Abstract: Theoretical Fluid Mechanics' has been written to aid physics students who wish to pursue a course of self-study in fluid mechanics. It is a comprehensive, completely self-contained text with equations of fluid mechanics derived from first principles, and any required advanced mathematics is either fully explained in the text, or in an appendix. It is accompanied by about 180 exercises with completely worked out solutions. It also includes extensive sections on the application of fluid mechanics to topics of importance in astrophysics and geophysics. These topics include the equilibrium of rotating, self-gravitating, fluid masses; tidal bores; terrestrial ocean tides; and the Eddington solar model.

1. Mathematical models of fluid motion -- 1.1. Introduction -- 1.2. What is a fluid? -- 1.3. Volume and surface forces -- 1.4. General properties of the stress tensor -- 1.5. Stress tensor in a static fluid -- 1.6. Stress tensor in a moving fluid -- 1.7. Viscosity -- 1.8. Conservation laws -- 1.9. Mass conservation -- 1.10. Convective time derivative -- 1.11. Momentum conservation -- 1.12. Navier-Stokes equation -- 1.13. Energy conservation -- 1.14. Equations of incompressible fluidflow -- 1.15. Equations of compressible fluid flow -- 1.16. Dimensionless numbers in incompressible flow -- 1.17. Dimensionless numbers in compressible flow -- 1.18. Fluid equations in Cartesian coordinates -- 1.19. Fluid equations in cylindrical coorinates -- 1.20. Fluid equations in spherical coordinates -- 1.21. Exercises

2. Hydrostatics -- 2.1. Introduction -- 2.2. Hydrostatic pressure -- 2.3. Buoyancy -- 2.4. Equilibria of floating bodies -- 2.5. Vertical stability of floating bodies -- 2.6. Angular stability of floating bodies -- 2.7. Determination of metacentric height -- 2.8. Energy of a floating body -- 2.9. Curve of buoyancy -- 2.10. Rotational hydrostatics -- 2.11. Equilibrium of a rotating liquid body -- 2.12. Maclaurin spheroids -- 2.13. Jacobi ellipsoids -- 2.14. Roche ellipsoids -- 2.15. Exercises

3. Surface tension -- 3.1. Introduction -- 3.2. Young-Laplace equation -- 3.3. Spherical interfaces -- 3.4. Capillary length -- 3.5. Angle of contact -- 3.6. Jurin's law -- 3.7. Capillary curves -- 3.8. Axisymmetric soap-bubbles -- 3.9. Exercises

4. Incompressible inviscid flow -- 4.1. Introduction -- 4.2. Streamlines, stream tubes, and stream filaments -- 4.3. Bernoulli's theorem -- 4.4. Euler's momentum theorem -- 4.5. d'Alembert's paradox -- 4.6. Flow through an orifice -- 4.7. Sub-critical and super-critical flow -- 4.8. Flow over a shallow bump -- 4.9. Stationary hydraulic jumps -- 4.10. Tidal bores -- 4.11. Flow over a broad-crested weir -- 4.12. Vortex lines, vortex tubes, and vortex filaments -- 4.13. Circulation and vorticity -- 4.14. Kelvin's circulation theorem -- 4.15. Irrotational flow -- 4.16. Exercises

5. Two-dimensional incompressible inviscid flow -- 5.1. Introduction -- 5.2. Two-dimensional flow -- 5.3. Velocity potentials and stream functions -- 5.4. Two-dimensional uniform flow -- 5.5. Two-dimensional sources and sinks -- 5.6. Two-dimensional vortex filaments -- 5.7. Two-dimensional irrotational flow in cylindrical coordinates -- 5.8. Flow past a cylindrical obstacle -- 5.9. Motion of a submerged cylinder -- 5.10. Inviscid flow past a semi-infinite wedge -- 5.11. Inviscid flow over a semi-infinite wedge -- 5.12. Two-dimensional jets -- 5.13. Exercises

6. Two-dimensional potential flow -- 6.1. Introduction -- 6.2. Complex functions -- 6.3. Cauchy-Riemann relations -- 6.4. Complex velocity potential -- 6.5. Complex velocity -- 6.6. Method of images -- 6.7. Conformal maps -- 6.8. Schwarz-Christoffel theorem -- 6.9. Free streamline theory -- 6.10. Complex line integrals -- 6.11. Blasius' theorem -- 6.12. Exercises

7. Axisymmetric incompressible inviscid flow -- 7.1. Introduction -- 7.2. Axisymmetric flow -- 7.3. Stokes stream function -- 7.4. Axisymmetric velocity fields -- 7.5. Axisymmetric irrotational flow in spherical coordinates -- 7.6. Uniform flow -- 7.7. Point sources -- 7.8. Dipole point sources -- 7.9. Flow past a spherical obstacle -- 7.10. Motion of a submerged sphere -- 7.11. Conformal maps -- 7.12. Flow around a submerged oblate spheroid -- 7.13. Flow around a submerged prolate spheroid -- 7.14. Exercises

8. Incompressible boundary layers -- 8.1. Introduction -- 8.2. No-slip condition -- 8.3. Boundary layer equations -- 8.4. Self-similar boundary layers -- 8.5. Boundary layer on a flat plate -- 8.6. Wake downstream of a flat plate -- 8.7. Von Kármán momentum integral -- 8.8. Boundary layer separation -- 8.9. Criterion for boundary layer separation -- 8.10. Approximate solutions of boundary layer equations -- 8.11. Exercises

9. Incompressible aerodynamics -- 9.1. Introduction -- 9.2. Kutta-Zhukovskii theorem -- 9.3. Cylindrical airfoils -- 9.4. Zhukovskii's hypothesis -- 9.5. Vortex sheets -- 9.6. Induced flow -- 9.7. Three-dimensional airfoils -- 9.8. Aerodynamic forces -- 9.9. Ellipsoidal airfoils -- 9.10. Simple flight problems -- 9.11. Exercises

10. Incompressible viscous flow -- 10.1. Introduction -- 10.2. Flow between parallel plates -- 10.3. Flow down an inclined plane -- 10.4. Poiseuille flow -- 10.5. Taylor-Couette flow -- 10.6. Flow in slowly-varying channels -- 10.7. Lubrication theory -- 10.8. Stokes flow -- 10.9. Axisymmetric Stokes flow -- 10.10. Axisymmetric Stokes flow around a solid sphere -- 10.11. Axisymmetric Stokes flow in and around a fluid sphere -- 10.12. Exercises

11. Waves in incompressible fluids -- 11.1. Introduction -- 11.2. Gravity waves -- 11.3. Gravity waves in deep water -- 11.4. Gravity waves in shallow water -- 11.5. Energy of gravity waves -- 11.6. Wave drag on ships -- 11.7. Ship wakes -- 11.8. Gravity waves in a flowing fluid -- 11.9. Gravity waves at an interface -- 11.10. Steady flow over a corrugated bottom -- 11.11. Surface tension -- 11.12. Capillary waves -- 11.13. Capillary waves at an interface -- 11.14. Wind-driven waves in deep water -- 11.15. Exercises

12. Terrestrial ocean tides -- 12.1. Introduction -- 12.2. Tide-generating potential -- 12.3. Decomposition of tide-generating potential -- 12.4. Expansion of tide-generating potential -- 12.5. Surface harmonics and solid harmonics -- 12.6. Planetary rotation -- 12.7. Total gravitational potential -- 12.8. Planetary response -- 12.9. Laplace tidal equations -- 12.10. Harmonics of the forcing term in the Laplace tidal equations -- 12.11. Response to the equilibrium harmonic -- 12.12. Global ocean tides -- 12.13. Non-global ocean tides -- 12.14. Useful lemma -- 12.15. Transformation of Laplace tidal equations -- 12.16. Another useful lemma -- 12.17. Basis eigenfunctions -- 12.18. Auxiliary eigenfunctions -- 12.19. Gyroscopic coefficients -- 12.20. Proudman equations -- 12.21. Hemispherical ocean tides -- 12.22. Exercises

13. Equilibria of compressible fluids -- 13.1. Introduction -- 13.2. Isothermal atmosphere -- 13.3. Adiabatic atmosphere -- 13.4. Atmospheric stability -- 13.5. Eddington solar model -- 13.6. Exercises

14. One-dimensional compressible inviscid flow -- 14.1. Introduction -- 14.2. Thermodynamic considerations -- 14.3. Isentropic flow -- 14.4. Sound waves -- 14.5. Bernoulli's theorem -- 14.6. Mach number -- 14.7. Sonic flow through a nozzle -- 14.8. Normal shocks -- 14.9. Piston-generated shock wave -- 14.10. Piston-generated expansion wave -- 14.11. Exercises

15. Two-dimensional compressible inviscid flow -- 15.1. Introduction -- 15.2. Oblique shocks -- 15.3. Supersonic flow in a corner or over a wedge -- 15.4. Weak oblique shocks -- 15.5. Supersonic compression by turning -- 15.6. Supersonic expansion by turning -- 15.7. Detached shocks -- 15.8. Shock-expansion theory -- 15.9. Thin-airfoil theory -- 15.10. Crocco's theorem -- 15.11. Homenergic homentropic flow -- 15.12. Small-perturbation theory -- 15.13. Subsonic flow past a wave-shaped wall -- 15.14. Supersonic flow past a wave-shaped wall -- 15.15. Linearized subsonic flow -- 15.16. Linearized supersonic flow -- 15.17. Flat lifting wings -- 15.18. Exercises

Appendices. A Vectors and vector fields -- A.1. Introduction -- A.2. Scalars and vectors -- A.3. Vector algebra -- A.4. Cartesian components of a vector -- A.5. Coordinate transformations -- A.6. Scalar product -- A.7. Vector area -- A.8. Vector product -- A.9. Rotation -- A.10. Scalar triple product -- A.11. Vector triple product -- A.12. Vector calculus -- A.13. Line integrals -- A.14. Vector line integrals -- A.15. Surface integrals -- A.16. Vector surface integrals -- A.17. Volume integrals -- A.18. Gradient -- A.19. Grad operator -- A.20. Divergence -- A.21. Laplacian operator -- A.22. Curl -- A.23. Useful vector identities -- A.24. Exercises

B. Cartesian tensors -- B.1. Introduction -- B.2. Tensors and tensor notation -- B.3. Tensor transformation -- B.4. Tensor fields -- B.5. Isotropic tensors -- B.6. Exercises

C. Non-Cartesian coordinates -- C.1. Introduction -- C.2. Orthogonal curvilinear coordinates -- C.3. Cylindrical coordinates -- C.4. Spherical coordinates -- C.5. Exercises

E. Calculus of variations -- E.1. Introduction -- E.2. Euler-Lagrange equation -- E.3. Conditional variation -- E.4. Multi-function variation -- E.5. Exercises

F. Solutions to exercises in chapter 1 -- G. Solutions to exercises in chapter 2 -- H. Solutions to exercises in chapter 3 -- I. Solutions to exercises in chapter 4 -- J. Solutions to exercises in chapter 5 -- K. Solutions to exercises in chapter 6 -- L. Solutions to exercises in chapter 7 -- M. Solutions to exercises in chapter 8 -- N. Solutions to exercises in chapter 9 -- O. Solutions to exercises in chapter 10 -- P. Solutions to exercises in chapter 11 -- Q. Solutions to exercises in chapter 12 -- R. Solutions to exercises in chapter 13 -- S. Solutions to exercises in chapter 14 -- T. Solutions to exercises in chapter 15 -- U. Solutions to exercises in appendix A -- V. Solutions to exercises in appendix B -- W. Solutions to exercises in appendix C -- X. Solutions to exercises in appendix D -- Y. Solutions to exercises in appendix E.