Scallop theorem

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 ${\displaystyle V}$ bounded by surface ${\displaystyle S}$ with a unit normal ${\displaystyle {\hat {\mathbf {n} }}}$. Suppose we have solutions to Stokes equations in the domain ${\displaystyle V}$ possessing the form of the velocity fields ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {u} '}$. The velocity fields harbor corresponding stress fields ${\displaystyle \mathbf {\sigma } }$ and ${\displaystyle \mathbf {\sigma } '}$ respectively. Then the following equality holds:

${\displaystyle \iint _{S}\mathbf {u} \cdot ({\boldsymbol {\sigma }}'\cdot {\hat {\mathbf {n} }})~\mathrm {d} S=\iint _{S}\mathbf {u} '\cdot ({\boldsymbol {\sigma }}\cdot {\hat {\mathbf {n} }})~\mathrm {d} S}$.

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, ${\displaystyle \mathbf {U} }$, of a swimmer subject to a force ${\displaystyle \mathbf {F} }$ to its swimming gait ${\displaystyle \mathbf {u} _{S}}$:

${\displaystyle {\hat {\mathbf {F} }}\cdot \mathbf {U} =-\iint _{S}\mathbf {u} _{S}\cdot ({\boldsymbol {\hat {\sigma }}}\cdot \mathbf {n} )~\mathrm {d} S}$.

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

${\displaystyle \mathbf {U} =\iint {\dot {\mathbf {r} }}_{S}\cdot \mathbf {g} (\mathbf {r} _{S})~\mathrm {d} S}$,

where ${\displaystyle \mathbf {u} _{S}\equiv {\dot {\mathbf {r} _{S}}}=\mathrm {d} \mathbf {r} _{S}/\mathrm {d} t}$ and ${\displaystyle \mathbf {r} _{S}}$ 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 ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$. The net displacement of the swimmer is

${\displaystyle \Delta X=\int _{t_{0}}^{t_{1}}\mathbf {U} ~\mathrm {d} t}$.

Now consider the swimmer deforming in the same manner but at a different rate. We describe this with the mapping

${\displaystyle t'=f(t),\quad \mathbf {r} _{S}(t)=\mathbf {r'} _{S}(t'),\quad {\dot {\mathbf {r} }}_{S}(t)={\dfrac {\mathrm {d} \mathbf {r'} _{S}(t')}{\mathrm {d} t}}={\dfrac {\mathrm {d} \mathbf {r'} _{S}(t')}{\mathrm {d} t'}}\cdot {\dfrac {\mathrm {d} t'}{\mathrm {d} t}}={\dot {\mathbf {r} }}_{S}'(t'){\dot {f}}(t)}$.

Using this mapping, we see that

${\displaystyle \Delta X'=\int _{t_{0}}^{t_{1}}\mathbf {U} '(t')~\mathrm {d} t'=\int _{t_{0}}^{t_{1}}\mathbf {U} '(f(t)){\dot {f}}~\mathrm {d} t=\int _{t_{0}}^{t_{1}}\iint {\dot {\mathbf {r} }}_{S}'{\dot {f}}\cdot \mathbf {g} (\mathbf {r'} _{S})~\mathrm {d} S\mathrm {d} t=\int _{t_{0}}^{t_{1}}\iint {\dot {\mathbf {r} _{S}}}\cdot \mathbf {g} (\mathbf {r} _{S})~\mathrm {d} S\mathrm {d} t}$

${\displaystyle =\int _{t_{0}}^{t_{1}}\mathbf {U} (t)~\mathrm {d} t\rightarrow \Delta X'=\Delta X}$.

This result means that the net distance traveled by the swimmer does not depend on the rate at 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 ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$. That means its shape at the start and end are the same, i.e. ${\displaystyle \mathbf {r} _{S}(t_{0})=\mathbf {r} _{S}(t_{1})}$. Next, we consider motion obtained by time-reversal symmetry of the first motion that occurs during the period starting and ending at times ${\displaystyle t_{2}}$ and ${\displaystyle t_{3}}$. using a similar mapping as in the previous section, we define ${\displaystyle t_{2}=f(t_{1})}$ and ${\displaystyle t_{3}=f(t_{0})}$ and define the shape in the reverse motion to be the same as the shape in the forward motion, ${\displaystyle \mathbf {r} _{S}(t)=\mathbf {r'} _{S}(t')}$. Now we find the relationship between the net displacements in these two cases:

${\displaystyle \Delta X'=\int _{t_{2}}^{t_{3}}\mathbf {U} '(t')~\mathrm {d} t'=\int _{t_{1}}^{t_{0}}\mathbf {U} (t)~\mathrm {d} t=-\int _{t_{0}}^{t_{1}}\mathbf {U} (t)~\mathrm {d} t=-\Delta X}$.

This is the second key result. Combining with our first key result from the previous section, we see that ${\displaystyle \Delta X'=\Delta X=-\Delta X\rightarrow \Delta X=0}$. We see that a swimmer that reverses its motion by reversing its sequence of shape changes leads to the opposite distance traveled. In addition, since the swimmer exhibits reciprocal body deformation, the sequence of motion is the same between ${\displaystyle t_{2}}$ and ${\displaystyle t_{3}}$ and ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$. Thus, the distance traveled should be the same independently of the direction of time, meaning that reciprocal motion cannot be used for net motion in low Reynolds number environments.

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:

• Use of a flagellum, which rotates, pushing the medium backwards — and the cell forwards — in much the same way that a ship's screw moves a ship. This is how some bacteria move; the flagellum is attached at one end to a complex rotating motor held rigidly in the bacterial cell surface.[4][5]
• Use of a flexible arm: this could be done in many different ways. For example, mammalian sperm have a flagellum which, whip-like, wriggles at the end of the cell, pushing the cell forward.[6] Cilia are quite similar structures to mammalian flagella; they can advance a cell like paramecium by a complex motion not dissimilar to breast stroke.

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 often migrate in elastic polymeric fluids. Non-Newtonian fluids have several properties that can be manipulated to produce small scale locomotion.[3]

First, one such exploitable property is normal stress differences. These differences will arise from the stretching of the fluid by the flow of the 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.[7]