The 2D microfluidics cookbook - modeling convection and diffusion in plane flow devices

A growing number of microfluidic systems operate not through networks of microchannels but instead through using 2D flow fields. While the design rules for channel networks are already well-known and exposed in microfluidics textbooks, the knowledge underlying transport in 2D microfluidics remains scattered piecemeal and is not easily accessible to experimentalists and engineers. In this tutorial review, we formulate a unified framework for understanding, analyzing and designing 2D microfluidic technologies. We first show how a large number of seemingly different devices can all be modelled using the same concepts, namely flow and diffusion in a Hele-Shaw cell. We then expose a handful of mathematical tools, accessible to any engineer with undergraduate level mathematics knowledge, namely potential flow, superposition of charges, conformal transforms and basic convection-diffusion. We show how these tools can be combined to obtain a simple "recipe" that models almost any imaginable 2D microfluidic system. We end by pointing to more advanced topics beyond 2D microfluidics, namely interface problems and flow and diffusion in the third dimension. This forms the basis of a complete theory allowing for the design and operation of new microfluidic systems.

Advective Trapping in the Flow Through Composite Heterogeneous Porous Media

We study the mechanisms of advective trapping in composite porous media that consist of circular inclusions of distributed hydraulic conductivity embedded in a high conductivity matrix. Advective trapping occurs when solute enters low velocity regions in the media. Transport is analyzed in terms of breakthrough curves measured at the outlet of the system. The curve’s peak behavior depends on the volume fraction occupied by the inclusions, while the tail behavior depends on the distribution of hydraulic conductivity values. In order to quantify the observed behaviors, we derive two equivalent upscaled transport models. First, we derive a Lagrangian trapping model using the continuous-time random walk framework that is parameterized in terms of volume fraction and the distribution of conductivities in the inclusions. Second, we establish a non-local partial differential equation for the mobile solute concentration by volume averaging of the microscale transport equation. We show the equivalence between the two models as well as (first-order) multirate mass transfer models. The upscaled approach parameterized by medium and flow properties captures all features of the observed solute breakthrough curves and sheds new light on the modeling of advective trapping in heterogeneous media.

Quantification of Advective Transport Phenomena to Better Understand Dispersion in the Field

Observation of dispersion in field situations has left three issues that may be better understood by applying advective transport phenomena. (1) In some experiments, the longitudinal dispersivity becomes constant with increasing pathlength and in other cases it remains growing. (2) Dispersivities reported from multiple comprehensive observations at a single site differ at similar pathlength in some cases more than a factor two. (3) The observed difference between the plume fronts and plume tails is not represented in the reported parameters. The analytic equations for advective transport phenomena at macroscale of De Lange (2020) describe the thickness of the affected flow-tube and the spread of the plume front and tail. The scale factor defines the size of the averaging domain and so of the initial phase. The new macroscale correlation coefficient relates the growth of the longitudinal dispersivity beyond the initial phase to the aquifer heterogeneity. Using stochastic parameters for the aquifer heterogeneity, the parameters are quantified at 14 field experiments in the United States, Canada and Europe enabling the comparison of calculated and reported final dispersivities. Using the quantified parameters, 146 reported and calculated dispersivities along the traveled paths show a good match. A dispersivity derived from the local plume growth may differ a factor of two from the aquifer-representative value. The growths of plume fronts and tails between two plume stages are assessed in 14 cases and compared to calculated values. Distinctive parameters for the plume front and tail support better understanding of field situations. A user-ready spreadsheet is provided.

The Effects of Periodicity Assumptions in Porous Media Modelling

The effects of periodicity assumptions on the macroscopic properties of packed porous beds are evaluated using a cascaded Lattice-Boltzmann method model. The porous bed is modelled as cubic and staggered packings of mono-radii circular obstructions where the bed porosity is varied by altering the circle radii. The results for the macroscopic properties are validated using previously published results. For unsteady flows, it is found that one unit cell is not enough to represent all structures of the fluid flow which substantially impacts the permeability and dispersive properties of the porous bed. In the steady region, a single unit cell is shown to accurately represent the fluid flow across all cases studied

Advective Transport Phenomena to Better Understand Dispersion in Field and Modeling Practice

The absence of recent research on dispersion in engineering applications indicates the need for a description that is more focused on field and modeling practice. Engineers may benefit from simple calculation tools allowing them to understand the processes encountered in the field. Based on a conceptual model for advective transport through an elongated conductivity zone, for example, in fluvial sediments, explicit expressions are presented for macro-scale phenomena: (1) the different travel distances of water particles traveling in laminar flow through and adjacent to a single zone with conductivity higher or lower than that of the aquifer; (2) the affected thickness of the bundle of flowlines; (3) the distinction of inflow, outflow, and through-flow sections; (4) the development of a plume front vs. that of a tail; (5) conservation of mass causing water particles to travel both slower and faster than the aquifer average velocity while passing a single zone. The spread derived from a spatial distribution in a field experiment relates to the geometric mean of the spreads of the plume front and tail. The results obtained for a single conductivity zone are expanded for a general aquifer that is characterized by stochastic parameters. A fundamental new expression describes the dispersive mass flux as the product of the advective volume shift and the related local concentration difference. Contrary to Fickian theory, the dispersive mass flux in both the front and tail of a plume in highly heterogeneous aquifers is limited. In modeling, the advective volume shift is proportional to the cell size.

Impact of the spatial structure of the hydraulic conductivity field on vorticity in three-dimensional flows

A material fluid element within a porous medium experiences deformations due to the disordered spatial distribution of the Darcy scale velocity field, caused by the heterogeneity of hydraulic conductivity. A physical consequence of this heterogeneity is the presence of localized kinematical features such as straining, shearing and vorticity in the fluid element. These kinematical features will influence the shape of solute clouds and their fate. Studies on the deformation of material surfaces highlighted the importance of stretching and shearing, whereas vorticity received so far less attention, though it determines folding, a deformation associated with the local rotation of the velocity field. We study vorticity in a three-dimensional porous formation exploring how its fluctuations are influenced by the spatial structure of the porous media, obtained by immersing spheroidal inclusions into a matrix of constant hydraulic conductivity. By comparing porous formations with the same spatial statistics, we analyse how vorticity is affected by the different shape and arrangement of inclusions, defined as the medium 'microstructure'. We conclude that, as microstructure has a significant impact on vorticity fluctuations, the usual second-order statistical description of the conductivity field is unable to capture local deformations of the plume.

Biogenic inputs to ocean mixing

Recent studies have evoked heated debate about whether biologically generated (or biogenic) fluid disturbances affect mixing in the ocean. Estimates of biogenic inputs have shown that their contribution to ocean mixing is of the same order as winds and tides. Although these estimates are intriguing, further study using theoretical, numerical and experimental techniques is required to obtain conclusive evidence of biogenic mixing in the ocean. Biogenic ocean mixing is a complex problem that requires detailed understanding of: (1) marine organism behavior and characteristics (i.e. swimming dynamics, abundance and migratory behavior), (2) mechanisms utilized by swimming animals that have the ability to mix stratified fluids (i.e. turbulence and fluid drift) and (3) knowledge of the physical environment to isolate contributions of marine organisms from other sources of mixing. In addition to summarizing prior work addressing the points above, observations on the effect of animal swimming mode and body morphology on biogenic fluid transport will also be presented. It is argued that to inform the debate on whether biogenic mixing can contribute to ocean mixing, our studies should focus on diel vertical migrators that traverse stratified waters of the upper pycnocline. Based on our understanding of mixing mechanisms, body morphologies, swimming modes and body orientation, combined with our knowledge of vertically migrating populations of animals, it is likely that copepods, krill and some species of gelatinous zooplankton and fish have the potential to be strong sources of biogenic mixing.

The concept of drift and its application to multiphase and multibody problems

The concept of drift is built around understanding how a rigid body moving in a straight line distorts a material sheet in an unbounded perfect fluid. As the body moves from infinity through a material sheet, which is initially perpendicular to the direction of translation of the body, the sheet is permanently distorted. Darwin showed that the 'drift' volume, D(f), formed between the distorted and undistorted sheet is equal to C(m)V, where the added-mass coefficient, C(m), characterizes the shape of the body whose volume is V. Darwin's result is important for two reasons: first, it provides a means of quantifying how dyed fluid is transported from one place to another and dispersed; second, it provides a fundamental Lagrangian coordinate system to study inhomogeneous inviscid problems. The aim of this article is to review Darwin's contribution to fluid mechanics. By drawing on recent experimental measurements of drift and the drift volume, we aim to demonstrate how Darwin's drift concept has developed and to describe its broader significance for multiphase and multibody problems.