Some Problems in the Steady Motion of Viscous, Incompressible Fluids ; with Particular Reference to a Variation Principle

Citation

Millikan, Clark Blanchard (1928) Some Problems in the Steady Motion of Viscous, Incompressible Fluids ; with Particular Reference to a Variation Principle. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/HZN5-8V87. https://resolver.caltech.edu/CaltechETD:etd-05212003-105955

Abstract

A discussion of the general equations for steady motion of a viscous incompressible fluid from the point of view of a minimum or variation principle has, as far as the writer is aware, never been given. Helmholtz, in a classical paper¹ "On the Theory of a Stationary Flow in Viscous Fluids", has shown that if the quadratic terms in velocity be neglected, if the velocities at t11e boundaries of a singly-connected region be kept constant, and if the external forces have a single-valued potential, then the motion is such that the variation of the energy dissipation in the region under consideration is zero. We shall refer to this work in more detail later. Korteweg subsequently showed that under the above conditions the steady motion which is set up is unique and is stable, i.e. the dissipation for this motion is an absolute minimum. Rayleigh³ still later showed that the dissipation is is an absolute minimum whenever

∇²‾q = ∇H

where ‾q is the vector velocity of the fluid, and H is a single-valued function subject to the condition

∇²H = 0

In this case there is no restriction upon the magnitude of the velocity. As far as the author is aware this represents practically the entire extent of the work to date on the application of a minimum principle to the steady motion of a viscous fluid.

In the present work we discuss the following problem: given an incompressible, viscous fluid with fixed (if any) boundaries, to find a function L, such that we set

δ∫LdT = 0 / V

the Eulerian, equations corresponding to this condition are exactly the Navier-Stokes equations for the motion of the fluid. The integral is (for three-dimensional cases) a volume integral and dT represents the volume element. We impose, of course, the customary restriction that the variation of the velocity is taken to be zero at the boundaries of the region considered. We shall for simplicity refer to such function L as a LaGrangian function, in spite of the fact that the term is not strictly accurate.

In the first section we set up a LaGrangian function by generalizing the considerations of Helmholtz relative to slow motion, and from the resulting Eulerian equations are led to a proof of the following theorem:

"If L be restricted to be a function of the velocity components and their first order space derivatives only; then it is impossible to find any such L which will give the general equations for steady flow of an incompressible fluid through the application of the variation principle described above". The conditions which must be imposed on the motion in order that it may correspond to a variation principle, involving a LaGrangian function or this type, are discussed, and it is shown that all the cases of steady motion which have thus far been discovered satisfy these conditions. The possible physical consequences relative to the existence of ateady motion are mentioned.

In the second section the LaGrangian functions are found for the cases of plane laminar motion, and Poiseuille flow through a circular tube.

The third and fourth sections do not strictly form a part of the general variation problem, but the results obtained in them are used in the subsequent section, and have also a certain amount of interest on their own account. Tn the third section formulae are given for tha transformation of certain expressions from vector to curvilinear coordinate form, and a vector expression independent of coordinate systems is deduced for the dissipation function. In the fourth section a new proof is given of a result obtained by Hamel in a very beautiful paper⁴ recently published. This paper deals with the two-dimensional flow of an incompressible viscous fluid where the streamlines coincide with logarithmic spirals, and the present considerations appear to be a little simpler than those used by Hamel.

In the fifth section the variation problem for logarithmic spiral flow is discussed, and the LaGrangian function is exhibited.

The vector method is used whenever it appears convenient, and in such cases the notation is that of Gibbs, involving the operator ∇. Vector quantities are denoted by a bar placed above the symbol. The magnitude of a vector is denoted by omitting the bar. We shall throughout consider no body forces, since, if such forces do occur and have a single-valued potential, then they may be taken account of in the pressure terms, without in any way altering the form of the equations.

Thesis Files