Hydrodynamic Ratchet for Tracer Transport in a Soft Microchannel: A Detailed Analysis

TL;DR

The study addresses surface-driven tracer transport in soft microchannels by developing a perturbative, low-$ ext{Re}$ analysis to obtain a self-consistent, inversion-symmetry-broken velocity field in a cylindrical, undulating channel. It couples this velocity field to an overdamped Langevin tracer model to demonstrate hydrodynamic ratcheting, revealing a net axial drift of approximately $0.15~oldsymbol{ m bc m/s}$ for micrometer-scale particles under room-temperature water and specific wall wavelengths. The work highlights how wall undulations, finite slip, and channel geometry govern ratcheting through analyzable $u_0$, $u_1$, $v_1$ components and a corresponding radius correction $R_1(z,t)$, with clear implications for microfluidic transport and filtration. Limitations include neglected hydrodynamic interactions with walls and boundary-condition complexity, suggesting avenues for extension toward more complete BBO-type unsteady-flow treatments and multi-particle dynamics.

Abstract

Understanding surface-driven transport is of paramount importance from the perspective of biological applications and the synthesis of microfluidic devices. In this work, we develop an analysis of a local inversion symmetry broken fluid flow model through an undulating microchannel. Surface undulations of a few tens of Hertz in a soft microchannel keep the fluid flow in a low Reynolds number regime, allowing the advantage of a perturbation analysis of fluid flow. Using this, we develop a detailed analysis of the relationship between the fluid velocity and surface undulations, which is crucial for the subsequent numerical analysis of tracer motion. We used this information to study the dynamics of a tracer particle in the velocity field of an undulating microchannel. We show that the tracer particle can undergo ratcheting (which we call the hydrodynamics ratchet effect) in very specific, physically meaningful circumstances. We observe a ratcheting velocity of $\sim 0.15 \;μ$m/sec for a micrometre-sized particle at room temperature in water when the undulations wavelength is of the order of 1 $μ$m.

Figures (4)

• Figure 1: The schematic diagram of a microchannel with undulating walls that are circularly symmetric around the $z$-axis. The wall geometry is generated by an external forcing, and a pressure gradient is applied along the length of the tube.
• Figure 2: Trajectory of a tracer particle in cylindrical coordinates. The simulation parameters used are: $\omega = 70~\text{rad s}^{-1}, \lambda = 1.0~\mu\text{m}, R_0 = 10~\mu\text{m}, D_0 = 1.0~\mu\text{m}^2\text{ s}^{-1}, v_{\text{slip}} = 1.0~\mu\text{m s}^{-1}, \phi = -\pi/2~\text{rad}.$ (a) Axial coordinate $z$ (in $\mu$m) as a function of time $t$ (in sec), showing the net drift of the particle. (b) Radial coordinate $r$ (in $\mu$m) as a function of time $t$ (in sec). (c) Angular coordinate $\theta$ (in radians) as a function of time $t$ (in sec).
• Figure 3: Variation of average velocity with the frequency of wall undulations ($\langle v \rangle$ is in $\mu\text{m s}^{-1}$ and $\omega$ is in $\text{rad s}^{-1}$). The simulation parameters used are: $R_0 = 10~\mu\text{m},~ v_{\text{slip}} = 1.0~\mu\text{m s}^{-1},~ \lambda = 1.0~\mu\text{m},~ \phi = -\pi/2,~ D_0 = 1.0~\mu\text{m}^2\text{ s}^{-1}$.
• Figure 4: Variation of average velocity with the diffusivity $D_0$ of the tracer particle ($D_0$ is in $\mu\text{m}^2\text{ s}^{-1}$ and $\langle v \rangle$ is in $\mu\text{m s}^{-1}$). The simulation parameters used are: $R_0 = 10~\mu\text{m},~ v_{\text{slip}} = 1.0~\mu\text{m s}^{-1},~ \lambda = 1.0~\mu\text{m},~ \phi = -\pi/2,~ \omega = 70~\text{rad s}^{-1}$.