Concluding Remarks and Further Work

In this paper, Saffman and Taylor used analytical results and experimental observations to develop a good description of certain features of the instability of the interface between two fluids of differing viscosities. However, the paper asks as many questions as it answers, in particular:

• analytically the equations yield an infinite family of solutions corresponding to different values of and although Saffman and Taylor observed experimentally that this value of is approximately 1/2 provided the flow is not very slow, it is unclear why the system selects this value of ;
• at the end of the paper Saffman and Taylor commented that the finger-shaped interface is unstable at zero surface tension, yet since the fingers were observed experimentally clearly the finger-shapes are stable in nature;
• the analysis is based on the assumption that curvature is approximately constant;
• the analysis is also based on the assumption that none of the receding fluid is deposited on the plates and while this assumption is shown to be reasonable by experimental evidence, it is not fully explained;
• the experimental observation that fingers ``compete'' and that a finger which is initially slightly larger than adjacent fingers will proceed to become significantly larger than its neighbors is not fully explained in the paper.

Many of these issues have been resolved only in the last ten years. The key to the selection of the value of and the stability of the finger shape was an understanding of the effects of surface tension. However, it was a long time before researchers were able to analyze the significance of the singular perturbation of the surface tension term successfully. Homsy [4] summarizes non-zero surface tension solutions as an ``infinite family of discrete solutions , all of which tend to the same shape with '' as , i.e. as surface tension effects decrease.

Researchers have come to understand other aspects of the system too. Park and Homsy [7] eliminated one source of disagreement between analysis of the Saffman-Taylor equations and experiment by considering the film flow in the system when the receding fluid is wetting. They used asymptotic methods to show that non-constant curvature should be accounted for by consideration of three-dimensional effects at the interface rather than two-dimensional effects as Saffman and Taylor used. This yields a pressure difference of

 

and a normal velocity jump (valid asymptotically as ). The effect of the pressure difference is stabilizing. Bensimon et. al. [1] found that when the effects of surface tension were included a different conformal mapping could be used to find the interface shape. Homsy [4] commented that the use of the modified boundary condition () yields better agreement between experiment and theory in the work of Park et. al. [7].

The system has been found to have a rich diversity of solutions. For example, Bensimon et. al. [1] observed that when is large (viscous effects are strong in comparison to surface tension), the fingers are unstable and chaotic patterns break out at the interface. Conversely when is small (viscous effects are weak in comparison to surface tension), the fingers are linearly stable yet nonlinearly unstable. The configuration has been modified also: a radial source flow has been studied ( [10],  [8]). Today researchers are considering the behavior of non-Newtonian and complex fluids in this system and it continues to be an area of active research ( [2],  [5],  [6]).