The Diffusion of Passive Tracers in Laminar Shear Flow
Summary
A protocol for the study of the diffusion of passive tracers in laminar pressure-driven flow is presented. The procedure is applicable to various capillary pipe geometries.
Abstract
A simple method to experimentally observe and measure the dispersion of a passive tracer in a laminar fluid flow is described. The method consists of first injecting fluorescent dye directly into a pipe filled with distilled water and allowing it to diffuse across the cross-section of the pipe to obtain a uniformly distributed initial condition. Following this period, the laminar flow is activated with a programmable syringe pump to observe the competition of advection and diffusion of the tracer through the pipe. Asymmetries in the tracer distribution are studied and correlations between the pipe cross-section and the shape of the distribution is shown: thin channels (aspect ratio << 1) produce tracers arriving with sharp fronts and tapering tails (front-loaded distributions), while thick channels (aspect ratio ~1) present the opposite behavior (back-loaded distributions). The experimental procedure is applied to capillary tubes of various geometries and is particularly relevant to microfluidic applications by dynamical similarity.
Introduction
In recent years, substantial efforts have been focused on developing microfluidic and lab-on-chip devices that can reduce the costs and increase the productivity of chemical preparation and diagnostics for a range of applications. One of the main features of microfluidic devices is the pressure-driven transport of fluids and dissolved solutes through microchannels. In this context, it has become increasingly important to better understand the controlled delivery of solutes at the microscale. In particular, applications such as chromatographic separation1,2 and microfluidic flow injection analysis3,4 require improved control and understanding of solute delivery. Researchers in microfluidics have studied and documented the influence of the channel's cross-sectional shape on solute spreading5,6,7,8, and the role of the channel's aspect ratio9,10.
Analytical and numerical studies of solute spreading along channels have recently lead to the identification of a correlation between the pipe cross-sectional geometry and the shape of the distribution9,10. At early timescales, the distribution strongly depends on the geometry: rectangular pipes break symmetry almost immediately, while elliptical pipes retain their initial symmetry much longer9. On the other hand, progressing into longer timescales the asymmetries in the solute distribution no longer differentiate ellipses from rectangles, and are set solely by the cross-sectional aspect ratio λ (ratio of the short to long side). Considering "pipes" of elliptical cross-sections and "ducts" of rectangular cross-sections, predictions from numerical simulations and asymptotic analysis were benchmarked with laboratory experiments. Thin channels (aspect ratio << 1) produce solutes arriving with sharp fronts and tapering tails, while thick channels (aspect ratio ~ 1) present the opposite behavior10. This robust effect is relatively insensitive to the initial conditions and can be used to help select the solute distribution profile required for any application.
The behavior outlined above of sorting thin versus thick domains happens before the classical "Taylor dispersion" regime is reached. Taylor dispersion refers to the enhanced spreading of passive solutes in laminar flow (stable at low Reynolds number, Re) with a boosted effective diffusivity, inversely proportional to the solute's molecular diffusivity κ11. This enhancement is observed only after long, diffusive timescales, when the solute has diffused across the channel. Such diffusive timescale is defined in terms of the characteristic length scale a of the geometry, as td = a2/κ. The Péclet number is a nondimensional parameter which measures the relative importance of fluid advection to diffusion effects. We define this parameter in terms of the shortest length scale as Pe = Ua/κ, where U is the characteristic flow speed. (The Reynolds number can be defined in terms of the Péclet number as Re = Pe κ/ν, where ν is the kinematic viscosity of the fluid.) Typical Péclet number values for microfluidic applications12 vary between 10 and 105, with molecular diffusivities ranging from 10-7 to 10-5 cm2/s. Hence, given the flow speeds and length scales of interest, it is critical to understand the behavior of solutes for intermediate-to-long timescales (relative to the diffusive timescale), well past the initial observations of geometry-driven behavior and into the cross-section-driven regimes universal for a large class of geometries.
Given the interest in microfluidic applications, the choice of a large scale experimental setup may at first seem unnatural. The experiments reported herein are at the millimeter scale, not at the microscale as in true microfluidic devices. However, the same physical behaviors characterize both systems and a quantitative study of the relevant phenomena can still be achieved by properly scaling the governing equations, just as scale models of aircraft are assessed in wind tunnels during the design phase. In particular, matching relevant nondimensional parameters (such as the Péclet number for our experiment) ensures the adaptability of the experimental model. Working at such larger scales, while maintaining a laminar pressure-driven flow, offers several advantages over a traditional microscale setup. In particular, the equipment required to manufacture, perform, and visualize the present experiments is easier to operate and less costly. Furthermore, other common challenges of working with microchannels, such as frequent clogging and the enhanced influence of manufacturing tolerances, are mitigated with the larger setup. Another possible use for this experimental setup is for studies of residence time distribution (RTD) in laminar flows13.
The asymmetries arising in the solute distribution downstream can be analyzed via its statistical moments; in particular, the skewness, which is defined as the centered, normalized third moment, is the lowest order integral statistic measuring the asymmetry of a distribution. The sign of the skewness typically indicates the shape of the distribution, i.e. if it is front-loaded (negative skewness) or back-loaded (positive skewness). Focusing on the aspect-ratios of the channels, there exists a clear correlation of thin geometries with front-loaded distributions, and thick geometries with back-loaded distributions10. Additionally, a critical aspect ratio separating these two opposite behaviors can be calculated for both elliptical pipes and rectangular ducts. Such crossover aspect ratios are remarkably similar for standard geometries, in particular, λ* = 0.49031 for pipes, and λ* = 0.49038 for ducts, suggestive of the universality of the theory10.
The experimental setup and method described in this paper are used to study the spreading of a pressure-driven passive solute in laminar fluid flows throughout glass capillaries of various cross-sections. The simplicity and reproducibility of the experiment defines a robust method of analysis for understanding the connection between a pipe's geometrical cross-section and the resulting shape of the injected solute distribution as it is transported downstream. The method discussed in this work has been developed to readily benchmark mathematical and numerical results in a physical laboratory setting.
A simple experimental procedure is described which highlights the definitive role played by a fluidic channel's cross-sectional aspect-ratio in setting the shape of a solute distribution downstream. The experimental setup requires a programmable syringe pump to produce a laminar steady flow, smooth glass pipes of various cross-sections, a second syringe pump to inject the diffusing solute (e.g. fluorescein dye) into the surrounding laminar flow, and UV-A lights and a camera to record the solute evolution. CAD files are provided for all the custom parts of the setup and such files can be used to 3D-print the experimental parts prior to assembly.
Protocol
Representative Results
Discussion
After injecting dye into the pipe, the bolus is transported away from the injection needle using a steady flow. Then, it is necessary to wait long enough for the dye to diffuse across the cross-section of the channel. In this way, a uniform Gaussian-like distribution is obtained and will serve as the initial condition for the experiment. Hence, a laminar background flow is created with the programmable syringe pump. The experimental run lasts for 5 min with photos taken every second.
The most common issues in the setup come from the connection of the parts and the pipes. The various 3D-printed parts need to be sealed properly when connected to avoid leaking. The glass pipes are very delicate and must be handled and installed with care.
An issue we encountered when transitioning from the thin rectangular pipe to the thick square pipe was related to the fact that the pipe volume was reduced by a factor of 10. To maintain the same cross-sectional average flow speed with the mounted 12 mL syringe, the plunger speed in syringe pump A would have needed to be extremely low. At this programmed speed, the plunger velocity was not uniform anymore and a steady flow cannot be guaranteed throughout the experimental run. Therefore, we switched to a much smaller 1 mL syringe when working with the thick square pipe in step 2.5.1.
Also, one should verify that the average intensity along the vertical dimension of the pipe in the initial condition is approximately uniform. If not, a filtering mask needs to be applied across all frames to account for this discrepancy.
The least repeatable part of the experiment is the dye injection (and consequently the width of the initial distribution). As illustrated earlier, it is not a concern for matching with the Monte Carlo simulations, as the experimental initial condition can be recreated using the analysis of the initial photograph. The dye injection and consequent manual withdrawing may not always produce dye plugs of precisely the same width. Particular care needs to be applied when setting up the initial dye bolus. The experiment becomes more repeatable as researchers gain experience in this part of the protocol, but future improvements could certainly be made.
When comparing the setup with microfluidic devices, the only parameter that appear in the governing equation when appropriately nondimensionalized is the Péclet number Pe if the tracer is passive, i.e. the tracer evolution is uncoupled from the flow. Dynamic similarity is implicit in the assumption of low Reynolds (Re << 1) which ensures stable laminar flows u(y,z). These two parameters are setting the full similarity between microfluidic setups and the scales of our experiment. In practice, the physical length of the pipe only restricts the nondimensional times we can safely reach with our setup. For very late non-dimensional times, the necessary length of the pipe could become prohibitively long for a fixed Péclet number in this large-scale setup.
An obvious limitation of this experimental protocol is that the data collected is a projected 2D representation of 3D geometry as the pictures are taken top-down on the pipe. The current process only allows to obtain the evolution of the cross-sectionally averaged dye distribution. Obtaining a distribution defined at each location in the tube rather than on its cross-sectional average and comparison with theoretical and numerical predictions are the subject of ongoing research.
All the experimental setup parts have technical drawings available for download which makes the setup easily accessible and customizable by any interested researcher. Building on the current results, the same setup will be used to study more complex and unexplored pipe geometries as well as different flow regimes.
Disclosures
The authors have nothing to disclose.
Acknowledgements
We acknowledge funding from the Office of Naval Research (grant DURIP N00014-12-1-0749) and the National Science Foundation (grants RTG DMS-0943851, CMG ARC-1025523, DMS-1009750, and DMS-1517879). Additionally, we acknowledge the work of Sarah C. Burnett who helped develop an early version of the experimental setup and protocol.
Materials
| Flourescein Dye | Flinn Scientific | LOT: 118362 CAS NO: 518-47-8 | |
| PhD ULTRA Hpsi Syringe Pump | Harvard Apparatus | 703111 | programmable digital syringe pump |
| Compact Infusion Pump Model 975 | Harvard Apparatus | 55-1689 | |
| Form 2 SLA 3D Printer | Formlabs | 100-240 | |
| Glass pipes | VitroCom | 4410 and 8100 | |
| PTFE sealing tape | Teflon | 4934A12 | |
| PVC tubing (1/8" ID) | McMaster | 5231K144 | 5 Foot Length |
| Reusable Stainless Steel Dispensing Needle 22 Gauge, .016" ID, .028" OD, 1/8" NPT Thrd, 2" Lg | McMaster | 7590A45 | 1 Required |
| RTV silicone rubber sealant | McMaster | 74945A69 | |
| Plastic Syringe Manual, w/ Luer Lock Connection, .34 oz Capacity, Packs of 10 | McMaster | 7510A653 | 1 required |
| Plastic Syringe Manual, w/ Luer Slip Connection, .034 oz Cap, Packs of 10 | McMaster | 7510A603 | 1 required |
| Plastic Syringe Manual, w/ Luer Lock Connection, 0.1 oz Capacity, Packs of 10 | McMaster | 7510A651 | 2 required |
| Plastic dispensing tip | McMaster | 6699A1 | 3 required |
| 6" C-Clamps | McMaster | 5133A18 | 2 required |
| Type 18-8 Stainless Steel Flat Washer Number 6 Screw Size, 0.156" ID, 0.312" OD, Packs of 100 | McMaster | 92141A008 | 8 required |
| 18-8 SS Pan Head Phillips Machine Screw 6-32 Thread, 2-1/4" Length, Packs of 50 | McMaster | 91772A167 | 4 required |
| Oil-Resistant Buna-N Multipurpose O-Ring 1/16 Fractional Width, Dash Number 016, Packs of 100 | McMaster | 9452K6 | 3 required |
| Type 18-8 Stainless Steel Hex Nut 6-32 Thread Size, 5/16" Wide, 7/64" High, Packs of 100 | McMaster | 91841A007 | 4 required |
| 18-8 SS Pan Head Phillips Machine Screw 6-32 Thread, 1/2" Length, Packs of 100 | McMaster | 91772A148 | 4 required |
| 24" Black Light Fixture with bulb | American DJ | B0002F5544 | 2 required |
| DSLR camera | Nikon | D300 | |
| 24-120 mm lens | Nikon | 2193 | |
| Remote programmable trigger | Nikon | 4917 | remote programmable trigger |
| Memory Card | SanDisk | SDCFX-032G-E61 | |
| Metric ruler | McMaster | 20345A35 |
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