There are numerous microfluidic particle separation applications in the literature such as spirals, micro-towers creating turbulence (like coin pusher machines) and box cascades that pinch flow, figure 1. The inertial aspect means no additional external control is required, magnetism or electric field, meaning separation occurs due to the flow dynamics alone. This is desirable for applications which do not want to foul the control fluid or impact the fluid contents by degradation. We first need to look at the flow velocity cross-section of a channel, Dean Drag forces, to understand the flow dynamics (Part 1) leading to inertial particle separation (Part 2 – coming soon).

Calculating the flow velocity has been achieved using computational fluid dynamics (CFD). This is an approximation of the expected fluid behaviour based on several parameters. The approximation is based on the partial differential equations (PDE) used to solve the case study. In this case, microfluidic flow is solved using the Navier-Stokes (NS) equation. When solving the Navier-Stokes (NS) equation, Dean Drag forces are expected to form. The Dean Drag forces are lift and drag forces within secondary flow perpendicular to the bulk direction of flow. The secondary flow dynamics are important for many inertia particle separation applications. The separation of particles occurs by the strength of the lift and drag forces acting on particles. Controlling the lift and drag forces can provide precise particle separation explored in Part 2 coming soon.

## Dean Drag Forces

Inertia microfluidics occurs in circular and straight microchannels, figure 2. Only lift forces are present. As a result, equilibrium positions will form where the shear lift gradient force and wall interaction force cancel out. As the wall interaction force scales with particle diameter, larger particles will remain in the microchannel centre compared to smaller particles.

Dean Drag forces is the name of the phenomena where secondary flow, cross-sectional flow perpendicular to bulk flow, forms vortexes which lift and drag within spiral or curved micro-channels, figure 1. The lift and drag forces are fundamental to inertial microfluidic separation where the forces can be highly controlled based on the geometry and flow properties. This control is due to the laminar flow at the microfluidic scale as explored with the Reynolds equation, equation 1.

The Reynolds number is almost always smaller than 1000 meaning flow will be laminar due to the characteristic linear length, *L*, at the micro-scale. Whilst the flow will likely be laminar, the flow dynamics will likely change due to their importance in the Navier-Stokes equation, equation 2. Unlike in *Understanding the Navier-Stokes Solver*, the characteristic linear length is not the radius as the micro-channel is rectangle / square. Therefore we need to calculate the hydraulic diameter, *D _{H}*, using the width,

*a,*and height,

*b,*of the channel, equation 2.

Additionally, Dean number can be calculated to validate the pair of Dean vortices formed, equation 3. A small Dean number can lead to unidirectional flow where there is no vortices forming (De < 40~60). As the Dean number increases, vortices develop until reaching a stable state (De > 64~75). However, a large Dean number can lead to secondary flow instabilities and unstable vortices (De > 200). The Dean number, similar to the Reynolds number, is an indication of the dynamics. The radius of curvature, R_{c}, is a range across the spiral of the device where the Dean number will be larger moving outwards from the centre. The Dean number can be controlled by repeating the curved microchannel geometries and adjusting fluid properties.

Although these Dean Drag ranges may be true for large pipes, at the microfluidic scale the Dean Drag range is commonly between 0-30 due to the relation of viscous forces to the characteristic linear length^{8-10}.

## How Navier-Stokes is Simulated

Several different Navier-Stokes (NS) solvers exist which make different assumptions when solving fluid flow. One difference is whether the solution should be solved with a compressible or incompressible NS solution which depends on the fluid properties. Additionally, whether the problem is time dependent or time independent. Time dependent studies, also known as continuous studies, are often required when studying turbulent flow.

Navier-Stokes cannot be solved directly due to the non-linear nature, equation 4. We treat the fluid as incompressible flow which is expected to be laminar where a steady solver could be used. However, due to the size of the mesh and computer resources, an unsteady solver may be appropriate due to the iterative solving method.

More can be read in *Understanding The Navier-Stokes Solvers* as well as the FEniCS documentation and the BERNAISE wiki.

## Problem Specific Variables

Inflow velocity, boundary domains and other flow properties set up the problem variables, table 1. Values are based on 21^{o}C water. These initial parameters act as a demo for the proof of principle before changing to other Newtonian fluids or more complex non-Newtonian fluids such as blood, a shear-thinning fluid.

Parameters | Value |

Water Dynamic Viscosity (kg / m.s) | 1.003e^{-3} |

Water Density (kg / m^{3}) | 998 |

Water Velocity (m / s) | 1.6e^{-2} |

Time Step (s) | CFL * 1e^{-2} |

The parameters were implemented into the BERNAISE script using the parameters presented, table 1. Example implementations are presented in the flow curved microchannel, the 2D spiral and 3D spiral demonstrations. It should be noted that 2D spiral model will not model Dean Drag forces and other secondary flow which require a 3D simulation.

## Software

BERNAISE, Binary ElectRohydrodyNAmIc SolvEr, is an open-source, free CFD solver created in FEniCS – a solver for partial differential equations (PDEs). BERNAISE is a flexible high-level solver of electrohydrodynamic flows in complex geometries. It is written in Python and built on the FEniCS project, which in turn effectively interfaces to optimized linear algebra backends such as PETSc. The solver is described in Asger Bolet’s, Gaute Linga’s and Joachim Mathiesen’s paper.

This particular version of BERNAISE has been modified by myself to use *.XDMF* computer aided design (CAD) meshes as well as has been focused towards microfluidic examples. Due to the large size of the simulation, additional computational power may be required. A less computationally demanding 2D spiral model is available however 3D is required to validate the Dean Drag forces.

The mesh used in this example was created in AutoDesk Fusion (non-commercial) / AutoDesk Inventor (University of Kent academic licence) and is available on GitHub.

## Step by Step

- Convert the CAD into a mesh file for FEniCS and BERNAISE to read and carry out simulations. A full description of the steps is available for 2D and 3D CAD conversion on the BERNAISE wiki. Briefly, produce the mesh in an appropriate CAD program and export as a
*.STEP*file. Load the Use meshio and convert the*.gmsh*file to*.XDMF*files for the mesh and boundary domain labels. - Assign parameters in the problem python file. In this case, the problem has been called
*flow_spiral*and the problem specific parameters are defined in the*problem()*function. More example problems can be found in the problems folder. - Run the problem from the command prompt
`python sauce.py problem=flow_spiral`

- View the exported
*.XDMF*results in ParaView.- Slice through 3D model. Attempt to slice where bulk flow is perpendicular to flow.
- Apply “Surface Vector” filter. I ensure that vectors are parallel to the surface and I am using the velocity vector field
- Apply “Glyph” field onto the “Surface Vector” slice. This will present Secondary Flow Visualisation in Paraview.

## Reviewing The Results

BERNAISE has solved secondary flow around the curved microchannel. Secondary flow visualisation requires precise placement of the slice. If the placement of the slice is not at the point where bulk flow is perpendicular to the slice, the bulk flow will dominate the secondary flow slice, figure 2. This is difficult to achieve but fortunately ParaView can create states which pre-loads data to visualise the secondary flow avaliable via on the GitHub.

The Reynolds (Re) and Dean (De) numbers where calculated as Re = 4.5 and De = 0.77, equations 1 and 3. The Dean number is at the lower end of the microfluidic Dean Drag forces range, 0 – 30. Based on the Dean number ranges, unstable Dean Drag forces were expected to form within the curved microchannel. Reviewing the secondary flow presents two clearly defined vortices above and below in the channel, figure 3. The strength of the secondary flow, maximum velocity magnitude of 1.9 m / s, is important to the lift and drag forces acting upon the particles and leading to the inertia separation. Increasing the fluid velocity, density and viscosity was investigated to test whether there was an effect on the vortices formed.

^{-3}m / s with arrow glyphs (left) and velocity streamlines (right). The velocity has been reduced by a factor of 10 compared to figure 3. The maximum velocity magnitude is 1e

^{-5}m / s. The glyph arrows present some instability with large glyphs at the boundaries but retains glyphs within the centre of the cross-section. Streamlines highlight some vortexes but flow instabilities with streamlines cutting into the vortexes.

Decreasing the fluid velocity by a factor of 10 will decrease the Reynolds number and Dean number, Re = 0.45 (4.5) & De = 0.077 (0.77). The decrease in velocity will in theory decrease the stability of the Dean Drag forces that develop in the channel. Reviewing the result, the arrow glyphs and streamlines present unstable secondary flow dynamics, figure 4. The arrow glyphs present relatively large vectors at the top and bottom of the microchannel with counter central flow. The streamlines presents secondary flow cutting into the vortexes disrupting the flow dynamics. There is a noticeable change in the strength and stability of the vortices compared to inflow velocity 1.6e^{-2} m / s, figure 3. The reduced velocity magnitude, 1.9e^{-5} m / s to 1e^{-5} m / s means particles will have reduced lift and drag forces which drive separation applications. The instability likely means inertia separation will fail and decrease in efficiency due to the lack of control.

^{3}with arrow glyphs (left) and velocity streamlines (right). The density has been increased to 1050 kg / m

^{3}from 988 kg / m

^{3}compared to figure 3. The maximum velocity magnitude is 1.5e

^{-5}m / s. The glyph arrows present some stable vortexes with vortexes above and below the centre of the channel. The glyphs present clear flow direction within the centre of the cross-section and the top and bottom of the channel. Streamlines highlights a small vortex in the top left expanding across the top where the bottom presents a tighter vortex to the right bottom corner. No flow instabilities can be seen with streamlines.

Increasing the fluid density to 1050 kg / m^{3} will increase the Reynolds number and Dean number, Re = 47 (4.5) & De = 7.9 (0.77). The change in density will in theory increase the stability of the Dean Drag forces that develop in the channel. The arrow glyphs and streamlines presented stable vortexes, figure 5. The arrow glyphs presented clear flow direction at the top, centre and bottom of the channel. The streamlines presented smaller vortexes in the corner of the cross-sections, top left and bottom right, when compared to 998 kg / m^{3} density flow. The smaller vortexes migration to the corner could be due to the increase in flow density but further tests are required to validate the hypothesis. The smaller, migrated vortexes could explain the slower velocity magnitude of 1.5e^{-5} m / s (1050 kg / m^{3} density flow) compared to 1.9e^{-5} m / s (998 kg / m^{3} density flow). The reduced magnitude is important to consider for lift and drag forces but equally is the stability of the vortexes.

^{-3}kg / m . s with arrow glyphs (left) and velocity streamlines (right). The velocity has been reduced by a factor of 10 compared to figure 3. The maximum velocity magnitude is 1e

^{-5}m / s. The glyph arrows present some instability with large glyphs at the boundaries but retains glyphs within the centre of the cross-section. Streamlines highlight some vortexes but flow instabilities with streamlines cutting into the vortexes.

Increasing the fluid dynamic viscosity to 6.71e^{-3} kg / m . s will decrease the Reynolds number and Dean number, Re = 0.76 (4.5) & De = 0.13 (0.77). The change in the fluid viscosity will in theory decrease the stability of the Dean Drag forces that develop in the channel. The vortexes formed were weak and unstable with a maximum velocity magnitude of 1e^{-5} m / s, figure 6. The arrow glyphs presented larger velocities at the top and bottom channel boundaries and retained flow through the centre plane. The streamlines presented flow cutting into the vortexes suggesting unstable secondary flow dynamics. The vortexes formed similar to the decrease in fluid velocity (1.6e^{-3} m / s) which is expected due to the smaller Reynolds and Dean number, Re = 0.45 (0.76) & De = 0.077 (0.13), figure 4.

Comparing all the results, a minimum Reynolds number of 1 and a minimum Dean number of approximately 0.2 is required to form stable vortexes based on the previous studies and supported by the literature^{8}. Additionally, the maximum Dean number is not explored here but is expected to be around 30. It should be noted that the Dean number range is dependent on the Reynolds as well as the geometry of the channel with regards to the characteristic linear length, hydraulic diameter, and the radius of curvature, equation 3. The work presented confirms the small Dean number for forming stable vortexes at the microscale important for microfluidics.

## Conclusion

Presented is a method for analysing secondary flow within curved and spiral microfluidic devices. Analysing the secondary flow provides understanding of the vortexes and Dean Drag forces driving inertial separation. The lift and drag forces can be optimised by adjusting the fluid properties as well as adjusting geometries which affects the Reynolds and drag number (coming soon, watch this space!). It was shown the Reynolds and Dean number are good indicators for Dean Drag forces forming regardless if one fluid parameter is adjusted independently of others. Shown in this model is a minimum Dean number of 0.2 for stable vortexes where the upper limit is likely to be around Dean number of 30^{8}. This work provides the foundation for separating particles of different sizes and densities using inertia forces using BERNAISE software package.

In part 2, we will explore simulating particles based on their size and density using Lagrangian Particle Tracking.

## References

- Herrmann, N., Neubauer, P., & Birkholz, M. (2019).
*Spiral microfluidic devices for cell separation and sorting in bioprocesses*. Biomicrofluidics, Vol. 13, p. 61501. https://doi.org/10.1063/1.5125264 - Hochstetter, A., Vernekar, R., Austin, R. H., Becker, H., Beech, J. P., Fedosov, D. A., … Inglis, D. W. (2020).
*Deterministic Lateral Displacement: Challenges and Perspectives*. ACS Nano, Vol. 14, pp. 10784–10795. https://doi.org/10.1021/acsnano.0c05186 - Bhagat, A. A. S., Hou, H. W., Li, L. D., Lim, C. T., & Han, J. (2011).
*Pinched flow coupled shear-modulated inertial microfluidics for high-throughput rare blood cell separation*. Lab on a Chip, 11(11), 1870–1878. https://doi.org/10.1039/c0lc00633e - Wang, X., Zandi, M., Ho, C. C., Kaval, N., & Papautsky, I. (2015).
*Single stream inertial focusing in a straight microchannel*. Lab on a Chip, 15(8), 1812–1821. https://doi.org/10.1039/c4lc01462f - Fluigent (2021)
*Basic Properties of Microfluidic Flows*. [Accessed on 25th August 2021] Physics of Microfluidics – Basic Properties of Microfluidic Flows | Fluigent - C.T. Shin U. Ghia, K.N. Ghia (1982)
*High-Resolution for incompressible flow using the Navier-Stokes equations and the multigrid method*. J. Comput. Phys., 48:387–411. - Ferry, M. S., Razinkov, I. A., & Hasty, J. (2011).
*Microfluidics for Synthetic Biology*. In Methods in enzymology (Vol. 497, pp. 295–372). https://doi.org/10.1016/b978-0-12-385075-1.00014-7 - Bayat, P., & Rezai, P. (2017).
*Semi-Empirical Estimation of Dean Flow Velocity in Curved Microchannels. Scientific Reports*, 7(1). https://doi.org/10.1038/s41598-017-13090-z - Bhagat, A. A. S., Kuntaegowdanahalli, S. S., & Papautsky, I. (2008).
*Continuous particle separation in spiral microchannels using dean flows and differential migration*. Lab on a Chip, 8(11), 1906–1914. https://doi.org/10.1039/b807107a - Russom, A., Gupta, A. K., Nagrath, S., Carlo, D. Di, Edd, J. F., & Toner, M. (2009).
*Differential inertial focusing of particles in curved low-aspect-ratio microchannels*. New Journal of Physics, 11, 075025. https://doi.org/10.1088/1367-2630/11/7/075025 - Ladyzhenskaya, O. A. (1975).
*Mathematical analysis of navier-stokes equations for incompressible liquids*. Annual Review Fluid Mechanics, v, 249–272. https://doi.org/10.1146/annurev.fl.07.010175.001341