Scallop theorem
In physics, the scallop theorem states that a swimmer that performs a reciprocal motion cannot achieve net displacement in a low-Reynolds number Newtonian fluid environment, i.e. a fluid that is highly viscous.
At low Reynolds number, time or inertia does not come into play, and the swimming motion is purely determined by the sequence of shapes that the swimmer assumes.
Edward Mills Purcell stated this theorem in his 1977 paper Life at Low Reynolds Number explaining physical principles of aquatic locomotion.
[1] The theorem is named for the motion of a scallop which opens and closes a simple hinge during one period.
Such motion is not sufficient to create migration at low Reynolds numbers.
The scallop theorem is a consequence of the subsequent forces applied to the organism as it swims from the surrounding fluid.
However, at the low Reynolds number limit, the inertial terms of the Navier-Stokes equations on the left-hand side tend to zero.
Plugging these quantities into the Navier-Stokes equations gives us: And by rearranging terms, we arrive at a dimensionless form: where
Redimensionalizing yields: The consequences of having no inertial terms at low Reynolds number are: In particular, for a swimmer moving in the low Reynolds number regime, its motion satisfies: This is closer in spirit to the proof sketch given by Purcell.
[1] The key result is to show that a swimmer in a Stokes fluid does not depend on time.
That is, a one cannot detect if a movie of a swimmer motion is slowed down, sped up, or reversed.
Since the instantaneous total force and torque on the swimmer is computed by integrating the stress tensor
To summarize, the linearity of Stokes equations allows us to use the reciprocal theorem to relate the swimming velocity of the swimmer to the velocity field of the fluid around its surface (known as the swimming gait), which changes according to the periodic motion it exhibits.
This relation allows us to conclude that locomotion is independent of swimming rate.
This is preferable to solving Stokes equations, which is difficult due to not having a known boundary condition.
denote the positions of points on the surface of the swimmer, we can establish that locomotion is independent of rate.
Consider a swimmer that deforms in a periodic fashion through a sequence of motions between the times
To illustrate the proof, let us consider a swimmer deforming during one period that starts and ends at times
Now we find the relationship between the net displacements in these two cases: This is the second key result.
In addition, since the swimmer exhibits reciprocal body deformation, the sequence of motion is the same between
The scallop theorem holds if we assume that a swimmer undergoes reciprocal motion in an infinite quiescent Newtonian fluid in the absence of inertia and external body forces.
[4] In one case, successful swimmers in viscous environments must display non-reciprocal body kinematics.
This simple swimmer possess two degrees of freedom for motion: a two-hinged body composed of three rigid links rotating out-of-phase with each other.
The flexible arm is a multi-dimensional swimmer, and it works because its motion is going around a circle in a square-shaped configuration space.
The assumption of a Newtonian fluid is essential since Stokes equations will not remain linear and time-independent in an environment that possesses complex mechanical and rheological properties.
Non-Newtonian fluids have several properties that can be manipulated to produce small scale locomotion.
Such time evolution of such stresses contain a memory term, though the extent in which this can be utilized is largely unexplored.
In other words, a swimmer would experience a different Reynolds number environment by altering its rate of motion.
Many biologically relevant fluids exhibit shear-thinning, meaning viscosity decreases with shear rate.
In such an environment, the rate at which a swimmer exhibits reciprocal motion would be significant as it would no longer be time invariant.
