Quasilinear hyperbolic systems, compressible flows, and waves / Vishnu D. Sharma
- Author
- Sharma, Vishnu D.
- Published
- Boca Raton, FL : CRC/Taylor & Francis, [2010]
- Copyright Date
- ©2010
- Physical Description
- xiii, 268 pages : illustrations ; 25 cm.
- Series
- Contents
- 1. Hyperbolic Systems of Conservation Laws -- 1.1. Preliminaries -- 1.2. Examples -- 1.2.1. Traffic flow -- 1.2.2. River flow and shallow water eqations -- 1.2.3. Gasdynamic equations -- 1.2.4. Relaxing gas flow -- 1.2.5. Magnetogasdynamic equations -- 1.2.6. Hot electron plasma model -- 1.2.7. Radiative gasdynamic equations -- 1.2.8. Relativistic gas model -- 1.2.9. Viscoelasticity -- 1.2.10. Dusty gases -- 1.2.11. Zero-pressure gasdynamic system -- 2. Scalar Hyperbolic Equations in One Dimension -- 2.1. Breakdown of Smooth Solutions -- 2.1.1. Weak solutions and jump condition -- 2.1.2. Entropy condition and shocks -- 2.1.3. Riemann problem -- 2.2. Entropy Conditions Revisited -- 2.2.1. Admissibility criterion I (Oleinik) -- 2.2.2. Admissibility criterion II (Vanishing viscosity) -- 2.2.3. Admissibility criterion III (Viscous profile) -- 2.2.4. Admissibility criterion IV (Kruzkov) -- 2.2.5. Admissibility criterion V (Oleinik) -- 2.3. Riemann Problem for Nonconvex Flux Function -- 2.4. Irreversibility -- 2.5. Asymptotic Behavior -- 3. Hyperbolic Systems in One Space Dimension -- 3.1. Genuine Nonlinearity -- 3.2. Weak Solutions and Jump Condition -- 3.3. Entropy Conditions -- 3.3.1. Admissibility criterion I (Entropy pair) -- 3.3.2. Admissibility criterion II (Lax) -- 3.3.3. k-shock wave -- 3.3.4. Contact discontinuity -- 3.4. Riemann Problem -- 3.4.1. Simple waves -- 3.4.2. Riemann invariants -- 3.4.3. Rarefaction waves -- 3.4.4. Shock waves -- 3.5. Shallow Water Equations -- 3.5.1. Bores -- 3.5.2. Dilatation waves -- 3.5.3. The Riemann problem -- 3.5.4. Numerical solution -- 3.5.5. Interaction of elementary waves -- 3.5.6. Interaction of elementary waves from different families -- 3.5.7. Interaction of elementary waves from the same family -- 4. Evolution of Week Waves in Hyperbolic Systems -- 4.1. Waves and Compatibility Conditions -- 4.1.1. Bicharacteristic curves or rays -- 4.1.2. Transport equations for first order discontinuities -- 4.1.3. Transport equations for higher order discontinuites -- 4.1.4. Transport eqations for mild discontinuities -- 4.2. Evolutionary Behavior of Acceleration Waves -- 4.2.1. Local behavior -- 4.2.2. Global behavior: The main results -- 4.2.3. Proofs of the main results -- 4.2.4. Some special cases -- 4.3. Interaction of Shock Waves with Weak Discontinuities -- 4.3.1. Evolution law for the amplitudes of C1 discontinuities -- 4.3.2. Reflected and transmitted amplitudes -- 4.4. Weak Discontinuities in Radiative Gasdynamics -- 4.4.1. Radiation induced waves -- 4.4.2. Modified gasdynamic waves -- 4.4.3. Waves entering in a uniform region -- 4.5. One-Dimensional Weak Discontinuity Waves -- 4.5.1. Characteristic approach -- 4.5.2. Semi-characteristic approach -- 4.5.3. Singular surface approach -- 4.6. Weak Nonlinear Waves in an Ideal Plasma -- 4.6.1. Centered rarefaction waves -- 4.6.2. Compression waves and shock front -- 4.7. Relatively Undistorted Waves -- 4.7.1. Finite amplitude disturbances -- 4.7.2. Small amplitude waves -- 4.7.3. Waves with amplitude not-so-small -- 5. Asymptotic Waves for Quasilinear Systems -- 5.1. Weakly Nonlinear Geometrical Optics -- 5.1.1. High frequency processes -- 5.1.2. Nonlinear geometrical acoustics solution in a relaxing gas -- 5.2. Far Field Behavior -- 5.3. Energy Dissipated across Shocks -- 5.3.1. Formula for energy dissipated at shocks -- 5.3.2. Effect of distributional source terms -- 5.3.3. Application to nonlinear geometrical optics -- 5.4. Evolution Equation Describing Mixed Nonlinearity -- 5.4.1. Derivation of the transport equations -- 5.4.2. The ε-approximate equation and transport equation -- 5.4.3. Comparison with an alternative approach -- 5.4.4. Energy dissipated across shocks -- 5.4.5. Application -- 5.5. Singular Ray Expansions -- 5.6. Resonantly Interacting Waves -- 6. Self-Similar Solutions Involving Discontinuities -- 6.1. Waves in Self-Similar Flows -- 6.1.1. Self-similar solutions and their asymptotic behavior -- 6.1.2. Collision of a C1-wave with a blast wave -- 6.2. Imploding Shocks in a Relaxing Gas -- 6.2.1. Basic equations -- 6.2.2. Similarity analysis by invariance groups -- 6.2.3. Self-similar solutions and constraints -- 6.2.4. Imploding shocks -- 6.2.5. Numerical results and discussion -- 6.3. Exact Solutions of Euler Equations via Lie Group Analysis -- 6.3.1. Symmetry group analysis -- 6.3.2. Euler equations of ideal gas dynamics -- 6.3.3. Solution with shocks -- 7. Kinematics of a Shock of Arbitrary Strength -- 7.1. Shock Wave through an Ideal Gas in 3-Space Dimensions -- 7.1.1. Wave propagation on the shock -- 7.1.2. Shock-shocks -- 7.1.3. Two-demensional configuration -- 7.1.4. Transport equations for coupling terms -- 7.1.5. The lowest order approximation -- 7.1.6. First order approximation -- 7.2. An Alternative Approach Using the Theory of Distributions -- 7.3. Kinematics of a Bore over a Sloping Beach -- 7.3.1. Basic equations -- 7.3.2. Lowest order approximation -- 7.3.3. Higher order approximations -- 7.3.4. Results and discussion -- 7.3.5. Appendices.
- Summary
- "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."--BOOK JACKET.
- Subject(s)
- ISBN
- 9781439836903 (hardcover : alk. paper)
1439836906 (hardcover : alk. paper) - Bibliography Note
- Includes bibliographical references (pages 249-263) and index.
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