Inertial microfluidics - Duct Flow
In inertial microfluidics a dispersion of paricles (e.g. blood cells in bood plasma) is flown through a microfluidics channel which can be centimeters long but only a few tens of micrometers wide. When the channel is straight and has a rectangular cross section this is also known as duct flow. In the channel, particles will be advected with the fluid due to stresses exerted by the fluid on the particle surface. Due to the small dimensions of the channel, however, the particles will also feel the channel walls. A lift force resulting from a pressure build up between the wall and the particles tends to push the particles away from the wall. On the other hand, a shear gradient force pushes the particles towards the walls. If not otherwise influenced, e.g. by the disturbance of the flow field due to the presence of other nearby particles, these two competing forces cause the particles to concentrate at so called equilibrium regions: a stationary point with respect to the channel wall at which the particles continue to flow through the channel . For circular channels, particles focus at a distance of ~0.6 times the channel diameter away from the wall. This is known as the Segre-Silberberg effect . For a squared channel the rotational symmetry is broken and the particles focus at the four wall faces.
Figure 1: Adapted from reference . (A) in a cylindrical channel, randomly distributed particles are known to focus to an annulus located between the center and wall of the channel. (B) In square channels, particles focus to four equilibrium regions centered at the faces of the channels for dilute suspensions of particles. (C) Two lift forces perpendicular to the flow direction act to create equilibrium positions in channels. The 'wall effect' lift directed away from the wall and the shear gradient lift that is directed down the shear gradient towards the wall.
Simulating inertial flow
There are many different shapes and sizes of channels that can be used to create the intertial effect that you need. In fact channel dimensions, channel aspect ratio, particle shape and diameter, and flow rate all play a role. The more the real case deviates from the ideal case described above, the less analytical solutions can be applied. In case of non-spherical particles or non-dilute systems it is difficult to predict particle focussing at all. Making a microfluidics device therefor takes a lot of time and effort and testing all possible combinations requires a lot of resources. That’s where simulations in which particle properties and two-way interaction between fluid and particle surfaces are explicitely computed can help. In such fully resolved simulations, the particle focussing and all deviations from it is a natural outcome. This renders designing your device not only easy and fast but also accurate even if it is a situation far from the ideal of the Segre-Silberberg setup such as in the case of non-diluted full blood.
In our simulations we use state-of-the-art models based on coupled Lattice Boltzmann methods to simulate the fluid flow. On top off that we use either a mesh model (e.g. for red blood cells) or a rigid body solver (e.g. for rigid particles). In essence, we are able to create virtual particles of any shape, size, and flexibility. Rotation and deformation of the particles in the fluid flow is intrinsically taken into account without using any simplifying assumptions, which is of great importance when working with red blood cells and other biological particles.
With our models we are able to accurately predict the fluid flow and equilibrium positions of particles in a variety of different 3-dimensional microfluidics channels. The hydrodynamic interactions, lift forces and flow patterns are a direct result from the simulation. No assumptions are put in with regard to these properties.
Below are some examples of our simulation results for the flow of a dispersion in a straight channel. Figure 2-left shows a simulated particle in flow in a 50 μm high microfluidics channel. The slice at the end of the channel indicates the lateral flow speed. The colors on the orthogonal slice indicate deviations from the mean fluid pressure due to movement and rotation of the particle. Figure 2-right shows the resulting particle trajectories in the flow projected on a plane perpendicular to the flow direction. Starting from different positions (colored points), particles of different diameters (4 and 8 μm) migrate to their distinct equilibrium positions. Note that the movement of the particles to the equilibrium distance from the wall (radial ordering, Segre-Silberberg) is fast (< 1 sec), while the migration towards the center of the faces is very slow (> 10 sec). This is in line with the experimental observations as schematically shown in figure 1.
Figure 2: Left: Simulated particle in duct flow. Right: Particle trajectories in duct flow. Particles with different sizes focus at different equilibrium distances from the channel wall, but both focus relative to the center of the faces.
If you are interested in using our methods to prototype your intertial microfluidics devices get in touch with us via firstname.lastname@example.org or give us a call.
 D. Di Carlo, Inertial Microfluidics, Lab on a Chip 9 (21), pp 3038, 2009.
 G Segre, A Silberberg, Radial particle displacements in Poiseuille ﬂow of suspensions. Nature 189, pp 209, 1961.