Time-dependent Taylor–Aris dispersion of an initial point concentration

Abstract

Based on the method of moments, we derive a general theoretical expression for the time-dependent dispersion of an initial point concentration in steady and unsteady laminar flows through long straight channels of any constant cross-section. We retrieve and generalize previous case-specific theoretical results, and furthermore predict new phenomena. In particular, for the transient phase before the well-described steady Taylor-Aris limit is reached, we find anomalous diffusion with a dependence of the temporal scaling exponent on the initial release point, generalizing this finding in specific cases. During this transient we furthermore identify maxima in the values of the dispersion coefficient which exceed the Taylor-Aris value by amounts that depend on channel geometry, initial point release position, velocity profile and Péclet number. We show that these effects are caused by a difference in relaxation time of the first and second moments of the solute distribution and may be explained by advection-dominated dispersion powered by transverse diffusion in flows with local velocity gradients.

Diffusive Promotion by Velocity Gradient of Cytoplasmic Streaming (CPS) in Nitella Internodal Cells

“…There seems to be conflicting explanations about the role of CPS, which would have enough mixing effect in large plant cells; however, such a Péclet number scale possesses fully developed diffusion to the whole length of the Nitella internodal cell, ~1 cm, of approximately 44 days! On the other hand, dispersion in a pipe flow with a radial shear gradient, which is well known as the Taylor–Aris dispersion [ 31 ], works as a radial dispersion, and consequently it effectively provides a longitudinal diffusion as the effective dispersion coefficient [ 32 ], . Here, γ is the geometric factor that depends on the flow distribution and the boundary condition in the flow field, discussed in detail in S1 Text , as being , because D eff = 1.57 × 10 −9 m 2 s −1 and s. This means that dispersion with a velocity gradient in the cytosol is 356 times faster than molecule dispersion with a flattened velocity field because of the dispersion ratio D eff / D Brown = 356.…”

Transient dispersion process of active particles

“…2014; Yariv & Schnitzer 2014; Sandoval & Dagdug 2014; Makhnovskii 2019) have experimentally and numerically investigated the transient active dispersion process in a corrugated channel without background flow, considering the application of sorting particles by their self-propelled speeds. Additionally, it is of considerable interest to systematically compare the transient active dispersion process with the classic dispersion of passive particles (Lighthill 1966; Foister & van de Ven 1980; Latini & Bernoff 2001; Camassa, Lin & McLaughlin 2010; Vedel, Hovad & Bruus 2014; Taghizadeh, Valdés-Parada & Wood 2020), to capture the difference in approaching the Taylor dispersion regime (Chatwin 1970; Wu & Chen 2014; Li et al. 2018).…”

Hydrodynamic dispersion in Hele-Shaw flows with inhomogeneous wall boundary conditions

“…(2004), Zhao & Bau (2007) and Adrover (2013). While the aforementioned works were concerned with spatial inhomogeneities, the role of time-dependent flow profiles was discussed by Vedel & Bruus (2012) and Vedel, Hovad & Bruus (2014). Here, the authors showed that oscillations can enhance dispersion, given that the oscillation frequency is lower than the momentum and solute diffusion time scale over the channel cross-section.…”