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Since these two animals can maintain a cruising speed of ca. 7.5 m s-1 for long periods, one may assume that they are then in a physiological steady state. The question is: how is it possible for two species with such different sizes to cruise at the same speed?

To answer this question, one must first consider the drag (F) on a streamlined body moving in a fluid. The equation is similar to eq. 3.8, except that the drag coefficient Cx is replaced here by the friction coefficient Cf

where p is the density of the fluid, V the velocity of the body, and A its total surface area. For laminar flow, Cf ~ 1.33 Re~1/2 whereas, for turbulent flow, Cf ~ 0.455 (logj0 Re)-2 58, Re being the Reynolds number. Low values of Re correspond to laminar flow, where resistance to motion is relatively weak, whereas high values of Re are associated with turbulent flow, which creates stronger resistance to motion. Normally, for a streamlined body, the flow is laminar over the front portion only and is turbulent towards the back.

The power developed by the muscles of moving cetaceans is calculated in three steps.

• Calculation of Re, for the animal under study:

Cfbeing computed from Re, using the equation for either laminar or turbulent flow.

• Calculation of power (P) developed during motion:

For the purpose of the calculation, consider (1) a dolphin with a length of 2 m, weighing 80 kg, whose surface area is 1.75 m2 and (2) a whale 25 m long, with a mass of 100 t and surface area of 250 m2.

(1) The value of Re for a dolphin moving at 7.5 m s 1 is of the order of 107, which seems to indicate highly turbulent flow. In the case of laminar flow,

and, for turbulent flow,

Cf = 0.455 (log10l07)-258 = 3 x 10-3 The drag (F) corresponding to these two flow regimes is:

F (laminar) = 0.5 (4.2 x 10-4) (1028 kg m-3) (7.5 m s-1)2 (1.75 m2) = 22 N F (turbulent) = 0.5 (3 x 10-3) (1028 kg m-3) (7.5 m s-1)2 (1.75 m2) = 155 N

The power (P = F x 7.5 m s-1) that a dolphin should develop, if its motion resulted in perfectly laminar flow, would be 165 W and, for turbulent flow, 1165 W. Since the size of a dolphin is of the same order as that of a man, it is reasonable to assume that the power it can develop under normal conditions is not higher than that of an athlete, i.e. a maximum power of 260 W. It follows that the flow must be laminar for the 9/10 front portion of the dolphin's body, with the rear 1/10 being perhaps turbulent. This conclusion is consistent with observations made in nature on dolphins. It is assumed that the absence of turbulence along the front part of the dolphin's body comes from the fact that the animal only uses its rear section for propulsion.

(2) The blue whale also swims at 7.5 m s-1, its Re being ca. 12.5 x 107 which corresponds to a turbulent flow regime. A laminar flow would lead to a value

and a turbulent flow to

Cf = 0.455 (log1012.5 x 107)-2 58= 2.1 x 10-3 The corresponding drag (F) would be:

F (laminar) = 0.5 (1.2 x 10-4) (1028 kg m-3) (7.5 m s-1)2 (250 m2) = 745 N F (turbulent) = 0.5 (2.1 x 10-3) (1028 kg m-3) (7.5 m s-1)2 (250 m2) = 13 kN. The power a whale should develop, if its motion at 7.5 m s-1 was accompanied by laminar flow, would be 5.6 kW and, in the case of turbulent flow, 100 kW. The maximum power developed by a 80 kg dolphin was estimated to be 260 W so that, if the maximum power of an animal was proportional to its mass, a 105 kg whale should be able to develop 325 kW. One should, however, take into account the fact that the available energy depends on blood flow. Since cardiac rate is proportional to (mass)-0.27, the heart of a whale beats at a rate (100/0.08)-027 = 1/7 that of a dolphin. The maximum power of a whale is thus ca. 1/7 of 325 kW, i.e. 46.5 kW. This leads to the conclusion that laminar flow takes place along the 2/3 front portion of the animal and that only the 1/3 rear part can sustain turbulent flow.

Ecological application 3.2b

A second study, taken from the same paper as the previous application (Hill, 1950), deals with land animals. It has been observed that several terrestrial mammals run more or less at the same speed and jump approximately the same height, even if their sizes are very different. Table 3.4 gives some approximate maximal values. The question is to explain the small differences observed between the performances of animals with such different sizes.

Table 3.4

Performances (maximal values) of five mammals.

Running speed (m s 1)

Height of jump (m)

Man Horse

Greyhound (25 kg)

Hare

Deer

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