A closed-form solution for streaming and Lagrangian transport in a deforming circular cavity

Abstract

Streaming from a deforming cavity wall serves micromixing, pumping and particle handling. We solve it in closed form in a two-dimensional circular cavity, for any azimuthal wall mode $m$, in the viscous-dominated limit $\mathrm{Wo}^2 \to 0$, where the Stokes layer spans the cavity. A biharmonic inversion against the Reynolds-stress forcing, corrected by the second-order slip a moving wall imposes, gives the Lagrangian mean a tracer follows as an elementary streamfunction for a deforming no-slip wall, $ψ_L = -[m(5m+4)a_m^2/(128(m+2)(2m+1))]\,r^{2m}(r^2-1)^2\sin 2mθ$, and a second for a shear-free interface. The factor $(5m+4)/(m+2)$ relating it to the auxiliary reference-boundary solution $ψ_2$ is universal across the prescribed-velocity family; at $m=2$ the physical Eulerian mean peaks an order of magnitude above $ψ_2$ and with opposite sign. At every $m$ the no-slip cell centers lie at $r^2 = m/(m+2)$, and at large $m$ the peak streamfunction falls as $m^{-2}$ and the peak speed as $m^{-1}$. The ranking over $m$ is set by the wall kinematics: an externally driven wall is largest at $m=1$, where rigidly translating the same circle drives nothing; an inextensible shell peaks at $m=3$. The inversion extends to mode superpositions without degenerating. At finite $\mathrm{Wo}$ the first order stays closed form in Bessel functions and the second reduces to quadrature; the construction recovers Rayleigh's coefficient $-3m/8$ on a separate tangentially driven boundary problem. An independent finite-element solver, written with the closed form withheld, reproduces $ψ_2$ with second-order convergence.