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# Kinematic Reversibility and the Scallop Theorem

Some of the most striking sequences of the movie were the demonstrations of reversibility of fluid flow. Blobs of dye and rigid and flexible bodies were suspended in an annulus of glycerin. The movie showed that after inducing motion on the bodies (by means of rotating one of the cylinder walls forming the annulus) the original shape and position of an object could be recovered by simply reversing the motion (rotation). Each particle retraced its path back to its original position. (The flexible body was the exception. The compressive forces made scrunch up - it did not recover its original shape. This can be viewed as applying boundary conditions that were not time-independent.)

Time reversibility affects the way an animal can propel itself through a fluid. In a low Reynolds number environment, a creature with a single joint (such as a fish) moves in a pattern that is time reversible. Thus the forward motion created by a swish of the tail is exactly canceled by the backward motion produced when the tail returns through its starting position. You can visualize this situation by thinking of running a film of a fish swimming and being unable to tell which direction the film is running based on the movement of the tail.

Purcell popularized an idea that is now called the ``scallop theorem''. As can be seen from the above demonstrations, an animal with only one joint cannot propel itself in a low Reynolds number environment. The only way for it to get moving is to move its appendage fast enough to get the Reynolds number higher, so that the inertial terms enter into the Navier-Stokes equation (propulsion due to inertial reaction). Purcell instead proposed two joints, jointed like so:
to allow for non-time-reversible movement. This is illustrated by the following diagram, with the arrows denoting time moving forward; the ``scallop's'' joints make the movements indicated:

This pattern of motion is not time-reversible. In analogy with the fish, with this motion you can which direction the film is playing. In conclusion we can see that: non-reversible motion net motion

Suppose you wanted to know what would happen to a microscopic animal with a curved tail:

In an imaginary film, you could tell which direction the tail was moving. The movement of a corkscrew is not time-reversible. The exact mechanism of movement comes from thinking of the tail of this creature as just a series of obliquely oriented thin cylinders (which will be discussed later).

The movie demonstrated these different methods of propulsion with wind-up models of two different creatures: our intrepid fish and the picturesque little creature above with the corkscrew tail. Taylor immersed both in water. The fish swam normally while the spiral tailed model was only able to move very slowly due to its ``feeble grip'' on the water. Imagine the difference in trying to move a coat hanger through some water in comparison to a canoe paddle. Then Taylor immersed both models in glycerin. The fish was ``dead in the glycerin'' whereas the microscopic creature was able to swim.

This discussion provides only a taste of the difficulties in getting ahead in a low Reynolds number regime. The topics of swimming will be covered in much more detail in the following lectures.

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brenner@math.mit.edu