Drag on the body
D = ½ ρ CD Sw U² · kswim · fwave, CD = Cf (1 + 1.5 (d/L)3/2 + 7 (d/L)3) · fapp
Cf = 0.075 / (log10Re − 2)², Re = U L / ν, Sw ≈ 0.78 π d L
Skin friction of a turbulent boundary layer (ITTC-1957 line) times a Hoerner shape factor for a streamlined body of fineness L/d. fapp adds the drag of legs, ears and a blunt snout for the early forms (up to 2.4x). kswim (slider, default 2) is the extra drag of a body that is flexing; measured values are debated, 1 to 3. fwave is a stylised surface-wave penalty that peaks near Froude number 0.45 and dies off within about two body diameters of depth (after Hertel): it is why dolphins leap clear of the water at speed.
Paddles, then flukes
Drag-based paddles (Pakicetus, hind feet of Ambulocetus and Rodhocetus): a limb stroked backward at speed vp pushes ½ρ Cd A (vp − U)² but only works when the foot moves faster than the body, and the return stroke costs drag. Efficiency is capped near U/vp, typically 15 to 30 %.
Oscillating foil (tail or fluke): Theodorsen's linear unsteady aerofoil theory for heave h = h0 sin ωt and a feathered pitch. With reduced frequency k = π f c / U and Theodorsen function C(k) = F + iG:
L = πρb²(h¨ + Uα̇ − baα¨) + 2πρU b C(k) [ḣ + Uα + b(½−a)α̇]
S = 2πρb [ C(k)(ḣ + Uα + b(½−a)α̇) − (b/2)α̇ ]², T = S − Lα, P = L ḣ − M α̇
(b = half-chord, a = pitch axis, S = leading-edge suction.) For pure heave this reduces to the classic η = (F²+G²)/F: 100 % at k → 0 and 50 % at k → ∞ (Knoller-Betz). Real flukes pitch so that the peak angle of attack stays below stall (slider, default 20°); the model chooses the pitch amplitude θ0 = π St − tan αmax that does so. Finite aspect ratio multiplies forces by AR/(AR+2) and a profile drag Cd0 = 0.012 is charged to the fluke.
Strouhal number
St = f A / U (A = peak-to-peak tail excursion)
Swimming and flying animals cruise between St = 0.2 and 0.4 (Taylor, Nudds & Thomas 2003). In this model thrust balance fixes the beat frequency for a given speed, so St is an output; the flume test shows the St that would maximise efficiency for that fluke. Where the two differ, the animal is trading efficiency for something else (agility, burst power, a limited stroke amplitude).
Cost of transport, oxygen and diving
Pmet = BMR + Phydro / (ηmuscle), BMR = 3.39 M0.75 × kB W, COT = Pmet / (M U)
O2 = Vblood Hb 1.34 · 0.65 + M fmusc [Mb] 1.34 · 0.8 + lung, ADL = O2 / (Pmet / 20.1 J mL-1)
Myoglobin (Mb) and blood volume are what turn a land mammal into a diver: values for extinct animals are extrapolated from the trend in living species (and from the electric charge of fossil-lineage myoglobin, Mirceta et al. 2013) and are illustrative. The aerobic dive limit shown is a model result that shortens when you swim faster.
Weight, buoyancy and size
Fnet = M g − ρ g (M/ρt + Vlung(z)), Vlung(z) = V0 / (1 + z/10.06 m)
Water carries almost all of a whale's weight. On land a skeleton's stress rises about as M1/3 with the same bone proportions, which is why the largest land animals stop near 20 to 70 tonnes; afloat, the 100-tonne blue whale carries only a few percent of that load. Dense (osteosclerotic) bone in Pakicetus and its relatives acts as ballast to keep waders down.
What is simplified, and what is uncertain
- Linear, quasi-2D foil theory with a flat plate, uniform flow and no body-fluke interaction. Real flukes are cambered, flexible and 3-D; the optimum St is broad and sits where the curve is flat.
- All propulsion is a fluke or paddles: body-wave thrust (Lighthill) is only represented through the tail amplitude.
- Masses, fluke sizes, limb proportions, muscle fractions, myoglobin and blood stores of extinct taxa are estimates with large error bars (shown on the species cards). Skin colour and fur cover of fossil forms are guesses.
- The skeletons are stylised reconstructions, not scans; vertebra counts follow the literature but bone shapes are simplified. Basilosaurus's swimming mode and fluke are debated.
- Fossil sites are plotted with modern coordinates; continents have moved.