PHYSICAL REVIEW A VOLUME 38, NUMBER 9 NOVEMBER 1, 1988 Pattern formation in the flow between two horizontal coaxial cylinders with a partially filled gap Innocent Mutabazi, * John J. Hegseth, and C. David Andereck Department of Physics, Ohio State Uniuersity, 174 West 18th Auenue, Columbus, Ohio 43210 Jose E. Wesfreid 10 rue Vauquelin, 75231 Paris Cedex 05, France (Received 29 February 1988) Flow between two horizontal coaxial cylinders with a partially filled gap is subject to several types of centrifugal instabilities which lead to the formation of a variety of spatial patterns. An ex- perimental investigation has shown that there are five distinct branches of primary instabilities occurring in the system and that four codimension-2 points are easily reached. Theoretical predic- tions are in qualitative agreement with the observations. I. INTRODUCTION The behavior of systems far from equilibrium has been the subject of intense investigation over the last several years. Of particular interest has been the manner in which spatial patterns arise. Pattern formation may be driven by a wide variety of mechanisms, including tem- perature gradients, concentration gradients, and centrifu- gal effects. Centrifugal instabilities' occur in flow with curved streamlines, and they play an important role in many problems of practical importance. Two of the best known examples are the Taylor-Couette instabilities, which occur in the flow between two concentric rotating cylinders, and the Taylor-Gortler instabilities, which occur in the boundary layer on a concave wall. The Taylor-Couette and Taylor-Gortler instabilities have been extensively investigated both theoretically and experi- mentally (see Refs. 3 — 5 and references therein). Howev- er, a third class, the Dean instabilities, which occur in the presence of a pressure gradient along a curved chan- nel (Poiseuille flow), has received much less attention. Here, we present a simple system in which it is possible to realize the main centrifugal instabilities by an ap- propriate choice of control parameters. We consider two horizontal coaxial cylinders of radii r; and r„which ro- tate independently with angular velocities 0; and 0, for the inner and the outer cylinder, respectively. When the gap between the cylinders is completely filled with fluid, we have the classical Taylor-Couette problem. When the gap is partially filled and both cylinders rotate there ex- ists a combination of Taylor-Couette flow (caused by the differential rotation of the cylinders) and Poiseuille flow (caused by the backflow induced by the presence of the horizontal free surfaces). As the system control parame- ters are varied, the base flow instabilities will change from those associated with Taylor-Couette to those asso- ciated with Dean. Under some conditions one might ex- pect to see flow patterns that result from competition be- tween instabilities. In other cases boundary layers on the curved surfaces may give rise to Taylor-Gortler instabili- ties. We will not discuss the last any further since boundary layer instabilities are apparently not dominant for the range of system parameters chosen. The relevant control parameters are the ratio of angular velocities p, =0, /0; and the Taylor number. In Sec. II we will summarize the few related experi- mental and theoretical results which have been reported so far. En Sec. III we describe our experimental pro- cedure and in Sec. IV we report the results obtained. Sec- tion V is a comparison of the experimental results with the present theory. Finally, we will conclude with sug- gested directions for future work. II. PREVIOUS WORK The flow between two horizontal coaxial cylinders with a partially filled gap was first investigated by Brewster and Nissan in 1958. They deduced approximate velocity fields for laminar flow with only the inner cylinder rotat- ing, and they measured the critical angular velocity and the wave number of the resulting rolls. In 1959, Brew- ster, Grosberg, and Nissan considered the critical condi- tions for the formation of vortices between the cylinders in three cases: when the gap is filled with fluid and the flow is caused by the rotation of the inner cylinder (Taylor-Couette problem), when the flow is produced by pumping around the annular space (Dean problem), and when the liquid is driven by the rotation of the inner cylinder and forced to reverse its flow at a free surface. Their results for the Dean problem were in satisfactory agreement with the theoretical values for the threshold of the instability and the wave number of the vortices. They considered also the combination of the pumping of fluid around the annular space and the rotation of the inner cylinder. They obtained the interesting result that, in the neighborhood of a particular value of the ratio of the pumping flow rate to the rotation flow rate, the critical value of the control parameter has an abrupt change, and 38 4752 1988 The American Physical Society