THE U N I V E R S I T Y OF M I C I G A N COLLEGE OF ENGINEERING Department of Engineering Mechanics Final Report STUDIES OF STRATIFIED FLOWS, ROTATING FLUIDS, AND FLUID-MECHANICS PROBLEMS RELATED TO GEOPHYSICAL PHENOMENA Chia-Shun Yih Chief Investigator ORA Project 07891 under contract with: U. S. ARMY RESEARCH OFFICE-DURHAM CONTRACT NO. DA-ARO-D-31-124-G717 DURHAM, NORTH CAROLINA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1970

Table of Contents Page I. Brief History of the Project 3 II. General Objectives ITI. Research Accomplishments 6 A. The First Period6 1. Basic mechanics of stratified flows G 2. Wave motion in stratified fluids 7 3. Stability of stratified flows or configurations 7 4. Stability of a revolving fluid 8 5. Large-amplitude motion of a swirling fluid or a fluid with vorticity 8 6. Stability of non-Newtonian fluids in rotation 8 7. Hydromagnetic studies 9 B. The Second Period 10 1. Large-amplitude motion of stratified fluids 10 2. Stability of stratified flows 11 3. Rotating fluids 11 4. Flow in porous media 11 5. Magnetohydrodynamics 12 C. The Third Period 13 1. Stratified flows 13 2. Stability of stratified flows 13 3. Free connection in a rotating medium 14 4. Surface-tension effects 14 5. Classical theory of hydrodynamic stability 14 IV. Doctoral Dissertations Completed 16 A. The First Period 16 B. The Second Period 16 C. The Third Period 17 V. Summary and Assessment of Total Accomplishment 18 VI. Publications of the Chief Investigator 20 VII. Publications of Former Research Assistants 22 2

I. Brief History of the Project Before I left the Iowa Institute of Hydraulic Research in 1956 to assume my duties in the University of Michigan, I had always been interested in the gravity effects on fluid flow, and therefore in the flow of stratified fluids, since for such fluids the specific weight changes from fluid particle to fluid particle. Shortly after I came to Michigan, a proposal was submitted to the Army;Research Office (I)urham), then the Office of Ordnance Research, to study the mechanics of stratified and rotating fluids. The proposal was accepted as the basis for a contract with The University of M.ichigan, and the contract (DA-20-018-ORD-15556) was granted and became effective on May 15, 1957, to rin for a year till May 14, 1958. The contract was renewed three times to terminate on M-ay 15, 1961. On March 1, 1961, a grant [DA-ARO(D)-31-124-G23] was made to The University of Ilichigan to continue the project, and the grant lasted till August 31, 1962. The period HMay 15, 1957 to August 31, 1.962 is a period of exuberance of the fluid. mechanics world whose keen interest in stratified and rotating fluids was awakened. M.uch. was accomplished in our research in this period, and many early contributions made to the field have proved of permanent value. The next grant by the Army Research Office hlad the number D)A-A'RO (T))31-124-G386, and covered the period OctTober 12, 1962, to October 11, 1!65, extended in time without additional funds to Miay 31, 1966. This four-voar period will he called the second period. This is the period during whicl we exploited the analogy between hydromagnetic flows, stratified flowss, and the flow of rotating fluids. Since fluid-mechanics problems related to geophysical phenomena were emphasized, we not only considered flows in the atmosphere and flows related to river flows, hut also ground-water flow with a free surface or with interfaces. Grant DA-ARO(D)-31-124-G717 was for the period March 1, 1966 to February 28, 1969, but was extended in time without additional funds to June 30, 1970. This four-year period will be called the third period, during which the main emphasis was on stratified flows and their stability, and related problems. This is a period of great activity. Very many papers of permanent value were published as a direct result of the project. We shall present the accomplishments of the three periods in the following sections. But we pause here to give a very brief summary of the history of the support of the Army Research Office, in the form of the following table: 3

Period 1 2 3 Gtant number DA-20-018-ORD-15556 DA-ARO(D)-31-124-G386 DA-ARO(D)-31-124-G717 DA-ARO(D)-31-124-G23 Total period May 16, 1957-May 15, 1961 Oct. 12, 1962-May 31, 1966 Mlarch 1, 1966-June 30, 1970 (after extensions) March 1, 1961-Aug. 31, 1962

II. General Objectives The general objectives of the project are to deepen the understanding of the mechanics of a stratified and of a rotating fluid, to broaden the knowledge in that field, and to apply the results to the explanation of geophysical phenomena which occur in the atmosphere, in the oceans, and in rivers, or to the solving of engineering problems. Although the scope of research was broadened somewhat in the course of the work to include certain related problems in classical hydrodynamics, the main effort htas always been directed toward the objectives stated above, 5

III, Research Accomplishments A, The First Period This period, extending from May 16, 1957, to August 31, 1962, is a period of expectation because the subjects of stratified fluids and rotating fluids, with its geophysical implications, had just caught the imagination of hydrodynamicists, meteorologists, oceanographers, and even astrophysicists. Periods of expectation are usually periods of exuberance. I had the good fortune to be a hydrodynamicist interested in gravity -effects on fluid flow, and when these subjects became attractive to the research world, was in a position to make early and I hope permanent contributions to the rise of geophysical fluid mechanics. The accomplishments in the first period will be described in the seven areas listed below: 1. Basic mechanics of stratified flows Thermoelectric plants often draw their cooling water from canals or rivers. After it has been used for cooling it becomes hotter, and it is discharged at a point downstream from the plant. But the hot water is not always carried downstream. Often it forms a wedge that floats on the cooler stream and intrudes upstream sometimes to the cooling-water intake, creating a serious problem in the economy of operation. I was attracted by the practical problem of how hot water can be "skimmed" and separated from the cold water below, and was thus led to the problem of stratified flow into a sink. In 1958, a study on this problem was finished (Yih, 1958b). In this study it was found that for Froude numbers which 1 are less than -, the upstream conditions cannot he maintained. To find what would happen for such low Froude numbers, experiments were performed (Debler 1959) and it was found that blocking occurred at low Froude numbers. This is in qualitative agreement with the result obtained in another study on the effects of gravitational or electromagnetic fields on fluid motion (Yih 1958a). These studies not only are of importance to engineering, as mentioned above, but also have meteorological significance, because they bear closely on the smog problem encountered in many cities. The prevention of blocking was then undertaken for study. Analytical results were obtained early in 1959, but experimental verification was accomplished only shortly 6

before Christmas of 1961 (Yih, O'Dell, Debler, 1961). It was then known that the presence or absence of blocking is also strongly dependent on geometry —a fact of practical importance. A discussion of the general features of stratified flows and their peculiar behavior at low Froude numbers was presented to the International Symposium on the Ionosphere, and published in the Journal of Geophysical Research (Yih 1959d), Another paper (Yih 1961a) dealt with the mechanics of stratified flows in pouous media, and opened a new area of investigation to fruitful attack. Had these been the only accomplishments in the five-year period, they would have looked substantial, but much more was accomplished. 2. Wave motion in stratified fluids That this is a field of tremendous importance to meteorology and oceanography is generally recognized. In 1959-60, while at the University of Cambridge on leave from Michigan, I undertook to study the linear theory of internal waves from a general point of view. A long paper (Yih 1960a) on this subject serves to explain many facts recognized but unexplained before, and to present the linear theory of internal waves from a unified point of view. Wave motion with large-amplitudes was then undertaken, and exact solutions for the flow of incompressible fluids of variable density were obtained (Yih 19601). Another paper (Yih 1960c) dealt with largeamplitude motion of a compressible fluid with variable entropy. In my opinion, these two papers represent a substantial contribution to the understanding of atmospheric waves in the lee of mountains. The latter work was continued by Alfons Claus in his doctoral research, culminating in a remarkable thesis in June, 1961. 3. Stability of stratified flows or configurations The study of stratified configurations was undertaken early in 1957. Walter Debler finished his thesis on the stability of a heatgenerating fluid in 1959. Two papers (Yih 1959a,c) dealt with the stability of a tube of fluid heated from below, with and without a circular magnetic field. The stability of a thermally stratified fluid in a tube in the presence of uniform magnetic fields was studied by Michael Bentwich as the subject of his dissertation, which was finished in the summer of 1959. Much was learned of the various ways in which magnetic fields can inhibit instability. 7

The study of the stability of a revolving fluid with variable density (Yih 1961b) brought out the dual role of viscosity and thermal diffusivity. The dual role of magnetic diffusivity was investigated successfully by Wei Lai, who used the result for one of the three parts of his doctoral dissertation, The stability of counter-flowing currents (Graebel 1960) was the subject for Graebel's dissertation, which was finished in 1959. 4. Stability of a revolving fluid In order to exploit the analogy between stratified fluids and rotating fluids, and to study the interplay or gravity and Coriolis forces, studies were made for revolving fluids with variable density. These have already been reported under Item 3 above (Yih 1961b and Wei Lai's thesis). I may add that the dual roles of diffusivities in the instability of fluids have not been widely recognized before, and give rise to unexpected and surprising results. 5. Large-amplitude motion of a swirling fluid or a fluid with vorticity Here the contribution has been (a) the discovery and mathematical description of corner eddies in fluid flows with rotation (Yihl 1958c). (b) the successful use of an inverse method for the study of vorticity flows and swirling flows (Wei Lai, second part of dissertation), and (c) the analysis on the prevention of blocking (hence stagnation) in a rotating fluid (Yih, O'Dell, Debler, 1962). The last study may have significance in the performance of fuels issuing from a rotating tank in a rocket. 6. Stability of non-Newtonian fluids in rotation This was suggested by me to Dr. William Graebel for investigation. He has successfully analyzed the stability of Weissenberg fluids in rotation (Graebel 1961) and of Bingham bodies in rotation (Graebel 1962). The study of instability of non-Newtonian fluids has since blossomed, and many papers have been written by various people in this field. 8

7. Hydromagnetic studies. In addition to the hydromagnetic studies mentioned under Item 3 above, a study of electromagnetically ring vortices was made. The result of the study (Yih 1959b) showed convincingly the analogy between stratified flows, flow with general rotation, and flows in a magnetic field, and the similarity of isopycnic lines, vorticity lines, and magnetic lines of force. This underlying analogy is a powerful suggestion for future research in all three fields, and gives them a certain unity which is appealing. 9

B. The Second Period The second period, extending from October 12, 1962 to May 31, 1966, is a period of intense exploitation of the direction pointed out by the work during the first period, as well as a period of venturing further into the fields of flow in porous media and of. magnetohydrodynamics. The accomplishments in this period can be conveniently divided into five main areas: 1. Large-amplitude motion of stratified fluids Since the phenomenon of blocking of the flow of a stratified fluid is so important, a doctoral research was undertaken by Timothy 1W, Kao to study it. The result of his research is published in the Journal of Fluid Mechanics (Kao 1965a) and in the Journal of Geophysical Research (Kao 1965b), the latter paper specifically deals with blocling of atmospheric flows by mountain ranges. The results of the first paper (Kao 1965a) have recently been improved and a paper is published recently in the Physics of Fluids (Kao 1970). This recent paper solves the problem of selective witn withdrawal aforementioned in connection the cooling-water problem. Large-amplitude motion of a compressible fluid in the Earth's atmosphere was investigated by Alfons Claus. lis results showl what effect compressihility has on atmospheric flow, and how the use of potential density can reduce the flow of a compressible fluid to the flow of an incompressible one, without appreciable error in the general flow pattern, This is a most welcome confirmation of the validity of the practice. In addition, specific methods were devised to solve the mountain-wave problem, giving specific results, (Clans 1964). During my sabbatical leave in 1964, iwhich I spent in Geneva, Switzerland, I finished a separately undertaken task of writing a book on stratified fluids. This book (Yih 1965a) contains many of the available results before 1965 (whether or not obtained by us for the project). Evidently our work on stratified fluids became known abroad. For I was invited to give one of the three general lectures at the Eleventh Congress of the International Association of Hydraulic Research, held in Leningrad, U.S.S.R. (Yih 1965d). 10

2. Stability of stratified flows Three dissertations by Roca, Lin, and Yuan (See IV,B) are in this area, resulting in the publications by Roca (1966) and Lin (1967). The dissertation done by W. Lai in the first period was published in this period (1962). The dissertation by Yuan was cited in a recent book I write (Fluid Mechanics, An Introduction to the Theory, McGraw-Hill, 1969), because, owing to his shift of interest (to astrophysics), he never published his dissertation. These works dealt with the stability of stratified fluids with or without rotation, and constitute a substantial contribution to knowledge in this area. In addition I wrote two papers (Yih 1963a and 1965b) on the stability of free-surface flows, which are flows with an extreme form of stratification. The method in that 1963 paper for studying the stability of long waves has now become generally adopted by the research world for studying longwave instability of many free-surface flots —a fact of some gratification to its author, 3. Rotating fluids Centrifugal waves, or waves in rotating fluids, were studied in this period also, leading to a paper on the formation of rings in a rotating fluid (Yih and Debler, 1962), and a paper on hydraulic jumps in a rotating fluid (Yih, Gascoigne, and Debler, 1964h). Both papers show how suggestive a fruitful gy betweeand fruitful the analogy between gravity force and centrifugal (and Coriolis) force is, giving a sense of unity in the understanding of the fields of stratified and of rotating fluids. 4. Flow in porous media In an effort to broaden the scope of geophysical applications of stratified flows, free-surface flows and the flow of nonhomogeneous fluids through porous media were studies in this period also. A paper (Yih 1963b) dealt with the velocity of fluid masses in another fluid flowing in porous mddia. When I gave a lecture on this subject at the California Institute of Technology in 1963, I was told of the applicability of my result to a capillary method in chemical analysis. This unexpected applicability was very satisfying to me. Another paper (Yih 1964a) gave a transformation to deal with the free-surface (or interfacial)condition in seepage flows-. which is the only source of difficulty in such flows. 11

5. Magnetohydrodynamics Acting on the suggestion of the analogy between gravity force, Coriolis force, and the Lorentz force, large-amplitude magnetohydrodynamics was studied, and a simple transformation greatly simplified the result of Long on this subject, rendering exact solutions possible (Yih 1965d). This is a significant advance on the magnetohydrodynamics of circular fields. Another paper (Yih 1965b) dealt with the stability of a fluid heated from below in the presence of a vertical magnetic field. 12

C. The Third Period The third and final period of sponsois.hip extended from March 1,1966, to June 30, 1970. In this period the main effort was concentrated on stratified flows, their stability, and related classical problems of hydrodynamic stability. In addition, flows induced by surface tension are Iso studied as a by-product. It is a period of intense activity, marked by many results of permanence and depth. The five areas of accomplishmont are: 1. Stratified flows In this period the equations governing steady three-dimensional flows of stratified fluids were derived for the first time (Yi!h 1967c). Unfortunately these equations are too difficult to solve. Howsever, a class of solutions for steady stratified flow.s was found (1969a) under the restriction of the shallow-water theory. In addition Gerstner's solution of 1.802 for surface waves of arbitrary amplitule was found to lead to edge waves of arbitrary amplitude in a stratified fluid (Yih 1966). Our work in stratified flows up to the point lead to an invited survey article (Yih 1969c) in the Annual Reviews of Fluid rlechanics. A dissertation by N. S. Hluang on the diffraction of gravity waves in a stratified fluid is finished in 1970. This lhas a bearing on protection of harbors against internal waves. 2. Stability of stratified flows. Two dissertations in this area are completed in this period: one by C-11. Li on the stability of steady or unsteady stratified flows, one by Charles E. Hickox on the stalility of axisymmetrically stratified fluids flowing through a pipe. Li's research was published in three articles (Li 1969a,b, Li 1970). In addition, articles based on their dissertations were published in this period by Roca (1966) and Lin (1967), which are also in the field of stability of stratified flows, 3. My own research in this area resulted in three articles (Yih 1967a, 1968b, 1968c). The first treated the instability due to viscosity 13

stratification. The second opens up research in the field of stability of unsteady flows. The third, treating the free-surface instability in a time-dependent vertical electric field, has a bearing on discharges in transformers. Activity in this area is poobably most intense in this period, 3. Free convection in a rotating medium The subject of free convection in a rotating fluid has meteorological implications, because the Earth is rotating, and the region from 600 to 900 North in latitude is on the average heated at the 60~ rim and cooled at the North Pole. Experiments done by Hide and Fultz in the fifties show the existence of wavy patterns for certain ranges of the Rossby number, In the analysis even the original axisymmetric mean flow is difficult to determine, Michael Samuels did a numerical calculation of this, and his work was published in the Physics of Fluids (Samuel 1968). 4. Surface-tension effects In a study of the interplay of gravity force and surface tension at a free surface, which is an extreme form of stratification, I was led to the study of surface-tension effects. Two articles were published on surface-tension-driven flows and their stability (Yih 1967c and 1968a). The results in the latter article was quoted by the Russian scientists Levich and Krylov in a survey article in the Annual Reviews of Fluid Mechanics in 1969 —with amazing speed. In addition, an article (Yih 1969b) treating three-dimensional flows driven by surface tension gave me much satisfaction. All three articles have bearing on liquid flow in space ships. S. Classical theory of hydrodynamic stability In my studies of the stability of stratified fluids, I was naturally led to certain problems in the classical theory of hydrodynamic stability, In this area one article (Yih 1969d) established the eigenvalue bounds for the Orr-Sommerfeld equation governing stability of parallel flows,.nd another (Yih 1970) presents the spectral theory of Taylor vortices. In 14

the latter article classical mathematical methods have been used to prove the principle of exchange of stabilities for Taylor- vortices in Couette flow between co-rotating cylinders. The spectral theory that evolved will, in my opinion, influence the direction and procedure of research in the theory of hydrodynamic stability in the next 5 to 10 years. I am very proud of this work, which took a good deal of thought, energy, and inventiveness, and is therefore a fitting conclusion to my work for the project, 1.5

IV. Doctoral Dissertations Completed In a university, education of talented young men through their participation in intellectually stimulating researches is an important reason why such researches are undertaken. The apprenticeship of Ph.I). candidates participating in such researches has proved to be of great value to their future development as leaders in science and industry —and in this way has increased the scientific asset -nd general power of this country. The dissertations partially supported by the Army Research Office in the three periods were: A. The first period D)ebler, W. R., "The onset of laminar natural convection in a fluid with homogeneously distributed heat sources," 1959. Graebel, h'. P., "The stalility of a stratified flow," 1959. Bentwich, 1., "Effect of electromagnetic fields on gravitational stability of a viscous fluid," 1959. Claus, A., "Large-amplitude motion of a compressible fluid in the atmosphere," 1961. Lai, W., "Thfree studies in fluid mechanics," 1961. B. The second period Kao, Timnothy iT., "The phenomenon of blocking in stratified fluids", 1963. I)avis, Philip K., "Motion of a sphere in a rotating fluid at small Reynolds numbers", 1963. Roca-Vila, Reclud, "Gravitational instability of a liquid layer heated from below," 1965. Lin, S. P., "Instability of layer of liquid down an inclined plane at large Reynolds numbers", 1965. 16

Yuan, Chi, "On the instability of thermally stratified flow", 1966. C. The third period. Li, C. Ii., "Instability of steady and time-periodic flows in fluids", 1968, Hickox, Charles E., "Two studies in fluid mecllanics", 1969. stratified and rotating fluids, Huang, N. S., "Diffraction of waves by straight boundaries", treats waves in stratified fluids. stratified This treats 1970. This In addition, Messrs. C. Y. Wang and C. I. Yang, who are still working on their Ph.D. theses, have been partially supported by the project. 17

V. Summary and Assessment of Total Accomplishment Fifty two articles, 35 by me and 17 by my research assistants, were published in prominant scientific journals as a result of the research project. These are in the fields of (a) stratified flows and their stability, and internal waves, (b) waves in and stability of rotating fluids, (c) flow in porous media, (d) magnetohydrodynamics, (e) surfacetension effects, (f) non-Newtonian fluids, and (g) classical theory of hydrodynamic stability. The emphasis has always been on stratified fluids, rotating fluids, and applications to fluid-mechanics problems related to geophysical phenomena. The contributions have been substantial, and have the quality of permanence. I think that a disinterested assessor may find many of the articles remarkable for their perception and depth. lWhether the research achievements are unusual and impressive is not for me to say. But recognitions from abroad and at home for our work have been generous. I have been invited to give general lectures at international meetings in Australia, Soviet Russia, Canada, and in meetings in the United States. In 1968 I was given the Stephen P. Timoshenko University Professorship at the University of Michigan, and in 1970-71 I shall be visiting professor at the University of Paris and the UIniversitv of Grenoble. These French universities have requested that my lectures he in two fields: stratified flows and hydrodynamic stability. In the 13-year run of the project 13 Ph.D.'s have obtained their degrees on the partial support provided through workr for the project.l leven of hese people are teaching andl/or doing research in universities in the United States, one is a prominent engineer in th!e IBell Research La boratories, one teaches in Israel, and one in Venezuela. Thus it is clear that not only have our general objectives (See II) been net, but contriibutions to the scientific man-power of this country have al.so leo )n m.ade. At th.is time of concern over our national expenditure and governmental support of basic research, may I venture to suggest that the power of a nation in general, and military power in particular, depend, amlongr othler factors (such as national productivityv of material goodr'), on th.e scientific knowledge at its disposal alnd scientific nan-powelr,rwitliin its realm. (ur project has dee!ened antd broadened th-ie former and fostered the latter. Thus, while we are grateful to the Army Research Office of Durham for its far-sighted and sustained support of our research and 18

educational efforts, we, on the other hand, feel we have made far more contributions than are normally expected, and have in our performance far transcended the mere discharging of our duties. 19

VI. Publications of the Chief Investigator 1. Yih, C.-S., "Effects of gravitational or electromagnetic fields on fluid motion," Q. Applied Math,, Vol 16, pp. 409-415, 1958a. 2. Yih, C.-S., "On the flow of a stratified tfluid," Proc. 3rd U.S. Nat'l Congress of Applied Meclh., pp. 857-861, 1958b. 3. Yih, C.-S., "Two solutions representing inviscid corner eddies," J. Fluid Mech., Vol. 5, pp. 36-40, 1958c. 4. Yih, C.-S., "Thermal instability of viscous fluids," Q. Applied Math., Vol. 17, pp. 25-42, 1959a. 5. Yih, C.-S., "Electromagnetically generated vortex rings," J. Fluid Mlech., Vol. 5, pp. 436-444, 1959b. 6. Yih, C.-S., "Inhibition of hydrodynamic instability by an electric current," Phys. of Fluids, Vol 2, pp. 125-130, 1959c. 7. Yih, C.-S,, "Effect of density variation on fluid flow," J. Geophys. Res., Vol. 64, pp. 2219-2223, 1959d. 8. Yih, C.-S., "Gravity waves in a stratified fluid," J, Fluid Mec.,, Vol. 8, pp. 481-508, 1960a. 9. Yih, C.-S., "Exact solutions for steady two-dimensional flow of a stratified fluid," J. Fluid Mech., Vol 9, pp. 161-174, 19601). 10. Yih, C.-S., "A transformation for non-lomentropic flows, with an application to large-amplitude motion in the atmosphere," J. Fluid Mech., Vol. 9, pp. 68-80, 1960c. 11. Yih, C.-S., "Flow of a non-homogeneous fluid in porous media," J. Fluid Mlech., Vol. 10, pp. 133-140, 1961a. 12. Yih, C.-S., "Dual role of viscosity in the instability of revolving fluids with variable density," Phys. of Fluids, Vol 4, pp. 806811, 1961b. 13. Yih, C.-S., "Prevention of stagnation zones in flows of a stratified or a rotating fluid," jointly with W. IV. O'Dell and V. R. Debler, to be presented at the 4th U.S. Nat'l Congress of Applied Mlech., Berkeley, Calif., 1962. 14. Yih, C.-S., "Formation of rings in a liquid film attached to the inside of a rotating cylinder" (with V. R. Debler), Journ. Aerospace Sci., Vol. 29, p. 364, 1962. 15. Yih, C.-S., "Stability of liquid flow down an inclined plane," Phys, of Fluids, Vol. 6, pp. 321-334, 1963a. 16. Yih, C.-S., "Velocity of a fluid mass imbedded in another fluid flowing in a porous medium," Phys. of Fluids, Vol. 6, pp. 1403-1407, 1963h. 20

17. Yih, C.-S., "A transformation for free-surface flow in porous media," Phys. of Fluids, Vol, 7, pp. 20-24, 1964a. 18. Yih, C.-S., "Hydraulic jump in a rotating fluid" (with H. E. Gascoigne and W. R. Debler), Phys. of Fluids, Vol 7, pp. 638-642, 1964b, 19. Yih, C.-S., Dynamics of Nonhomogeneous Fluids, New York: The Macmillan Co., 1965a. 20. Yih, C.-S., "Gravitational instability of a viscous fluid in a magnetic field,"' ournal of Fluid Mechanics, Vol. 22, pp. 579-586, 1965b, 21. Yih, C.-S., "Stability of a non-Newtonian liquid film flowing down an incldned plane," Physics of Fluids, Vol, 8, pp, 1257-1262, 1965c, 22. Yih, C.-S., "Large-amplitude motion of a nonhomogeneous fluid," General Lecture, Eleventh Congress, International Association of Hydraulic Research, held in Leningrad, U.S.S.R., 1965d. 23. Yih, C.-S., "On large-amplitude magnetohydrodynamics," Journal of Fluid Mechanics, Vol. 23, pp. 261-271, 1965e. 24. Yih, C.-S,, "Note on edge waves in a stratified fluid," Journ. Fluid Mech., Vol. 24, pp. 765-767, 1966. 25. Yih, C.-S., "Instability due to viscosity stratification," Journ. Fluid Mlech., Vol. 27, pp. 337-352, 1967a. 26. Yih, C.-S, "Instability of laminar flows due to a film of adsorption," lourn. Fluid Mech., Vol. 28, pp. 493-500, 1967b. 27. Yih, C.-S., "Equations governing steady three-dimensional large-amplitude motion of a stratified fluid," Journ. Fluidl Mlec.ch, Vol 29, pp. 5319-544, 1967c. 28. Yih, C.-S., "Fluid motion induced by surface-tension variation", Plysics of Fluids, Vol. 11, pp. 477-480, 1968a. 29. Yih, C.-S., "Instability of unsteady flows or configurations, Part I. Instability of a horizontal liquid laver on an oscillating plane", Journ. Fluid Mech., Vol 31, pp. 737-751, 1968b. 30. Yih, C.-S., "Instability of a free surface in a time-dependent vertical electric field", Physics of Fluids, Vol. 11, pp 1447-1449, 1968c. 31. Yih, C.-S., "A class of solutions for steady stratified flows," Journ. Fluid Mech., Vol, 36, pp. 75-85, 1969a. 32. Yih, C,-S., "Three-dimensional motion of a liquid film induced by surface-tension variation or gravity," Physics of Fluids, Vol. 12, pp. 1982-1987, 1969b. 33. Yih, C.-S., "Stratified flows," Annual Reviews of Fluid M!echanics, Vol 1, pp. 73-108, 1969c. 34. Yih, C.-S., "Note on eigenvalue bounds for the Orr-Sommerfeld equation," Journ. Fluid Mech., Vol. 38, pp. 373-378, 1969d. 35. Yih, C.-S,, "Spectral theory of Taylor vortices," General Lecture, Seventh Ieat Transfer and Fluid Mechanics Institute, Honterey, California, June, 1970. 21

VII. Publication of Former Research Assistants (Chronologically arranged) A. The First Period: 1. Debler, W, R., "Stratified flow into a line sink," Proc. A S*C E., Vol. 85, pp. 51-65, 1959. 2. Graebel, 1, P., "The stability of a stratified flow," J. Fluid Mech., Vol. 8, pp. 321-336, 1960. 3. Graebel, W. P., "Stability of a Stokesian fluid in Couette flow," Phys. of Fluids, Vol. 4, pp. 362-368, 1961. 4. Graebel, W. P., "The stability of a Bingham fluid in Couette flow," to be presented to Symposium on second order effects in Elasticity, Plasticity, and Fluid Dynamics, I.U.T.A.M., April 21-28, 1962, B. The Second Period: 5. Lai, W,, "Stability of a revolving fluid with variable density in the presence of a circular magnetic field," Physics of Fluids, Vol, 5, pp. 560-566, 1962. 6. Lai, W., "Flow of an inviscid fluid past a sphere in a pipe," J. Fluid Mech., Vol. 18, pp. 587-594, 1964. 7. Claus, A., "Large amplitude motion of a compressible fluid in the atmosphere," J. Fluid Miech,, Vol 19, pp. 267-289, 1964, 8. Kao, Timothy W., "A free-streamline solution for stratified flow into a sink," J. Fluid Mech., Vol. 21, pp. 535-543, 1965a. 9. Kao, Timothy W., "The phenomenon of blocking in stratified flows," J. Geophys. Research, Vol, 70, pp. 815-822, 1965b. 10. Kao, Timothy W., "Free-streamline theory for inviscid stratified flow into a line sink", Physics of Fluids, Vol. 13, pp. 558-564, 1970, (This is chronologically out of order, but is listed here because it is an outcome of further efforts originating in this period.) 11. Davis, Philip K., "Mlotion of a sphere in a rotating fluid at small Reynolds numbers," Physics of Fluids, Vol. 8, pp. 560-567, 1965. C. The Third Period: 12. Roca, Reclus, "Gravitational instalility of a liquid layer flowing down an plane and heated from below," Journal de Mlecanique, Vol. 5, pp. 117-147, 1966. 13. Lin, S. P., "Instability of a liquid film flowing down an inclined plane," Physics of Fluids, Vol. 10, pp. 308-313, 1967. 22

14. Samuels, Michael, "Free convection in a rotating medium," Physics of Fluids, Vol. 11, pp. 1889-1894, 1968. 15. Li, C. H., "Stability of two superposed elasticoviscous liquids in plane Couette flow," Physics of Fluids, Vol. 12, pp. 531-538, 1969a. 16. Li, C. IH., "Instability of three-layer viscous stratified flow," Physics of Fluids, Vol. 12, pp. 2473-2481, 1969b. 17. Li, C. H., "Instability of tirme-periodic flows of stratified fluids," Physics of Fluids, Vol. 13, pp. 1121-1134, 1970. 23