Scallop Theorem: Fundamentals of Motion and Efficiency in Microfluidic Environments

Chapter 1: Scallop theorem

In the field of physics, the scallop theorem asserts that a swimmer who engages in a reciprocal motion is unable to produce net displacement in a Newtonian fluid environment with a low Reynolds number, which is another way of saying that the fluid in question is highly viscous. One type of swimmer is able to distort their body into a specific shape by performing a series of motions, and then they are able to return to their original shape by performing the sequence in reverse. The swimming motion is solely dictated by the sequence of shapes that the swimmer assumes when the Reynolds number is low. Time and inertia do not play a role in this situation.

This theorem was presented by Edward Mills Purcell in his work titled Life at Low Reynolds Number in 1977. The publication included an explanation of the fundamental processes that govern aquatic mobility. The action of a scallop that opens and shuts a simple hinge throughout one period is the inspiration for the theorem's name. This kind of motion is not enough to produce migration when the Reynolds number is low. When it comes to motion, the scallop is an example of a body that only has one degree of freedom to work with. The bodies that have a single degree of freedom deform in a manner that is reciprocal, and as a consequence, bodies that have one degree of freedom are unable to achieve locomotion in an environment that is highly viscous.

The scallop theorem is a result of the successive forces that are exerted to the creature as it swims from the fluid that is surrounding it. When the flow of an incompressible Newtonian fluid with density

{rho} and dynamic viscosity

{eta} is considered, the Navier–Stokes equations are satisfied. The velocity of the fluid is denoted by

{bold u}, which is the equation that describes the flow. On the other hand, when the Reynolds number is low, the inertial terms of the Navier-Stokes equations on the left-hand side tend to become zero. Through the process of nondimensionalizing the Navier–Stokes equations, this reality becomes more obvious. In order to cast our variables into a dimensionless form, we must first define a characteristic velocity and length, which we will refer to as

{u 0} and

{upper L}, respectively. This will allow us to scale the dimensionless pressure properly for flow that has considerable viscous effects. The following results are obtained by plugging these quantities into the Navier-Stokes equations:

In addition, by rearranging the terms, we are able to arrive at a form that is dimensionless: where Re

{Re equals rho u 0 upper L divided by eta} represents the Reynolds number. A dimensionless version of Stokes equations is reached when the LHS tends to zero in the low Reynolds number limit (as

{upper R e right arrow 0}). This is the case because the LHS approaches to zero. A result of redimensionalization is:

As a result of the absence of inertial terms at low Reynolds numbers, the following effects occur:

One of the consequences ensures that the swimmer is subjected to almost little net force or torque.

-! There is a second implication that tells us that the force is directly related to the velocity (the same can be said about the relationships between torque and rotational velocity).

-! At this point, the Stokes equations are linear and not dependent on the passage of time.

In specifically, the motion of a swimmer who is moving in the regime of low Reynolds number satisfies the following conditions:

The Stokes equations would be satisfied even if the same motion were to be sped up or slowed down. This is because the motion being described is independent of time. Additionally, this indicates that the velocity of a swimmer in the low Reynolds number regime is solely dictated by the form of its trajectory in configuration space. This is a more geometric interpretation of the situation.

- Kinematic reversibility: the ability to reverse the motion that is being described. It is just the direction of the flow that will be altered by any momentary reversal of the forces that are operating on the body; the nature of the fluid flow that is around the body will remain unchanged. Motion is brought about by these factors, which are responsible for it. In situations when a body only possesses one degree of freedom, the action of reversing forces will result in the body deforming in a manner that is reciprocal. As an illustration, a scallop that opens its hinge will simply close it in an effort to achieve propulsion from the opening. As a result of the fact that the reversal of forces does not alter the nature of the flow, the body will travel in the other direction in the same manner, which will result in no net displacement. It is through this method that we are able to arrive at the results of the scallop theorem.

This is more similar to the proof sketch that Purcell provided in terms of aesthetics. The most important finding is that the success of a swimmer in a Stokes fluid is not dependent on the passage of time. To put it another way, it is impossible to tell whether a movie footage of a swimmer's action has been slowed down, sped up, or reversed. Therefore, the remaining results are straightforward corollaries.

It is

{sigma Subscript i j Baseline equals minus p delta Subscript i j Baseline plus mu left parenthesis partial differential Subscript i Baseline u Subscript j Baseline plus partial differential Subscript j Baseline u Subscript i Baseline right parenthesis} that serves as the stress tensor for the fluid.

Assume that

{r} is a real constant that is not zero. In the event that we have a swimming motion, we are able to perform the scaling that is as follows:Utilize

{p right arrow from bar r p semicolon u right arrow from bar r u semicolon sigma right arrow from bar r sigma} in order to acquire an additional solution to the Stokes equation. In other words, even if we scale the hydrostatic pressure, flow velocity, and stress tensor all by

{eta}0, we are still able to derive a solution to the Stokes equation.

Inertial forces are insignificant because the motion is in the low Reynolds number zone. Additionally, the instantaneous total force and torque on the swimmer must both balance to zero in order for the swimmer to be considered in equilibrium. As a result of the fact that the instantaneous total force and torque on the swimmer are computed by integrating the stress tensor

{eta}1 across its surface, the instantaneous total force and torque rise by

{eta}2 as well, despite the fact that they remain zero. 

Therefore, by scaling the motion of the swimmer as well as the motion of the fluid that is surrounding them by the same factor, we are still able to produce a motion that is consistent with the Stokes equation.

The proof of the scallop theorem can be expressed in a manner that is mathematically sophisticated. In order to accomplish this, it is necessary for us to first know the mathematical repercussions that result from the linearity of Stokes equations. To recap, the linearity of Stokes equations enables us to apply the reciprocal theorem to establish a connection between the swimming velocity of the swimmer and the velocity field of the fluid around its surface (often referred to as the swimming gait), which varies in accordance with the periodic motion that the fluid demonstrates. Because of this link, we are able to draw the conclusion that the rate of swimming does not affect locomotion. Subsequently, this leads to the realization that the reversal of periodic motion is the same as the forward motion due to symmetry, which enables us to arrive at the conclusion that there is no possibility of a net displacement.

The reciprocal theorem is a mathematical expression that expresses the relationship between two Stokes flows that are in the same geometry and where the viscous effects are more substantial than the inertial effects.  Take into consideration a region

{eta}3 that is filled with fluid and is surrounded by a surface

{eta}4 and a unit normal

{eta}5.  Consider the following scenario: we have solutions to Stokes equations in the domain

{eta}6 that have the form of the velocity fields

{eta}7 and

{eta}8.

{eta}9 and

{bold u}0 are the stress fields that correspond to the velocity fields, where they are located accordingly. Therefore, the equivalence that follows is valid:

Through the application of the reciprocal theorem, we are able to procure information regarding a certain flow by utilizing information from another flow. When compared to solving Stokes equations, which is a challenging task because there is no known boundary condition, this is a more acceptable alternative. For example, if one wants to understand flow from a complicated problem by examining the flow of a smaller problem in the same geometry, this is a particularly beneficial method.

In order to establish a connection between the swimming velocity,

{bold u}1, of a swimmer who is subject to a force, one can make use of the reciprocal theorem. From the swimming gait of

{bold u}2 to that of

{bold u}3:

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 follows the general form, where

{bold u}4 and

{bold u}5 denote the positions of points on the surface of the swimmer, we are able to establish that locomotion is independent of rate. This opens up a lot of possibilities for us. Take into consideration a swimmer who undergoes a series of actions that cause them to deform in a periodic manner between the