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 . The pressure is the nearly the same at the bump interface 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 (R). The same pressure difference over a shorter difference means greater flow velocity. As a result, the bump expands faster than the rest of the circular interface. This is why the interface is unstable, bumps grow, and 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.


Figure 4.13: Bump on circular interface of injected fluid. R', the distance from the bump to the edge of the cell, is less than R, the distance from the circular interface to the edge of the cell. Because the velocity depends inversely on this distance, the bump grows faster.

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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