Swimming at Low Reynolds Number

The characteristic length of many microorganisms is such that the hydrodynamic interaction with their surroundings is governed by the equation for Stokes flow. In this paper, I present a computational model for calculating the flow generated by a simple microorganism utilizing a rotating motor and a number of connected, rigid rods to induce locomotion. The efficiency of several flagella combinations is presented. Introduction Since the introduction of the now famous Scallop Theorem , a number of interesting questions about low Reynolds number swimming have been asked. One of the more interesting questions is: what types of flagellar tail combinations induce locomotion? Let’s start with the famous Naviér-Stokes equation for incompressible fluids (∇ · u = 0) ∇p = μ∇u− ρ ( ∂u ∂t + (u · ∇)u ) , where p is the pressure, μ is the fluid viscosity, ρ the fluid density (inertial force multiplier) and u is the velocity of the flow. This equation can be made dimensionless through the introduction of the Reynolds number R ≡ ρLU μ , where L and U represent the characteristic length and velocity, respectively, for a given problem. For microorganisms, the characteristic lengths and velocities are very small, meaning that the inertial term in the Naviér-Stokes equation becomes negligible. The resulting equation governing low Reynolds number flow becomes ∇p = μ∇u, which is known as the Stokes equation. Since the ∇ operator acting on u is linear, the viscous force exerted on a fluid by a low Reynolds number swimmer scales directly with the stroke velocity applied by the swimmer. 1 The consequence is that the rate at which a flagellar tail rotates does not enable it to move faster through a viscous fluid. Take, for example, a simple scallop. The scallop opens and closes via a spring mechanism connecting its upper and lower halves. This represents a one degree of freedom system. The scallop can either increase or decrease the rate at which it opens and closes and yet it will be unable to propel itself through a viscous fluid since the fluid will apply an equal and opposite force to balance. In order to move, a system must utilize multiple degrees of freedom. One simple organism to consider would be a rigid rod connected by two motors each capable of applying torques in any direction; see Figure 1. Figure 1: Example of a low Reynolds number swimmer The fluid flow will arrange itself to balance viscous forces with those generated by torques applied at each endpoint. The resulting flow, since the organism has two degrees of freedom, will exert different forces at different points along the organism which, after integration, result in net displacement of the organism from its original position. To extend the argument further, it is possible to change the magnitudes and directions of the torque so that the organism can move in any desired direction. While this simple example illustrates an organism which can move in any direction, there may be more efficient combinations; i.e. organisms which can move farther in a shorter amount of time. This paper focuses on the numerical simulation of different flagella that are composed of multiple rods interacting with one another and the surrounding fluid via Hooke’s law forces and applied torques. Each rod introduces a new degree of freedom to the system, as does each applied torque. The focus is specifically on an organism with torque application at one point (the head of the flagella) and connected rods. This simple microorganism is simulated using different flagellar combinations. The output of this paper is the efficiency of each flagellar combination; quantified as the distance traveled as a function of the number of motor cycles underwent.