Colloid particles in microfluidic inertial hydrodynamic ratchet at moderate Reynolds number

František Slanina

Phys. Rev. E 102, 052601 – Published 2 November 2020

Abstract

The movement of spherical Brownian particle carried by an alternating fluid flow in a tube of periodically variable diameter is investigated. On the basis of our previous results [Phys. Rev. E 99, 012604 (2019)] on the hydrodynamics of the problem, we look at the competition of hydrodynamics and diffusion. We use the method of Fick-Jacobs mapping on an effective one-dimensional problem. We calculate the ratchet current and show that is is strictly related to finite size of the particles. The ratchet current grows quadratically with particle radius. We also show that the dominant contribution to the ratchet current is due to inertial hydrodynamic effects. This means that Reynolds number must be at least of order one. We discuss the possible use for separation of particles by size and perspectives of optimization of the tube shape.

Received 22 May 2020
Accepted 8 October 2020

DOI:https://doi.org/10.1103/PhysRevE.102.052601

Physics Subject Headings (PhySH)

Diffusion Particle-laden flows

Colloids Microfluidic devices

Polymers & Soft MatterStatistical Physics & Thermodynamics

Authors & Affiliations

František Slanina^*

Institute of Physics, Czech Academy of Sciences, CZ-18221 Praha, Czech Republic

^*slanina@fzu.cz

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Issue

Vol. 102, Iss. 5 — November 2020

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Images

Figure 1
Schematic picture of the system investigated here. The fluid flows through axially symmetric tube of periodically varying diameter $d (z)$ . Within the fluid, there is a colloid particle of spherical shape and radius $R$ .
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Figure 2
Dependence of the average ratchet velocity of the particle on the reduced particle radius. The parameters of the tube are $Ω = 10^{5} m^{- 1}, \bar{d} = 0.3, A = 0.15$ , and $B = 0.2$ . The Reynolds number is $Re = 3$ (solid line), $Re = 2$ (dotted line), and $Re = 1$ (dashed line). In the inset, average ratchet velocity for the same tube parameters and $Re = 3$ , in the hypothetical case of neglected inertial effects.
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Figure 3
Dependence of the average ratchet velocity of the particle on the Reynolds number. The parameters of the tube are $Ω = 10^{5} m^{- 1}, \bar{d} = 0.3, A = 0.15$ , and $B = 0.2$ . The reduced radius of the particle is $\bar{R} = 0.02$ (solid line), $\bar{R} = 0.015$ (dotted line), and $\bar{R} = 0.01$ (dashed line). In the inset, average ratchet velocity for the same tube parameters and $\bar{R} = 0.05$ . The result computed to the lowest (i.e., first) order in $β^{- 1}$ is drawn by dot-dashed line. The solid line shows the result computed up to second order in $β^{- 1}$ . Including third and fourth order in $β^{- 1}$ brings difference smaller than the width of the line.
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Figure 4
Dependence of the average ratchet velocity of the particle on the parameter $b$ determining the shape of the tube. Other parameters of the tube are $Ω = 10^{5} m^{- 1}, \bar{d} = 0.3, a = 0.1$ (solid line), $a = 0.08$ (dotted line), and $a = 0.06$ (dashed line). Reynolds number is $Re = 2.5$ , particle radius is $\bar{R} = 0.02$ . The symbol $\times$ at $b = 1.5$ indicates the ratchet velocity for the same $Re, \bar{R}, \bar{d}$ , and $Ω$ , but in the tube with shape (43) with parameters $A = 0.15$ , and $B = 0.2$ . In the inset, function describing the tube wall. In dot-dashed line, we show the shape according to (43) with parameters $A = 0.15$ , and $B = 0.2$ . Other three lines are shapes according to (44) with $\bar{d} = 0.3, b = 1.5$ , and $a = 0.1$ (solid line), $a = 0.08$ (dotted line), and $a = 0.06$ (dashed line).
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Figure 5
Dependence of the average ratchet velocity of the particle on the parameter $a$ determining the shape of the tube. Other parameters of the tube are $Ω = 10^{5} m^{- 1}, \bar{d} = 0.3, b = 1.5$ (solid line), $b = 1.0$ (dotted line), and $b = 0.5$ (dashed line). Reynolds number is $Re = 2.5$ , particle radius is $\bar{R} = 0.02$ . In the inset, function describing the tube wall according to (44) with $\bar{d} = 0.3, a = 0.1$ , and $b = 1.5$ (solid line), $b = 1.0$ (dotted line), and $b = 0.5$ (dashed line).
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