Octopus - 2D side view of jet propulsion

What is modelled

An octopus swims by jet propulsion: the muscular mantle squeezes water out through a funnel, the recoil pushes the body the other way, then the mantle refills while the animal glides with its arms trailing. The page models one such pulse and the glide that follows as a real equation of motion, then lets you fire jets, steer the funnel and release an ink cloud.

Equations

• The pulse expels a volume Ve through a funnel of area Af in a half-sine of duration tj, so volume is conserved: U(t) = Umax sin(πt/tj) with Umax = πVe / (2Aftj).
• Thrust is the momentum flux of the jet relative to the body: T = ρ Af U².
• Newton's second law for the body plus the water it drags along: (m + ma) dv/dt = T − ½ ρ Cd S |v| v. The added mass is ma = k ρ V, with k from the potential flow around a prolate spheroid (Lamb): 0.5 for a sphere, 0.21 for a 2:1 body.
• Propulsive efficiency η = ∫T v dt / ∫½ ρ Af U³ dt (useful work over the jet's mechanical power) and cost of transport = jet energy per kilogram per metre.
• Reynolds number Re = v L / ν shows why the body coasts: at body-length speeds of order one metre per second it is of order 10⁵.
• The ink cloud spreads by eddy diffusion: σ² = σ₀² + 2Dt.

Checks run in the page

• Without drag, the speed after one pulse equals the impulse divided by the mass plus added mass.
• Under a steady thrust the speed approaches the terminal speed from the thrust-drag balance.
• The volume pushed out equals the integral of the jet speed times the funnel area.
• The added-mass coefficient of a sphere found by integrating the kinetic energy of the flow equals one half of the displaced mass; the spheroid formula reproduces Lamb's 0.21.
• The full model with drag converges when the time step is halved.

Honest simplifications

• Motion is a single point mass with a constant added mass and a quadratic drag law (the drag coefficient is a rough function of arm spread, not a measurement); the body is a spheroid 0.55 of its length wide with a mantle cavity of 30 % of its volume. Real octopuses change shape during a jet, and the jet-speed profile is a clean half-sine.
• Refilling the mantle costs momentum in a real animal; here the refill is treated as momentum-neutral. The 2D view has no steering; the 3D page lets the funnel swivel.
• Numbers such as the peak speed and the efficiency are outputs of these assumptions, not data about a particular species; they are plausible for an animal of this size but have not been fitted to measurements. Buoyancy is taken as neutral.
• The ink cloud is a Gaussian blob with a fixed eddy diffusivity; real ink forms a structured mass that depends on the squirt. Skin colour change is a simple tint, not a model of chromatophores.

What to try

• Choose Escape jet: a short strong pulse, higher peak speed, lower efficiency.
• Widen the funnel at a fixed volume and watch the jet slow down and the efficiency rise (third graph).
• Spread the arms and compare the distance of one cycle with the streamlined body.
• Compare the two lower graphs: distance per jet against funnel diameter and against arm spread.

Independent educational model; not data from a laboratory study.