potential Vortex with Rotational Core (Rankine; Hallock and
Burnham 1997) ..........................................................................................................62
2.2.1 Vorticity Concentrated in a Finite Core (Rankine)....................................62
2.2.2 Smooth Radial Variation of the Vorticity (Hallock and Burnham
1997) .................................................................................................................65
2.2.3 Momentum Equation for an Inviscid Vortex .............................................68
2.2.4 Pressure and Core Radius for a Vortex .......................................................70
2.2.5 Pressure in the Core of the Generalized Rankine Vortex ........................71
2.2.6 Pressure for the Generalized Hallock–Burnham Vortex .........................73
2.2.7 Radial Variation of the Stagnation Pressure ..............................................74
2.2.8 Vorticity and the Gradient of the Stagnation Pressure ............................76
2.2.9 Vortical and Local Inertia Forces .................................................................77
2.2.10 Columnar Vortex in the Gravity Field ........................................................78
Contents ix
2.2.11 Shape of the Free Surface of a Swirling Liquid .........................................79
2.2.12 Free Surfaces, Tornadoes, and Wake Vortices ...........................................80
2.3 Minimum Energy (Thomson 1849) and Intrinsic Equations of Motion..............81
2.3.1 Momentum Equation in Intrinsic Coordinates .........................................81
2.3.2 Vorticity in Terms of Intrinsic Coordinates ...............................................83
2.3.3 Variation of the Stagnation Pressure across Streamlines due
to the Vorticity ................................................................................................83
2.3.4 Potential Flow as the Minimum of the Kinetic Energy (Kelvin 1849) .........84
2.4 Laplace/Poisson Equations in Complex Conjugate Coordinates .........................85
2.4.1 Incompressible, Irrotational, and Potential Flows ....................................86
2.4.2 Complex Conjugate Coordinates .................................................................87
2.4.3 General Integral of the Laplace Equation ...................................................88
2.4.4 Complete Integral of the Poisson Equation ................................................89
2.4.5 General/Simple Unidirectional Shear Flow ..............................................90
2.4.6 Complex Conjugate Velocity and Cartesian Components .......................92
2.4.7 Uniform Flow with Angle of Attack ...........................................................93
2.4.8 Relation between Polar and Complex Conjugate Coordinates ...............94
2.4.9 Compound Derivates and Laplacian in Polar Coordinates .....................96
2.4.10 Polar Coordinates and Components of the Velocity .................................97
2.4.11 Potential Vortex, Source/Sink, and Monopole ..........................................98
2.5 Second Forces/Moment and Circle Theorems......................................................100
2.5.1 Forces and Pitching Moment due to the Pressure ..................................100
2.5.2 Forces and Moment in an Incompressible Rotational Flow ..................101
2.5.3 Second Circle Theorem in a Rotational Flow ..........................................103
2.6 Cylinder in a Unidirectional Shear Flow ...............................................................104
2.6.1 Unidirectional Shear Flow with Angle of Attack ...................................105
2.6.2 Free Stream, Vorticity, and Circulation Effects .......................................107
2.6.3 Lift and Shear Forces and Coefficients .....................................................108
2.6.4 Correspondence between the Circulation and the Vorticity .................110
2.7 Monopole Interactions and Equilibrium Positions ..............................................113
2.7.1 Centroid of an Ensemble of Line Monopoles ..........................................114
2.7.2 Radial Motion of a Pair of Sources/Sinks ................................................115
2.7.3 Azimuthal Motion of Two Vortices ...........................................................118
2.7.4 Source/Sink, Vortex, or Monopole near a Wall .......................................122
2.7.5 Pair of Conjugate Monopoles in a Parallel-Sided Duct ..........................126
2.7.6 Boundary Conditions at Rigid Impermeable Walls ................................130
2.7.7 Velocity of the Monopoles between Parallel Walls .................................130
2.7.8 Uniform Convection and Equilibrium at Rest .........................................132
2.7.9 Paths of Vortices, Sources, and Sinks in a Duct.......................................133
2.8 Cylinder in a Stream with Two Trailing Monopoles (Föppl 1913) .....................135
2.8.1 Pair of Conjugate Monopoles as the Wake of a Cylinder .......................137
2.8.2 Rigid Wall Boundary Condition on the Impermeable Cylinder ...........140
2.8.3 Circulation and Flow Rate for Static Equilibrium ..................................141
2.8.4 Direct Equilibrium Problem and Downforce Coefficient ......................143
2.8.5 First Inverse Problem: Pair of Opposite Vortices at Rest
(Föppl 1913) ...............................................................................................144
2.8.6 Second Inverse Problem: Pair of Equal Sources/Sinks at Rest ..............147
2.8.7 Comparison of Equilibria for Vortices and Sources/Sinks ....................149
2.8.8 Small Perturbations from Positions of Rest .............................................152