Table Of ContentVector Fields with
Applications to
Thermodynamics and
Irreversibility
Mathematics and Physics for Science and Technology
Series Editor: L.M.B.C. Campos
former Director of the Center for Aeronautical
and Space Science and Technology
Lisbon University
Volumes in the series:
TOPIC A – THEORY OF FUNCTIONS AND POTENTIAL PROBLEMS
Volume I (Book 1) – Complex Analysis with Applications to Flows and Fields
L.M.B.C. Campos
Volume II (Book 2) – Elementary Transcendentals with Applications to Solids and Fluids
L.M.B.C. Campos
Volume III (Book 3) – Generalized Calculus with Applications to Matter and Forces
L.M.B.C. Campos
TOPIC B – BOUNDARY AND INITIAL-VALUE PROBLEMS
Volume IV – Ordinary Differential Equations with Applications to Trajectories and
Vibrations
L.M.B.C. Campos
Book 4 – Linear Differential Equations and Oscillators
L.M.B.C. Campos
Book 5 – Non-Linear Differential Equations and Dynamical Systems
L.M.B.C. Campos
Book 6 – Higher-Order Differential Equations and Elasticity
L.M.B.C. Campos
Book 7 – Simultaneous Differential Equations and Multi-Dimensional Vibrations
L.M.B.C. Campos
Book 8 – Singular Differential Equations and Special Functions
L.M.B.C. Campos
Book 9 – Classification and Examples of Differential Equations and their Applications
L.M.B.C. Campos
Volume V – Partial Differential Equations with Applications to Waves and Diffusion
L.M.B.C. Campos & L.A.R. Vilela
Book 10 – Vector Fields with Applications to Thermodynamics and Irreversibility
L.M.B.C. Campos & L.A.R. Vilela
Book 11 – Compressible Flow with Application to Shocks and Propulsion
L.M.B.C. Campos & L.A.R. Vilela
For more information about this series, please visit: http://www.crcpress.com/Mathematics-
and-Physics-for-Science-and-Technology/book-series/CRCMATPHYSCI
Vector Fields with
Applications to
Thermodynamics and
Irreversibility
Luis Manuel Braga da Costa Campos
and
Luís António Raio Vilela
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2023 Luis Manuel Braga da Costa Campos and Luís António Raio Vilela
CRC Press is an imprint of Taylor & Francis Group, LLC
The right of Luis Manuel Braga da Costa Campos and Luís António Raio Vilela to be identified as authors
of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs
and Patents Act 1988.
Reasonable efforts have been made to publish reliable data and information, but the author and publisher
cannot assume responsibility for the validity of all materials or the consequences of their use. The authors
and publishers have attempted to trace the copyright holders of all material reproduced in this publication and
apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright
material has not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter
invented, including photocopying, microfilming, and recording, or in any information storage or retrieval
system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.com
or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. For works that are not available on CCC please contact [email protected]
Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used
only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Names: Campos, Luis Manuel Braga da Costa, author. | Vilela, Luís António Raio, author.
Title: Partial differentials with applications to thermodynamics and compressible flow / Luis Manuel
Braga da Costa Campos and Luís António Raio Vilela.
Description: First edition. | Boca Raton, FL : CRC Press, 2021.
| Series: Mathematics and physics for science and technology | Includes bibliographical
references and index.
| Summary: “This book is part of the series “Mathematics and Physics Applied to Science and
Technology.” It combines rigorous mathematics with general physical principles to model practical
engineering systems with a detailed derivation and interpretation of results”– Provided by publisher.
Identifiers: LCCN 2021005670 (print) | LCCN 2021005671 (ebook) | ISBN 9781032029870
(v. 1; hardback) | ISBN 9781032030838 (v. 1; paperback) | ISBN 9781032030838 (v. 2; hardback)
| ISBN 9781032030814 (v. 2; paperback) | ISBN 9781003186595 (v. 1; ebook) | ISBN 9781003186588
(v. 2; ebook) Subjects: LCSH: Thermodynamics. | Fluid dynamics. | Gases, Compressed—Mathematical
models. | Differential equations, Partial.
Classification: LCC QC168 .C29 2021 (print) | LCC QC168 (ebook) | DDC 515/.353—dc23
LC record available at https://lccn.loc.gov/2021005670
LC ebook record available at https://lccn.loc.gov/2021005671
ISBN: 978-1-032-02987-0 (hbk)
ISBN: 978-1-032-03083-8 (pbk)
ISBN: 978-1-003-18659-5 (ebk)
DOI: 10.1201/9781003186595
Typeset in Times
by SPi Technologies India Pvt Ltd (Straive)
To Sir Michael James Lighthill
Contents
List of Notes, Tables, and Diagrams ........................................................................xv
Series Preface .........................................................................................................xvii
Preface to Volume V ................................................................................................xix
About the Authors .................................................................................................xxiii
Acknowledgments ..................................................................................................xxv
List of Mathematical Symbols .............................................................................xxvii
List of Physical Quantities ...................................................................................xxxv
Chapter 1 Classes of Equations and Similarity Solutions......................................1
1.1 Hierarchy of Partial Differential Equations ................................1
1.1.1 Single Equations and Simultaneous Systems ...............1
1.1.2 Partial Differential Equation in N Variables .................2
1.1.3 One Dependent and Two Independent Variables ..........4
1.2 General Integral and Arbitrary Functions ...................................6
1.2.1 First-Order P.D.E. and One Arbitrary Function............6
1.2.2 P.D.E. of Order N and N Arbitrary Functions ..............7
1.3 Unforced P.D.E. with First-Order Derivatives............................7
1.3.1 Classification of P.D.E.S of First Order .......................8
1.3.2 Family of Hypersurfaces Tangent to a Vector Field .....9
1.3.3 Characteristic Curve Tangent to a Vector Field ..........10
1.3.4 Characteristic Variables as Solutions of the
Characteristic Equations .............................................10
1.3.5 Plane Curve Tangent to a Vector Field .......................11
1.3.6 Example of the Family of Surfaces Tangent to
the Position Vector ......................................................12
1.4 Quasi-Linear and Forced First-Order P.D.E.s ..........................13
1.4.1 Quasi-Linear P.D.E. of the First-Order.......................13
1.4.2 Solution as Implicit Function of N Variables .............14
1.4.3 Linear Forced P.D.E. with First-Order Derivatives ....15
1.4.4 Plane Curves with Unit Projection on the
Position Vector ............................................................15
1.5 Differentials of First-Degree in Three Variables ......................16
1.5.1 Exact, Inexact and Non-Integrable Differentials ........17
1.5.2 Linear and Angular Velocities and Helicity ................18
1.5.3 Exact Differential and Immediate Integrability ..........20
1.5.4 Inexact Differential with an Integrating Factor ..........21
1.5.5 Inexact Differential without Integrating Factor ..........21
1.5.6 Irrotational/Rotational and Non-Helical/Helical
Vector Fields ...............................................................22
1.5.7 Irrotational or Conservative Vector Field as the
Gradient of a Scalar Potential .....................................23
vii
viii Contents
1.5.8 Rotational Non-Helicoidal Vector Field and Two
Scalar Potentials .........................................................24
1.5.9 Helical Vector Field and Three Scalar Potentials .......25
1.5.10 Existence and Determination of the Three Scalar
Potentials ....................................................................27
1.5.11 Scalar Poisson Equation Forced by the
Divergence of a Vector Field ......................................28
1.5.12 Vector Poisson Equation Forced by the Curl of a
Vector Field ................................................................29
1.5.13 Potential Vector Field and Laplace Equation .............29
1.5.14 Scalar and Vector Potentials for a General Vector
Field ............................................................................30
1.5.15 Three Alternative Scalar Euler or Clebsch
Potentials ....................................................................31
1.5.16 Eight Cases of Three-Dimensional Vector Fields .......32
1.5.17 Potential Vector Field with Scalar or Vector
Potentials ....................................................................32
1.5.18 Irrotational Vector Field and One Scalar Potential .....36
1.5.19 Solenoidal Vector Field and Vector Potential .............37
1.5.20 Non-Solenoidal, Rotational Vector Field and
Scalar and Vector Potentials .......................................39
1.5.21 Rotational, Non-Helical Vector Field and Two
Euler Scalar Potentials................................................40
1.5.22 Helical Vector Field and Three Scalar Euler
Potentials ....................................................................41
1.5.23 Rotational, Non-Solenoidal Vector Field and
Three Scalar Clebsch Potentials .................................42
1.5.24 Partial Differential Equations and Characteristic
Systems .......................................................................44
1.5.25 Unicity of Scalar and Vector Potentials ......................46
1.5.26 Characteristic Systems for Euler and Clebsch
Potentials ....................................................................47
1.6 P.D.E.s with Constant Coefficients and All Derivatives of
Same Order ...............................................................................48
1.6.1 P.D.E. of Constant Order and Characteristic
Polynomial .................................................................48
1.6.2 Similarity Solutions for a Linear Combination
of Variables .................................................................49
1.6.3 General Integral for Distinct Roots ............................49
1.6.4 Single or Multiple Roots of the Characteristic
Polynomial .................................................................50
1.6.5 Method of Variation of Parameters .............................51
1.6.6 Method of Parametric Differentiation ........................51
1.6.7 General Integral for Multiple Roots ...........................52
1.6.8 Linear P.D.E. with Constant Coefficients and
Second-Order Derivatives ..........................................53
Contents ix
1.7 Harmonic and Biharmonic Functions on the Plane ..................54
1.7.1 Laplace Equation in Cartesian Coordinates ...............55
1.7.2 Real/Complex Harmonic Functions and
Boundary Conditions ..................................................56
1.7.3 Biharmonic Equation in the Cartesian Plane ..............57
1.7.4 Real/Complex Biharmonic Functions and Forcing ....58
1.8 Forced Linear P.D.E. with of Derivatives Constant Order .......60
1.8.1 Forcing of P.D.E. by a Similarity Function ................60
1.8.2 Similarity Forcing in the Non-Resonant Case ............61
1.8.3 Parametric Differentiation for Multiple Resonance ...62
1.8.4 Multiple Resonance via L’Hôspital Rule ...................62
1.8.5 Comparison of Resonant and Non-Resonant Cases ...63
1.8.6 Forcing by an Exponential Similarity Function .........63
1.8.7 Forcing by a Sinusoidal Similarity Function ..............65
1.8.8 General, Particular and Complete Integrals................66
1.8.9 Complete Integrals with Simple/Double
Resonances .................................................................67
1.9 Forced Harmonic and Biharmonic Equations ..........................68
1.9.1 General Forcing of the Laplace Equation ...................69
1.9.2 Complete Integral of the Harmonic Equation ............69
1.9.3 General Forcing of the Double Laplace Equation ......70
1.9.4 Complete Integral of the Biharmonic Equation .........70
1.9.5 Method of Similarity Variables for Arbitrary
Forcing. .......................................................................71
1.9.6 Fourth-Order Equation with Two Independent
and Three Similarity Variables ...................................72
1.9.7 Two Methods for Exponential Forcing .......................74
1.10 Conclusion ................................................................................93
Chapter 2 Thermodynamics and Irreversibility ...................................................95
2.1 Work, Heat, Entropy and Temperature .....................................96
2.1.1 Work of Conservative and Non-Conservative
Forces .........................................................................97
2.1.2 Internal Energy and First Principle of
Thermodynamics ........................................................98
2.1.3 Inertia Force and Kinetic Energy ...............................99
2.1.4 Mass and Gravity Field, Potential, Force and
Energy ......................................................................100
2.1.5 Electric Charge, Field, Scalar Potential and Force ...101
2.1.6 Electric Displacement, Work and Energy .................102
2.1.7 Dielectric Permittivity Tensor and Scalar. ................103
2.1.8 Work in Terms of the Electric Field and
Displacement ............................................................104
2.1.9 Electric Current and Magnetic Vector Potential .......105
2.1.10 Magnetic Field, Induction and Energy .....................107
