The Diffusion of Passive Tracers in Laminar Shear Flow

The overall goal of this experiment is to observer the diffusion of passive tracers in capillary tubes of various geometric cross sections. This method can help answer key questions in the field of small scale fluid dynamics such as the spreading of chemicals in laminar flows and capillary tubes. The main advantage of this technique is that it is simple and cheap to manufacture and suitable for capillary tubes of an arbitrary cross section.

Though this method can provide insight into small scale laminar flow dynamics, it can also be applied to other systems such as microfluidic scale experiments, thanks to dynamic similarities. We first attempted these experiments after obtaining a theoretical result and performing numerical simulations. These suggested a relation between asymmetries in solute spreading and the pipe's cross sectional geometry.

Construct the experimental setup on a bench. Some components of the system are 3D printed, including the reservoir and the injector post equipped with an injection needle. There is also a hexagonal connector and pipes, in which the flow takes place.

The pipe geometry is a variable in the experiment. These are two sample cross sections used in this experiment. The ratio of the short to long side of the cross section is what varies.

Each pipe has two 3D printed plates to support it in the setup. The experiment requires two syringe pumps, one of them programmable. For fluids, have a fluorescein dye solution ready and a source of distilled water.

To set up the programmable pump, fill a 12 milliliter plastic syringe with distilled water. Next insert a plastic dispensing tip on the syringe. Then mount the syringe in the syringe pump and connect a 30 centimeter long hose to its tip.

Fill a three milliliter plastic syringe with a dispensing tip with the fluorescein dye solution. Manually push the syringe plunger to fill the internal chamber of the injector post. Stop when the chamber is completely full and there is no air trapped inside.

Then mount the syringe into a second syringe pump. Next clamp the injector post to the lab bench so that the tube remains connected to the syringe. Insert four long screws with washers into the four holes surrounding the injector's dispensing needle.

The next step is to work with the hexagonal connector. Identify the circular cutout on each side of the connector and place an O-ring in each one. Then take the hexagonal connector to the injector post, orient it so its larger hole faces the injector post, align its holes with the four screws and insert the screws.

When done, ensure the O-rings have not moved out of place. Get the tube from the syringe pump, attach it to the fitting on the bottom of the hexagonal connector. Next select the pipe geometry for the experiment.

The pipe in this case is a tube with a one millimeter by 10 millimeter rectangular cross section. At the hexagonal connector use an end-plate with the pipe and ensure the injection needle will enter the pipe. Then insert the screws that extend from the connector into the end-plate.

Now get the reservoir. It should have its O-ring in place. Place it at the other end of the pipe and attach the pipe to the reservoir using an end-plate, screws and washers.

Again make sure the O-rings remain in their correct position. Align the reservoir with the injector post and clamp it to the table. At the top of the hexagonal connector insert a three milliliter syringe with a plastic dispensing tip.

For data gathering place two 61 centimeter long UVA tube lights on either side of the pipe. Arrange for a camera above the pipe to take a photo of the pipe every second. Begin by filling the reservoir with distilled water.

It should be filled to slightly above the pipe. Next push on the 12 milliliter distilled water syringe. This will fill the pipe with water.

Turn on the UVA tube lights and darken the room. Flush the pipe with the syringe pump before taking a single reference image of the pipe filled with pure distilled water. Then set up the programmable pump to inject water at a slow rate but do not run it.

Push the plunger of the dye syringe. This injects a dollop of the solution approximately three millimeters thick. Then run the programmable syringe pump for five minutes.

After this the bolus is away from the needle. Now manually pull the dye syringe plunger backwards. This draws in water to ensure the dye does not reach the needle.

Wait for the dye bolus to diffuse across the cross section of the pipe. For this thin pipe the wait time is more than 12 hours. The dye solution needs to be injected and then pulled back with great care.

These are the most delicate steps in the protocol. If an error occurs, start the procedure again. Practice makes it easier and more repeatable.

With the initial conditions established, study flow using a slow pump rate. Start the pump and camera simultaneously. Allow them both to run for five minutes.

When done hold a ruler next to the pipe and take a calibration image. The images from the experiment are above the inferred concentration curves at three different non-dimensional times during the experiment. These data are for a thin pipe, for which the ratio of short to long sides of the pipe cross section is small.

The shape of the distribution changes as the dye bolus moves downstream. The initial Gaussian like symmetry in the longitudinal direction is quickly broken. Monte Carlo simulations, with a matching initial distribution and flow rate, confirm the experimental results at two different dimensionless times.

The solid lines are experimental data and the dashed lines are simulation results. This comparison is for a thin pipe with a small ratio of short to long sides of the cross section. Monte Carlo simulations also provide confirmation in the case of a thick pipe, where the ratio of cross sectional lengths is near one.

Note the difference in the evolution of the concentration between the thin and thick pipes. Over time the thin pipe has solute with a sharp front and a tapering tail. The thick pipe has the opposite profile.

Once mastered this technique can be done in about one hour if it is performed properly. This is excluding the wait time for the dye concentration to diffuse across the cross section. While attempting this procedure it's important to remember to flush the pipe slowly before the experimental run and to be extremely careful when creating the initial conditions.

After watching this video you should have a good understanding of how to observe the diffusion of solute in laminar flow through capillary tubes of arbitrary cross sections.