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E-raamat: Quasilinear Hyperbolic Systems, Compressible Flows, and Waves

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  • Formaat: 282 pages
  • Ilmumisaeg: 29-Apr-2010
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781439836910
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Quasilinear Hyperbolic Systems, Compressible Flows, and WavesSuurem pilt
  • Formaat: 282 pages
  • Ilmumisaeg: 29-Apr-2010
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781439836910
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Filled with practical examples, Quasilinear Hyperbolic Systems, Compressible Flows, and Waves presents a self-contained discussion of quasilinear hyperbolic equations and systems with applications. It emphasizes nonlinear theory and introduces some of the most active research in the field.





After linking continuum mechanics and quasilinear partial differential equations, the book discusses the scalar conservation laws and hyperbolic systems in two independent variables. Using the method of characteristics and singular surface theory, the author then presents the evolutionary behavior of weak and mild discontinuities in a quasilinear hyperbolic system. He also explains how to apply weakly nonlinear geometrical optics in nonequilibrium and stratified gas flows and demonstrates the power, generality, and elegance of group theoretic methods for solving Euler equations of gasdynamics involving shocks. The final chapter deals with the kinematics of a shock of arbitrary strength in three dimensions.





With a focus on physical applications, this text takes readers on a journey through this fascinating area of applied mathematics. It provides the essential mathematical concepts and techniques to understand the phenomena from a theoretical standpoint and to solve a variety of physical problems.
Preface ix
About the Author xiii
1 Hyperbolic Systems of Conservation Laws
1(14)
1.1 Preliminaries
1(1)
1.2 Examples
2(13)
1.2.1 Traffic flow
2(1)
1.2.2 River flow and shallow water eqations
3(1)
1.2.3 Gasdynamic equations
4(1)
1.2.4 Relaxing gas flow
5(2)
1.2.5 Magnetogasdynamic equations
7(3)
1.2.6 Hot electron plasma model
10(1)
1.2.7 Radiative gasdynamic equations
11(1)
1.2.8 Relativistic gas model
11(1)
1.2.9 Viscoelasticity
12(1)
1.2.10 Dusty gases
12(1)
1.2.11 Zero-pressure gasdynamic system
13(2)
2 Scalar Hyperbolic Equations in One Dimension
15(24)
2.1 Breakdown of Smooth Solutions
15(10)
2.1.1 Weak solutions and jump condition
17(4)
2.1.2 Entropy condition and shocks
21(1)
2.1.3 Riemann problem
22(3)
2.2 Entropy Conditions Revisited
25(5)
2.2.1 Admissibility criterion I (Oleinik)
25(1)
2.2.2 Admissibility criterion II (Vanishing viscosity)
25(1)
2.2.3 Admissibility criterion III (Viscous profile)
26(2)
2.2.4 Admissibility criterion IV (Kruzkov)
28(1)
2.2.5 Admissibility criterion V (Oleinik)
29(1)
2.3 Riemann Problem for Nonconvex Flux Function
30(2)
2.4 Irreversibility
32(2)
2.5 Asymptotic Behavior
34(5)
3 Hyperbolic Systems in One Space Dimension
39(36)
3.1 Genuine Nonlinearity
39(1)
3.2 Weak Solutions and Jump Condition
40(1)
3.3 Entropy Conditions
41(3)
3.3.1 Admissibility criterion I (Entropy pair)
41(1)
3.3.2 Admissibility criterion II (Lax)
42(1)
3.3.3 k-shock wave
43(1)
3.3.4 Contact discontinuity
43(1)
3.4 Riemann Problem
44(10)
3.4.1 Simple waves
44(1)
3.4.2 Riemann invariants
45(1)
3.4.3 Rarefaction waves
46(1)
3.4.4 Shock waves
46(8)
3.5 Shallow Water Equations
54(21)
3.5.1 Bores
55(2)
3.5.2 Dilatation waves
57(2)
3.5.3 The Riemann problem
59(2)
3.5.4 Numerical solution
61(4)
3.5.5 Interaction of elementary waves
65(1)
3.5.6 Interaction of elementary waves from different families
66(2)
3.5.7 Interaction of elementary waves from the same family
68(7)
4 Evolution of Week Waves in Hyperbolic Systems
75(58)
4.1 Waves and Compatibility Conditions
75(9)
4.1.1 Bicharacteristic curves or rays
77(1)
4.1.2 Transport equations for first order discontinuities
78(3)
4.1.3 Transport equations for higher order discontinuites
81(1)
4.1.4 Transport eqations for mild discontinuities
82(2)
4.2 Evolutionary Behavior of Acceleration Waves
84(10)
4.2.1 Local behavior
85(1)
4.2.2 Global behavior: The main results
86(3)
4.2.3 Proofs of the main results
89(2)
4.2.4 Some special cases
91(3)
4.3 Interaction of Shock Waves with Weak Discontinuities
94(6)
4.3.1 Evolution law for the amplitudes of C1 discontinuities
94(2)
4.3.2 Reflected and transmitted amplitudes
96(4)
4.4 Weak Discontinuities in Radiative Gasdynamics
100(6)
4.4.1 Radiation induced waves
101(2)
4.4.2 Modified gasdynamic waves
103(1)
4.4.3 Waves entering in a uniform region
104(2)
4.5 One-Dimensional Weak Discontinuity Waves
106(6)
4.5.1 Characteristic approach
106(3)
4.5.2 Semi-characteristic approach
109(1)
4.5.3 Singular surface approach
110(2)
4.6 Weak Nonlinear Waves in an Ideal Plasma
112(8)
4.6.1 Centered rarefaction waves
116(2)
4.6.2 Compression waves and shock front
118(2)
4.7 Relatively Undistorted Waves
120(13)
4.7.1 Finite amplitude disturbances
122(1)
4.7.2 Small amplitude waves
123(7)
4.7.3 Waves with amplitude not-so-small
130(3)
5 Asymptotic Waves for Quasilinear Systems
133(32)
5.1 Weakly Nonlinear Geometrical Optics
133(4)
5.1.1 High frequency processes
134(2)
5.1.2 Nonlinear geometrical acoustics solution in a relaxing gas
136(1)
5.2 Far Field Behavior
137(3)
5.3 Energy Dissipated across Shocks
140(6)
5.3.1 Formula for energy dissipated at shocks
140(2)
5.3.2 Effect of distributional source terms
142(2)
5.3.3 Application to nonlinear geometrical optics
144(2)
5.4 Evolution Equation Describing Mixed Nonlinearity
146(11)
5.4.1 Derivation of the transport equations
147(2)
5.4.2 The ε-approximate equation and transport equation
149(3)
5.4.3 Comparison with an alternative approach
152(1)
5.4.4 Energy dissipated across shocks
152(3)
5.4.5 Application
155(2)
5.5 Singular Ray Expansions
157(3)
5.6 Resonantly Interacting Waves
160(5)
6 Self-Similar Solutions Involving Discontinuities
165(40)
6.1 Waves in Self-Similar Flows
167(9)
6.1.1 Self-similar solutions and their asymptotic behavior
168(5)
6.1.2 Collision of a C1-wave with a blast wave
173(3)
6.2 Imploding Shocks in a Relaxing Gas
176(20)
6.2.1 Basic equations
177(1)
6.2.2 Similarity analysis by invariance groups
178(3)
6.2.3 Self-similar solutions and constraints
181(7)
6.2.4 Imploding shocks
188(1)
6.2.5 Numerical results and discussion
189(7)
6.3 Exact Solutions of Euler Equations via Lie Group Analysis
196(9)
6.3.1 Symmetry group analysis
197(1)
6.3.2 Euler equations of ideal gas dynamics
198(4)
6.3.3 Solution with shocks
202(3)
7 Kinematics of a Shock of Arbitrary Strength
205(44)
7.1 Shock Wave through an Ideal Gas in 3-Space Dimensions
206(17)
7.1.1 Wave propagation on the shock
210(2)
7.1.2 Shock-shocks
212(2)
7.1.3 Two-demensional configuration
214(1)
7.1.4 Transport equations for coupling terms
215(3)
7.1.5 The lowest order approximation
218(2)
7.1.6 First order approximation
220(3)
7.2 An Alternative Approach Using the Theory of Distributions
223(7)
7.3 Kinematics of a Bore over a Sloping Beach
230(19)
7.3.1 Basic equations
231(3)
7.3.2 Lowest order approximation
234(2)
7.3.3 Higher order approximations
236(1)
7.3.4 Results and discussion
237(6)
7.3.5 Appendices
243(6)
Bibliography 249(16)
Index 265
Vishnu D. Sharma is chair professor in the Department of Mathematics at the Indian Institute of Technology, Bombay (IITB). Dr. Sharma is also president of the Indian Society of Theoretical and Applied Mechanics.
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