 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-73/1/b/iv/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Put

W_{2} = F_{2} / ζ

. To the needed accuracy the Faxén translation law gives

U_{2} = W_{2} + M V + O (ϵ^{3} ∣ V ∣)

. Enforcing its prescribed velocity therefore gives

F_{2} = 6 π μ a (I - M) V + O (6 π μ a ϵ^{3} ∣ V ∣) .

Here the force on sphere 1 remains prescribed; no extra second-order inversion term is needed in

F_{2}

. Substitute this force into the incident field at sphere 1:

U_{1} = (I + M - M^{2}) V + O (ϵ^{3} ∣ V ∣), M^{2} = \frac{9 a ^{2}}{16 R ^{2}} (I + 3 nn) .

The negative second-order term reflects the reduced force required on sphere 2 because sphere 1 already entrains it.

 New to topics? Read the docs here!