Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 73 / 1 / b / iv

Put ${\mathbf{W}}_2={\mathbf{F}}_2/\zeta$. To the needed accuracy the Faxén translation law gives ${\mathbf{U}}_2={\mathbf{W}}_2+\mathsf{M}\mathbf{V}+O({ϵ}^3|\mathbf{V}|)$. Enforcing its prescribed velocity therefore givesHere the force on sphere 1 remains prescribed; no extra second-order inversion term is needed in ${\mathbf{F}}_2$. Substitute this force into the incident field at sphere 1:The negative second-order term reflects the reduced force required on sphere 2 because sphere 1 already entrains it.