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Full Description
Introduction to the Potential Theory for the Time-Dependent Stokes System is made up of two parts. The first part deals with a careful presentation of the principles on which the physical problems are based. The fluids under consideration are assumed to be incompressible and the equations so obtained are nonlinear. The linear problems are obtained by introducing characteristic parameters and so determining which terms can be neglected. The authors feel it is important that when a mathematical problem is solved, one knows precisely which problem has actually been solved. The second part deals with the mathematical treatment of the problems derived in the first part. These equations are linear and time dependent. The first step is the construction of a fundamental solution for the equations involved. They are analogous to the fundamental solutions for the potential and heat equations commonly found in the mathematical and engineering literature. The fundamental solution is used as in classical potential theory to construct solutions to initial and certain boundary value problems for the linear Stokes equations.
Features
Careful presentation of the kinematics of fluid dynamics
Derivation of the basic equations from first principles
Rigorous treatment of the linearization of the equations leading to Reynolds and Euler numbers
Derivation of the fundamental solutions for the Stokes and Oseen equations
Explicit solutions to the Stokes and Oseen equations for initial value problems
Potential theory for the Stokes system
Comparison of compressible and incompressible fluids.
Contents
Part 1: Background 1. Kinematics 2. Material Dynamics 3. Density and Stress 4. Recapitulation, Vorticity, Initial and Boundary Conditions 5. Scaling and Linearization Part 2: Stokes and Oseen Systems - Initial Value Problems 6. The Three-Dimensional Fundamental Solutions 7. The Two-Dimensional Fundamental Solutions 8. The Cauchy problem for the time dependent Stokes and Oseen systems 9. The Existence and Uniqueness of Solutions to the Cauchy Problem Part 3: Boundary Value Problems 10. Uniqueness Theory 11. Outline Recalling Classical Potential Theory 12. Boundary Value Problems for the Unsteady Stokes Equations 13. The Half Space Problem for the Dirichlet Problem 14. The Half Space Problem for the Neumann Problem Part 4: Compressible Fluids 15. Compressible Liquids 16. Temperature Dependent, Compressible Fluids
