The bell radius R(t) sweeps between Rmax and Rmin = Rmax(1−strength) on an asymmetric cycle: a fast active power-stroke (duty-cycle share of the period) followed by a slower passive elastic recoil, matching how real scyphomedusae swim — a quick muscular contraction and a longer relaxation.
V(R) = (2/3)πR³ subumbrellar cavity volume
dV/dt = 2πR²·dR/dt rate of water displaced
v_jet = |dV/dt| / A_o A_o = π(k·R)², k≈0.6 (orifice ratio)
F_thrust = ρ·A_o·v_jet² momentum-flux thrust, active only while dV/dt<0
F_drag = ½·ρ·Cₜ·A·v² opposes motion, A = πR_max²
m·dv/dt = F_thrust − F_drag
Thrust exists only while the bell is contracting (water is being expelled through the bell margin) — during the slower refill/relaxation half of the stroke the cavity draws water back in and net thrust drops to zero, so the animal glides on momentum while drag brakes it. That produces the characteristic pulse-glide swimming pattern visible in the speed and thrust strips: sharp acceleration spikes separated by decelerating coasts.
- Effective inertial mass m = ρ·Vmax·(Ca+tissue fraction), with added-mass coefficient Ca≈0.5 for an oblate body accelerating through water (Daniel 1983) — medusae are documented to be dominated by added mass rather than their own (mostly water) tissue mass, which is why bell size affects acceleration so strongly.
- Because F_thrust ∝ R²·(dR/dt)², both a larger bell and a faster/stronger contraction raise thrust quadratically — try the frequency and bell-diameter sliders together.
- Jet (vortex-ring) propulsion is the most metabolically efficient known swimming mode in the animal kingdom, which is what this thrust/drag balance is modelling in simplified 2D form.