 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-77/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

sin θ cos θ = θ + O (ϵ^{3})

and

\overset{x}{˙} = - ϵ aω sin ω t

. Since

⟨ θ \overset{x}{˙} ⟩ = - ϵ^{2} aω ⟨ cos (ω t + ϕ) sin ω t ⟩ = \frac{ϵ ^{2} aω}{2} sin ϕ,

the mean transverse force from a rocking rod, on the fluid in the tilt convention of part (a), is

⟨ F_{y} ⟩ = - \frac{1}{2} (c_{⊥} - c_{∥}) L ϵ^{2} aω sin ϕ + O (ϵ^{4}) .

The parallel and perpendicular drag coefficients of a slender filament are per unit length here. Their difference is essential: isotropic local drag gives zero transverse force. The hydrodynamic force on the rod is the negative of the displayed answer.

 New to topics? Read the docs here!