变形圆形空腔中流动与拉格朗日输运的闭式解
A closed-form solution for streaming and Lagrangian transport in a deforming circular cavity
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中文总结 AI 辅助
针对变形圆形空腔中的声流，推导出任意壁模式的闭式拉格朗日流函数，揭示涡胞位置与峰值标度，并通过有限元验证其收敛性。
中文摘要 AI 辅助
变形空腔壁产生的声流可用于微混合、泵送和颗粒操控。我们在二维圆形空腔中，针对任意方位角壁模式 $m$，在黏性主导极限 $\mathrm{Wo}^2 \to 0$（此时斯托克斯层跨越整个空腔）下，给出了该声流的闭式解。通过对雷诺应力强迫项进行双调和求逆，并修正移动壁面施加的二阶滑移，我们得到了示踪粒子所遵循的拉格朗日平均流，其以基本流函数形式表示：对于变形无滑移壁面，$\psi_L = -[m(5m+4)a_m^2/(128(m+2)(2m+1))]\\,r^{2m}(r^2-1)^2\sin 2m\theta$；对于无剪切界面，则有另一个流函数。将 $\psi_L$ 与辅助参考边界解 $\psi_2$ 关联的因子 $(5m+4)/(m+2)$ 在给定速度族中具有普适性；当 $m=2$ 时，物理欧拉平均流峰值比 $\psi_2$ 高一个数量级，且符号相反。对于每个 $m$，无滑移涡胞中心位于 $r^2 = m/(m+2)$；当 $m$ 较大时，峰值流函数按 $m^{-2}$ 衰减，峰值速度按 $m^{-1}$ 衰减。不同 $m$ 的排序由壁面运动学决定：外部驱动壁在 $m=1$ 时最大，此时刚性平移同一圆不产生任何流动；不可伸展壳在 $m=3$ 时达到峰值。该求逆方法可推广至模式叠加而不退化。在有限 $\mathrm{Wo}$ 下，一阶解仍以贝塞尔函数闭式表示，二阶解简化为求积；该构造在独立的切向驱动边界问题上恢复了瑞利系数 $-3m/8$。一个独立的有限元求解器（编写时未使用闭式解）以二阶收敛精度重现了 $\psi_2$。
英文摘要
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, $ψ_L = -[m(5m+4)a_m^2/(128(m+2)(2m+1))]\,r^{2m}(r^2-1)^2\sin 2mθ$, 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 $\mathrm{Wo}$ 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 $ψ_2$ with quadratic mesh convergence.
发表机构
- University of Illinois Chicago（伊利诺伊大学芝加哥分校）
- University of Rhode Island（罗德岛大学）
- University of Memphis（孟菲斯大学）
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
