Date of Award
Doctor of Philosophy
F. N. Peebles
W. O. Harms, Robert Marsh Boarts, H. F. Johnson, E. Eugene Stansbury, Hilton A. Smith
The optical characteristic exhibited by certain liquids known as flow double refraction offers attractive features in the analysis of two-dimensional laminar flow situations, especially those in which measuring probs are undesirable. Such liquids become anisotropic to the passage of light when set in motion; and when they are viewed in polarized light while flowing through a transparent channel, visible interference patterns are observed. These patterns may be related to the magnitude and direction of the shearing stresses set up by two-dimensional laminar flow, and from suitable calibration under down flow conditions they may be used to determine the stress distribution in an unknown flow situation.
Attempts made by previous investigators to apply this technique to the analysis of flow have not yielded promising results. The doubly refracting test liquids used have been either very unstable or extremely viscous. Also, results have been analyzed under the assumption that doubly refracting properties could be correlated directly in terms of streamline directions and velocity gradients, rather than in terms of the direction and magnitude of the shearing stresses in the liquid. This assumption appears questionable in the light of certain experimental facts noted in this investigation, except for flow cases involving parallel streamlines.
The objectives of the present investigation were two-fold: (a) the development of methods for predicting stream-line directions and velocity gradients from the stress distributions obtained from double refraction measurements, thus avoiding the necessity of any direct correlation between the gradients themselves and the doubly refracting properties, and (b) the experimental application of these methods to two-dimensional laminar flow situations, and the checking of results by independent means whenever possible.
The desired analytical methods were developed, and experimental studies were carried out for the following situations: (a) flow between parallel plates, (b) converging and diverging flow between flat, non-parallel plates, and (c) flow about a cylindrical obstacle in a bounded channel. In case (a), streamlines were determined by the parallel channel walls. In case (b), streamlines were assumed to be straight lines emanating radically from the point of intersection of the non-parallel walls. In case (c), streamlines were determined from information on stress directions obtained from the flow double refraction patterns. The doubly refracting liquids employed were aqueous solutions of a commercial dye, milling yellow, which remain stable over a period of approximately two weeks, and which have viscosities in the neighborhood of twenty centipoises. They show Newtonian behavior under shearing stresses less than 5 dynes per swuare centimeter, the range of interest of this investigation. Correlation of doubly refracting properties with stress was obtained by observation of the known flow situation in the annular space between fixed and rotating concentric cylinders. The flow test channel was made of Plexiglass and was of rectangular cross section with a high aspect ratio (ratio of width to depth) in order to approximate two-dimensional flow. The channel was five inches wide and contained 1-inch and 1/2-inch deep sections joined by a converging section.
A velocity profile obtained from double refraction measurements for plane parallel flow was checked with that predicted by the theoretical formula for flow between plane parallel plates, and all points agreed with 3.0 percent. Velocity profiles in plane parallel, converging, and diverging flow were integrated to yield total discharges for each case. Agreement was within a maximum deviation of 10.0 percent and a mean square deviation of 4.2 percent for all plane parallel flow cases, and within a maximum deviation of 13.5 percent and a mean square deviation of 7.8 percent for all converging and diverging flow cases. The experimental discharge was required in the analysis of flow about a cylindrical obstacle, and could not be used for comparison. However, the viscous drag coefficient for the cylinder was calculated from flow double refraction measurements and compared with theoretical and experimental literature values. Agreement was found within 2.0 percent with the theoretical value and within 10.5 percent with the experimental value. The range of flow rates covered was from 20 to 200 cu.cm./sec., and corresponded to Reynolds numbers from approximately 1 to 10.
In conclusion, it appears that the technique offers attractive possibilities for the analysis of complex two-dimensional laminar flow situations, and with certain experimental refinements, should be capable of even better results that those reported above.
Prados, John William, "The Analysis of Two-Dimensional Laminar Flow Utilizing a Doubly Refracting Liquid. " PhD diss., University of Tennessee, 1957.