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 Wo2 → 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, ψL = −[m(5m+4)am2/(128(m+2)(2m+1))] r2m(r2−1)2sin 2mθ, 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 ψ2 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 ψ2, with opposite sign. The no-slip cell centers lie at r2 = 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 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 ψ2 with quadratic mesh convergence.
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BibTeX
@misc{liu2026closedform,
title = {{A closed-form solution for streaming and Lagrangian transport in a deforming circular cavity}},
author = {Zijian Liu and Julian Olszewski and Yang Lin and Yuan Gao and Mengren Wu and Jie Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24368},
eprint = {2609.24368},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24368},
}
RIS
TY - UNPB
TI - A closed-form solution for streaming and Lagrangian transport in a deforming circular cavity
AU - Zijian Liu
AU - Julian Olszewski
AU - Yang Lin
AU - Yuan Gao
AU - Mengren Wu
AU - Jie Xu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.24368
UR - https://arxiv.org/abs/2609.24368
AB - 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 Wo^2 → 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, ψ_L = −[m(5m+4)a_m^2/(128(m+2)(2m+1))] r^2m(r^2−1)^2sin 2mθ, 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 ψ_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 ψ_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 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 ψ_2 with quadratic mesh convergence.
ER -