Biophysics, Fluids & Geoscience
Neural excitation, porous flow, atmospheric and ocean dynamics, and solid-Earth wave cartoons
Hodgkin–Huxley Action Potential
Squid-axon HH ODEs: membrane voltage V(t) and gating (m, h, n); step stimulus and Na⁺/K⁺ conductance traces.
Cable Equation on an Axon
1D passive cable ∂V/∂t = D∂²V/∂x² − (V−V_r)/τ + I; compare uniform fiber vs myelinated (higher D in internodes).
Coupled FitzHugh–Nagumo Neurons
Two excitable FHN units with diffusive coupling: phase plots, time traces, and synchronization vs coupling strength.
Nernst Potentials & Membrane
Nernst E for K⁺ and Na⁺ from inside/outside concentrations; optional Goldman–Hodgkin–Katz resting estimate.
Kelvin–Helmholtz Shear Instability
Two parallel flows with slip U: Fourier-mode growth rate sketch and animated interface billows (linear toy model).
Taylor–Couette Flow (Criterion)
Gap d, cylinders Ω_i, Ω_o, kinematic ν: Taylor number Ta and critical Ta_c for toroidal rolls — cartoon + formula.
Darcy Flow & Penetration Depth
1D head h(z,t): ∂h/∂t = K ∂²h/∂z²; recharge at surface, fixed head at depth — relaxation time and diffusion length.
Quadrupole / Penning-Style Trap
2D motion in a hyperbolic electric potential Φ ∝ x²−y² with axial magnetic field B: epicycloid-like bounded orbits.
Layered Medium: P/S Fronts & Hodograph
P and S in a two-layer straight-ray toy: reflected legs and optional head branches when the lower layer is faster; hodograph plus minimum-time first arrivals (no conversions or amplitudes).
Geomagnetic Reversals (Toy Dynamo)
Low-order α–Ω cartoon: axial dipole B(t) from coupled amplitudes with symmetry breaking — stochastic flips, not MHD.
Greenhouse: One vs Two Slabs
Compare gray-atmosphere surface temperature for one IR slab vs two coupled slabs; albedo α and absorptivities.
Ekman Spiral in the Ocean
Wind stress, Coriolis f, eddy viscosity A_z: classical Ekman spiral of horizontal velocity with depth; deflection angle.