Are bacterial flagella the most efficient propulsion system at the microscopic scale? Bacterial flagella can be regarded as microscopic propellers. The rotor is a long, helical protein filament powered by proton gradient. When protons pass through the gap between the rotor and stator, they drive the rotor to spin (in much the same way of the ATP synthase), which pushes bacteria forward through the fluids.
What I don’t understand why the rotor is a long helical filament instead of a multi-blade propeller or an Archimedes screw even though it won’t be difficult to evolve structures like that. Is it because at microscopic levels, long filaments are the most efficient propeller shape? Is there a mathematical model of microscopic fluid dynamics (in analogy to the Navier-Stokes equation of macroscopic fluid dynamics) to prove that?
 A: The most important difference between bacteria and everyday propulsion is size. Bacteria are microscopic.
There are two sources of resistance when moving through water. First, flowing water generates friction forces. These are called viscosity. Second, a moving object must push water out of its way. The reaction forces push the object back. These are called inertial forces.
In most cases, one of these is so much bigger that you might as well ignore the other.
Inertial forces are big when the object is big. Lots of water gets pushed around. Also when the projectile is fast, water must move fast and a larger acceleration is needed to get it out of the way. When the fluid is dense, mass goes up, and so do the forces.
Viscous forces are big when the fluid has more internal friction. Forces are bigger when moving through honey than water.
Viscous forces are more important when the object is small. Most of the fluid motion takes place near the object. Far away, the fluid is not disturbed. There is a layer right next to the object that moves with the object. Just above this, there is a layer that moves a little slower. Just above that, a little slower still. The thickness of the moving region depends on the viscosity. Fluids with a lot of friction drag more fluid with them. For a small object, the layer might be as big as the object. For a microscopic one, it might be bigger.
The ration of inertial forces to viscous forces is called the Reynolds number. While it is difficult to calculate exactly, it is easy to approximate. And an approximation is all you need when one force is overwhelmingly bigger than the other.
$$Re = \frac{\rho uL}{\mu}$$
where

*

*$\rho$ is the density of the fluid

*$u$ is the velocity

*$L$ is the size of the projectile

*$\mu$ is the viscosity of the fluid

We can plug in a few numbers. See Speed of a Bacterium
$$Re = \frac{(10^6 kg/m^3)(50 \cdot 10^{-6} m/s)(0.5 \cdot 10^{-6} m)}{(10^{-3} kg/{ms})} = 0.025$$
Viscous forces are 40 time bigger.
A ship's propeller pushes large volumes of water backward, creating a large inertial force forward. A bacterium must do something different.
You might think of a flagellum as closer to the threads on a screw, pushing the bacterium forward through a fluid that flows much less than you would expect.
A: As mentioned in comment it will be crazy to discuss why organism have bodyparts in such numbers, but also flagellas not just work in rotary motion they perform other moments to and if needed organism have evolved have multiple flagella at one end, also evolution occur as needed not just as to do ideal thing.
A: Bacterial flagella is not a result of evolution exploring all the possibilities and coming up with the most optimal one (it can be mathematically proven that such an exploration couldn't have realistically taken place). Rather, the evolution has taken a random direction (see more about it in this answer). It is fair to say that the existing bacteria are more efficient than most of their evolutionary competitors and predecessors, since they won in the evolutionary competition, but it doesn't mean that nothing more efficient is possible. Note that some more efficient organisms might have existed, but turned out to be less lucky (in terms of survival).
Interestingly, the efficiency of organism is used to reconstruct the evolution in cases where no fossil record is available: e.g., we can characterize organisms as more ancient or more advanced, depending on the number of stages in their cellular respiration cycle (which is directly related to the efficiency of energy extraction from food).
