Applied Mathematical Sciences

I. Compressible Fluids.- 1. Qualitative differences between linear and nonlinear waves.- A. General Equations of Flow. Thermodynamic Notions.- 2. The medium.- 3. Ideal gases, polytropic gases, and media with separable energy.- 4. Mathematical comments on ideal gases.- 5. Solids which do not satisfy Hooke's law.- 6. Discrete media.- 7. Differential equations of motion.- 8. Conservation of energy.- 9. Enthalpy.- 10. Isentropic flow. Steady flow. Subsonic and supersonic flow.- 11. Acoustic approximation.- 12. Vector form of the flow equations.- 13. Conservation of circulation. Irrotational flow. Potential.- 14. Bernoulli's law.- 15. Limit speed and critical speed.- B. Differential Equations for Specific Types of Flow.- 16. Steady flows.- 17. Non-steady flows.- 18. Lagrange's equations of motion for one-dimensional and spherical flow.- Appendix-Wave Motion in Shallow Water.- 19. Shallow water theory.- II. Mathematical Theory of Hyperbolic Flow Equations for Functions of Two Variables.- 20. Flow equations involving two functions of two variables.- 21. Differential equations of second order type.- 22. Characteristic curves and characteristic equations.- 23. Characteristic equations for specific problems.- 24. The initial value problem. Domain of dependence. Range of influence.- 25. Propagation of discontinuities along characteristic lines.- 26. Characteristic lines as separation lines between regions of different types.- 27. Characteristic initial values.- 28. Supplementary remarks about boundary data.- 29. Simple waves. Flow adjacent to a region of constant state.- 30. The hodograph transformation and its singularities. Limiting lines.- 31. Systems of more than two differential equations.- 32. General remarks about differential equations for functions of more than two independent variables. Characteristic surfaces.- III. One-Dimensional Flow.- 33. Problems of one-dimensional flow.- A. Continuous Flow.- 34. Characteristics.- 35. Domain of dependence. Range of influence.- 36. More general initial data.- 37. Riemann invariants.- 38. Integration of the differential equations of isentropic flow.- 39. Remarks on the Lagrangian representation.- B. Rarefaction and Compression Waves.- 40. Simple waves.- 41. Distortion of the wave form in a simple wave.- 42. Particle paths and cross-characteristics in a simple wave.- 43. Rarefaction waves.- 44. Escape speed. Complete and incomplete rarefaction waves.- 45. Centered rarefaction waves.- 46. Explicit formulas for centered rarefaction waves.- 47. Remark on simple waves in Lagrangian coordinates.- 48. Compression waves.- Appendix to Part B.- 49. Position of the envelope and its cusp in a compression wave.- C. Shocks.- 50. The shock as an irreversible process.- 51. Historical remarks on non-linear flow.- 52. Discontinuity surfaces.- 53. Basic model of discontinuous motion. Shock wave in a tube.- 54. Jump conditions.- 55. Shocks.- 56. Contact discontinuities.- 57. Description of shocks.- 58. Models of shock motion.- 59. Discussion of the mechanical shock conditions.- 60. Sound waves as limits of weak shocks.- 61. Cases in which the mechanical shock conditions are sufficient to determine the shock.- 62. Shock conditions in Lagrangian representation.- 63. Shock relations derived from the differential equations for viscous and heat-conducting fluids.- 64. Hugoniot relation. Determinacy of the shock transition.- 65. Basic properties of the shock transition.- 66. Critical speed and Prandt's relation for polytropic gases.- 67. Shock relations for polytropic gases.- 68. The state on one side of the shock front in a polytropic gas determined by the state on the other side.- 69. Shock resulting from a uniform compressive motion.- 70. Reflection of a shock on a rigid wall.- 71. Shock strength for polytropic gases.- 72. Weak shocks. Comparison with transitions through simple waves.- 73. Non-uniform shocks.- 74. Approximate treatment of non-uniform shocks of moderate strength.- 75. Decaying shock wave. N-wave.- 76. Formation of a shock.- 77. Remarks on strong non-uniform shocks.- D. Interactions.- 78. Typical interactions.- 79. Survey of results.- 80. Riemann's problem. Shock tubes.- 81. Method of analysis.- 82. The process of penetration for rarefaction waves.- 83. Interactions treated by the method of finite differences.- E. Detonation and Deflagration Waves.- 84. Reaction processes.- 85. Assumptions.- 86. Various types of processes.- 87. Chapman-Jouguet processes.- 88. Jouguet's rule.- 89. Determinacy in gas flow involving a reaction front.- 90. Solution of flow problems involving a detonation process.- 91. Solution of flow problems involving deflagrations.- 92. Detonation as a deflagration initiated by a shock.- 93. Deflagration zones of finite width.- 94. Detonation zones of finite width. Chapman-Jouguet hypothesis.- 95. The width of the reaction zone.- 96. The internal mechanism of a reaction process. Burning velocity.- Appendix-Wave Propagation in Elastic-Plastic Material.- 97. The medium.- 98. The equations of motion.- 99. Impact loading.- 100. Stopping shocks.- 101. Interactions and reflections.- IV. Isentropic Irrotational Steady Plane Flow.- 102. Analytical background.- A. Hodograph Method.- 103. Hodograph transformation.- 104. Special flows obtained by the hodograph method.- 105. The role of limiting lines and transition lines.- B. Characteristics and Simple Waves.- 106. Characteristics. Mach lines and Mach angle.- 107. Characteristics in the hodograph plane as epicycloids.- 108. Characteristics in the (u,v)-plane continued.- 109. Simple waves.- 110. Explicit formulas for streamlines and cross Mach lines in a simple wave.- 111. Flow around a bend or corner. Construction of simple waves.- 112. Compression waves. Flow in a concave bend and along a bump.- 113. Supersonic flow in a two-dimensional duct.- 114. Interaction of simple waves. Reflection on a rigid wall.- 115. Jets.- 116. Transition formulas for simple waves in a polytropic gas.- C. Oblique Shock Fronts.- 117. Qualitative description.- 118. Relations for oblique shock fronts. Contact discontinuities.- 119. Shock relations in polytropic gases. Prandtl's formula.- 120. General properties of shock transitions.- 121. Shock polars for polytropic gases.- 122. Discussion of oblique shocks by means of shock polars.- 123. Flows in corners or past wedges.- D. Interactions-Shock Reflection.- 124. Interactions between shocks. Shock reflection.- 125. Regular reflection of a shock wave on a rigid wall.- 126. Regular shock reflection continued.- 127. Analytic treatment of regular reflection for polytropic gases.- 128. Configurations of several confluent shocks. Mach reflection.- 129. Configurations of three shocks through one point.- 130. Mach reflections.- 131. Stationary, direct and inverted Mach configuration.- 132. Results of a quantitative discussion.- 133. Pressure relations.- 134. Modifications and generalizations.- 135. Mathematical analysis of three-shock configuration.- 136. Analysis by graphical methods.- E. Approximate Treatments of Interactions. Airfoil Flow.- 137. Problems involving weak shocks and simple waves.- 138. Comparison between weak shocks and simple waves.- 139. Decaying shock front.- 140. Flow around a bump or an airfoil.- 141. Flow around an airfoil treated by perturbation methods (linearization).- 142. Alternative perturbation method for airfoils.- F. Remarks about Boundary Value Problems for Steady Flow.- 143. Facts and conjectures concerning boundary conditions.- V. Flow in Nozzles and Jets.- 144. Nozzle flow.- 145. Flow through cones.- 146. De Laval nozzle.- 147. Various types of nozzle flow.- 148. Shock patterns in nozzles and jets.- 149. Thrust.- 150. Perfect nozzles.- VI. Flow in Three Dimensions.- A. Steady Flow with Cylindrical Symmetry.- 151. Cylindrical symmetry. Stream function.- 152. Supersonic flow along a slender body of revolution.- 153. Resistance.- B. Conical Flow.- 154. Qualitative description.- 155. The differential equations.- 156. Conical shocks.- 157. Other problems involving conical flow.- C. Spherical Waves.- 158. General remarks.- 159. Analytical formulations.- 160. Progressing waves.- 161. Special types of progressing waves.- 162. Spherical quasi-simple waves.- 163. Spherical detonation and deflagration waves.- 164. Other spherical quasi-simple waves.- 165. Reflected spherical shock fronts.- 166. Concluding remarks.- Index of Symbols.