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 $m$, as ${\mathrm{Wo}}^2\to 0$. A biharmonic inversion against the Reynolds stress, corrected by the second-order slip a moving wall imposes, gives the Lagrangian mean a tracer follows for a deforming no-slip wall, ${\psi }_L=-[m(5m+4)a_m^2/(128(m+2)(2m+1))]r^{2m}(r^2-1{)}^2\sin 2m\theta$, with a companion form for a shear-free interface. For a single mode the factor $(5m+4)/(m+2)$ relating it to the auxiliary solution ${\psi }_2$ is the same for every member of the co-phased prescribed-velocity family; at $m=2$ the physical Eulerian mean peaks an order of magnitude above ${\psi }_2$, with opposite sign. The no-slip cell centers lie at $r^2=m/(m+2)$, and at large $m$ the peak streamfunction falls as $m^{-2}$, the peak speed as $m^{-1}$. At fixed radial wall-velocity amplitude the ranking over $m$ follows the wall kinematics: an externally driven wall peaks at $m=1$, a wall with zero first-order surface strain at $m=3$. Mode superpositions invert without degenerating, each harmonic carrying its own correction. At finite $\mathrm{Wo}$ the first-order field stays closed form in Bessel functions and the mean flow reduces to quadrature; the construction approaches the $m=2$ Rayleigh limit on a separate tangentially driven boundary problem. An independent finite-element solver, with the closed form withheld, reproduces ${\psi }_2$ with quadratic mesh convergence.

24 pages, 2 figures, 3 tables. v2: the (5m+4)/(m+2) relation is stated for a single mode; the Wo^-1 decay law applies to the reference-boundary field, and the Lagrangian mean decays as Wo^-2; adds a moving-domain Navier-Stokes comparison, an appendix and references