Left: radial velocity profile u_z(r, t) in a circular pipe under prescribed flow Q(t) = Q₀ + Q₁ cos(ωt). Right: the imposed Q(t); the dot marks the current phase. As α → 0 the profile collapses to Poiseuille's parabola; as α ≫ 1 (aortic systole) a thin Stokes boundary layer develops and the core flattens. When Q₁/Q₀ > 1 the flow reverses (backflow) during part of the cycle — try it at low α to see the parabola flip.