At low Reynolds number, viscous forces overwhelm inertia and the Navier–Stokes equations reduce to the linear, time-independent Stokes equations. This lab evaluates the exact closed-form solution for uniform flow past a rigid sphere and uses it in three ways:
u_x = U[1 − (3a)/(4r)(1+x²/r²) − a³/(4r³)(1−3x²/r²)]
u_y = U[−(3a)/(4r)(xy/r²) + a³/(4r³)(3xy/r²)]
- Field mode — hundreds of tracers are advected by this exact field; the streamlines are fore-aft symmetric and independent of Re, the signature of creeping flow.
- Reversibility — dye is deformed by an exactly invertible Couette shear map; because Stokes flow has no inertia to scramble the pattern, reversing the shear snaps every particle back to its start.
- Swimmer — a two-link swimmer's thrust is the shape-space loop area ∮a·db; a reciprocal stroke (phase shift 0) traces a zero-area loop, so it cannot swim (the scallop theorem), while a non-reciprocal stroke (90° shift) encloses area and advances.
The drag and settling readouts apply Stokes' law F=6πμaU and the Stokes settling velocity v=2Δρga²/(9μ) (Δρ, g fixed at illustrative model values) to the current radius, viscosity and speed sliders.