Harnessing random low Reynolds number flow for net migration

Random noise in low Reynolds number flow has rarely been used to obtain net migration of microscale objects. In this study, we numerically show that net migration of a microscale object can be extracted from random directional fluid forces in Stokes flow, by introducing deformability and inhomogeneous density into the object. We also developed a mathematical framework to describe the deformation-induced migration caused by noise. These results provide a basis for understanding the noise-induced migration of a microswimmer and are useful for harnessing energy from low Reynolds number flow.

Figure 1

Vertical propulsion of the microcapsule by random fluid forces ($a_1/a_2=3,A=10,{\tau }^{*}=3.15\times {10}^{-3}\mathrm{and}\mathrm{Bo}=1$) (a) Schematic diagram of the microcapsule and problem settings. The gravitational acceleration $g̠$ is imposed in the $-z$ direction. The dotted line on the capsule indicates the area under which the spring forces were considered. (b) Time change of the $x,y$, and $z$ components of the volume center. The broken line indicates the slope of $z_c^{*}$, which is calculated by the method of least squares in the range $15\le t^{*}\le 50$. (c) Box plots of the propulsion velocities in the x, y, and $z$ directions. Sixteen independent simulation cases were examined; the symbol ‘ $\times$ ’ indicates the outliers. (d) Sample snapshots of the deformed microcapsule from $15\le t^{*}\le 19$ (please see Supplemental Material Movie 1).

Figure 2

Migration mechanism (a) Time change of the $z$ component of the volume center under different random force conditions ($a_1/a_2=3,{\tau }^{*}=3.15\times {10}^{-3}\mathrm{and}\mathrm{Bo}=1$). In the figure, $A_y=A_z=0$ indicates the case with only $x$-directional random forces ($A_y=A_z=0$, $A_x=10$), $A_z=0$ indicates the case with horizontal random forces in the $x-y$ plane ($A_z=0$, $A_x=A_y=10$), and $3D$ indicates the case with full three-dimensional random forces ($A=10$). (b),(c) Vertical propulsion of the microcapsule under uniaxial sinusoidal oscillations in the $x$ direction ($a_1/a_2=3,A_x^{\prime }=10,T^{*}=1\mathrm{and}\mathrm{Bo}=1$). (b) Schematic diagram of the problem settings. The acceleration $\underline{\alpha }$ of the oscillating fluid is horizontal, sinusoidal, and unidirectional in the $x$ direction. (c) The propulsion mechanism of the microcapsule. The red curve indicates the actual change of $z_c^{*}$ during a half period of oscillation in the simulation. The blue curve indicates the vertical propulsion velocity $V_z^{*}$, which was obtained by assuming rigid body motion under a given shape and torque. Inset figures show the shape of the capsule at the peak velocity. Red and blue curves indicate the shapes at red and blue points; the difference in shape is shown in yellow.

Figure 3

Effects of parameters on the average vertical migration velocity $\overline{u_z^{*}}$. Each plot consists of 12 independent simulation cases. Effect of (a) amplitude $A$ ($\mathrm{Bo}=1$, $a_1/a_2=3$ and ${\tau }^{*}=3.15\times {10}^{-3}$), (b) Bond number $\mathrm{Bo}$, which indicates the deformability of the capsule ($A=10$, $a_1/a_2=3$ and ${\tau }^{*}=3.15\times {10}^{-3}$), (c) the aspect ratio $a_1/a_2$ ($A=10$, $\mathrm{Bo}=1$ and ${\tau }^{*}=3.15\times {10}^{-3}$), and (d) the relaxation time ${\tau }^{*}$($\mathrm{Bo}=1$, $a_1/a_2=3$), where blue indicates $A=10$, red indicates $A=5$, and green indicates $A=2$.

Figure 4

The average vertical migration velocity $\overline{u_z^{*}}$ based on the proposed theory. (a) Effect of amplitude $A$ on the magnitude of $\overline{u_z^{*}}$, plotted in log-log scale ($\mathrm{Bo}=1$, $a_1/a_2=3$ and ${\tau }^{*}=3.15\times {10}^{-3}$). Each plot is a median value of 12 independent simulation cases. Slope 2 is drawn for comparison. (b) Effect of the relaxation time ${\tau }^{*}$ on $\overline{u_z^{*}}/A^2$ ($\mathrm{Bo}=1$, $a_1/a_2=3$). A line passing through the origin is plotted using the least squares method (slope $c_0=-1.47\times {10}^{-2}$, $r=0.967$). (c) Effect of relaxation time ${\tau }^{*}$ on $\overline{u_z^{*}}$ under the condition of $A^2\tau =0.315$ ($\mathrm{Bo}=1$, $a_1/a_2=3$). Function $c_0{A}^2\tau /\left(1+\tau /{\tau }_d\right)$ is plotted by the least squares method (${\tau }_d=0.098$, $r=0.882$).