Introduction to the Potential Theory for the Time-Dependent Stokes System
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.
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Persons
Ronald B. Guenther is an emeritus professor in the Department of Math-ematics at Oregon State University. His career began at the Marathon Oil Company where he served as an advanced research mathematician at its Den-ver Research Center. Most of his career was spent at Oregon State University, with visiting professorships at the Universities of Hamburg and Augsburg, and appointments at research laboratories in the United States and Canada, and at the Hahn-Meitner and Weierstrass Institutes in Berlin. His research inter-ests include mathematically modeling deterministic systems and the ordinary and partial differential equations that arise from these models.
Ernest Roetman (1936 - 2023) earned his Ph.D. in applied mathematics from Oregon State University in 1963. After a post doc at the University of Aachen, Germany, he took a position at Bell Labs. He began his academic ca-reer at Stevens Institute of Technology, New Jersey. Subsequently, he moved to the University of Missouri in Columbia, Missouri, with brief visiting positions at the University of Aachen, Germany, Oregon State University in Corvallis, Oregon, and the Marathon Oil Co. Research Center in Denver, Colorado. In 1980 he joined the Boeing Co. in Seattle, Washington as a researcher and manager. He retired from Boeing in 2003. He then taught mathematics and engineering courses at Henry Cogswell College, Everett, Washington until his final retirement in 2006.
Content
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
