Cross-flow microfluidic devices were a natural progression from Taylor’s four-roll mill, and resolved the limitations of Akbaridoust’s miniature four-roll mill. The cross-slot microchannel, seen in
Figure 3e, creates a very similar flow field. The device consists of two opposing inlets channels perpendicular to two opposing outlet channels. Fluid is injected into the inlets and can either be withdrawn at a defined rate or simply allowed to exit though the outlets. This bounded flow provides better flow stability allowing higher strain rates to be achieved, providing an advantage over the four-roll mill
^{[35]}. The device creates a planar extensional flow similar to Taylor’s four-roll mill with a similar velocity field, hyperbolic streamlines, and a stagnation point in the center where pure planar extensional flow is experienced
^{[20]}^{[36]}. Objects are trapped in the stagnation point for a finite amount of time and experience compression along the inlet axis and extension along the outlet axis as in the four-roll mill
^{[36]}^{[37]}. The extensional rate at the stagnant point is inversely related to the channel dimensions according to
where
U is the average flow velocity in the inlet/outlet channels and
w is the channel width
^{[35]}^{[38]}. Theoretically, this device can achieve high extensional rates, easily controlled by adjusting the volumetric flow rates. Akbaridoust et al. achieved strain rates up to 142 s
^{−1} in their cross slot microchannel, which was significantly greater than that achieved in their miniature four-roll mill
^{[20]}. Similar to the different types of constriction devices discussed above, cross-slot channels only have a small region of pure extensional flow with shear stresses existing to varying degrees outside of this area. Therefore, it is important to determine the area of uniform extension rate, where cell deformation can be observed. The geometries of cross-slot channels are often described by using a dimensionless number (α), where α is the ratio of the channel depth (d) and width (w). Two-dimensional (2D) numerical simulations of a cross-slot channel with an infinite α show that in a radius of w/16 from the stagnation point the extension rate changes less than 5%
^{[36]}. Flow velocimetry measurements show an even larger radius (w/4) of uniform extension rate in channels with an α of 0.53
^{[39]}. Microparticle image velocimetry (micro-PIV) measurements on a channel with α = 0.1 showed that a central region of 0.6 w × 0.6 w resulted in an extension rate variation of 2%, as well as stagnation point variations limited to 1 µm
^{[20]}. Clearly, the area of uniform extension rate depends on the channel dimensions and on the flow rates. While these 2D simulations are advantageous, it should be kept in mind that the flow is never truly 2D, and the boundaries on the top and bottom of the device will have effects through the depth of the channel, even for larger values of α which do ensure a more uniform extension rate through the z-axis
^{[35]}. As with all the devices discussed thus far, at high enough flow rates or with highly viscoelastic fluids, instabilities occur along with asymmetric flows fields. The asymmetric flow field, characterized as a forward bifurcation, was also confirmed by numerical simulations
^{[40]}. Lower values of α had a stabilizing effect for these instabilities
^{[35]}, but this must be balanced with the fact that a larger α better approaches a 2D planar extensional flow field. Another limitation of cross-slot microchannels is that the cell trajectory through the channel has significant effects on the strain rates experienced and thus the extent of deformation measured. Henon et al. performed numerical simulations of RBCs flowing through cross-slot microchannels and found three deformation modes depending on the entry position
^{[19]}. The first occurs in cells flowing along the centerline of the inlets, which experience very little shear stress and upon reaching the center, near the stagnation point, show large deformations along one axis and remain largely symmetrical. Cells in between the centerline and the walls make up the second mode of deformation, experiencing limited levels of shear stress in the channel arms prior to extensional stresses in the center of the channel leading to asymmetric deformations. The last deformation mode occurs for cells traveling near the wall that experience the highest shear in the channel arms as well as additional shear in the area located between the channel center and the corners resulting from boundary effects, this results in highly asymmetrical cell deformation. Limiting observations to cells in the first category or unifying the cell trajectories, providing a more uniform kinematic history, would yield more accurate deformability measurements. Several studies have used inertial or viscoelastic focusing in cross-slot microchannels to achieve limited cell trajectories
^{[37]}^{[41]}^{[42]}. Using inertial flow (Re ~
O10
^{−1}) to focus cell trajectories the kinematics of the flow field significantly changed compared to an inertia-less field
^{[37]}. Under inertial flow, particles decelerated closer to the stagnation point and had dramatically changed streamlines in the flow field
^{[37]}. The strain rate gradient gradually increased with increasing Re, thus shrinking the region with uniform pure extensional flow, and at Re ≥ 40 vortices developed near the curved walls where the channel arms intersected (i.e., the intersection had rounded corners)
^{[37]}. Viscoelastic focusing results in an almost identical flow field to the inertia-less Newtonian case, producing similar velocity fields, streamlines, and regions of uniform strain
^{[37]}. Although, using a viscoelastic focusing method, one must consider the issues discussed above for highly viscoelastic fluids. Using such cell focusing methods would produce more accurate measurements of cell deformability and more accurate bulk measurements of cellular damage.