2.2. Inertia and Dean Flow

In fluid dynamics, secondary flow is a flow pattern, which is relatively weaker than the primary flow. The secondary flow can be controlled by the fluidic forces and the shape, size, and position of inserts [74]. In the study of macroscopic rigid spheres in Poiseuille flow by Segre and Silberberg particles migrated away from the wall and then accumulated at an equilibrium position of 0.6 from the axis around the tube radius due to lateral forces [75]. When a particle moves along a straight microchannel, two inertial lift forces are acting on the particle: shear-gradient-induced lift force, and wall-effect induced lift force [76]. Deformable particles contained in biomedical suspensions are underlying deformability-induced lift forces which lead to differences in dynamics [77]. The motion of a deformable particle in shear flow was studied by Bayareh and Mortazavi [78,79,80] with neglecting the gravity influence. Their results demonstrated that the equilibrium position of suspended particles is affected by the wall effect, deformability and sizes of particles, Reynolds number, density and viscosity ratio, etc. The nonlinear effects in finite-Reynolds-number flow were investigated, including the tubular pinch effect in cylindrical pipes [75]. Liu et al. [81] explored the focusing positions of different particle sizes in four focusing configurations for the separation of plasma, red blood cells, and cancer cells from the blood. The wall-induced inertia is significant in the thin layers near the walls where the lift is close to that calculated for linear shear flow, which increases dramatically with increasing Re above about 100 [82]. By analyzing the spatial distributions of spherical particles, Kim et al. [83] concluded the lateral migration of particles are induced by the high shear rate due to the small-scale effect and the particle equilibrium position as a function of Re. They observed the migration of particles markedly occurs at a very low Reynolds number and the critical Re when in the range of 20 to 30. Moreover, the inertial migration of spherical particles in a circular Poiseuille flow was numerically investigated with a Re smaller than 2200 [84]. A conclusion was drawn that the hydrodynamic interactions between the particles in different periodic cells have significant effects on the migration of the particles. The lateral migrations of viscous capsules [85], liquid drops, and vesicles [86] were also investigated.

Inertial microfluidics was applied in deformability-based cell classification and enrichment to reduce the complexity and costs of clinical applications [87]. Dean flow is a kind of secondary flow that can be generated by the fact that when a fluid flow in a curved pipe with a small radius of curvature, the flow has helical streamlines [88]. Focusing of particles suspended in solutions is largely independent of centrifugal forces, which suggests that Dean drag is the dominant lateral force to balance the influence of lift forces [89]. Di Carlo et al. [90] evaluated the migration attributed to lifting forces on particles in microfluidic devices by fabricating straight and curved microchannels under laminar flow conditions, when ordering is observed to be independent of particle buoyant direction. They developed a theoretical description of the underlying forces and a semiempirical relationship of cutoff and the channel geometry [91]. Inertia and Dean flow fractionation were applied in microfluidic separation and sorting of biochemical sample mixtures [40,75,92].

The concept of inertial microfluidics was used in continuous separation of a multiparticle mixture in a simple spiral microchannel coupled with rotational Dean drag [93]. In inertial microfluidic experiments, the particle diameters cannot be very small compared to the characteristic channel length scale, and the Reynolds number of the particle is in order of 10 [94]. A spiral lab-on-a-chip (LOC) was used for size-dependent focusing of particles at distinct equilibrium positions across the microchannel cross-section from a multiparticle mixture [95], which exhibited 90% separation efficiency. Lee at al. [96] developed a spiral microchannel system for the synchronization and selection of cancer cells at different phases of cell cycle of blood to predict the condition of disease as shown in Figure 2b. Yousuff et al. [97] proposed a new configuration of spiral channel, where collection outlets are a series of side-branching channels perpendicular to the main channel of egress in which closely spaced particle streams can be collected separately. A novel inertial separation technique using spiral microchannel having a stair-like cross-section was introduced for size-based particle separation [98]. A spiral microfluidic chip was also employed for continuous separation of CTCs [99] and sperm-like-particles (SLPs) [100] from blood.

(a) Schematic diagram of multiorifice flow fractionation (MOFF) device. The figure has been reproduced with permission from the American Chemical Society [76]. (b) Schematic illustration of the spiral microfluidic design developed for cell-cycle synchronization. The figure has been reproduced with permission from the Royal Society of Chemistry [96].

The secondary flow induced by a microchannel with arc-shaped groove arrays was studied by Zhao et al. [101] with numerical approaches, and their results showed the secondary flow can guide different-size particles to the corresponding equilibrium positions. In the experiments, the performance of particles focusing was relatively insensitive to the variation of flow rate, which proves the availability of flow-insensitive microfluidic separation method in a reliable biosample preparation processing step for downstream bioassays. Yoon et al. [95] developed a size-selective separation system for microbeads by using secondary flow induced by centrifugal effects in a curved rectangular microchannel. The effects of curvature angles and channel heights on inertial focusing of microparticles in curvilinear microchannels were also investigated by Özbey et al. [102], and an optimum condition/configuration was obtained with a curvature angle of 280° at Re of 144 in the transition region.

Inertial size separation can be achieved in a contraction–expansion array (CEA) microchannel by a force balance between inertial lift and Dean drag forces in fluid regimes in which inertial fluid effects are significant [103]. In CEA systems, similar effects compared to Dean flows are produced by an abrupt change of the cross-sectional area, which is balanced by inertial lift forces throughout the contraction regions [81]. The CEA microchannels are applied for high-yield blood plasma separation with a level of 62.2% yield [104]. A fishbone-shaped microchannel was proposed by Kwak et al. [105] to separate platelets, erythrocytes, and leukocytes from human blood.

Sim et al. [76] developed a novel separation method named as multiorifice flow fractionation (MOFF), where a microparticle moves laterally driven by the hydrodynamic inertial forces due to a multiorifice structure (Figure 2a). To improve the low efficiency of single-stage multiorifice flow fractionation (SS-MOFF) in separation for large particles, multistage multiorifice flow fractionation (MS-MOFF) was developed to isolate rare cells from human blood with a recovery increased from 73.2% to 88.7% while the purity slightly decreased from 91.4% to 89.1% [106]. A parallel multiorifice flow fractionation (p-MOFF) chip was developed and used for high-throughput size-based CTC separation, where CTCs can be focused at the center of the channel due to the wall-effect-induced lift force [107].

Separation of suspension in symmetric and asymmetric serpentine microchannels is also driven by inertial and Dean effects. Yuan et al. [108] investigated particle focusing under Dean flow coupled with elasto-inertial effects in symmetric serpentine microchannels, which demonstrated acceleration of particle focusing and reduction of channel length.

Compared with PFF, techniques based on inertia and Dean flow can be applied in higher Reynolds number flow since they are based on the balance of inertial shear-gradient-induced lift force and wall-effect-induced lift force, where the Reynolds number is generally in the range of 10–270 [34].