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Inertial microfluidic physics† Cite this: Lab Chip, 2014, 14, 2739

Hamed Amini,‡ab Wonhee Leec and Dino Di Carlo*ab Microfluidics has experienced massive growth in the past two decades, and especially with advances in

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rapid prototyping researchers have explored a multitude of channel structures, fluid and particle mixtures, and integration with electrical and optical systems towards solving problems in healthcare, biological and chemical analysis, materials synthesis, and other emerging areas that can benefit from the scale, automation, or the unique physics of these systems. Inertial microfluidics, which relies on the unconventional use of fluid inertia in microfluidic platforms, is one of the emerging fields that make use of unique physical phenomena that are accessible in microscale patterned channels. Channel shapes that focus, concentrate, order, separate, transfer, and mix particles and fluids have been demonstrated, however physical underpinnings guiding these channel designs have been limited and much of the development has been based on experimentally-derived intuition. Here we aim to provide a deeper understanding of mechanisms and underlying physics in these systems which can lead to more effective and reliable designs with less iteration. To place the inertial effects into context we also discuss related fluid-induced forces present in particulate flows including forces due to non-Newtonian fluids, particle asymmetry, and particle deformability. We then highlight the inverse situation and describe the effect of the suspended particles acting on the fluid in a channel flow. Finally, we discuss the importance of Received 29th January 2014, Accepted 27th May 2014 DOI: 10.1039/c4lc00128a

structured channels, i.e. channels with boundary conditions that vary in the streamwise direction, and their potential as a means to achieve unprecedented three-dimensional control over fluid and particles in microchannels. Ultimately, we hope that an improved fundamental and quantitative understanding of inertial fluid dynamic effects can lead to unprecedented capabilities to program fluid and particle flow

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towards automation of biomedicine, materials synthesis, and chemical process control.

Introduction The belief that all practically achievable and useful flows in microfluidic systems operate not only in the laminar flow regime but also in a Stokes flow regime (Re → 0, Re = ρUH/μ where U is the average velocity, H is the channel dimension and ρ and μ are fluid density and dynamic viscosity, respectively) was widespread in the microfluidic community until recent years.1 Following this belief, the inertia of the fluid is ignored in most microfluidic platforms and contributions of fluid momentum are omitted from the Navier–Stokes equations resulting in linear, and thus time-reversible, equations of

a

Department of Bioengineering, University of California, 420 Westwood Plaza, 5121 Engineering V, P.O. Box 951600, Los Angeles, CA, 90095, USA. E-mail: [email protected]; Fax: +1 310 794 5956; Tel: +1 310 983 3235 b California NanoSystems Institute, 570 Westwood Plaza, Building 114, Los Angeles, CA, 90095, USA c Graduate School of Nanoscience and Technology, Korea Advanced Institute of Science and Technology, N5-2322 291 Daehak-Ro, YuseongGu, Daejeon, South Korea † Electronic supplementary information (ESI) available. See DOI: 10.1039/ c4lc00128a ‡ Current address: Illumina Inc., 25861 Industrial Blvd, Hayward, CA 94545, USA.

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motion for Newtonian fluids. However, the importance of intermediate range flow (~1 < Re < ~100) was emphasized recently2–4 in which nonlinear and irreversible motions are observed for fluid and particles in microchannels. This regime, in which both the inertia and the viscosity of the fluid are finite, still lies within the realm of laminar flow (Re ≪ 2300) which provides a deterministic nature and thus controllability of fluid and particles within. The velocity difference across the particle or obstacle length scale (and therefore velocity gradient) is especially important in the emergence of inertial effects as it impacts the inertial lift force via the lift coefficient magnitude or secondary flow magnitude respectively, which will be discussed in following sections. It should be noted that the necessity for large velocity gradients to manipulate particles and fluids in inertial microfluidic platforms has likely prevented observation of similar behaviors in larger channels filled with fluids with viscosity similar to water at room temperature and pressure. We show this with an example: in a typical microchannel with a diameter of 100 μm and water as a working fluid ( ρ ~ 1000 kg m−3 and μ ~ 0.001 Pa s) a flow rate of 150 μL min−1 results in a maximum velocity of ~0.375 m s−1, which is achieved at the channel center only 50 μm away from

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Critical review

the wall, and corresponds to Re ~ 25 (a common set of easily achievable conditions in inertial microfluidic systems). If one were to scale up all the elements to achieve a similar velocity gradient in a channel that is only 1 cm in diameter (i.e. 100× wider) the required maximum velocity would be ~37.5 m s−1, which corresponds to Re ~ 250 000! Even though such a flow rate is achievable (since the pressure to drive such a flow actually decreases due to the increased channel dimension), the flow would be highly chaotic and turbulent (Re ≫ 2300) not allowing the precise quantitative control achievable with laminar flow. In this paper we aim to revisit the fluid physics in such microscale systems operating in a laminar but finite Re regime and demonstrate how integration of the physical phenomena occurring in this scale introduces a quantitative framework to passively control and manipulate fluid and particles in channels, similar to the framework semiconductor physics provided for controlling the flow of electrons that revolutionized electronics. We first discuss straight microchannels and introduce the different mechanisms that can be used to cause lateral migration and control of particles due to fluid inertia, fluid viscoelasticity, particle shape and particle deformability. We also review the inter-particle interactions in these systems and discuss how particles can interestingly be used as “dynamic/moving structures” to manipulate fluid in straight microchannels. We then argue how channels can be modified to control the fluid and particles flowing inside. A straight channel with no boundary irregularities can only be used to manipulate particles, but not fluid. However, “structured” channels (i.e. channels with irregular geometry in a streamwise direction) can be used to manipulate both particles and fluid. Any deviation from a straight channel can be considered a structured channel, for instance the presence of curvature in a channel, or introduction of pillars and obstacles in the channel or grooves on the walls of the channel. We discuss how any geometrical deviation can induce net secondary flows (i.e. net recirculation perpendicular to the main flow direction) and present the different types of fluid/particle manipulation that can be achieved by engineering channel irregularities and structures. We discuss the importance of inertia in each of these cases and show how the two regimes, namely Stokes and inertial flow, can be used to achieve unprecedented control over fluid and particles in a completely passive manner. Throughout, we emphasize the differences between inertial and Stokes flow and connect the reversibility associated with Stokes flow to the “mirror symmetry time reversal” theorem,5 which leads to the conclusions that with fore-aft asymmetries even Stokes flow can cause net twisting of fluid streamlines while inertial flow can inherently lead to such flow deformations even in symmetric geometries. In combination with introducing the governing fluid physics for each of these systems we also introduce simplified “rules of thumb” (Tutorial Boxes 1–5) that we hope will aid novices in the design of inertial microfluidic systems.

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i. Particles in a straight channel A straight channel with a square/rectangular cross-section is the simplest form of conduit that can be used in microfluidics (although the concepts introduced in this section can be generalized to other cross-sections and geometry scales as well). Introduction of different forms of nonlinearity into these simple systems can produce lift forces (i.e. forces directed perpendicular to the main flow) that can be used to manipulate flowing particles.

a) Inertial migration and focusing Dominant inertial lift forces. In order to achieve spatial control of particles in microchannels, a mechanism of lateral migration is required; a manipulation tool that can be used to introduce order into an otherwise randomly distributed set of particles in a flow. Such a behavior was first reported by Segre and Silberberg6 where they observed that randomly distributed millimeter-sized particles migrated laterally to focus on an annulus with a radius ~0.6 times the radius from the center of a one-centimeter-diameter pipe. The lateral motion was unexpected at the time and proved the existence of some form of lift force acting on these particles. Additionally, since the particles reach a stable dynamic equilibrium at the annulus, there are possibly two (or more) types of opposing lift forces acting on the particles in the Poiseuille flow. Later theoretical analyses suggested7–9 that there are two opposing forces that dominate in this situation for neutrally buoyant particles: 1) the wall-induced lift force (or wall effect lift force), due to the interaction between the particle and the adjacent wall, which directs the particle away from the wall and 2) the shear gradient induced lift force, due to the curvature of the velocity profile, which directs the particle away from the channel center (Fig. 1A). Note that the finite and wall-directed gradient in the shear rate (second derivative of velocity) in Poiseuille flow (which is due to its natural parabolic shape) plays an important role leading to stable dynamic equilibrium positions. Weaker inertial lift forces. Additional lift forces also arise when the particle leads, lags, or rotates in the fluid, but these forces play a dominant role only in special circumstances. Saffman showed that a particle lagging/leading the fluid in Poiseuille flow experiences a channel center/wall-directed lift which is an inertial effect (slip-shear). Note that as a result of the presence of the walls, particles tend to lag the undisturbed flow velocity at the particle centerline in general in Poiseuille flow, however, these slip-shear effects scale with an order of magnitude higher a/H (ratio of particle diameter, a, to channel dimension, H).8,10 Hence, this type of lift is not generally applicable to inertial focusing systems given small a/H ≪ 1. However, in special circumstances, such as an additional force acting on a particle (e.g. electrical, gravitational, or magnetic), slip-shear lift can begin to dominate particle behavior.11 In another study, Rubinow and Keller described that the flow about a spinning sphere moving in

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Fig. 1 Inertial lift in straight channels. (A) Two lift forces act on a particle at finite Re (Rp > ~1) in Poiseuille flow, (i) a wall effect lift that pushes a particle away from the wall, and (ii) a shear gradient lift which pushes a particle away from the channel center. The balance between the two forces defines the equilibrium position for the particle. This phenomenon is called inertial focusing.3 (B) In square channels, inertial focusing creates four symmetric equilibrium positions. The force map acting on a particle in a channel cross-section obtained by finite element method solutions yields an equilibrium position approximately at ~0.6 of the distance from center to channel wall.14 (C) If the channel cross-section is rectangular, there are usually two preferred equilibrium positions. This can be explained by the force distribution across the channel, which almost always leads a randomly distributed particle to the stable equilibrium positions near the wider channel faces.16 (D) Increasing Re causes the focusing position to shift slightly away from the channel center. This presumably also leads to stabilization of the two previously unstable equilibrium positions in rectangular channels. (E) Based on the theory proposed by Ho and Leal8 the direction of the shear gradient induced lift depends on the shear rate and its gradient. For Poiseuille flow with a parabolic velocity profile, this force always points away from the channel center. However, in an arbitrary velocity profile with an inflection point, the direction of the force could change across the channel.

an unbounded and stationary fluid exerts a lift force on it (slip-spin),12 originating from rotational differences between the sphere and underlying flow.8 Should this lift be applied to a particle in Poiseuille flow the particle would be directed towards the channel center. Following the analysis of Ho and Leal, slip-spin effects contribute minimally compared to wall-effect and shear-gradient lift in Poiseuille flow, scaling with three orders higher a/H8,12 given that the particle rotation will generally match the undisturbed rotation in the fluid. The origins and directions of action for weaker and dominant lift are tabulated in Table S1.† Inertial lift scaling. The scaling of inertial lift forces has recently become more clearly elucidated to depend on the finite size of particles. Following Hood, Lee and Roper, for small particles (a/H ≪ 1) lift scales as FL∝ρU 2a 4/H 2 but shifts to scale less strongly with particle diameter as the particle size becomes more comparable to the channel dimension (a common occurrence for many microfluidic flows).13 For a finite-size particle (a/H between 0.05 and 0.2), at Re of 20 and 80, finite element method simulations yielded a net lift acting on the particle scaling as FL∝ρU 2a 3 /H near the channel center and as FL∝ρU 2 a 6/H 4 near the channel wall. The dimensionless lift coefficient fL remains a weak function of Re and strongly depends on position within the channel.14

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Focusing in square and rectangular channels. As a result of these lift forces, for a particle Reynolds number, Rp of order of equal or larger than 1 (where Rp = Re(a/H)2 = ρUa 2/μH) the randomly dispersed particles in a square microchannel inertially focus to four symmetric equilibrium positions near the center of each channel wall face and, in agreement with Segre and Silberberg's observation, ~0.6(H/2) away from the channel center (Fig. 1B) as confirmed experimentally.2,15 The finite element solutions also provide a clear map of the lift imposed on the particle in these channels which is in agreement with experimental and theoretical work (Fig. 1B). Interestingly, when particles are introduced into a channel with rectangular cross-section (without the four-fold symmetry of a square) the number of favored dynamic equilibrium positions is decreased to two, located near the wider faces of the channel (Fig. 1C). Numerical solutions show that the force field present in the channel results in a set of stable equilibrium positions and saddle points in the channel.16 To understand how this can lead to two points of dominant equilibrium, we can think of releasing a particle at a point in a quadrant of the channel and assessing the probability of it approaching either of the two reachable equilibrium positions (i.e. on the long and the short face). In such a cross-section (Fig. 1C) the shear gradient lift force, which is

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Critical review

dominant anywhere away from the walls, is stronger in the z-direction but much weaker in the y-direction (due to the blunted velocity profile in the y-direction). Therefore, a particle that is randomly released in this channel has a greater chance of being pushed away from channel's y-centerline (blue line). Consequently, the particle will approach the long channel face (green line) in the z-direction first and then due to dominant wall-effect lift is directed towards the centerline (red line) in the y-direction until it settles at its stable equilibrium position (Fig. 1C). Along the channel face (green line), wall-effect lift dominates likely because of the lower fluid velocity – and local fluid inertia – in this region, as described further below. Overall, for a moderate Re the lift forces in this cross-section lead to limited particle paths that arrive at the other equilibrium positions at the short channel faces. Significant biasing to two focusing positions can be observed for even slight deviations from square geometries (e.g. aspect ratio of 3/2). Effect of Re on focusing positions. At higher Re (up to ~150) the equilibrium positions in square and rectangular channels tend to shift slightly towards the walls (Fig. 1D). This also can be explained by taking the two competing lift forces into account. It is clear that both the wall effect lift and the shear gradient lift increase with increasing velocity or Re. However, the increase in shear gradient lift is relatively larger for a given increase in Re.7,16 Therefore, increasing Re causes the shear gradient lift to become more dominant and, consequently, the particle equilibrium positions shift closer to the wall. A similar behavior occurs in rectangular channels, in which the two previously unstable positions on the short faces now collect additional particles, as the wall effect becomes less dominant in directing particles away along the long face, leading to an increase in the total number of focusing positions to four (Fig. 1D).17,18 Therefore, the distribution of lift forces within the channel is altered such that the competing y-directed shear gradient lift force can direct particles into the previously unstable/unreachable equilibrium position. This was found to occur at above Rp ~ 4.5–6 experimentally.17,18 Above Rp ~ 8, equilibrium positions were observed to shift again but now towards streamlines closer to the center of the channel.18 We propose a similar argument can provide intuition about the local relative importance of wall-effect and sheargradient lift within a channel in which the cross-sectional flow shape remains the same. The local Re, i.e. local relative importance of inertial to viscous effects, can be greatly variable in a quarter of the channel (Fig. 1C) due to the large velocity differences across a channel. Our conjecture is that due to relatively higher velocities near the channel center (blue line), shear gradient lift is dominant in this region, thus directing particles towards the wall in the y-direction. On the other hand, flow is slower near the walls (green line) due to the local no-slip condition. Therefore, wall effect lift dominates directing the particles back towards the channel center (negative y-direction). Effect of a/H on focusing positions. Particle size can also affect the focusing position. For a/H ≪ 1 the focusing

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position approaches ~0.6(H/2) as in the original studies of Segre and Silberberg. But both experimental and numerical results indicate that the equilibrium position of particles shift towards the channel center as a/H approaches 1.14,17,19 This is largely due to the fact that steric effects become more important as a/H increases, especially above ~0.4. For instance, in the limiting case when a/H = 1 the particle is geometrically forced to pass through the channel centerline. Particle concentration effects. It has also been shown that both the location and the number of focusing positions depend on the number of particles per unit length along the channel,20 such that at high length fractions, i.e. φ > ~75% (φ, length fraction, defined as the fraction of particle diameters per channel length)3 multiple streams are observed across the channel. At these increased concentrations, if the particles are to occupy two to four focused streams they are forced to stay in close proximity to each other. In this case, there will be a large increase in particle–particle hydrodynamic interactions which have been shown to be important once particles are within a short distance of one another (< ~10a).21 As a result, a portion of the particles are pushed out of the lift-specified focusing streams and form new nearby focusing streams. These particle–particle interactions act as one limit on achieving precision focusing of all particles in a channel. Building physical intuition for inertial lift. Despite multiple works discussing the lift forces on particles in channel flow,7,8,22 there is no simple intuitive explanation for the physical origins of inertial lift. In their theoretical work on the lift forces exerted on infinitely small particles, Leal and Ho8 conclude that the lateral force mainly originates from the shear field acting on the sphere and the induced disturbance flow (i.e. flow difference due to the presence of a particle in the stream) rather than the presence of wallinduced lag velocity (Saffman lift/slip-shear) or slip-spin, in agreement with our previous description. They decompose net lift into “wall-effect” which is the interaction of the disturbance stresslet and its wall reflection interacting with the bulk shear, and “shear-gradient lift” which is the interaction between the stresslet and the curvature of the bulk velocity profile. A stresslet, with its characteristic orthogonal “flowin–flow-out” quadrapolar pattern, can be shown to be related to the disturbance around a particle in a flow with finite rate of strain. Rigid particles induce stresslets in such flows because they do not deform appreciably with the underlying flow. To maintain their shape the particle must apply a force field back on the flow that resists the underlying flow deformation, resulting in a stresslet.23 Furthermore, they conclude that the direction of the net lateral force depends on the direction of both the shear rate and its gradient, and arises from the interaction of the local disturbance velocity field with both the shear rate and its gradient. For instance, in a Poiseuille flow in a rectangular channel the shear rate is always positive and its gradient negative as an inherent characteristic of the shape of the velocity profile (Fig. 1E).

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Therefore, the shear gradient induced lift is always directed towards the wall. However, if the suspension was to flow through a channel with a more complex cross-sectional shape, such as the one shown in Fig. 1E (bottom), the velocity profile correspondingly becomes more complex and even possesses an inflection point. In this case, the direction of shear rate and shear rate gradient could alter across the channel width which, based on Leal and Ho's analysis, leads to shear gradient lift forces that change signs across the channel. Numerical predictions and experimental results with additional channel shapes will likely clarify the importance of this effect, and have the potential to open up a whole new set of design tools for particle manipulation and control based on velocity field sculpting. Length required for focusing. Considering shear-gradient lift alone we have previously presented an expression based on the cross-streamline motion of finite-sized particles3 that proscribes a channel length required for particles to reach lateral equilibrium positions (Lf) in rectangular channels Lf 

 h 2 , where Um is the maximum channel U ma2 fL

velocity (~1.5U, the mean channel velocity). The average fL is about 0.02–0.05 for channel aspect ratios (height/width) between 2 and 0.5. For instance, let us consider focusing of 10 μm particles in a 40 × 60 μm2 rectangular channel with water as working fluid, flowing at 150 μL min−1. In this case Rp ~ 3.3 which is not high enough to give rise to four stable equilibrium positions. Therefore, particles focus along the

longer face, i.e. along w = 60 μm. A simple calculation shows Um ~ 1.56 m s−1 and h/w = 0.66. Therefore, assuming fL ~ 0.04, the channel length required for particle focusing is Lf ~ 2 cm. Entry length required for focusing has also been explored experimentally using holographic imaging of particles,24 obtaining general agreement with this basic equation. Also in agreement with our discussions above, Choi et al.24 found that migration down the shear-gradient, for which Lf is calculated, preceded wall-induced migration to the center of channel faces, suggesting that to describe complete focusing further work to develop a more accurate approximation for Lf will be required. Applications of inertial focusing. The simplicity of the phenomenon suggests its ability to impact many applications. For instance, since there are no extra force fields or devices required (only the channel itself is needed for alignment of particles), this method can be easily parallelized by branching off many channels from a single inlet and arraying them in close proximity. This leads to extremely high throughput particle focusing (Fig. 2A) which has been used for highthroughput sheath-less flow cytometry.17,18,25 This may be particularly useful in reducing the size and cost of operation of flow cytometers, which normally require a large bulk of aqueous sheath fluid to pinch cell streams prior to optical interrogation. Remaining challenges to overcome to speed adoption for this application include focusing of cell sized particles in channels with larger channel dimensions26 (currently precise focusing requires channels with dimensions approximately less than 5–7 times particle diameters).

Fig. 2 Applications of inertial microfluidics. (A) Massive parallelization of particle focusing channels is possible due to the passive nature of inertial focusing, which enables extremely high-throughput flow cytometry.17 (B) Kinetic size-based separation of smaller pathogenic bacteria from larger blood cells is achieved by controlling the separation distance to focus larger blood cells only.27 (C) Lateral migration of particles to reach stable equilibrium positions in low aspect ratio channels is used for particle exchange across fluid streams.16

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Critical review

Achieving this would require less frequent replacement of the focusing chip and lower pressure pumping systems. Going beyond the use of straight rectangular channels, as discussed below, can aid in this direction. Inertial focusing in straight channels has also been used to separate particles or pathogenic bacteria cells from diluted blood based on differential migration rates (kinetic separation, Fig. 2B).27–29 In this simple design, the larger blood cells or particles become focused in a shorter channel length following the aforementioned equation for Lf, while the smaller bacteria or particles are less affected by the inertial lift forces and remain dispersed throughout the channel (due to their smaller sizes). These approaches also all rely on the reduced number of equilibrium positions in high-aspect ratio rectangular channels which enable two outer collection outlets. In the work from Mach et al. a gradual expansion of the channel width is also used such that the focused cells are then moved to a position closer to the side walls (while maintaining their focused state), leaving a large portion of the channel cross-section free of blood cells. The focused streams are then captured with side outlets while the remaining fluid that contains most of the bacteria is captured separately. An advantage of separations in straight channel designs is the ability to radially parallelize the system which has been shown to process up to 240 mL h−1 with a throughput of 400 million cells min−1 and with >80% bacteria removal from blood after two passes.27 As with focusing, inter-particle interactions at high blood cell concentrations can prevent precise focusing and reduce the accuracy of separation, such that separations using inertial lift forces operate best in semi-dilute solutions with length fractions below ~50–60%. Lateral inertial migration across laminar co-flowing streams has also been used to achieve fast fluid exchange around cells and particles.16 This technique relies on the intelligent use of the presence of two (rather than four) stable equilibrium positions in low aspect ratio channels. Here, the particle suspension is co-flowed with an exchange solution that has a higher flow rate (~2-fold) into a low aspect ratio channel (i.e. h/w smaller than 1 with w in the direction orthogonal to the line of contact of streams). Once in this main channel, particles start to migrate towards new stable equilibrium positions in the center of the new channel, which due to the asymmetry in the rates of the co-flows is located along streamlines occupied by the exchange solution (Fig. 2C). Consequently, with an appropriate outlet design the particles can be captured in the exchange solution with high purity, high yield and at a high rate (which is a general characteristic of inertial microfluidic platforms). The high rate necessary for operation is not ideal for exchanging solution around one or a few cells from a rare sample, which would require much more delicate instrumentation, but is ideal for applications requiring millisecond solution exchange around hundreds to thousands of cells or particles per second, with the capability of downstream inline analysis.

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High throughput is a primary advantage of inertial microfluidic platforms which makes them compatible with other high-speed platforms. For instance, in a recent development, cancer and blood cells inertially focused in straight channels were analyzed by a novel line-scan based high-speed imaging technology (serial time-encoded amplified microscopy – STEAM) and achieved nearly 100 000 particle s−1 real-time image-based screening (compared to the state of the art ~1000 particle s−1).30 In this system, particles, including blood cells and budding yeast, are first ordered and focused in a inertial microfluidic channel fabricated in thermosetting polyester (TPE),31 to prevent fluctuations in channel dimensions in softer polymer channels that affect the optical signal quality, and achieve higher pressure flows without device failure. The typically high operating flow rates of inertial microfluidics platforms leads to very high pressure drops in these channels. In practice, the inability of Polydimethylsiloxane (PDMS)-based devices, which are the gold standard for rapid and cheap device fabrication, to withhold such high pressure drops acts as one of the limiting factors, especially in prototyping and research development. Therefore, the development of stiff material that can withstand higher pressure drops for rapid prototyping of microfluidic devices is important and could push the limits on both research and application fronts.31 Inertial microfluidics can also be integrated with other microfluidic schemes to offer more comprehensive solutions. For instance, Toner et al. combined different effects in their device where, first, small red blood cells were separated from blood using size-based deterministic lateral displacement,32 followed by inertial focusing of remaining cells which are further distinguished downstream using magnetophoresis.33 Although inertial focusing is well-suited for dilute solutions a useful frontier is investigating focusing and separation in concentrated particulate solutions to address the challenge of focusing and separation of cell populations present in whole blood. Development of novel techniques and evaluation approaches that are better compatible with high cell concentrations has aided in this endeavor. For instance, conventional imaging techniques used for studying inertial focusing – including high-speed bright-field imaging and long exposure fluorescence imaging – are generally limited in complex fluids such as whole blood due to interference from the large numbers of red blood cells. In a recent study, Toner et al. applied particle trajectory analysis (PTA) as a means to observe inertial migration of particles in dilute and whole blood. In this technique, fluorescently labeled particles are suspended in the biological fluid, and images taken at multiple vertical positions in the channel, using extremely short (~10 ns) but intense pulses, are used to find in-focus particles and determine their diameter and two-dimensional spatial coordinates in the channel cross-section. This led to uncovering a radical shift in focusing behavior of PC-3 prostate cancer cells in whole blood.34 Tanaka et al. also studied migration of cancer cells in concentrated blood cell suspensions up to 40% hematocrit, but results suggested minimal or

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abnormal migration, as determined by reduced separation purity downstream, until hematocrit was decreased to about 10%.35 An understanding of inertial focusing of rigid spherical particles in straight rectangular channels provides a solid foundation for applications in focusing, separation, enrichment, and ordering of cells and particles. Our current understanding is summarized in Table S2,† including what has been proven analytically, numerically, and experimentally. A more complete understanding of migration behavior for realworld particles and applications requires additional knowledge of how fluid properties, particle shape and deformability, and channel structures contribute to overall motion.

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b) Non-Newtonian fluids Lateral migration in non-Newtonian fluid. In addition to finite-Re flow conditions, non-Newtonian fluids have long been known to cause lateral migration of single particles in a pressure-driven channel flow. The focusing positions for particles in a non-Newtonian flow depend on the rheological properties of the fluid. We will briefly touch upon the viscoelastic lift force that can be used (or must be taken into account for some fluids) to manipulate particles in inertial microfluidic systems. Particles migrate typically towards the center or the wall of a pressure driven channel flow, depending on rheological properties of fluid (i.e. shear thinning or shear thickening) and the channel blockage ratio (α = a/H).36,37 Ho and Leal predicted that a non-uniform normal stress distribution in a second-order fluid results in lateral particle migration.38 A second order fluid is defined to have stresses in the fluid that are not linearly dependent on the shear rate – as in a Newtonian fluid. Particles migrate in the direction of decreasing absolute shear rate, which is towards the center of the channel for a Poiseuille flow.  The non-dimensional Weissenberg number ( Wi   where  is the shear rate and λ is the fluid relaxation time) is often used to characterize the viscoelasticity of a fluid.

Critical review

For a rectangular channel, average shear rate becomes U 2U 2Q and Wi can be expressed as Wi  where Q is  h/ 2 h wh 2 the flow rate. The Elasticity number, El is the ratio between Wi and Re, and indicates the relative strength of elastic forces to inertial effects. For a rectangular channel El 

Wi   h  w  .   h2w Re

As both Wi and Re are proportional to the flow rate, El becomes independent of the flow rate. This assumption would be valid if the viscosity is constant. However, in many cases viscosity is not constant due to shear thinning and thickening. In recent years, experimental and numerical studies have demonstrated the ability to apply viscoelasticity-induced forces for particle focusing and separation.39–44 These forces have also been combined with inertial lift forces to enable higher throughput manipulation of particles.42–44 Applications of elasto-inertial flows. In a circular tube, (e.g. glass capillary43) particles in viscoelastic flow migrate toward the center of the tube for a wide range of Wi and α, these parameters can be tuned to achieve a bi-stable condition where particles migrate both towards the channel center and the walls depending on the initial position of the particle. In this study, Re was sufficiently small that inertial effects could be ignored, which also makes this technique more suitable for applications requiring small volume operations. With typical microfluidic channels of rectangular cross-section, elastic force focuses particles to the center and the four corners instead.42,44 The addition of inertial lift forces that focus particles near the walls (Fig. 3A) can lead to a reduction in the number of focusing positions. By controlling flow rate, inertial lift forces and viscoelastic lift forces can be carefully balanced to achieve an Elasticity number that yields a single line of focusing along the channel center-line (Fig. 3A and B).42 Single-line focusing without external fields or sheath fluid has advantages in application areas such as miniaturization of flow cytometers. The viscoelastic flows that yield single-line focusing have Re ~ O(1).

Tutorial Box 1 – Practical design rules and considerations for inertial microfluidic systems: inertial focusing in straight channels Due to the complexity of inertial microfluidic physics and interplay of multitude of physical parameters, design of an effective inertial microfluidic platform could be challenging for the beginner. However, here we provide simplified rules and general considerations for these designs in five Tutorial Boxes. These estimates are meant to provide the reader with a reasonable idea of how and to what extent different physical parameters in their experimental design affect particle and fluid behavior. Note that the accuracy of these assumptions have not been rigorously and universally tested experimentally, yet the proposed rules serve as a decent starting point. The channel length required for particles to reach lateral equilibrium positions (Lf) in rectangular channels is

Lf 

 h 2 U m a 2 f L

, where U m is the

maximum channel velocity (~1.5U, the mean channel velocity). The average f L is about 0.02–0.05 for channel aspect ratios (height/width) between 2 and 0.5. Consequently, in order to achieve inertial focusing over a length of L, a flow rate of

Q

2 wh 2 3 La 2 f L

, where the 2/3 multiplier comes from the

conversion of the maximum velocity to the average velocity in the flow through the straight channel.

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Fig. 3 Elasto-inertial flows. (A) While inertial effects create four focusing positions near the center of channel walls (left), elastic properties of non-Newtonian fluids are shown to cause focusing near the corners as well as the center of the channel (middle). The combined effect of the two, called elasto-inertial flow (i.e. presence of inertial effects in non-Newtonian fluids), can lead to single stream focusing of particles (right).42 (B) Inertial and viscoelastic lift forces are dependent on particle size and the combined force is predicted to have a quadratic dependence of migration velocity on particle size. This effect has been used for kinetic size-based separation of microparticles.44

Considering inertial focusing (without viscoelastic effects) applied to flow cytometry applications have operated at Re ~ O (10–100),45,46 viscoelastic focusing provides 1–2 orders of magnitude slower flow rates. Increases in flow rate will result in higher Re, which in turn requires higher relaxation time materials or viscosity to balance inertial lift forces and viscoelastic forces. In practice, high viscosity fluid limits the flow rate due to high pressure for driving flow, however, using very dilute DNA (0.0005 (w/v)%) solutions that possess a longer relaxation time47 may be applied for particle focusing over a wide range of flow rates. The λ-DNA that was used in Kang et al. has a relaxation time of 140 ms, while poly(ethyleneoxide) (PEO) that is water-soluble synthetic polymer with similar contour length with λ-DNA

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(16.4 μm) has a relaxation time of 0.7 ms. Viscoelastic focusing can also be applied to separate particles of different sizes using the fact that lift forces (inertial and viscoelastic) are dependent on particle size39,44 leading to a predicted quadratic dependence of migration velocity on particle size. Small particles (2.4 μm) are relatively less focused and the larger particles are tightly focused at the center of the channel. Similarly, 1 μm and 5 μm particles were separated by introducing particles (Fig. 3B)44 in a side channel and relying on differences in the migration rate towards the center of the channel (a kinetic separation48). Effect of fluid viscosity on throughput. High flow rate (typically >10 μL min−1) is one important feature of inertial

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microfluidic systems but is more challenging with viscoelastic focusing systems. Addition of polymers to utilize viscoelastic lift forces leads to an increase in fluid viscosity and the high viscosity limits maximum flow rates and potential applications of the manipulation scheme in areas that can benefit from high-throughput. The change in viscosity of shear-thinning fluids and shear-thickening fluids will also affect the maximum throughput. Shear-thinning and -thickening fluids are defined as fluids having a viscosity that decrease or increase in magnitude with increasing shear stress respectively. Use of shear thinning fluids such as blood and typical polymer solutions will result in lower fluidic resistance at higher flow rate which could help to increase maximum throughput. For the same reason, use of shear thickening fluid will be disadvantageous for high throughput applications. On the contrary, these viscoelastic focusing approaches may be better suited to deal with small samples, as the sample would not be wasted during the start-up time required to achieve the focusing flow rate. Microchannels that can withstand high pressure31 may also be used to allow higher maximum flow rates with non-Newtonian fluids. Experimental and theoretical studies of non-Newtonian fluids in microchannels are expected to attract more attention over the coming years to (i) develop a systematic understanding of viscoelastic effects acting on particles for fluids that can be viscoelastic in many different ways (with a range of non-Newtonian constitutive equations), and (ii) engineer structured or shaped channels49 that can introduce additional forces to achieve further separation and focusing control.

c) Shaped particles (with or without fluid inertia) Non-spherical particles in real systems. A sphere is the ideal shape for a particle in many cases and studies owing to its ultimate symmetry and lack of singularities, which lead to mathematical simplicity. Therefore, in a large portion of the work describing particulate flows (theoretical, numerical and experimental), a sphere has been used as a simplified case, an approximation. However, in a large portion of real systems non-spherical particles exist. For instance, a human red blood cell (RBC) adopts a discoid shape in the body, far from being a sphere. Shape represents an important factor to specifically identify a bioparticle,50 including bacteria, viral particles, budding yeast, and marine micro-organisms. Synthesized barcoded particles represent another example of particles that would be helpful to operate on and manipulate.51–53 Therefore, it is important to improve our understanding of systems that contain shaped particles and the corresponding lift forces and dynamics of these particles. One of the classic works in this area was the investigation of the motion of ellipsoidal particles in unbounded shear flow in a Stokes flow regime by Jeffery54 which led to the discovery of the so-called “Jeffery orbit” in which a particle rotates in a closed repeating cycle around any of an infinite set of axes that depend on its initial orientation.55 Here we discuss how particle shape and fluid inertia combined can affect particle

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motion, focusing and separation dynamics in microchannel systems. Mirror symmetry time reversal theorem. An interesting point about shaped particles is that they can be subject to lift forces even in the absence of inertia (i.e. Stokes flow) which can be intuitively described via an exception to Bretherton's “mirror symmetry time reversal” theorem (MSTR theorem).5 Simply put, the theorem indicates that if a sufficiently symmetric particle moves in a uni-directional viscous shear flow without inertia and we momentarily reverse time, the resulting configuration (in terms of the flow field, pressure field, forces, etc.) is mirror symmetric to another configuration of the particle that is rotated, at some angle, around the rotational axis defined by its Jeffery orbit. We will define what a “sufficiently symmetric” particle entails in the following discussion. This is due to the linearity of the Stokes equations, and a full proof can be found in Bretherton's paper.5 The MSTR theorem, can also apply to a channel flow in which a velocity gradient is also present leading to particle rotation.56 For a rigid spherical particle, the MSTR theorem provides a proof that there can be no lift forces acting on the particle in Stokes flow for a simple channel (Fig. 4A). If we assume  there is a net lift on the particle ( FL ), then due to the linearity of the equations of motion there is an opposite directed   force ( FL   FL ) if we reverse the time. However, the initial and time-reserved configurations are exactly the same in every way (due to the fore-aft symmetry of the system). There  fore the two forces need to be equal ( FL   FL ) which is only  satisfied for FL  0 Hence, there can be no lift on a spherical particle in Stokes flow through a channel. If the particle has an ellipsoidal shape, the scenario is slightly different (Fig. 4B): an ellipsoidal particle follows an orbit similar to a Jeffery orbit (Fig. 4B.i). If time is reversed, the direction of the instantaneous lift force is also reversed (Fig. 4B.ii). The configuration that is mirror symmetric (Fig. 4B.iii), however, is in fact a rotation of the original configuration. Therefore, instantaneous lift can possibly exist during the particle's rotational cycle. However, the lift force at any point has a complete counterpart during the cycle with opposite directed lift, thus there can be no net lift on the particle during a full orbit. This result requires that the particle is sufficiently symmetric, that is, the particle is mirror symmetric to a rotation around its rotational axis. Classes of shapes that can satisfy this condition include disks, cylinders, rectangular prisms, triangular prisms, and cones. For a rotationally asymmetric particle, the MSTR theorem cannot be applied (Fig. 4C) and a net lift force is possible for a rotating particle. For example if we have an h-shaped particle in the channel (i) and we reverse time (ii), the mirrorsymmetric configuration is impossible to achieve by any degree of rotation. Therefore, the MSTR theorem does not apply to such asymmetric particles, and cannot be used to

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Fig. 4 Lift forces on shaped particles in straight channels. A, B and C are in the frame of reference of the channel. (A) The mirror symmetry time reversal (MSTR) theorem teaches that there is no lift possible on a spherical particle in Stokes flow. (B) The MSTR theorem also applies for ellipsoidal particles that are rotationally symmetric. In this case, while instantaneous lift is possible, no net lift can be induced over a complete rotation of the particle. (C) However, the MSTR theorem does not apply to classes of asymmetric shaped particles such as an h-shaped particle, since the mirror symmetry of the time-reversed configuration cannot be achieved through rotation of the original particle. Therefore, a net lift is allowed in this case. (D) Lift forces and the focusing behavior of various shapes of rigid particles such as symmetric disks (ii), cylinders (iii) and asymmetric disks (iv) have been experimentally studied.59 (E) These studies reveal that most shaped particles follow the focusing behavior of a spherical particle of the equivalent rotational diameter, with the exception of the rotationally asymmetric h-shaped particle.59 (F) Ellipsoidal particles with higher aspect ratio focus closer to the center line compared to spheres of the same volume. The dependence of focusing position on the shape and aspect ratio of particles has been used for shape-based separation of budding yeasts.57

eliminate the possibility of a net lift force acting on the particle throughout its rotational cycle, even in Stokes flow. Clearly, the rotational axis of the particle is important in defining the lift forces and subsequent motion of a particle, and this preferred rotational axis can be set due to the presence of fluid inertia. In Stokes flow we know that particles can occupy an infinite set of rotational axes that depend on initial conditions. This situation changes due to the presence of fluid inertia.57,58 The addition of inertia and channel confinement leads to the precession of particle orbits to a stable rotational axis in the parabolic flow. Inertial focusing of shaped particles. In recent works, Hur et al. and Masaeli et al. used straight microchannels (Fig. 4D.i) to experimentally study the focusing behavior of multiple types of particles such as symmetric disks (Fig. 4D.ii), cylinders (Fig. 4D.iii), ellipsoids (Fig. 4F), and asymmetric disks (Fig. 4D.iv).57,59 Above a critical Reynolds number (Re ~ 50) particles predominantly adopted preferred

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equilibrium positions and “tumbling” rotational axes. For particles symmetric about this rotational axis focusing position was found to be independent of cross-sectional shape and mainly depends on rotational diameter (Dmax). During the tumbling motion, when the major axis rotates to be perpendicular to the wall, wall effect lift increases substantially due to the closer distance. As a result, the integrated wall effect (over time) on these particles is higher than that of spheres of the same volume. Interestingly, only the highly asymmetric h-shaped disk did not follow this trend and focused closer to the walls than expected for its Dmax, in agreement with the MSTR theorem allowing an additional lift force acting on this rotationally asymmetric particle (Fig. 4E). Interestingly, if complete particle rotation is prevented (due to channel confinement or other forces) lift acting on shaped particles can be employed to direct particles to locations in the channel cross-section in a shapedependent manner.59,60

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Shape-based differential focusing has potential applications, and has already been used to sort and separate budding yeast at the high rate of 1500 cells s−1, an order of magnitude improvement from previous methods.57

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d) Deformable particles Models for particle deformability. In many studies of particulate systems, solid rigid particles are used as simple components to study system behavior. However, bioparticles such as cells and vesicles, or two-phase emulsion droplets are not rigid, but deformable. Importantly, deformability of particles can create nonlinearity in these systems which can lead to additional lift forces on these particles61 (Table S1†). Three types of models are generally used to approximate complex bioparticles or more simple emulsions when studying their motion in flow. Models approximating a deformable particle in a shear flow are (i) an elastic solid particle, (ii) a deformable drop, and (iii) a deformable capsule (Fig. 5A). (i) An elastic solid particle:62 the whole particle is considered as a single solid body which can deform with a particular elastic constant. In this case, for the flowing particle to retain a steady-state shape it needs to constantly deform or “tanktread” as it is rotating under the shear flow. (ii) A deformable drop:61,63–66 the particle is assumed to be a drop with fluid on the inside and a boundary that can slip under the driving flow. In this case, recirculation of fluid can occur inside the droplet due to the outer flow and the elasticity of the droplet is defined by its surface tension. (iii) A deformable capsule:67–69 the inner fluid is covered by a deformable but solid shell with a specified elastic constant. The solid shell undergoes a tank-treading motion to maintain a steady-state deformed shape while fluid recirculation can occur inside the capsule. Effect of particle deformability on lateral migration. These studies suggest that in a shear or Poiseuille flow, regardless of the approach to approximate deformability, there is a component of force perpendicular to the free-stream direction that causes the particle to migrate across undisturbed streamlines. This lateral migration is argued to be due to the shape-change of the particle and can occur even in the absence of inertia.62 This links with our discussion of shaped particles in which instantaneous lift can act on a particle in Stokes flow (Fig. 4B). The difference in this case is that the deformable particle in many cases does not tumble to occupy mirror symmetric configurations but rather can perform a tank-treading motion such that it maintains a steady-state shape as it flows downstream. Therefore, in contrast with a rigid elongated rotational symmetric particle, a deformable particle can have a net lift leading to its lateral migration. In addition to lift caused by the deformed shape, lateral migration of deformable particles is reported to also arise from nonlinearities caused by the matching of velocities and stresses at the deformable particle/droplet interface.70 In general this means that the magnitude of lateral drift is related again to the deformed shape of the particle or gradients in

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Fig. 5 Deformable particles. (A) Three models for deformable particles include solid elastic particles, deformable droplets (with fluid interior and a slipping interface) and deformable capsules (with fluid interior but an elastic solid boundary). (B) Deformability of a particle in Poiseuille flow generally leads to an additional lift force which directs the particle away from the wall. Balance of these forces creates a focusing position closer to the channel center, compared to rigid solid spherical particles.61 (C) Experimental data shows that, in general, increasing deformability corresponds with a focusing position closer to the center (for λd from ∞ to ~4.6). There is a shift in the trend when drop viscosity approaches the underlying fluid viscosity (λd < ~4.6).61

tension at the interface. For droplets with a surface tension, σ, the Weber number, We 

U 2 a (inertial stress vs. surface 

Ua (viscous stress vs. h surface tension) provide dimensionless parameters that characterizes the relative deformation for a droplet. The viscosity

tension), or capillary number, Ca 

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ratio, λd = μd/μ where μd is the dynamic viscosity of fluid inside the droplet, is another important parameter in defining the shape of the deformed drop. By measuring the shifting balance between deformability-induced lift and inertial lift, Hur et al. experimentally demonstrated that deformabilityinduced lift increased with increasing drop deformation. Droplets shifted towards the channel center as λd decreases from 970 to 4.6 (i.e. more deformable particle). Counterintuitively, as the viscosity ratio decreased below λd < 4.6, the focusing position shifted back towards the wall (Fig. 5C), in agreement with theoretical predictions70 that may also implicate strong internal circulations within a drop affecting lift. A recent study by Stan et al.66 also suggests that interfacial effects (e.g. gradients in surface tension that develop in the shear field) yield lift forces that can play a dominant role especially in flows of bubbles or droplets. In fact, lift arising from surface tension gradients is starting to be actively controlled and used to steer droplets.71 Similar to droplets, numerical studies on deformable capsules in inertial flows69 suggest that particle equilibrium position depends on the shell compliance and particles with more flexible shells move closer to the channel center. Scaling of deformability-induced lift. As Ca increases and Rp decreases deformability-induced lift begins to dominate motion of drops and bubbles in microchannels. Chan and Leal give the conditions λd < 1 and λd > 10 for deformabilityinduced lift force being directed towards center of the channel. Their analytical study suggests that when a drop or bubble is not too close to the walls (e.g. the separation between the drop and the wall is larger than a) the deformability3

induced lift force scales as FL,d  CaUa  a   d  f  d  , H H where d is the distance between drop and center of the channel.72 Experimental measurements of inertial lift force in microchannels by Di Carlo et al.14 have also been used to extract an equation for inertial force near the center of the channel.66 For d/H < 0.1 the negative sign indicates that this force is directed towards the walls and fits d  center FL,inertial  10 R p Ua   . When both drop deformation and H inertial forces are small (for Ca < 0.01 and Rp < 0.01), Stan et al.66 suggest an empirical formula for net lift force on bub3

a d ble or droplet as FL,empirical  CL Ua     , where CL is an H H empirical factor which has to be determined experimentally for a given pair of carrier liquid and drop or bubble fluid. However, for order-of-magnitude estimates CL ~30 can be used. This formula is applicable for drops or bubbles that are (nearly) spherical (since Ca < 0.01) and positioned near the channel center (d < ~0.15H). A summary of our current understanding concerning deformable particles is provided in Table S2.† In Poiseuille flow, the direction of deformability-induced lift force is towards the center of the channel under most

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operating conditions, owing to the parabolic velocity profile.62 This lift force is attributed as a factor responsible for the cell-free layer in channel flow of red blood cells even with negligible fluid inertia.63 In inertial flow, deformabilityinduced lift leads to equilibrium positions closer to the channel center for deformable particles in comparison with rigid particles61 (Fig. 5B). If the force balance is calibrated precisely this effect can be used to measure the mechanical properties of the suspended cells as it correlates with equilibrium position in a microchannel.61 Furthermore, particle deformability can also affect the focusing dynamics and the velocity of lateral migration in inertial flows. Center-directed deformability-induced lift combined with wall-directed shear-gradient lift leads to an overall wall-directed lift that is lower in magnitude. This would lead to slower migration to equilibrium positions which was reported in recent studies that found that the migration of cancer cells (which are deformable) required a longer channel length than for rigid beads of the same size.35

ii. Effects of particles on flow The two main elements of particle-laden fluidic systems are the fluid and the particles with several possible interactions between them. There have been multiple studies on the effect of the fluid on particles, and we also discussed the interactions between particles. The other important interaction is the effect of particles on the fluid flow which we discuss here. In general, particles significantly perturb the fluid around them and cannot be considered passive components of a channel system. a) Reversing streamlines Reversing streamlines are one of the distinguishing features of the flow field around a freely-rotating particle placed in a shear flow with finite Reynolds number. By reversing streamlines we refer to fluid that approaches and then moves away from a particle in its reference frame (Fig. 6). Normally one would expect fluid to be diverted around a rigid particle and return to a similar location upon passing the particle and continue in the same direction as shown in Fig. 6A, however, under conditions described in this section the fluid can change flow direction near the particle and form reversing streamlines as shown in Fig. 6B. The reversing streamlines can lead to repulsive particle–particle interactions.21,73,74 Acrivos et al. showed that streamlines near a cylinder in a simple 2D shear flow differs in the case of inertial flow and Stokes flow.75 The region of closed streamlines diminishes with increasing Reynolds number and reversing streamlines appear due to fluid inertia (Fig. 6A, B). Later they experimentally confirmed the presence of reversing streamlines around a cylinder and a sphere.76 Similar reversing streamline formation for finite Reynolds number flows were also found in later studies.74,77–79 Fluid inertia leading to reversing streamlines also resulted in breaking down of closed stream lines such that fluids around the particle spiral outwards

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Fig. 6 Effect of particles on flow. (A) There will be only closed streamlines near a particle in a simple shear flow with no confinement (i.e. no reversing streamlines) under Stokes flow conditions.78 (B) However, the addition of fluid inertia to the system (without confinement, unbounded flow) leads to creation of reversing streamlines and saddle points near the particle.78 (C) Channel confinement, without fluid inertia, also leads to the similar creation of reversing streamlines. Interestingly, the reversing flow fore and aft of the particle are somewhat mirrored on the other half of the channel.21 (D) Real systems usually include both inertia and confinement, which inevitably create the reversing streamlines as well. (E) Presence of reversing streamlines has been experimentally demonstrated for very low Re as well as finite Re flows.21 (F) The reversing streamlines act as a repulsive mechanism between particle pairs and contribute to their inertial ordering.21 (G) Inertially focused particles create strong net secondary flows in the channel, as they rotate and flow downstream.56 (H) The presence of particle-induced convection has also been demonstrated experimentally with applications for fluid mixing.56 (I) Type, direction and strength of the induced secondary flows by the particle highly depend on lateral position of the particle. Therefore, inertial focusing plays an important role in the output of the induced secondary flow.56

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(Fig. 6A, B). The spiraling streamlines are expected to enhance mass and heat transfer.78 Zurita–Gotor et al. suggested another mechanism of reversing streamline formation that does not require fluid inertia.73 Numerical studies in a confined geometry revealed that the presence of a wall near a particle leads to breaking of closed streamlines and generation of open and reversing streamlines. Reflection of the stresslet flow at the near wall leads to this disturbance and lateral forces that can act on another nearby particle. In Stokes flow, this results in two particles moving along streamlines with swapping trajectories. Note that the stresslet disturbance flow decays with 1/r 2 such that this is expected to be a short range interaction. Reversing streamlines can also be found in inertial microfluidic systems in which both fluid inertia and confining walls are present.21 Microfluidic channels typically have rectangular cross-sections and the reversing streamlines are almost identical to those in a simple shear flow, except the symmetry in the vertical direction is broken due to the parabolic base flow (Fig. 6C, D). As in linear shear flow, reversing streamlines appear in a channel flow for Stokes flow conditions. This formation of reversing streamlines at both finiteReynolds numbers and zero-Reynolds number was confirmed with numerical simulation and experiments in microchannel flows (Fig. 6C, D, E). The reversing streamlines act to create repulsive particle– particle interactions in the process of inertial ordering (Fig. 6F). These particle-induced reversing streamlines initiate the self-assembly of ordered structures by pushing particles in transverse directions (vertical direction in schematics, Fig. 6F), which directs particle centers onto streamlines with opposite flow directions. The subsequent particle pair's motion resembles harmonic oscillations with damping. Pair dynamics that lead to self-assembled states have several steps. First, particles are directed vertically due to reversing streamlines and move away from each other. Then, inertial lift forces direct particles back towards the focusing position. The particles approach but the interactions repeat if particles overshoot the focusing position. Note this self-assembled particle system is not in equilibrium, therefore inter-particle spacing can be tuned with

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simple expansion–contraction channels.21 Since the interaction strength of the reversing streamlines increases rapidly as particles approach each other in the expansion, particles are pushed off the focusing position again and find a new self-assembled state which is further expanded upon reduction of the channel cross-section downstream. The capability to achieve inter-particle spacing control, in addition to inertial focusing, allows particle manipulation in all directions, which can be useful for applications such as flow cytometry and controlling cell numbers for tissue printing.80 In this section, we focused on inertial microfluidic physics and the effects of reversing streamlines in such systems, however, reversing streamlines appear in a broader range of scenarios in which there is Stokes flow with confinement, and particle–particle interactions due to reversing streamlines should be widely considered for any two-phase flows in microfluidic channels. Table S2† summarizes our current understanding of these particle-induced disturbances and the need for additional numerical and analytical understanding especially for how they contribute to particle ordering.

b) Particle-induced convection Particles flowing in confined channels with finite inertia not only produce reversing disturbance flows that occupy locations up and downstream of the particle but also induce net secondary flows in the channel cross-section. This effect, termed “particle-induced convection,” results in the creation of a set of net flows that span across the channel from a focused particle and recirculate around the top and bottom of the channel to conserve mass56 (Fig. 6E). The net secondary flow, which is not present in a confined particulate flow in Stokes flow, is mostly created by flow differences upstream of the particle for inertial flow. The intensity of this mode of lateral fluid transport scales strongly with particle size (~a 3), exhibits an inertial scaling with flow velocity (~U 2) and linearly scales with fluid density. Numerical study of the physically constrained system suggests that both translational and rotational components of particle motion affect the magnitude of the net secondary flow: increasing

Tutorial Box 2 – Practical design rules and considerations for inertial microfluidic systems: interparticle interactions While individual particles migrate towards focusing positions, particle induced disturbance flows can influence motion of other particles. These interparticle interactions decay rapidly (~1/r 2), thus the effect is local and more easily noticeable in flows with high particle concentrations. However, initially random particle positions lead to faster particles catching up to slower particles, which means that a large portion of particles experience inter-particle interactions before they migrate to the focusing positions even at low concentration. Inter-particle interactions in inertial microfluidic systems lead to regular interparticle spacing (inertial ordering). The interparticle spacing has been shown to be tunable with a simple expansion–contraction channel geometry.21 Particles are forced to rearrange in the expansion channel due to inter-particle interactions. Then when the particles enter the contraction channel interparticle spacings become larger due to mass conservation. In principle, the ratio of inter-particle spacing before and after an expansion–contraction channel should be an inverse of the ratio of channel cross-section area. However, particles in expansion channels may not rearrange with sufficient regularity depending on local particle concentration, expansion channel length and Re: high particle concentration (e.g. φ > ~25%) can limit tunability because there is a lack of void spaces that neighboring particles can occupy. Note the final inter-particle spacing, ds is related to the length fraction and particle size by the following formula: ds = a(2 − φ)/φ if particles are focused in a single line. Repeating contraction–expansion structure can improve the regularity, or completeness, of the inter-particle spacing to fill voids.

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Tutorial Box 3 – Practical design rules and considerations for inertial microfluidic systems: particle affecting flow

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Particles are observed to considerably affect and disturb the flow around themselves via creation of net secondary flows, an effect referred to as “particleinduced convection”. Particle size, flow conditions and particle concentration are important parameter in the creation of particle-induced convection. At a/H > 0.1 the particle is large enough to create strong net secondary flows and disturb the flow field. Flow conditions (including flow rate, channel geometry and particle size) leading to Rp > ~2 lead to significant particle-induced convection. As each particle acts as a dynamic convection site, the effect becomes more prevalent at higher length fractions, e.g. φ > 35% to achieve considerable convection over a few centimeters in a straight channel. This effect is cumulative over channel length, so the overall disturbance of the main flow depends on the distance over which particles have affected it. For instance, for 10 μm cells in a 100 × 100 μm2 channel, flowing ~4 × 106 cells mL−1 (i.e. φ = 40%) at 1.5 mL min−1 (i.e. Rp = 2.5) leads to cross-stream motion that spans the channel ~4 cm downstream.

with increasing drag force on the particle (i.e. increasing slip of the particle relative to the fluid velocity), and also with decreasing torque (i.e. increasing particle rotation rate), with the former having a more significant effect. It should be noted however that these are difficult conditions to physically achieve unless significant body forces act on the particle and for the majority of cases particle motion is passively adjusted by the system such that the particle flow force- and torque-free. Particle-induced convection can have significant impact on the flow field at Rp > ~2. The behavior is the result of an intricate interplay between geometry and flow conditions – the direction and magnitude of the flow depend heavily upon fluid inertia, the location of the particle in the channel, particle rotation rate and the extent to which the particle lags or leads the underlying channel flow. As shown in Fig. 6I, if the lateral position of the particle is slightly shifted the secondary flows created alter drastically, to the point of changing direction. Consistent with the importance of velocity gradients discussed elsewhere, the weakest secondary flow and a reversal in direction are observed when particle is closest to the center which corresponds with the smallest local velocity differences across the particle. Experimental results confirm that particles occupying inertial focusing equilibrium positions have uniform flows that each act as a dynamic site for mass convection while flowing downstream and constructively add to effectively transfer fluid across channels (Fig. 6H). Consequently, a high concentration of particles in the channel, e.g. φ > ~35%, which equates to more convection sites can also contribute to the dominance of the effect. Interestingly, the particles remain focused and are not disturbed by the self-induced secondary flow.56 Therefore, the unique lateral positioning of the particles due to inertial focusing plays an important role in creating a strong directed secondary flow. As a result, at Rp > ~2 the particle-induced convection dominates the diffusion completely as a means of lateral fluid transport, especially at high enough concentrations of particles, φ > ~35%. This mechanism has been used for applications such as simple fluid switching and mixing around beads and cells. Other potential applications include enhanced transport in microcooling systems and bead surface functionalization. Similar to Dean flow, this secondary flow accumulates along the channel, so one needs to characterize and time it appropriately (via geometrical design, flow conditions, etc.) to achieve

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the desired transport rather than rotating the fluid around the channel fully.

iii. Effects of microstructures on flow and particles – flow programming Thus far, we have focused on how different forms of nonlinearities or particle asymmetry in straight channel flows lead to lift forces and unique flow disturbances. In this section, we discuss the effects of channel geometric changes on the flow and how these changes of boundary conditions induce a range of secondary flows that can be utilized to control fluid and particles. The deviations from a straight channel can be imposed in different forms such as through channel curvature, grooves on channel walls or obstacles within the channel. Theoretical analysis of the perturbations introduced by channel geometry is generally difficult due to the complex set of varying boundary conditions imposed on the already complex and nonlinear Navier–Stokes equations. However, we provide some insight into the physics of these systems, as well as present their applications. a) Curving channels – Dean flow Secondary flow in curving channels. Dean flow – the secondary flow induced in curving channels is a well-known inertial effect which is due to the momentum mismatch of fluid parcels at different cross-sections of a channel as these parcels pass around a curve (Fig. 7A). In this case, fluid parcels near the center flow faster and have a larger inertia directed outwards compared to the fluid parcels near the walls. In order to conserve mass, the fluid near the top and bottom walls move inwards, resulting in a set of secondary flows. The magnitude and qualitative features of the secondary flow is characterized by the dimensionless Dean number De = Re(H/2R)0.5, where R is the radius of channel curvature. The strength of the secondary flow depends both on the strength of the underlying downstream flow (Re), as well as the geometrical ratio H/2R which, intuitively, means that a faster turn in the channel (i.e. smaller radius of curvature R) leads to a stronger Dean flow. As an inertial effect the secondary flow velocity is expected to scale with downstream velocity squared as UD ~ De2μ/(ρH)1. Berger et al. also noted that increases in Dean number are associated with changes in shape of the secondary flow, with centers of the symmetric

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Fig. 7 Dean flow and its applications. (A) Dean flow, an inertially-induced secondary flow, is created due to the velocity mismatch of fluid parcels in a curving channel.85 (B) Dean flow was employed to increase the interfacial surface between two fluid streams to enhance microscale mixing.4 Two major classes of curving channels are generally employed in inertial microfluidic platforms: spiraling channels (C)85 and alternating asymmetric curving channels (D).2 (E) Spiraling channels have been used for differential focusing and separation of particles.89 (F) Alternating channels have been employed for accurate positioning of cells for deformability cytometry.97

vortices moving towards the outer wall and development of boundary layers and additional vortices within the crosssection with increasing De.81 Applications of Dean flow. Traditionally, secondary flows in microfluidic systems have been utilized to achieve mixing. Ugaz et al. employed Dean flow to control the interface of two fluids and perform fast mixing via a splitand-recombine arrangement (Fig. 7B).4 The ability to control fluid streams with Dean flow has also been used for 3D hydrodynamic focusing,82 and for creating a variable inertially modulated focal length lens.83 More recently, there has been an increasing interest in using curving channels to control particles in microchannels, as the developed secondary flow enhances the lateral motion of particles across the channel and alters inertial focusing equilibrium positions by imposing a drag force proportional to the Dean flow velocity, UD (Fig. 7A).2,84,85 The two major roles channel curvature plays in inertial microfluidic platforms are: (i) it allows for mixing of the

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flow and particles to sample equilibrium positions faster. Gossett et al. demonstrated that at the sufficiently high Dean number (De ~ 17 in their example) distance required for particle focusing is nearly 5 times shorter compared to the case of a straight channel of the same cross-sectional dimensions.86 (ii) It allows for unique equilibrium positions for particles with different properties due to Rf, the lift to dean drag force ratio (FL/FD) varying with particle dimensions and locations across the channel. Dean drag follows the underlying secondary velocity field, directed outwards near the channel center, and inwards near the wall (Table S1†). Details of how system parameters can affect the equilibrium position in curved channels can be found in previous work.86,87 Spiral and asymmetric curving channels. The two major classes of curving channels used to enhance inertial focusing are spiraling channels, with one direction of curvature (Fig. 7C),85,88–96 and sigmoidal channels with alternating curvature (Fig. 7D).2,86,97 A comprehensive set of studies of inertial focusing in spiral microchannels of varying widths in

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Tutorial Box 4 – Practical design rules and considerations for inertial microfluidic systems: inertial focusing in curving channels Because curved microchannels easily perturb flowing particles from unstable positions, they can be used to enhance the speed of lateral particle migration to stable equilibria. This, however, requires considerable secondary flow in the curved channel. For instance for similar flow conditions, when Re or De are low (22 or 4), focusing length was reported to be the same. However, at increased Re or De (89 or 17), focusing length of the curving channel (with repeating features) was nearly 5 times shorter than the required length on the straight channel. One important practical advantage of shorter focusing distance is decreased fluidic resistance, which directly affects the upper limit of flow rate, thus throughput, in a channel. Therefore, the length required for focusing in curving channels can be obtained by multiplying the straight channel result by a factor such that f = 0.2–1 depending on the curving channel design.

Lf  f

 h 2 U m a 2 f L

where

Furthermore, the ratio of inertial lift and Dean drag (Rf) must be taken into consideration. If this ratio is too small, Dean drag can be so strong as to lead to mixing and disrupt particle focusing. As a result, the following inequality is proposed to be satisfied when designing these systems:

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Rf 

2a 2 R  ~ 0.08 H3

Evidently, if Rf ≫ 1 inertial lift is dominant and the user is not benefiting from the effect of Dean flow to achieve shorter focusing length.

which Re, De, and a/H were independently controlled has provided many of the missing pieces to the overall inertial focusing parameter space.85,87 This work has highlighted the transitions in focusing behavior as a function of increasing channel curvature ratio for Re up to 400, in which top and bottom particle equilibrium positions in a low aspect ratio channel (here aspect ratio is height/width, where width is in the direction of channel radius of curvature) shift inwards, yield a single stable equilibrium position and then move back outwards as the balance between lift and drag changes (Fig. 7A). As the velocity profile becomes flattened for wider channels, the shear gradient lift in this wide direction decreases (Fig. 7A.i), as discussed previously. Simultaneously, the Dean drag on the particle increases with channel size and becomes stronger in wider channels (Fig. 7A.iii). Therefore, at a critical channel width, Dean drag will become the dominant mechanism for lateral migration of the particles (Fig. 7C). This, as mentioned above, modifies the focusing behavior of the particle by shortening focusing length and altering the equilibrium position. Recent work further modified the balance between inertial lift and Dean drag within the cross-section of a channel by using a nonrectangular slanted geometry. This had an unexpected effect of creating particle size-dependent equilibrium positions that switch quickly with increases in Re allowing larger separation in space within the channel cross-section, which is useful to improve separation accuracy.95 Note that unlike for spiral channels which approach a steady-state Dean flow, incorporation of alternating curvature may prevent the secondary flow from reaching steady state and leads to more complex inertial focusing behavior as a function of Reynolds number.85,86 In using the alternating curvature design it is important to note that opposing channel segments should be asymmetric, such that the curvatureinduced secondary flows will not act counter to each other and reduce the overall effect of the Dean drag or mixing action (Fig. 7D).

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Spiral and asymmetric curving channels each have unique situations to switch they are better suited. Dean flow in spiraling channels can be employed for differential focusing and separation of microparticles (Fig. 7E)88,94,95 which has been used for high-throughput cell cycle synchronization89 and isolation of circulating tumor cells.91 Spiral channels have achieved higher performance separations than asymmetric curving channels.98 Spiral channels have also been used for ordering and deterministic cell-in-droplet encapsulation.90 Alternating curvature channels were initially introduced to reduce the number of focused streams from straight channels2 and have also been found to enhance inertial focusing (tighter focusing at shorter downstream distances) compared to straight channels.86 These designs are easier to layout as a segment within a microfluidic circuit because they occupy the same footprint as a straight channel and can be easily parallelized in a radial or array layout. For example an asymmetric focusing geometry has been inserted inline and used to achieve accurate localization of cells in microchannels to enable uniform hydrodynamic stretching in order to characterize single cells at high rates using “deformability cytometry” (Fig. 7F).97 Density mismatch effect (inertia of particles). Another mechanism that can lead to cross-stream migration in particulate systems is the density mismatch between the particle and fluid. If the fluid suddenly changes direction due to changing channel curvature or structures, the particle may continue for a time on a straight trajectory due to its inertia. For example, particles with larger density than the fluid will experience a centrifugal force directed radially outward in its frame of reference when the channel curves. Because of this relatively simple physical mechanism, particle migration in spiraling channels can often be understood mistakenly as an effect of centrifugal force due to density mismatch rather than Dean flow. The Stokes number, Stk = (2ρp + ρ)a 2ΔU/36μdc (where ρp is density of the particle, ΔU is the velocity change and dc is the length scale of the change), is a dimensionless parameter

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that compares the flow timescale – a distance divided by velocity difference – to the particle acceleration time scale, and provides a measure of the fidelity of the particle's ability to follow fluid streamlines which is especially important in particle image velocimetry (PIV). However, this number is usually quite small (Stk < 0.01) in microfluidic flows and even inertial microfluidic flows (unless there is an extremely large density mismatch or abrupt changes in velocity). As a result, fluid inertia, particle shape, or particle deformability most often dominate effects from particle inertia.2

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b) Grooves and herringbones on channel walls Addition of grooves and herringbones onto the wall surfaces of straight microchannels has also been known as a simple means for inducing strong secondary flows which was introduced in the seminal work of Stroock et al. (Fig. 8A).99 These secondary flows may be used instead of Dean flow to modify inertial focusing behavior,19 however, most of such systems have been operated in a low-Re flow regime, where inertia is typically negligible. In Stokes flow, in order for the phenomenon to be effective, the grooves/herringbones must lack foreaft symmetry otherwise, based on the MSTR theorem, there should be no net deformation to the fluid flow as it passes the groove which explains the need for slanted structures in these systems. Applications in which herringbone-induced flow deformations are used provide a good starting point for identifying applications of inertia-induced secondary flows and flow deformations. The herringbone structure has been used to create hydrodynamically focused sheath flow100 to perform flow cytometry101 (Fig. 8B). Numerical simulations also suggest that these systems can be used for size-based sorting of microparticles in combination with inertial lift forces.19 In this scheme, particles of different sizes are inertially focused near the top and bottom walls in a low aspect ratio channel, where smaller particles have equilibrium positions closer to channel walls. Periodical ridges on top and bottom walls generate two symmetrical secondary flows. Due to their different initial z-positions by inertial focusing, larger and smaller particles are forced to move in opposing y-directions by the secondary flows, resulting in their distinct lateral locations. Stott

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et al. have also employed the microvortices created with these structures to mix blood, allow cells to better sample and interact with surfaces, and effectively isolate circulating tumor cells.102 However, similar to most other secondary flow inducing mechanisms, this technique has most widely been utilized to achieve fast mixing.99,103 c) Pillars and other structures With the addition of fluid inertia, even symmetric structures in channels – like circular pillars – can create net secondary flows. Lauga et al. previously showed that flow in a channel with varying cross-section is three-dimensional even in Stokes flow,104 however, the secondary flows are completely reversed and there is no net deformation when the channel crosssection returns to its original shape symmetrically downstream (Fig. 9A). With finite inertia symmetric changes in channel geometry, such as the presence of a circular pillar, yield a net deformation of fluid streamlines instead.105 In brief, pillars inside a channel can induce a net secondary flow which causes the fluid to inertially deform around these obstacles and not return to its original configuration downstream. We demonstrated, with numerical simulations and experiments, that the pressure gradient created by the presence of a pillar irreversibly pushes fluid parcels laterally which lead to creation of a set of secondary flows. This behavior has features in common with Dean flow. However, in contrast with Dean flow, here one can use lateral position of the pillar to tune the lateral location of the secondary flows. Furthermore, while Dean flow presents a steady-state behavior along the curvature, this phenomenon creates a net motion with additional complexities. Importantly, four dominant modes have been identified for this phenomenon based on the number and direction of the secondary flows (Fig. 9B) which is determined based on flow conditions and channel geometry. For instance, Mode 1 occurs in low aspect ratio channels and Re ~ 0.3 to achieve motion that would span the channel with less than 10–15 pillars. Also, Re > 10 is recommended to achieve significant lateral displacement per pillar. For instance, in a 50 × 200 μm2 channel (i.e. h/w = 0.25) operating in mode 1 (i.e. pushing fluid away from the pillar at the channel center and back towards the pillar near the walls) with 100 μm pillars (i.e. D/w = 0.5) deformation that spans the channel width following ~15 pillars can be observed at Q > 150 μL min−1, corresponding to Re > ~20.

with similar structured channels in which the secondary flow along with steric pinching of larger cells allowed separation of larger rare cells from blood cells.108 Operating in Mode 1 structures introduced in these works would lead to a net center directed secondary flow that is useful in mixing and centering larger cancer cells at inertial lift equilibrium positions at the channel centerline. A limitation of this approach, like other inertial focusing-based approaches, is that high cell concentrations prevent accurate separations. Herringbone structures also offer a unique ability to deterministically manipulate flow and shape fluid cross-section, especially at low-Re and Stokes flow which is starting to be explored and applied.100

iv. Conclusions and perspectives In this work we reviewed a variety of mechanisms by which system nonlinearity (mainly inertia) and asymmetry in channel or particle geometry can be used to control particles and flows in microfluidic platforms. Importantly, based on accumulating evidence we conclude that generation of lift or secondary flows is not the exception, but the rule for any

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perturbation in channel geometry or particle properties beyond the simplest cases especially in the inertial flow regime (see Table S1† and S2 for summaries). In other words, any type of irregularity will give rise to some form of lift or secondary flow. Although most of the applications of such secondary flows in microsystems to date have focused on “mixing” and creating chaos, we suggest that due to their deterministic nature in laminar flow these techniques can be exploited to move in the opposite direction: to induce order into an otherwise random system by utilizing flows transverse to downstream fluid streamlines to offer a strong tool for controllably positioning fluid parcels and particles inside microchannels. The ability to make the most of these effects will require libraries of pre-simulated structured channels for accurate and fast numerical prediction of total transformations. We have already demonstrated such a capability for pillar-induced flow deformations112 and similar strategies are applicable for any structured channel such as herringbones and curving channels.100 We anticipate that the complimentary use of these different types of structures should enable unprecedented 3D control of a fluid stream and engineering its shape inside channels.

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Inertial microfluidics is still a nascent field despite the increasing number of research articles published each year. Even though a basic tool kit of operations that can be performed on particles and fluids exists, the underlying nonlinear fluid physics is extremely rich with a multitude of fundamental directions that are still unexplored. For instance, the physics of how different modes of operation are created in pillar-induced secondary flows is yet to be understood, predictions for how combinations of Dean drag and lift forces interact within a channel cross section require painstaking experiments, and unexpected changes in modes of inertial focusing in straight channels alone as Re increases require improved physical intuition. Channel irregularities and changes in boundary conditions that create flow deformations offer a vast number of other areas to be explored. Emerging applications of this field are being discovered at a fast pace and are only recently finding their way to translate from academic research settings into real-world and commercial devices. Examples include the “CTC-iChip” from Harvard Medical School and Massachusetts General Hospital which is en route to be commercialized presumably by Johnson & Johnson,33 and uses inertial focusing to sort rare CTCs from whole blood at 107 cell s−1. Another example is the mechanophenotyping platform developed at UCLA and being commercialized by CytoVale, which employs inertial focusing alongside deformability cytometry to offer a new class of biomarkers. By measuring a range of biophysical cell markers at rates above 2000 cells s−1, CytoVale could offer a highthroughput, low-cost method of detecting disease.113 Clearbridge BioMedics, a spin-off from National University of Singapore, is also using inertial focusing in spiral channels to separate larger cancer cells from other blood components in their ClearCell FX system.91 Still a better fundamental understanding of all these systems will be necessary to enable the integration into more sophisticated and useful platforms towards a range of novel applications in biomedicine, industrial filtration, and beyond.

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103 104 105

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Lab Chip, 2014, 14, 2739–2761 | 2761

Inertial microfluidic physics.

Microfluidics has experienced massive growth in the past two decades, and especially with advances in rapid prototyping researchers have explored a mu...
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