Variational Principles of Fluid Mechanics and Electromagnetism: Imposition and Neglect of the Lin Constraint
Date of Award:
Doctor of Philosophy (PhD)
W. Farrell Edwards
W. Farrell Edwards
Variational principles in classical fluid mechanics and electromagnetism have sprinkled the literature since the eighteenth century. Even so, no adequate variational principle in the Eulerian description of matter was had until 1968 when an Eulerian variational principle was introduced which reproduces Euler's equation of fluid dynamics. Although it successfully produces the appropriate equation of motion for a perfect fluid, the variational principle requires imposition of a constraint which was not fully understood at the time the variational principle was introduced. That constraint is the Lin constraint. The Lin constraint has subsequently been utilized by a number of authors who have sought to develop Eulerian variational principles in both fluid mechanics and electromagnetics (or plasmadynamics). How-ever, few have sought to fully understand the constraint.
This dissertation first reviews the work of earlier authors concerning the development of variational principles in both the Eulerian and Lagrangian nomenclatures. In the process, it is shown rigorously whether or not the Euler-Lagrange equations which result from the variational principles are equivalent to the generally accepted equations of motion. In particular, it is shown in the case of several Eulerian variational principles that imposition of the Lin constraint results in Euler-Lagrange equations which are equivalent to the generally accepted equations of motion. On the other hand, it is shown that neglect of the Lin constraint results in Euler-Lagrange equations restrictive of the generally accepted equations of motion.
In an effort to improve the physical motivation behind introduction of the Lin constraint a new variational constraint is developed based on the concept of surface forces within a fluid. The new constraint has the advantage of producing Euler-Lagrange equations which are globally correct whereas the Lin constraint itself allows only local equivalence to the standard classical equations of fluid motion.
Several additional items of interest regarding variational principles are presented. It is shown that a quantity often referred to as "the canonical momentum" of a charged fluid is not always a constant of the motion of the fluid. This corrects an error which has previously appeared in the literature. In addition, it is demonstrated that there does not exist an unconstrained Eulerian variational principle giving rise to the generally accepted equations of motion for both a perfect fluid and a cold, electromagnetic fluid.
Allen, Ross Roundy Jr., "Variational Principles of Fluid Mechanics and Electromagnetism: Imposition and Neglect of the Lin Constraint" (1987). All Graduate Theses and Dissertations. 7540.
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