Inertial hydrodynamic ratchet: Rectification of colloidal flow in tubes of variable diameter

We investigate analytically a microfluidic device consisting of a tube with a nonuniform but spatially periodic diameter, where a fluid driven back and forth by a pump carries colloidal particles. Although the net flow of the fluid is zero, the particles move preferentially in one direction due to the ratchet mechanism, which occurs due to the simultaneous effect of inertial hydrodynamics and Brownian motion. We show that the average current is strongly sensitive to particle size, thus facilitating colloidal particle sorting.

Figure 1

Schematic sketch of the flow carrying a spherical particle with radius $R$ within a tube of varying diameter. The total volumetric flow of the fluid is $Q$.

Figure 2

Streamlines of the difference flow according to (16) in a tube with a profile given by (15), where $A=0.15$ and $B=0.2$. The orientation of the flow is indicated by arrows. The thick curve indicates the tube wall. Using heavy dots, we plot also the results of the exact solution using openfoam, with $\nu ={10}^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ (water), $d=0.3\mathrm{mm}$, and $U=5\mathrm{mm}{\mathrm{s}}^{-1}$. (This means $Q=0.245...{\mathrm{mm}}^3{\mathrm{s}}^{-1},{\mathrm{Re}}_t=1.5$.) Note that the vertices would be totally absent if the flow were described by the Stokes equation.

Figure 3

Streamlines of the difference flow of particles, according to formula (18), in a tube with identical parameters as in Fig. 2.

Figure 4

Graph of the function $\mathrm{\Phi }\left(r\right)$ defined by (23). The approximation (24) is drawn by the dashed line.