Inertial Particle Focusing in Curved Ducts: Bifurcation and Dynamics

So you can use this to separate particles by size. In this case, the tumor cells from smaller healthier cells. Now, in this example, the cross-sectional shape shown is a trapezoid, but a similar phenomenon can also be observed for rectangular cross sections, as shown on the right. Curved ducts with rectangular cross sections are quite common in these inertial microfluidic experiments. The current advances in this field are mainly driven by experimental trial and error, where different designs of microfluidic devices can be assembled and tested within a few days. However, the ability to predict and optimize Inertial Focusing behaviors for different applications based on first principles is still not well understood. Motivated by this, we decided to systematically investigate particle focusing behaviors in curved rectangular channels. Now, I'll hand over to Brandon who will discuss the theoretical model used in our work and present some key results for constant curvature circular ducts.

Good day, I'm going to explain the theoretical model used in this work. Before I begin, I want to acknowledge Yon Stokes and Andrea Poi, who were collaborators on the development of this model, which was published in JFM in 2019. I'll also mention that Yvonne gave a very informative talk on the model as part of the Journal of Engineering Math seminar series on Cassini, and I will provide a link to this at the end of the video.

Now, let's discuss our model. We have a curved duct with a fixed bend radius around the z axis, and the cross section is also fixed with a height H and a width W. In this talk, we will mainly focus on rectangular cross sections, but it could be extended to other shapes. Additionally, there is a particle located at an angle θ P. Before considering the particle, let's examine the fluid flow without it. A steady fluid flow develops through the duct, and we denote its pressure and velocity field as P and U bar, respectively. It is known, thanks to Dean's work, that the fluid flow in a curved duct develops a secondary flow in the cross-sectional plane consisting of two rotating vortices. This secondary flow will perturb the particle motion.

With the presence of the particle, the pressure and velocity fields are modified and denoted as P and U, respectively. The particle changes the flow locally around it. Instead of analyzing the full flow field, it is more convenient to focus on the disturbance caused by the particle. The disturbance pressure is denoted as Q and the disturbance velocity as V. This is simply the difference between the full fields and the background fields. We focus on the disturbance because we expect it to be localized around the particle and smaller in magnitude compared to the full flow field.

To simplify the analysis, we use a rotating reference frame where the particle maintains a fixed angular position as it moves around the ducts. This allows us to apply a quasi steady approximation within the rotating frame. This approximation is possible because the axial velocity is not changing significantly, and the changes in the cross section are small enough to neglect acceleration effects.