Allometry - Allometric Scaling in Fluid Locomotion

Allometric Scaling in Fluid Locomotion

The mass/density of organism has a large effect on organisms locomotion through a fluid. For example tiny organisms use flagella and can effectively move through a fluid it is suspended in. Then on the other scale a Blue Whale that is much more massive and dense in comparison with the viscosity of the fluid, compared to a bacterium in the same medium. The way in which the fluid interacts with the external boundaries of the organism is important with locomotion through the fluid. For streamlined swimmers the resistance or drag determines the performance of the organism. This drag or resistance can be seen in two distinct flow patterns. There is Laminar Flow where the fluid is relatively uninterrupted after the organism moves through it. Turbulent flow is the opposite, where the fluid moves roughly around an organisms that creates vortices that absorb energy from the propulsion or momentum of the organism. Scaling also affects locomotion through a fluid because of the energy needed to propel an organism and to keep up velocity through momentum. The rate of oxygen consumption per gram body size decreases consistently with increasing body size. (Knut Schmidt-Nielson 2004)

In general, smaller, more streamlined organisms create laminar flow (R<0.5x106), whereas larger, less streamlined organisms produce turbulent flow (R>2.0×106). Also, increase in velocity (V) increases turbulence, which can be proved using the Reynolds equation. In nature however, organisms such as a 6‘-6” dolphin moving at 15 knots does not have the appropriate Reynolds numbers for laminar flow R=107, but exhibit it in nature. Mr. G.A Steven observed and documented dolphins moving at 15 knots alongside his ship leaving a single trail of light when phosphorescent activity in the sea was high. The factors that contribute are:

• Surface area of the organism and its effect on the fluid in which the organism lives is very important in determining the parameters of locomotion.
• The Velocity of an organism through fluid changes the dynamic of the flow around that organism and as velocity increases the shape of the organism becomes more important for laminar flow.
• Density and viscosity of fluid.
• Length of the organism is factored into the equation because the surface area of just the front 2/3 of the organism has an effect on the drag

The resistance to the motion of an approximately stream-lined solid through a fluid can be expressed by the formula: C_fρ(total surface)V2/2
V = velocity
ρ = density of fluid
C_f = 1.33R-1 (laminar flow) R= Reynolds number

Reynolds number =VL/ν
V = velocity
L = Axial length of organism
ν = kinematic viscosity (viscosity/density)

Notable Reynolds numbers:
R<0.5x106 = Laminar Flow threshold
R>2.0x106 = Turbulent Flow threshold

Scaling also has an effect on the performance of organisms in fluid. This is extremely important for marine mammals and other marine organisms that rely on atmospheric oxygen to survive and carry out respiration. This can affect how fast an organism can propel itself efficiently and more importantly how long it can dive, or how long and how deep an organism can stay underwater. Heart mass and lung volume are important in determining how scaling can affect metabolic function and efficiency. Aquatic mammals, like other mammals, have the same size heart proportional to their bodies.

Mammals have a heart that is about 0.6% of the total body mass across the board from a small mouse to a large Blue Whale. It can be expressed as: Heart Weight =0.006Mb1.0. Where Mb is the body mass of the individual. (Knut Scmidt-Nielson 1997) Lung volume is also directly related to body mass in mammals (slope = 1.02). The lung has a volume of 63 ml for every kg of body mass. In addition, the tidal volume at rest in an individual is 1/10 the lung volume. Also respiration costs with respect to oxygen consumption is scaled in the order of Mb.75. (Knut Scmidt-Nielson 1997) This shows that mammals, regardless of size, have the same size respiratory and cardiovascular systems and it turn have the same amount of blood: About 5.5% of body mass. This means that for a similarly designed marine mammals, the larger the individual the more efficiently they can travel compared to a smaller individual. It takes the same effort to move one body length whether the individual is one meter or ten meters. This can explain why large whales can migrate far distance in the oceans and not stop for rest. It is metabolically less expensive to be larger in body size. (Knut Scmidt-Nielson 1997) This goes for terrestrial and flying animals as well. In fact, for an organism to move any distance, regardless of type from elephants to centipedes, smaller animals consume more oxygen per unit body mass than larger. This metabolic advantage that larger animals have make it possible for larger marine mammals to dive for longer durations of time than their smaller counterparts. The fact that the heart rate is lower means that larger animals can carry more blood, which carries more oxygen. Then in conjuncture with the fact that mammals reparation costs scales in the order of Mb.75 shows how an advantage can be had in having a larger body mass. More simply, a larger whale can hold more oxygen and at the same time demand less metabolically than a smaller whale.

Traveling long distances and deep dives are a combination of good stamina and also moving an efficient speed and in an efficient way to create laminar flow, reducing drag and turbulence. In sea water as the fluid, it traveling long distances in large mammals, such as whales, is facilitated by the fact that they are neutrally buoyant and have their mass completely supported by the density of the sea water. On land animals have to expend a portion of their energy during locomotion to fight the effects of gravity.

It should be mentioned that flying organisms such as birds are also considered moving through a fluid. In scaling birds of similar shape it has also been seen that larger individuals have less metabolic cost per kg than smaller species. This would be expected because it holds true for every other form of animal. Birds also have a variance in wing beat frequency. Even with the compensation of larger wings per unit body mass, larger birds also have a slower wing beat frequency. This allows larger birds to fly at higher altitudes, longer distances, and faster absolute speeds than smaller birds. Because of the dynamics of lift-based locomotion and the fluid dynamics, birds have a U-shaped curve for metabolic cost and velocity. Because flight, in air as the fluid, is metabolically more costly at the lowest and the highest velocities. On the other end, small organisms such as insects can make gain advantage from the viscosity of the fluid (air) that they are moving in. A wing-beat timed perfectly can effectively uptake energy from the previous stroke. (Dickinson 2000) This form of wake capture allows an organism to recycle energy from the fluid or vortices within that fluid created by the organism itself. This same sort of wake capture occurs in aquatic organisms as well, and for organisms of all sizes. This dynamic of fluid locomotion allows smaller organisms gain advantage because the effect on them from the fluid is much greater because of there relatively smaller size.