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# Inertial migration of deformable capsules and droplets in oscillatory and pulsating microchannel flows

Studying the motion of cells and investigating their migration patterns in inertial microchannels have been of great interest among researchers because of their numerous biological applications such as sorting, separating, and filtering them. A great drawback in conventional microfluidics is the inability to focus extremely small biological particles and pathogens in the order of sub-micron and nanometers due to the requirement of designing an impractically elongated microchannel, which could be in the order of a few meters in extreme cases. This restriction is because of the inverse correlation between the cube of the particle size and the theoretically required channel length. Exploiting an oscillatory flow is one solution to this issue where the total distance that the particle needs to travel to focus is virtually extended beyond the physical length of the device. Due to the present symmetry in such flow, the directions of the lift forces acting on the particle remain the same, making the particle focusing feasible.

Here, we present results of simulation of such oscillatory flows of a single capsule in a rectangular microchannel containing a Newtonian fluid. A 3D front-tracking method has been implemented to numerically study the dynamics of the capsule in the channel of interest. Several cases have been simulated to quantify the influence of the parameters involved in this problem such as the channel flow rate, capsule deformability, frequency of oscillation, and the type of applied mechanism for inducing flow oscillations. In all cases, the capsule blockage ratio and the initial location are the same, and it is tracked until it reaches its equilibrium position. The capability to focus the capsule in a short microchannel with oscillatory flow has been observed for capsule deformabilities and mechanisms to induce the oscillations used in our study. Nevertheless, there is a limit to the channel flow rate beyond which, there is no single focal point for the capsule. Another advantage of having an oscillatory microchannel flow is the ability to control the capsule focal point by changing the oscillation frequency according to the cases presented in the current study. The capsule focusing point also depends on its deformability, flow rate, and the form of the imposed periodic pressure gradient; more deformable capsules with lower maximum velocity focus closer to the channel center. Also, the difference between the capsule equilibrium point in steady and oscillatory flows is affected by the capsule stiffness and the device flow rate. Furthermore, increasing the oscillation frequency, capsule rigidity, and system flow rate shorten the essential device length.

Although the oscillation frequency can provide us with new particle equilibrium positions, especially ones between the channel center and wall that can be very beneficial for separation purposes, it has the shortcoming of having a zero net throughput. To address this restriction, a steady component has been added to the formerly defined oscillatory flow to make it pulsating. Furthermore, this type of flow adds more new equilibrium points because it behaves similarly to a pure oscillatory flow with an equivalent frequency in that regard. They also enable the presence of droplets at high Ca or Re that could break up in the steady or a very low-frequency regime. Therefore, we perform new numerical simulations of a deformable droplet suspended in steady, oscillatory, and pulsating microchannel flows. We have observed fluctuations in the trajectory of the drop and have shown that the amplitude of these oscillations, the average of the oscillatory deformation, and the average migration velocity decrease by increasing the frequency. The dependence of the drop focal point on the shape of the velocity profile has been investigated as well. It has been explored that this equilibrium position moves towards the wall in a plug-like profile, which is the case at very high frequencies. Moreover, due to the expensive cost of these simulations, a recursive version of the Multi Fidelity Gaussian processes (MFGP) has been used to replace the numerous high-fidelity (or fine-grid) simulations that cannot be afforded numerically. The MFGP algorithm is used to predict the equilibrium distance of the drop from the channel center for a wide range of the input parameters, namely Ca and frequency, at a constant Re. It performs exceptionally well by having an average R^2 score of 0.986 on 500 random test sets.

The presence of lift forces is the main factor that defines the dynamics of the drop in the microchannel. The last part of this work will be dedicated to extracting the active lift force profiles and identify their relationships with the parameters involved to shed light on the underlying physics. This will be based on a novel methodology that solely depends on the drop trajectory. Assuming a constant Re, we then compare steady lift forces at different Ca numbers and oscillatory ones at the same constant Ca. We will then define analytical equations for the obtained lift profiles using non-linear regression and predict their key coefficients over a continuous range of inputs using MFGP.