Relations between morphology, buoyancy and energetics of requiem sharks

Sharks have a distinctive shape that remained practically unchanged through hundreds of millions of years of evolution. Nonetheless, there are variations of this shape that vary between and within species. We attempt to explain these variations by examining the partial derivatives of the cost of transport of a generic shark with respect to buoyancy, span and chord of its pectoral fins, length, girth and body temperature. Our analysis predicts an intricate relation between these parameters, suggesting that ectothermic species residing in cooler temperatures must either have longer pectoral fins and/or be more buoyant in order to maintain swimming performance. It also suggests that, in general, the buoyancy must increase with size, and therefore, there must be ontogenetic changes within a species, with individuals getting more buoyant as they grow. Pelagic species seem to have near optimally sized fins (which minimize the cost of transport), but the majority of reef sharks could have reduced the cost of transport by increasing the size of their fins. The fact that they do not implies negative selection, probably owing to decreased manoeuvrability in confined spaces (e.g. foraging on a reef)

Longitudinal Stability Criteria for a Propeller-Driven Aircraft

The Routh criterion is used to assess longitudinal dynamic stability of a propeller-driven aircraft. Under a few plausible assumptions on possible ranges of the pertinent stability derivatives, it reduces to a pair of simple conditions imposing a traditional aft limit (the forward of the maneuver and the neutral-speed-stability points) on the center-of-gravity position and an upper limit on the longitudinal moment of inertia. It is demonstrated that most aircraft have sufficiently small inertia to remain stable as long as their center-of-gravity is properly placed. At the same time, sailplane-like aircraft (as, e.g., long endurance UAVs), with an engine installed at the rear extremity of the aircraft, may have sufficiently high inertia to become unstable regardless of their center-of-gravity placement

The reference frame and the notation.

<p>The reference frame and the notation.</p

Simulated snapshots of the caudal fin during swimming.

<p>Direction of motion is from right to left. A dot marks the caudal peduncle. Conditions are those of case 1 in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0163517#pone.0163517.t001" target="_blank">Table 1</a>. Particular data is shown to the left of the respective figures. To emulate the fish motion, it was assumed that the length of the fish <i>l</i> equals 2<i>πl</i>_<i>t</i>–under this assumption, the stride length, , is one body length at .</p

Propulsion efficiency as a function of reduced stiffness at three values of (shown next to respective lines).

<p>Conditions are those of cases 2 and 3 in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0163517#pone.0163517.t001" target="_blank">Table 1</a>. In (a), <i>s</i>_<i>t</i> is constant; in (b), <i>s</i>₂ is constant. The thick lines are the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0163517#pone.0163517.g002" target="_blank">Fig 2</a>. Short lines adjacent to the left and right margins mark asymptotic values from Appendices E and F.</p

Supplementary 1: Underlying data from Relations between morphology, buoyancy and energetics of requiem sharks

Sharks have a distinctive shape that remained practically unchanged through hundreds million years of evolution. Nonetheless, there are variations of this shape that vary between and within species. We attempt to explain these variations by examining the partial derivatives of the cost of transport of a generic shark with respect to buoyancy, span and chord of its pectoral fins, length, girth and body temperature. Our analysis predicts an intricate relation between these parameters, suggesting that ectothermic species residing in cooler temperatures must either have longer pectoral fins and/or be more buoyant in order to maintain swimming performance. It also suggests that, in general, the buoyancy must increase with size, and therefore, there must be ontogenetic changes within a species, with individuals getting more buoyant as they grow. Pelagic species seem to have near optimally sized fins (which minimize the cost of transport), but the majority of reef sharks could have reduced the cost of transport by increasing the size of their fins. The fact that they do not imply negative selection, likely, owing to decreased manoeuvrability in confined spaces (e.g. foraging on a reef)

Supplementary 2: Wind tunnel experiments from Relations between morphology, buoyancy and energetics of requiem sharks

Sharks have a distinctive shape that remained practically unchanged through hundreds million years of evolution. Nonetheless, there are variations of this shape that vary between and within species. We attempt to explain these variations by examining the partial derivatives of the cost of transport of a generic shark with respect to buoyancy, span and chord of its pectoral fins, length, girth and body temperature. Our analysis predicts an intricate relation between these parameters, suggesting that ectothermic species residing in cooler temperatures must either have longer pectoral fins and/or be more buoyant in order to maintain swimming performance. It also suggests that, in general, the buoyancy must increase with size, and therefore, there must be ontogenetic changes within a species, with individuals getting more buoyant as they grow. Pelagic species seem to have near optimally sized fins (which minimize the cost of transport), but the majority of reef sharks could have reduced the cost of transport by increasing the size of their fins. The fact that they do not imply negative selection, likely, owing to decreased manoeuvrability in confined spaces (e.g. foraging on a reef)