Controlled particle displacement by hydrodynamic obstacle interaction in non-inertial flows

TL;DR

The paper demonstrates that in zero-Reynolds-number flows, purely hydrodynamic interactions with fore-aft symmetry-breaking obstacles can cause lasting deflections of force-free microparticles. By developing a comprehensive 2-D Stokes framework with wall-corrected velocity components, multiple near-wall expansions, and an analytical treatment of the dive phase, the authors derive scaling laws for the maximum net displacement $\Delta\psi_{\max}$, showing it scales as $\sim a_p^3$ with a leading $\sin 2\alpha$ dependence and a strong $\beta$-dependent prefactor for eccentric obstacles. The work further provides analytical and numerical validation, compares hydrodynamic displacement to short-range roughness effects, and identifies conditions under which particles may approach walls closely enough to stick due to attractive forces. These results offer rigorous, geometry-driven guidelines for hydrodynamic particle manipulation and filtering in microfluidic devices, and suggest design principles for DLD-like separation using asymmetric obstacles rather than relying solely on inertia or contact interactions.

Abstract

Systematic deflection of microparticles off of initial streamlines is a fundamental task in microfluidics, aiming at applications including sorting, accumulation, or capture of the transported particles. In a large class of setups, including Deterministic Lateral Displacement and porous media filtering, particles in non-inertial (Stokes) flows are deflected by an array of obstacles. We show that net deflection of force-free particles passing an obstacle in Stokes flow is possible solely by hydrodynamic interactions if the flow and obstacle geometry break fore-aft symmetries. The net deflection is maximal for certain initial conditions and we analytically describe its scaling with particle size, obstacle shape, and flow geometry, confirmed by direct trajectory simulations. For realistic parameters, separation by particle size is comparable to what is found assuming contact (roughness) interactions. Our approach also makes systematic predictions on when short-range attractive forces lead to particle capture or sticking. In separating hydrodynamic effects on particle motion strictly from contact interactions, we provide novel, rigorous guidelines for elementary microfluidic particle manipulation and filtering.

Figures (10)

• Figure 1: (a) Schematic of a small spherical particle of radius $a_p$ located at $\boldsymbol{x}_p=(x_p,y_p)$ in the vicinity of an obstacle boundary of radius of curvature $R(\boldsymbol{x}$). The particle is immersed in a density-matched Stokes background flow $\boldsymbol{u}(\boldsymbol{x})$. (b) sketches of fore-aft symmetric obstacle-flow geometries with a circular cylinder or a symmetrically placed elliptic cylinder. This fore-aft symmetry breaks in (c), where the flow has non-trivial angle of attack $\alpha$. Also sketched is the elliptic coordinate system $(\xi,\eta)$. (d) An example of wall-normal particle velocity as a function of the gap coordinate $\Delta$ at an angular position $\eta=20^\degree$ using variable expansion modeling (cf. \ref{['eq vpVE']}) with $\Delta_E=3$ showing a smooth transition from particle expansion for large $\Delta$ to the wall expansion (inset) for small $\Delta$.
• Figure 2: (a) Fore-aft asymmetric Stokes flow (dashed blue streamline contours from \ref{['eq psiB']}) impacting on an elliptic obstacle ($\beta=1/2$) under an inclination $\alpha=30\degree$. $W_\perp$ is positive in the purple shaded zone and is negative in the orange shaded zone indicating particle repulsion from the obstacle and attraction towards the obstacle, respectively. The separating streamlines $\psi=0$ are indicated in dashed brown. (b) The sign changes of the quantity $\partial_{\perp} u_{\perp}$ reflect, to leading order, those of $W_{\perp}$ (shading). This quantity is measured along the green dashed line in (a), locations at a distance $\Delta = 1$, here for $a_p=0.1$, where $\partial_{\perp} u_{\perp}$ dominates the corrections of particle motion. (c) The background flow curvature $-\kappa=-\partial^2_{\perp}u_{\perp}$ evaluated at the obstacle wall determines normal particle motion in the wall expansion model \ref{['eq wall xpnsn']} for $\Delta\ll 1$. The upstream zero of this quantity, $\eta_c$, indicates a point of closest approach to the obstacle as discussed in section §\ref{['sec closest approach']}.
• Figure 3: (a) Two computed trajectories (orange and red) of an $a_p=0.1$ particle above an obstacle ($\beta=0.5$) and $\alpha=30\degree$ inclined Stokes flow ($\psi$ isolines in dashed blue). Red shading indicates the exclusion zone for the hard-sphere particle center. (b) Magnified downstream view shows that the final streamline the particle asymptotes to is lower than its initial streamline ($\psi_f<\psi_i$). (c) Particle-wall gap $\Delta$ as a function of travel time. The orange trajectory stays far from the wall ($\Delta>1$), while the red trajectory remains very close ($\Delta\ll1$). The inset shows good agreement around the minimum with computations using the wall expansion model \ref{['eq wall xpnsn']} (red dashed line). (d) Local value of stream function on the trajectories as a function of angular position $\eta$. The orange trajectory remains essentially undeflected, while the red trajectory shows the effects of strong deflection away (1-2) and towards (2-3) the obstacle. Asymmetry of the flow ensures a net change in streamfunction $\Delta\psi=|\psi_i-\psi_f|$.
• Figure 4: Particle net displacement results for $(\alpha,\beta,a_p) = (30\degree,0.5,0.1)$. (a) Plot of displacement $\Delta\psi$ for particle trajectories released from different initial streamlines $\psi_i$. The inset magnifies the region around the maximum value $\Delta\psi_{max}$. (b) Variation of $\psi$ with $\Delta$ along the trajectories corresponding to the colored points in (a), including the $\Delta\psi_{max}$ trajectory in purple. (c) The solid line depicts $\Delta\psi(\psi_i)$ from the analytical model equation \ref{['eq etafinal etaini']} for small $\psi_i$. Both the magnitude of $\Delta\psi_{max}$ and its location $\psi_{i,max}$ are in good agreement with the empirical calculations.
• Figure 5: Scaling of $\Delta\psi_{max}$ with (a) flow angle $\alpha$$(a_p=0.1,\beta=0.5)$, (b) particle size $a_p$$(\alpha=30\degree,\beta=0.5)$, and (c) aspect ratio $\beta$$(a_p=0.1,\alpha=10\degree$ and $30\degree$). The red lines are obtained from the analytical scaling theory \ref{['eq scaling psiB']} for $\beta\ll 1$. In (c), the small-$\beta$ theory is successful even for $\beta=0.5$, but to capture displacements for near-spherical obstacles ($\beta\lesssim 1$), the complete theory of \ref{['eq DeltaEtaf full']} -- \ref{['eq Deltapsimax full in delta1 Deltaetaf']} is needed (blue lines).