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. Such a swimmer deforms its body into a particular shape through a sequence of motions and then reverts to the original shape by going through the sequence in reverse. 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. 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 is an example of a body with one degree of freedom to use for motion. Bodies with a single degree of freedom deform in a reciprocal manner and subsequently, bodies with one degree of freedom do not achieve locomotion in a highly viscous environment.
See main article: Stokes flow.
The scallop theorem is a consequence of the subsequent forces applied to the organism as it swims from the surrounding fluid. For an incompressible Newtonian fluid with density
\rho
η
\rho\left(\dfrac{\partial}{\partialt
where
u
u0
L
\tilde{u
where the dimensionless pressure is appropriately scaled for flow with significant viscous effects. Plugging these quantities into the Navier-Stokes equations gives us:
\dfrac{\rho
| 2 | |
| u | |
| 0 |
And by rearranging terms, we arrive at a dimensionless form:
Re\left(\dfrac{\partial}{\partial\tilde{t}}+\tilde{u
where
Re=\rhou0L/η
Re → 0
0=-\nablap+η\nabla2u, \nabla ⋅ u=0.
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. 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. The other results then are simple corollaries.
The stress tensor of the fluid is
\sigmai=-p\deltai+\mu(\partialiuj+\partialjui)
Let
r
r
Since the motion is in the low Reynolds number regime, inertial forces are negligible, and the instantaneous total force and torque on the swimmer must both balance to zero. Since the instantaneous total force and torque on the swimmer is computed by integrating the stress tensor
\sigma
r
Thus, scaling both the swimmer's motion and the motion of the surrounding fluid scales by the same factor, we still obtain a motion that respects the Stokes equation.
The proof of the scallop theorem can be represented in a mathematically elegant way. To do this, we must first understand the mathematical consequences of the linearity of Stokes equations. 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. Subsequently, this leads to the discovery that reversal of periodic motion is identical to the forward motion due to symmetry, allowing us to conclude that there can be no net displacement.
The reciprocal theorem describes the relationship between two Stokes flows in the same geometry where inertial effects are insignificant compared to viscous effects. Consider a fluid filled region
V
S
\hat{n
V
u
u'
\sigma
\sigma'
\iintSu ⋅ (\boldsymbol{\sigma}' ⋅ \hat{n
The reciprocal theorem allows us to obtain information about a certain flow by using information from another flow. This is preferable to solving Stokes equations, which is difficult due to not having a known boundary condition. This particularly useful if one wants to understand flow from a complicated problem by studying the flow of a simpler problem in the same geometry.
One can use the reciprocal theorem to relate the swimming velocity,
U
F
uS
\hat{F
Now that we have established that the relationship between the instantaneous swimming velocity in the direction of the force acting on the body and its swimming gait follow the general form
U=\iint
| r |
S ⋅ g(rS)~dS,
where
| u | |||
|
=drS/dt
rS
t0
t1.
\Delta
| t1 | |
| X=\int | |
| t0 |
U~dt.
Now consider the swimmer deforming in the same manner but at a different rate. We describe this with the mapping
t'=f(t), rS(t)=r'S(t'),
| r |
S(t)=\dfrac{dr'S(t')}{dt}=\dfrac{dr'
|
|
(t).
Using this mapping, we see that
\Delta
| t1 | |
| X'=\int | |
| t0 |
| t1 | ||
| U'(t')~dt'=\int | U'(f(t)) | |
| t0 |
| f |
~dt=
| t1 | |
| \int | |
| t0 |
\iint
| r |
|
⋅ g(r'S)~dSdt=\int
| t1 | |
| t0 |
\iint
| rS |
⋅ g(rS)~dSdt
| t1 | |
| =\int | |
| t0 |
U(t)~dt → \DeltaX'=\DeltaX.
This result means that the net distance traveled by the swimmer does not depend on the rateat which it is being deformed, but only on the geometrical sequence of shape. This is the first key result.
If a swimmer is moving in a periodic fashion that is time invariant, we know that the average displacement during one period must be zero. To illustrate the proof, let us consider a swimmer deforming during one period that starts and ends at times
t0
t1
rS(t0)=rS(t1).
t2
t3.
t2=f(t1)
t3=f(t0)
rS(t)=r'S(t').
\Delta
| t3 | |
| X'=\int | |
| t2 |
| t0 | |
| U'(t')~dt'=\int | |
| t1 |
| t1 | |
| U(t)~dt=-\int | |
| t0 |
U(t)~dt=-\DeltaX.
This is the second key result. Combining with our first key result from the previous section, we see that
\DeltaX'=\DeltaX=-\DeltaX → \DeltaX=0.
t2
t3
t0
t1.
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. However, there are instances where the assumptions for the scallop theorem are violated. In one case, successful swimmers in viscous environments must display non-reciprocal body kinematics. In another case, if a swimmer is in a non-Newtonian fluid, locomotion can be achieved as well.
In his original paper, Purcell proposed a simple example of non-reciprocal body deformation, now commonly known as the Purcell swimmer. 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. However, any body with more than one degree of freedom of motion can achieve locomotion as well.
In general, microscopic organisms like bacteria have evolved different mechanisms to perform non-reciprocal motion:
Geometrically, the rotating flagellum is a one-dimensional swimmer, and it works because its motion is going around a circle-shaped configuration space, and a circle is not a reciprocating motion. 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. Notice that the first kind of motion has nontrivial homotopy, but the second kind has trivial homotopy.
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. It is also common knowledge that many living microorganisms live in complex non-Newtonian fluids, which are common in biologically relevant environments. For instance, crawling cells oftenmigrate in elastic polymericfluids.Non-Newtonian fluids have several properties that can be manipulated to produce small scale locomotion.
First, one such exploitable property is normalstress differences. These differences will arise from the stretching of the fluid by the flow ofthe swimmer. Another exploitable property is stress relaxation. Such time evolution of such stresses contain a memory term, though the extent in which this can be utilized is largely unexplored. Last, non-Newtonian fluids possess viscosities that are dependent on the shear rate. 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. This is in stark contrast to what we established where the rate in which a swimmer moves is irrelevant for establishing locomotion. Thus, a reciprocal swimmer can be designed in a non-Newtonian fluid. Qiu et al. (2014) were able to design a micro scallop in a non-Newtonian fluid.