This problem concerns the power that an organism of length L must expend to swim underwater

This problem concerns the power Φ that an organism of length L must expend to swim underwater at a constant velocity U. In general, Φ will be affected by ρ and μ of water, in addition to L and U.

(a) If N₁ has Φ in the numerator and ρ in the denominator, what can you conclude from dimensional analysis?

(b) It has been found that, from microbes to whales, the overall rate of metabolic energy production varies as body weight to the 3⁄4 power. Observed originally for mammals ranging from mice (20 g) to cattle (600 kg) (Kleiber, 1947), this is called Kleiber's law. When combined with certain assumptions, Kleiber's law suggests that Φ = CL9⁄4, where C is constant within an aquatic family. What are the necessary assumptions?

(c) Suppose that a modern fish with L = 0.5 m and U = 1 m/s had a prehistoric relative with L = 5 m. Show that Re for the modern fish is already quite large. For many objects at such large Re, the drag force F_D (the fluid-dynamic force that opposes the motion) is approximately F_D = K₁L²ρU², where K1 is a dimensionless coefficient that depends only on the shape of the object. Assume that this holds at any Re greater than or equal to that of the modern fish. Given that Φ ∝ F_DU, what might have been the swimming speed of the now-extinct giant?

(d) Suppose that a bacterium with L = 1 μm propels itself at U = 20 μm/s by movement of flagella. For Re ≪ 1, the drag force is F_D = K₂μUL, where K₂ is another dimensionless constant that depends only on the shape of the object. If there were a geometrically similar bacterium with L = 2 μm, how fast would you expect it to swim?