An experimental investigation of initial oscillations in a radial Hele-Shaw cell

Hele-Shaw cell is a laboratory device consisting of two parallel plates of glass separated by a thin gap. In this cell, in the flow of two immiscible fluids, when a fluid of higher viscosity is displaced by a fluid of lower viscosity, the less viscous fluid is observed to form “fingers” into the more viscous one due to the unstable interface. The Saffman-Taylor or viscous finger instability has been examined and modeled for over forty years for the rectilinear Hele-Shaw cell and about half as long for the radial Hele-Shaw cell. In this paper, we study, in detail, the early development of viscous instabilities in a radial Hele-Shaw cell. This source flow configuration has been chosen so that the instability can be monitored precisely. The objective of this study is to examine the onset of fingering, i.e. initial number of fingers that form, and the evolution of interface instability. Our experiments suggest that there may be some order in this formation process and one can model this aspect by considering the unsteady velocity components and predicting temporal changes in wavenumber responsible for the initial number of fingers and may be later accounting for the fingertip oscillations and splitting. We injected a water-based fluid into an oil in a radial Hele-Shaw cell at constant flow rate and recorded the movement of the less viscous droplet as it evolved. The relative curvature changes on the expanding droplet boundary was plotted with the angular positions about the interface and subtracting out the average radius, resulting in a plot of the change in amplitude with respect to time for the interface configuration. Three unstable configured tests at kinematic viscosity contrast (v ^O) of 0.34, 0.68, and 0.94 were run at approximately the same flow rate (2π cm²/s). The droplet exhibited oscillatory movement for these unstable configuration. The amplitude and the rate of oscillations were measured from digitized data. The smaller the viscosity difference, the smaller was the amplitude growth rate and resulted in a longer time to form visible finger initiation. This work was supported by National Science Foundation, grant number EID-9017555. We also like to thank Dr. Len Schwartz, Professor of Mechanical Engineering at the University of Delware for his insight and helpful suggestions.