Laminar dispersion at low and high Peclet numbers in a sinusoidal microtube: Point-size versus finite-size particles

Abstract

This paper adopts Brenner’s homogenization theory to investigate dispersion properties, over a wide range of Peclet values, of point-size and finite-size particles in sinusoidal cylindrical microchannels in the presence of a pressure-driven Stokes flow field. The periodic alternation of entropic barriers/traps can unexpectedly increase the effective finite-size particle velocity as well as decrease the effective dispersion coefficient for both point-size and finite-size particles, for large values of the radial Peclet number. While this phenomenon has a simple explanation for tracer particles, its understanding for finite-size particles is not trivial and goes through the analysis of the localization feature of the equilibrium unit-cell particle density w0(x) and how this spatial nonuniformity impacts upon the effective particle velocity and on the solution of the so-called b field, controlling the large scale axial dispersion coefficient. Unfortunately, dispersion reduction cannot be exploited for the sake of the separation of particles having different radii because the separation performance of a hydrodynamic sinusoidal column turns out to be worse than that of a standard straight column for experimentally feasible Peclet values. Interesting analytical results for long-wavelength sinusoidal channels are obtained by a long-wave asymptotic expansion. Both zero-order and first-order terms for the asymptotic expansion of the w0(x) measure and of the b field are obtained, thus exploring a wide range of Peclet values and deriving an analytical expression for the Taylor dispersion coefficient.

Exact moment analysis of transient/asymptotic dispersion properties in periodic media with adsorbing/desorbing walls

“…18,30 For both systems, the transient analysis shows how the tortuous fluid motion, induced by the sinusoidal walls or by the presence of pillars, causes wide and persistent temporal oscillations of the effective velocity and dispersion coefficient even for a steady (non-pulsating) Stokes flow. Moreover, the transient analysis shows that the phenomenon of the overshoot for the effective dispersion coefficient, already observed numerically and experimentally for non-reactive straight and sinusoidal tubes, 60,[76][77][78][79] is largely amplified by the adsorption/desorption process.…”

“…By replacing the asymptotic expressions Eqs. ( 48), (60), and (61) for first-order and second-order local moments into Eq. ( 32) for the centered third-order moments, the following compact expressions for d t l 3;0 and d t l 0;3 are obtained:…”

“…Indeed, sinusoidal tubes represent one of the most investigated geometries, introduced to describe the influence of section narrowing/widening on transport phenomena in a variety of contexts as fractures, nanotubes, 56 zeolites, 57 cell biology, 58 blood vessels, 59 entropic barrier/entropic trap. 3,[60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75] Moreover, sinusoidal tubes are also widely investigated as models for stenosed arteries in hemodynamics and the presence of reactive walls is relevant to describe oxygen transport and atherogenesis (see Refs. 35 and 55 and references therein).…”

Shape-Enhanced Open-Channel Hydrodynamic Chromatography

“…Here, we consider conduits having an angular cross section, that is, shapes marked by the presence of distinct corners, as these can be expected to lead to steeper velocity gradients (see Figure for an overview of considered shapes). The present study makes use of Brenner’s macro-transport formalism to compute the effective velocity (via Brenner’s first order moment) and the effective dispersion (via the second order moment), which can be quantified by the Taylor-Aris dispersion coefficient − Previously, this theory has been applied to study effective macrotransport in a diverse range of fields. − As this is a first exploration into the present topic, the solutes are treated as hard, impermeable spheres, and the hydrodynamic effects induced by the interaction of the particles with the wall and the velocity profile (wall friction, particle rotation, concomitant lift forces, and slip) are not taken into account. Hence, results have been limited to the range of dimensionless particle size λ = d p / l ≤ 0.2 (where d p is the particle size, and l is the cross-sectional characteristic size parameter, see Figure for further details).…”

Generalized Taylor dispersion for translationally invariant microfluidic systems

“…The first method, as implemented by Taylor [1] and then Aris [2], is based on the averaging of the Fokker-Planck equation describing the full advection-diffusion problem. Subsequently, methods based on the direct computation of the moments of the particle dispersion along the channel were developed [4][5][6][7]. These methods have the advantage that for simple systems the full temporal dependence of the moments can be obtained.…”