4.5 - Why Do Viscous Fingers Branch?

Why do bumps grow faster than the rest of the interface? The answer to this question is fundamental to understanding why the Hele-Shaw experiment gives rise to a treelike pattern when a less viscous fluid is injected into a more viscous one. Recall from our brief discussion above that the flow velocity is inversely proportional to the distance over which the pressure difference occurs. (You drink more water when you suck just as hard on a shorter straw.)

Consider a bump on a circular interface as in Figure 4.13. The pressure is nearly the same at the bump as on the circular portion of the interface. On the other hand, the top of the bump is closer to the edge of the cell (R') than the rest of the interface is (R). The same pressure difference over a shorter distance produces a greater flow velocity. As a result, the bump grows, flowing outward faster than the rest of the circular interface. This is why the interface is unstable, and why viscous fingers develop. In a self-similar fashion, subsequent bumps grow on the fingers themselves causing the fingers to branch, and so on: a treelike structure develops.

Q4.27: Can you make an analogy between this process of branching by viscous fingers and the branching process for electrochemical deposits? What corresponds to the pressure in the case of electrochemical deposition?

Q4.28: What determines the thickness of the viscous fingers? Why are the viscous fingers in the carrageenan experiment narrower than those in the glycerol experiment? Predicting the minimum thickness of a viscous finger is still a matter for current research. A key factor is the difference in surface tension between the two fluids. Are your results consistent with your understanding of surface tension? Write a short argument saying why viscous fingers in glycerol should be thinner or thicker when you inject air rather than water, and explain your reasons. Try this experiment.

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