Conduction through the blubber (cylindrical shell)
The body is a cylinder of length L and radius rs = sqrt(M / (rho pi L)); the blubber occupies rc < r < rs. Steady Fourier conduction gives Q = 2 pi k L (Tcore - Tskin) / ln(rs/rc), so the thermal resistance is R = ln(rs/rc) / (2 pi k L) with k about 0.2 W/m/K for blubber. Doubling thickness does less than doubling R because the log grows slowly for thin layers and the area grows with radius.
Boundary film and Newton cooling
Heat leaves the skin as Q = h A (Tskin - Tambient). Here h comes from the Churchill-Bernstein cross-flow correlation Nu = 0.3 + 0.62 Re^1/2 Pr^1/3 [1 + (0.4/Pr)^2/3]^-1/4 [1 + (Re/282000)^5/8]^4/5, h = Nu kfluid / D. Water conducts about 25 times better than air (0.57 vs 0.024 W/m/K) and carries about 3,500 times more heat per volume, so the same skin loses heat far faster in water. Total resistance is the series sum Rblubber + Router layer + 1/(h A).
Heat made and the lower critical temperature
Basal heat production uses Kleiber's law, P = 3.4 M^0.75 W, multiplied by a species factor (marine mammals run roughly 1.4 to 2 times the terrestrial value, approximate) and by activity. The lower critical temperature solves P = (Tcore - Tw) G with G the total conductance of the fully constricted animal. Below it the animal must shiver or burn more fuel.
Skin vasomotor control and flippers
A proportional controller moves a vasodilation state d from 0 to 1 as the core rises above about 37 C. Dilation raises the blubber's effective conductivity (blood bypasses the layer) and the flippers' blood flow. A counter-current exchanger returns most of the arterial heat to the veins before the flipper: heat to water is Gfl = Gmax (0.25 + 0.75 d)(1 - 0.9 (1 - d)) in this model (the 0.9 retention is a stated simplification).
Diving reflex
While diving the model forces peripheral vasoconstriction, lowers heart rate (bradycardia, species factor) and cuts metabolic rate by a species fraction (approximate). Oxygen use is P / 20.1 J per ml against a store of ml per kg times mass; the aerobic dive limit is store divided by use.
Surface-area-to-volume scaling
For geometrically similar animals SA/V falls as M^-1/3, and heat loss per kilogram as M^-2/3 (loss grows like length, mass like length cubed), while basal production per kilogram falls only as M^-1/4. Large seals therefore tolerate much colder water, small pups cannot.
Numerics and assumptions
The shell is solved with a finite-volume radial scheme, backward Euler in time (unconditionally stable), coupled to a lumped core. Verified against the analytic shell solution (Experiment A). Simplifications: cylinder geometry (no end caps), uniform core temperature, constant k, fixed 12 percent flipper area, no wet-fur or haul-out contact conduction, no seasonal blubber change, no solar heating. Species data are approximate and marked as such.