(a) E.coli swims at about 20 μm / s by rotating a bundle of helical flagella. If the motors were

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(a) E.coli swims at about 20 μm / s by rotating a bundle of...

(a) E.coli swims at about $20 \mu \mathrm{m} / \mathrm{s}$ by rotating a bundle of helical flagella. If the motors were to turn 10 times faster than normal, what would their swimming speed be? If their fluid environment were made 10 times more viscous, but the motors were to turn at the same rate, what would the swimming speed be? How does the power output of the motor change in these two hypothetical situations? (b) Two micron-sized spheres, one made of silver and the other gold, sediment (that is, fall under gravity) in a viscous fluid. The silver sphere has twice the radius of the gold one. Which sediments faster? (c) The left ventricle of the human heart expels about $50 \mathrm{cm}^{3}$ of blood per heartbeat. Assuming a pulse rate of I heartbeat per second and a diameter of the aorta of about $2 \mathrm{cm},$ what is the mean velocity of blood in the aorta? What is the Reynolds number? (d) What is the Reynolds number of a swimming bacterium? A tadpole? A blue whale?

(a) E.coli swims at about $20 \mu \mathrm{m} / \mathrm{s}$ by rotating a bundle of helical flagella. If the motors were to turn 10 times faster than normal, what would their swimming speed be? If their fluid environment were made 10 times more viscous, but the motors were to turn at the same rate, what would the swimming speed be? How does the power output of the motor change in these two hypothetical situations?
(b) Two micron-sized spheres, one made of silver and the other gold, sediment (that is, fall under gravity) in a viscous fluid. The silver sphere has twice the radius of the gold one. Which sediments faster?
(c) The left ventricle of the human heart expels about $50 \mathrm{cm}^{3}$ of blood per heartbeat. Assuming a pulse rate of I heartbeat per second and a diameter of the aorta of about $2 \mathrm{cm},$ what is the mean velocity of blood in the aorta? What is the Reynolds number?
(d) What is the Reynolds number of a swimming bacterium? A tadpole? A blue whale?

Key Concepts

Flagellar Propulsion

This concept explains how microorganisms such as bacteria generate thrust by rotating helical filaments (flagella). The rotational motion converts into forward movement through interaction with the surrounding fluid, and changes in the rotation rate can directly alter swimming speed. Understanding how flagella work involves considerations of mechanics at low Reynolds numbers where viscous forces dominate.

Viscosity Effects

Viscosity is a measure of a fluid’s resistance to deformation and flow. It plays a critical role in determining the drag experienced by moving objects in fluid. In contexts like bacterial swimming or sedimentation, an increase in viscosity leads to higher drag forces, which can slow down movement even when the driving forces (such as flagellar rotation or gravity) remain constant.

Motor Power in Viscous Environments

The power output of biological motors, like those driving bacterial flagella, depends on both the rotational speed and the load imposed by the fluid’s viscosity. Adjustments in rotation rate or environmental viscosity affect the mechanical work done against viscous drag. Changes in these parameters alter the efficiency and power consumption of the motor in overcoming resistance.

Stokes’ Law and Sedimentation Velocity

Stokes’ law describes the drag force experienced by spherical objects moving through a viscous fluid at low Reynolds numbers. It relates the sedimentation velocity to the radius of the sphere, the viscosity of the fluid, and the density difference between the object and the fluid. This concept is crucial for understanding why differences in size (and sometimes density) influence the speed at which particles settle under gravity.

Cardiac Output and Flow Velocity

Cardiac output is the volume of blood expelled by the heart per unit time, and it can be related to the mean velocity of blood in a vessel by dividing by the vessel's cross-sectional area. This relationship helps in understanding hemodynamics in circulatory systems, including how changes in flow rate or vessel diameter affect blood velocity, which can in turn influence shear stresses on vessel walls.

Reynolds Number

The Reynolds number is a dimensionless quantity that compares inertial forces to viscous forces in a fluid flow. It provides insight into the flow regime—whether it is laminar or turbulent—and is key to understanding how the size and speed of different organisms (from bacteria to whales) affect their movement through fluid. Low Reynolds numbers indicate viscous-dominated, laminar flows typical for small organisms, while high Reynolds numbers indicate significant inertial effects and the possibility of turbulent flow in larger systems.

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