Computational methods for inertial microfluidics: recent advances and future perspectives

Abstract

Numerical modeling has played a pivotal role in advancing inertial microfluidics, tracing its development from inception and offering deeper insights into the microscale phenomena governing inertial focusing. These computational approaches have simultaneously supported the proliferation of on-chip technologies. Initially adopted across diverse industries for passive and high-throughput operations such as trapping, separation, and sorting of particles, the greatest potential of inertial microfluidics lies in biomedical applications, where it serves as a cornerstone for processing cells in clinical and research settings. As the range of applications continues to expand, microfluidic devices are evolving into increasingly complex systems capable of handling diverse cell types and particles within miniature chip architectures. This growing complexity necessitates the enhancement of conventional numerical techniques and the integration of innovative computational approaches to address these emerging challenges. This review aims to provide an overview of the available numerical techniques, highlighting their advantages and limitations. We explore recent strides in computational inertial microfluidics, emphasizing advancements within the last four years and the emergence of innovative methodologies such as smoothed particle hydrodynamics. Furthermore, we describe the nascent role of machine learning in inertial microfluidics, noting its limited adoption compared to conventional microfluidics and highlighting the potential to transform the field, as well as challenges that need to be overcome.

Introduction

Inertial microfluidics is an effective approach for manipulating cells in microchannels through their positioning in flow using carrier fluid inertia^1,2,3. This method is widely used in applications such as flow cytometry, liquid biopsy, blood fractionation, and cell sorting. Research has extensively explored the principles and hydrodynamic forces involved in inertial focusing, particularly the lift forces acting on particles. Ho and Leal⁴ initially developed a theoretical model to describe these lift forces, identifying key forces such as the wall-induced and shear-gradient forces. Later studies by Di Carlo et al. ⁵ and Asmolov et al. ⁶ refined this understanding, showing different scaling behaviors in regions near the channel wall. Recent work by Zhou and Papautsky⁷ highlighted the significance of rotation-induced lift forces at the channel walls, proposing a two-stage model to explain asymmetric equilibrium positions in rectangular microchannels. It is now well-accepted that inertial focusing is generally the result of a balance between the wall-induced and the shear-gradient forces (Fig. 1). Addition of channel curvature or non-Newtonian fluid medium can modulate this force balance and alter the cross-sectional locations of the equilibria^1,2,3,8.

Fig. 1: Inertial focusing and forces responsible for cross-streamline migration.

a Two lift forces orthogonal to the flow direction act to equilibrate particles near the wall at Re_p > 1. The shear-induced lift force F_s is directed down the velocity gradient and drives particles toward channel walls. The wall-induced lift force F_w directs particles away from the walls and drives particles toward the channel centerline. The balance of these two lift forces causes particles to equilibrate. Reproduced from ref. ⁷. Copyright © Royal Society of Chemistry. b Five focal points are influenced by elastic force (F_E) in viscoelastic flows with negligible inertia and without shear thinning (e.g., Re < 0.01 and Wi > 0). Interaction between F_E and inertial force results in the disappearance of corner positions in viscoelastic flows where inertia is significant (e.g., Re > 0.01 and Wi > 0); Reproduced from ref. ⁸, licensed under CC BY 4.0

The understanding of inertial and viscoelastic focusing has largely been developed through simplified models, such as rigid spherical particles in straight or curved channels with common cross-sectional shapes such as circular, square, or rectangular. However, current experimental techniques face limitations in achieving a specific level of precision when investigating scenarios that involve more complex channel geometries, deformable particles, or combining inertial and viscoelastic effects. As applications using these combined effects and intricate devices become more common, there is a growing need to improve our understanding of the fundamental principles governing inertial migration dynamics in microfluidic systems involving non-Newtonian fluids, non-spherical and deformable particles, and curved channels.

Historically, numerical models such as the finite element method (FEM) and lattice Boltzmann method (LBM) have been used to explore the underlying physics of microfluidic systems. These techniques have been instrumental in solving velocity and pressure fields and investigating particle-fluid interactions, often validated through experimental data. However, as microfluidic systems evolve to include more complex channel geometries and are applied to more complex biological samples, standard numerical techniques are becoming inadequate for managing the increased complexity, and novel computational approaches are needed to effectively address these challenges.

In this review, we aim to provide an overview of current numerical techniques, highlighting their advantages and limitations for specific applications. Instead of delving deeply into the underlying physics, we will offer references for those seeking more detailed studies. We refer the interested reader to the reviews by Amini et al. ⁹, Martel et al. ¹⁰, Zhang et al. ², and Zhou et al. ³ for the fundamentals and applications of inertial microfluidics. Our discussion herein centers on recent advancements in computational inertial microfluidics, particularly in the past five years. The most recent reviews addressing this topic are the 2020 article by Bazaz et al. ¹¹ and the 2023 review by Shi¹². We begin by summarizing the most commonly used numerical techniques for simulating inertial focusing dynamics in microfluidic channels and highlighting recent innovations, such as the application of Smoothed Particle Hydrodynamics (SPH). The latter part of the review explores the use of artificial intelligence (AI) and machine learning (ML) algorithms in inertial microfluidics and the associated challenges.

Model selection and rationale

The selection of numerical methods for modeling inertial microfluidic devices is largely determined by the specific application and the complexity of the physical model required to accurately represent the relevant mechanisms. Generally, as the complexity of the physical model increases (i.e., incorporating additional physical phenomena), the complexity of the numerical model and the computational resources needed also increase. Figure 2 illustrates the minimum modeling requirements based on application complexity. For example, including particle deformation and the number of particles under consideration dictate the physical model, and consequently, the numerical scheme. In most cases, an inertial microfluidic model necessitates a fluid solver. Several numerical methods have been extensively used in inertial microfluidic studies, including FEM, LBM, and SPH. Regardless of the specific numerical method, the fundamental equations being solved are the Navier–Stokes equations, which govern the conservation of mass and momentum in fluid dynamics. These equations are:

$$\begin{array}{c}\nabla \cdot {\boldsymbol{u}}=0\\ \rho \left[\frac{\partial{\boldsymbol{u}}}{{\partial{t}}}+({\boldsymbol{u}}\cdot \nabla ){\boldsymbol{u}}\right]=-\nabla p+\mu {\nabla }^{2}{\boldsymbol{u}}+{\boldsymbol{f}}\end{array}$$

(1)

where u fluid velocity field, ρ is fluid density, p is pressure, µ is fluid dynamic viscosity, and f is the body force density. For non-Newtonian fluids, these equations are modified by incorporating either the divergence of the additional elastic stress term or by making the viscosity (µ) a function of the shear rate. Solving the Navier–Stokes equations accurately and efficiently using numerical methods is challenging because of the incompressibility constraint and the nonlinear convective term¹³, especially at finite Reynolds numbers (Re). Greater difficulties arise with viscoelastic flows due to the high Weissenberg (Wi) number problem related to numerical instability¹⁴. When selecting a numerical method for modeling inertial microfluidic devices, it is also important to consider the user’s experience and the availability of models. In the following section, we outline the minimum models required to represent a problem. However, more complex models may be preferable if the user has greater expertise with a specific model and it is readily accessible.

Fig. 2: The minimum modeling requirements.

The minimum capability required to model an inertial microfluidic device is based on the application type, separating rigid and deformable particles in dilute or concentrated flows

Point particle models

The simplest physical model for a particle in an inertial device is a point particle model, which uses an inertial lift force model to govern the particle’s lateral motion. Inertial lift models for rigid particles of various confinements have been developed^4,15. This model has shown good agreement with experimental data for dilute suspensions where particle deformation is small¹⁶, although it neglects interactions between different particles as well as the effect of the particle on the fluid. The computational resource requirements of such a model are relatively low, increasing its applicability in a device design process where extensive parameter studies may be required with multiple iterations. Such models are readily available in commercial software, such as COMSOL Multiphysics and Ansys Fluent, further increasing their utility.

Rigid fully resolved particle models

When modeling devices with multiple particles, particle interactions can significantly alter the overall flow conditions. Existing inertial lift models in point particle simulations do not consider these interactions. Consequently, it is essential to fully resolve the particles, which may necessitate a structural solver, depending on the extent to which particle deformation impacts the overall system. For rigid particles, a fluid solver coupled with appropriate boundary conditions can represent the interactions between rigid particles and their effect on the fluid. The specific boundary condition used depends on the numerical method, but must satisfy the no-slip condition at the particle surface¹⁷.

Deformable fully resolved particle models

Particle deformation can significantly influence lateral migration behavior, particularly for soft particles such as red blood cells (RBCs). To account for particle deformability, structural models are coupled with fluid solvers to create fluid-structure interaction (FSI) models. Various FSI models with different coupling strategies are discussed below, but generally, deformable particles are modeled as capsules that consist of a thin membrane encasing an internal fluid¹⁸. While capsule modeling has been extensively studied for non-inertial microfluidic applications¹⁹, recent advancements have been applied to inertial microfluidics. The capsule membrane can be modeled as either elastic (using Hooke’s law) or hyperelastic, which is commonly used for biological cells (neo-Hookean). A comprehensive discussion of physical particle models is available in a recent tutorial review by Owen et al. ²⁰.

Fluid-structure interaction methods

Fluid-structure interaction (FSI) methods can be generally categorized as either monolithic or partitioned. A monolithic method uses the same numerical approach to solve both the fluid and structural governing equations simultaneously. In contrast, a partitioned method uses different numerical methods for the fluid and the structural governing equations, necessitating a coupling scheme to manage their interaction. A critical difference between these approaches lies in how they ensure the kinematic and dynamic interface conditions at the fluid-structure boundary. Monolithic methods inherently satisfy these conditions by solving the governing equations synchronously, while partitioned methods solve the equations sequentially, requiring explicit measures to ensure the interface conditions are met²¹. Monolithic methods are also generally more stable and accurate than partitioned approaches, but can be challenging to implement for large deformations in mesh-based numerical methods, where discretizing the interface is difficult²². Additionally, using the same numerical method for both fluid and structure can impose significant constraints if the method is not suitable for both. In contrast, partitioned approaches allow for the coupling of different numerical methods tailored to the fluid and structure. However, since they are not solved synchronously, they can face accuracy and stability issues without careful handling, particularly in meeting interface conditions. More complex coupling schemes can enhance accuracy and stability²³, but are inherently more computationally demanding and challenging to implement. A detailed discussion of partitioned FSI coupling schemes is provided by Owen²¹.

Numerical solvers in inertial microfluidics

The selection of a numerical method often depends on the specific problem and the available computational resources. The most commonly used numerical solvers in inertial microfluidics are the FEM and LBM. Additionally, Smoothed Particle Hydrodynamics (SPH) is gaining popularity for its potential in modeling more complex inertial microfluidic systems. These methods are briefly summarized below.

Finite element method

FEM is a well-established numerical technique widely used for solving partial differential equations across various engineering fields, including fluid dynamics, although it was originally developed for structural analyses²⁴. FEM has different formulations, each designed for specific applications, with Eulerian FEM being the most commonly used in microfluidics. In Eulerian FEM, the physical domain is discretized into small finite elements, each with a specific shape and a fixed number of nodes, where the solutions are approximated using piecewise shape functions. The equations describing behavior at each element are then assembled into a larger system of equations representing the entire domain. A mesh dependence test is typically conducted to evaluate the accuracy and convergence of the simulation results concerning mesh size and quality, ensuring accurate and reliable results while optimizing computational resources.

In inertial microfluidics, FEM is frequently used to calculate the steady-state lift forces on particles^5,25,26,27. This method, known as Direct Numerical Simulation (DNS), solves the Navier–Stokes equations in a simulation box representing a channel segment with appropriate periodic boundary conditions. A spherical ‘void’ in the domain represents the particle, with boundary conditions on the surface to account for the translational and rotational velocity of a freely moving particle (Fig. 3a). The lift forces are then calculated by integrating the stress on the particle surface. DNS is one of the most accurate and robust methods for obtaining inertial and elasto-inertial lift force fields.

Fig. 3: Main numerical methods and coupling strategies for fluid flow and fluid-particle interactions.

a Example of a FEM simulation setup for DNS showing the domain and cross-section of a rectangular channel. Finer mesh is used at the surface of the particle, surrounded by several boundary layer meshes to capture sharp stress and velocity gradients in the vicinity of the particle; Reproduced from ref. ²⁶, licensed under CC BY 4.0. b Schematic of SPH showing the kernel function operating on a particle and extending over a certain smoothing length; Reproduced from ref. ³⁶, licensed under CC BY 4.0. c Schematic diagram of the LBM-IBM-FEM method. LBM is used to solve fluid flow; FEM is used to capture the motion of rigid or deformable particles, and IBM is used for the fluid-particle interaction. Reproduced with permission from ref. ⁹⁴. Copyright © 2022, American Chemical Society

Several commercial FEM packages, such as COMSOL Multiphysics, are popular in inertial microfluidics. These software packages offer user-friendly interfaces that allow researchers to create and simulate microfluidic models quickly and easily. Their popularity is due to their versatility, accuracy, and ability to model a wide range of physical phenomena. Overall, FEM has several advantages in inertial microfluidics, particularly its ability to handle complex geometries and provide accurate flow fields, which help reveal the underlying physical mechanisms governing particle motion and are valuable in device design and optimization.

Lattice Boltzmann method

LBM is a numerical simulation technique for modeling fluid dynamics²⁸. Here, the fluid is represented as a lattice of discrete points, with probability distributions describing the velocity and density at each point. These distributions evolve over time according to the Boltzmann transport equation, which describes fluid movement and interactions. The particle populations in these distributions have velocities belonging to sets denoted as DdQq. The number of velocities in the set affects the computational power required, with D3Q15 and D3Q27 commonly used for discretizing the Navier–Stokes equations. The D3Q19 set is optimal for 3-D simulations in inertial microfluidics, balancing accuracy and memory cost²⁹. In LBM, particle populations (f_i) propagate and collide following the evolution equation

$${f}_{i}\left(\boldsymbol{x}+\boldsymbol{c}_{i}\Delta t,t+\Delta t\right)={f}_{i}^{* }\left(x,t\right)$$

(2)

where $\Delta t$ is the time step, c_i are the discrete velocities, and f_i^* are the post-collision populations. This involves a two-step process: during the streaming phase, particles move to cells based on their velocities, and in the collision phase, distribution functions within each cell adjust towards local thermodynamic equilibrium, conserving particle number and momentum. The lattice Boltzmann equation is derived by discretizing velocity space, physical space, and time, enabling efficient and parallel computations on modern architectures. Comprehensive descriptions of collision models³⁰ and general LBM methods can be found in various works^29,31,32, including an extensive tutorial review by Owen et al. on LBM’s application to inertial microfluidics³³.

Smoothed particle hydrodynamics method

SPH is a Lagrangian method developed by Gingold and Monaghan^34,35 initially for astrophysical problems, and now applied across various fields³⁶. Unlike mesh-based methods, SPH discretizes the computational domain using particles, which serve as interpolation nodes. Each SPH particle has its own equation of motion:

$${\rho }_{i}\Delta {v}_{i}\frac{{d}^{2}{{\boldsymbol{r}}}_{i}}{d{t}^{2}}=-\Delta {v}_{i}{\boldsymbol{\nabla }}P+{\rho }_{i}\Delta {v}_{i}{{\boldsymbol{F}}}_{i}$$

(3)

where ${\rho }_{i}$, $\Delta {v}_{i}$ and $\boldsymbol{r}_{i}$ are the density, volume, and center of mass of the i^th SPH particle, $\nabla P$ is the pressure gradient at $\boldsymbol{r}_{i}$, and $\boldsymbol{F}_{i}$ is the body force acting on each SPH particle i, which possesses its own mass, velocity, and other field variables. Each field variable f is averaged at the position ${r}_{i}$ as:

$$f\left(\boldsymbol{r}_{i}\right)=\sum _{j}{m}_{j}\frac{{f}_{j}}{{\rho }_{j}}W\left(\boldsymbol{r}_{i}-\boldsymbol{r}_{j}\right)$$

(4)

where W is a smoothing kernel function, normalized so its integration equals 1 within its smoothing length, and ${m}_{j}$ and $\boldsymbol{r}_{j}$ are the mass and position of the neighboring particle j.

In SPH simulations, the kernel support domain size is defined by the smoothing length h (Fig. 3b). As h approaches zero, the kernel function approximates the Dirac delta function. Commonly used kernel functions in SPH include the Gaussian, quintic spline, and Lucy kernels. SPH has proven effective for simulating complex fluid flows with free surfaces and has been successfully applied to inertial microfluidics^37,38.

Most SPH formulations assume weakly compressible fluid properties, approximating the incompressible limit. Monaghan used an Equation of State (EOS) to relate fluid pressure to density through a power law^39,40, providing a simple and efficient method for computing pressure based on density. The general form of this EOS is:

$$P(\rho)=B\left[{\left(\frac{\rho }{{\rho }_{0}}\right)}^{\gamma }-1\right]$$

(5)

where P is the fluid pressure, B is the effective bulk modulus, $\rho$ and ${\rho }_{0}$ are the current and initial densities, and $\gamma$ is a power coefficient. A popular choice, Tait’s EOS, used to model water at ambient conditions, is

$$P(\rho)=\frac{{c}_{0}^{2}{\rho }_{0}}{7}\left[{\left(\frac{\rho }{{\rho }_{0}}\right)}^{7}-1\right]$$

(6)

where ${c}_{0}$ and ${\rho }_{0}$ are the speed of sound and density at zero applied stress.

Our research group pioneered the use of weakly compressible SPH to model inertial microfluidics, employing a formulation similar to the original method by Gingold, Monaghan, and Lucy⁴¹. By setting the sound speed in the simulation to be more than 20 times higher than the maximum velocity, we limited flow compressibility to within 3%. We used Morris’s formula⁴² to incorporate viscous stress, applying the viscosity of water at room temperature. To ensure sufficient nearby particles, we used a Lucy smoothing kernel with an appropriate smoothing length. Two critical components in our simulations were the generation and the implementation of non-penetration and non-slip boundary conditions. We adopted the approach by Morris et al. ⁴² to extrapolate velocity to wall particles based on their distance from the boundary. To generate flow, we applied a constant body force to all SPH particles in the simulation domain, modeling the pressure gradient in actual microfluidic devices. Periodic boundary conditions (PBCs) allowed fluid particles exiting the channel outlet to re-enter from the inlet, simulating a long channel. The body force approach, combined with PBCs, was preferred for its simplicity and robustness⁴³, though more complex processes could also create a pressure-driven flow in the channel^44,45.

Coupling strategies for fluid-structure interaction

Immersed boundary method and fictitious domain methods

The Immersed Boundary Method (IBM) and fictitious domain methods (FDMs) facilitate seamless coupling between fluid mechanics and solid/structure mechanics solvers, enabling accurate simulations of complex fluid-structure interactions. In FDMs, the Navier–Stokes equations are solved within a computational domain that includes both the actual fluid domain and a fictitious domain representing the immersed structure. This approach allows fluid flow to penetrate the fictitious domain while imposing appropriate boundary conditions on the immersed structure, simplifying the treatment of complex geometries and moving boundaries by eliminating the need for explicit tracking within the fluid mesh. IBM, a widely used FDM-type approach in inertial microfluidics, is not a standalone solver but a method that couples fluid mechanics solvers with solid/structure solvers. IBM employs two meshes: a Lagrangian mesh for the immersed structure, which moves with deformations and interactions, and a fixed Eulerian mesh for the fluid. The key aspect of IBM is the interpolation and mapping of displacement, velocity, and forces between these meshes to facilitate fluid-structure interaction.

The force f exerted by the structure on the fluid is calculated using a forcing term F (s,t) in the momentum equation:

$$\boldsymbol f\left(x,t\right)=\int \boldsymbol F\left(s,t\right)\delta \left(\boldsymbol x- \boldsymbol X\left(s,t\right)\right){\text{d}s}$$

(7)

where $\delta$ is the Dirac delta function and ${\boldsymbol{ X}}\left(s,t\right)$ represents the set of Lagrangian points with Lagrangian coordinate s at time t. Simulating high Re number flows using IBMs can be challenging due to the difficulty of resolving thin boundary layers near the structure surface. However, since inertial microfluidics typically operates in a laminar regime, IBM is well-suited for investigating inertial migration dynamics in microchannels⁴⁶.

IBM can also bridge the lattice Boltzmann and the finite element methods in the LBM-IBM-FEM approach (Fig. 3c). In this method, FEM elements capture the motion of rigid or deformable particles, while the LBM computes the fluid flow. This combined solver enables the study of deformable particles, while the LBM-IBM can only resolve rigid particle dynamics. Another notable FDM for capturing the boundary of a moving rigid particle is the Interpolated Bounce-Back (IBB) scheme. This method constructs the unknown distribution functions at boundary nodes using the known ones while observing the hydrodynamic constraints^47,48.

Arbitrary Lagrangian–Eulerian method

Most Eulerian-based FEM simulations in inertial microfluidics are limited to modeling spherical particles as rigid bodies or using point particle assumptions. To address fluid-particle interactions involving transient particle motion, the Arbitrary Lagrangian–Eulerian (ALE) approach is employed. ALE-FEM combines Eulerian and Lagrangian FEM, using a blend of meshes that move and deform with the structure, making it suitable for fluid-structure interaction problems in inertial microfluidics. For instance, Esposito et al. ⁴⁹ used ALE-FEM to study cell deformability in cylindrical and square microchannels. This study neglected particle rotation to reduce the need for re-meshing during the simulation. As particles rotate, the fluid domain and its mesh change over time, complicating FEM use even with ALE-FEM due to excessive re-meshing requirements. One solution to this challenge is the Immersed-FEM (IFEM) method⁵⁰, where finite element formulations are solved for both fluid and solid domains, eliminating the need for re-meshing. In IFEM, the background fluid mesh does not need to follow the motion of the flexible fluid-particle interface. Other numerical techniques, such as SPH, may be more suitable for simulating the transient motion of rotating spherical and non-spherical particles in microfluidic systems, as discussed below.

Meshless monolithic method

As previously mentioned, our group pioneered the use of the SPH method in inertial microfluidics⁵¹, employing a meshless or monolithic scheme to couple fluid and particle dynamics. This approach solves the governing equations for both fluid and particles simultaneously, eliminating the need for separate coupling strategies. Both fluid and particle domains are represented as SPH particles, with non-penetration and no-slip boundary conditions imposed through specific pair interactions. We utilized this technique alongside high-speed imaging at the single-particle level to fully resolve inertial migration in the channel cross-section for spherical particles⁵¹. Additionally, we conducted a parametric investigation of prolate particles in straight rectangular channels using this method³⁷. Meshless coupling techniques are also effective for modeling deformable particles, such as RBCs.

Soleimani et al. ⁵² developed a 3-D numerical method entirely based on SPH, capable of simulating RBCs through interactions between a shell-like solid structure and fluid. They modeled the deformation of a single RBC passing through a stenosed capillary, using the fluid-solid coupling approach developed by Adami et al. ⁵³. While this work demonstrated the robustness of SPH in handling fluid-particle interactions for deformable cells, to our knowledge, SPH has not yet been employed to model deformable particles for inertial focusing purposes. Table 1 summarizes the most commonly used numerical methods in studies of inertial microfluidics.

Table 1 Commonly used numerical methods in inertial microfluidics based on model complexity

Reduced-order and surrogate methods

In certain scenarios, modeling particle motion within the channel using fully resolved fluid-particle coupling results in substantial computational overhead and decreased efficiency. In such instances, surrogate models offer effective alternatives and can significantly reduce computational time and complexity. These models calculate hydrodynamic lift forces in the first step, and particle trajectories are obtained in the second step using a straightforward particle tracing algorithm, typically by solving Newton’s second law:

$${m}_{p}{\ddot{{\boldsymbol{x}}}}_{{p}}={{\boldsymbol{F}}}_{{\text{drag}}}+{{\boldsymbol{F}}}_{{\text{lift}}}+{{\boldsymbol{F}}}_{{\text{extra}}}$$

(8)

where ${m}_{p}$ and ${\ddot{{\boldsymbol{x}}}}_{{{p}}}$ are the particle’s mass and acceleration vector, respectively. The drag force ${{\boldsymbol{F}}}_{{\text{drag}}}$ is generally calculated using Stokes’ drag formula, while the lift force ${{\boldsymbol{F}}}_{{\text{lift}}}$ are obtained from analytical lift force equations. Additional forces ${{\boldsymbol{F}}}_{{\text{extra}}}$, such as the elastic lift for viscoelastic fluids, may be incorporated into the equation. For curved channel geometries, such as spirals or serpentines, the additional Dean drag force is typically included in the ${{\boldsymbol{F}}}_{{\text{drag}}}$ term.

Alternatively, the lift forces may be calculated using machine learning data-driven algorithms, which will be discussed in a subsequent section. Lift force data can also be extracted using the Direct Numerical Simulation Particle Tracing (DNS-PT) approach. In DNS-PT, a fully resolved fluid-particle algorithm is employed to calculate lift forces, while particle trajectories are obtained using the point particle assumption, thereby reducing model complexity. Additionally, surrogate methods may incorporate various analytical and semi-empirical techniques aimed at reducing model complexity.

Modeling focusing on Newtonian fluids

The initial observation of inertial migration was reported by Segré and Silberberg⁵⁴ in circular pipes. Saffman later proposed a force in linear shear flow, independent of particle rotation and based solely on the velocity differences on each side of a particle⁵⁵. Ho & Leal⁴ and Vasseur & Cox⁵⁶ applied similar mathematical approaches to quadratic flows, identifying a force directed toward channel walls due to shear rate changes. These developments have shaped the current understanding of inertial lift forces. Fundamental studies by McLaughlin⁵⁷, Asmolov^6,15, and Matas et al. ⁵⁸ progressively considered finite Re numbers and particle sizes, offering a more realistic description of this phenomenon. Zhou and Papautsky⁷ proposed a two-stage migration model that clarified equilibrium positions in rectangular channels, incorporating shear-induced, wall repulsive, and rotation-induced lift forces.

Numerical methods have become crucial for overcoming experimental limitations, providing deeper insights into inertial focusing. The use of DNS in 2-D flow by Yue et al. ⁵⁹ marked a significant milestone, with Di Carlo et al. ⁹ later using FEM to propose scaling laws for inertial lift forces. The wall-induced lift was found to scale as a⁶/H⁴ and the shear-gradient lift as a³/H, where a is the particle diameter, and H is the channel height. The shear-gradient force dominates the inertial migration process, with the Saffman and rotational forces generally neglected in many inertial focusing applications. In-depth review of the physics and applications of inertial microfluidics is given in previous reviews^1,3,10. Instead, the following sections aim to highlight the numerical methods and emerging simulation techniques applied to inertial microfluidics.

Wall-bounded flows and straight channels

Much of the research on inertial focusing has centered on rigid spherical particles. However, biomedical applications often involve suspensions of soft particles, such as cells, leading to increased interest in bubbles, drops, and the impact of particle deformability and shape. Soft particles are typically modeled as capsules, which are elastic membranes enclosing a fluid interior. These capsules deform under hydrodynamic forces and stress from the enclosed fluid. The capillary number (Ca) and Laplace number (La) are common metrics for characterizing capsule behavior in microfluidic systems. The capillary number (Ca = μV/σ, where μ is the fluid viscosity, V is the characteristic velocity, and $\sigma$ is the surface tension) establishes a relationship between viscous forces to surface tension, with a small Ca indicating a less flexible capsule or a strong external flow. The Laplace number (La = σρL/μ², where L is the length,$\,\rho$ is the density, and $\mu$ is the fluid viscosity) relates the viscous stress to the surface tension force. Particles with low La are considered “soft” and more susceptible to deformation, while those with high La are “stiff” and maintain their shape. Indeed, deformability-based inertial microfluidic devices use La to differentiate particles, with values between 1 and 500^33,60,61.

Bubbles and drops in microchannels have been used to investigate deformation-induced forces. Studies at finite Re numbers by Mortazavi & Tryggvason⁶² and Rivero-Rodríguez & Scheid⁶³ analyzed the dynamics of dispersed objects like bubbles and drops in microchannels. They examined how bubble size, Re number, and capillary number affect velocity and lateral positioning. Hadikhani et al. ⁶⁴ demonstrated that bubble trajectories in microchannels can be controlled by adjusting the balance of forces acting on them. At low Re numbers (Re < 40) and capillary numbers (Ca < 1), the bubble diameter and the channel aspect ratio critically influence the observed trajectories.

Exploring rigid and soft particle pairs and trains can yield insights into the physics of multi-particle systems, such as clusters of RBCs. Aouane et al. ⁶⁵ investigated the pairing mechanism of two particles due to hydrodynamic interaction, while Rivero-Rodríguez & Scheid^62,63 examined bubble train dynamics. Patel & Stark⁶⁰ conducted a numerical investigation on particle pairing dynamics in a square microchannel. In these studies, the LBM-IBM-FEM technique was used (LBM was used to simulate fluid flow in the microchannel, and FEM was used to model the particles’ deformation and dynamics, while IBM was then used to couple the flow field and the particle model). More recently, Huet et al. ⁶⁶ investigated the dynamics of 3-D elastic capsules flowing through a square microchannel with a sharp corner, analyzing single capsules, pairs, and trains of ten. They found that capsule deformation and membrane stress increase significantly in the inertial regime, scale nearly linearly with Re and Ca numbers, and that hydrodynamic interactions can induce capsule repulsion and separation—providing practical insights for microfluidic capsule control.

Studies on the deformability of spherical particles in channel flows have primarily used IBM-LBM-FEM techniques. Huang et al. ⁶⁷ explored the distribution of deformable particles in 2-D inertial channel flow, while Rezghi et al. ⁶⁸ investigated the 3-D lateral migration of viscoelastic particles in tube flow. Esposito et al. ⁴⁹ used FEM-ALE simulations to study deformable cells, finding that softer cells tend to migrate closer to the center in cylindrical ducts, and, more recently, investigated the behavior of two initially spherical, neutrally buoyant elastic particles suspended in a Newtonian fluid that were subjected to pressure-driven flow within a cylindrical microchannel⁶⁹. Figure 4a illustrates the interaction between the fluid and deformable capsules.

Fig. 4: Effects of particle shape, pair formation, or particle train on inertial migration of rigid and deformable particles.

a LBM-IBM-FEM was used to capture the dynamics of trains of deformable capsules. A ‘+’ marker in (i) and (ii) indicates the capsule being pulled outwards, A ‘−’ shows the capsule is being pushed inwards. Reproduced with permission from ref. ⁶⁷. Copyright © AIP Publishing. b The formation and stability of homogeneous pairs of soft particles were studied using LBM-IBM-FEM. (i) ‘swap and scatter’ and (ii) ‘swap and capture’ interactions, listed as two of the six interaction types observed; Reproduced from ref. ⁷⁰, licensed under CC BY 4.0. c LBM was used to model the self-organization of a single-line particle train with different shapes; Reproduced with permission. Copyright © AIP Publishing. d Comparison of high-speed imaging and SPH simulations of a cancer cell aggregate and a prolate particle undergoing logrolling motion, respectively. The snapshots show the top view of a straight rectangular channel at 3 different moments; Reproduced with permission from ref. ³⁷. Copyright © AIP Publishing. e The SPH simulation setup was used to study transient pathways of particles in a rectangular channel. Reproduced from ref. ⁵¹, licensed under CC BY 4.0. f SPH and FEM were used in conjunction to study inertial migration of ellipsoidal particles in inertial shear flow between two walls. (i) Bifurcation diagram showing focusing position of prolate particles as a function of Re_P. (ii) Streamwise vorticity in the particle plane is responsible for the reversal in the bifurcation. Reproduced from ref. ⁷⁹ JFM © Cambridge Univ. Press

The LBM-IBM-FEM technique has also been used to investigate the formation and stability of multiple deformable objects in flow. Owen & Kruger ⁷⁰ studied the formation of soft particle pairs in a straight channel, identifying six distinct interaction types. These are illustrated in Fig. 4b. Thota et al. ⁷¹ explored the impact of size on stable pair formation while maintaining constant particle softness. However, despite the valuable insights into particle-particle interactions, both studies concentrate solely on slightly deformable particles within a straight duct configuration.

In addition to particle deformability, several studies have explored the influence of particle shape. Non-spherical particles, such as ellipsoidal and non-ellipsoidal particles, exhibit distinct behaviors in microfluidic flows. However, the behavior of these shaped particles in microfluidic flows remains inadequately understood. Jeffery⁷² initially investigated the rotational dynamics of ellipsoidal particles in simple shear flow, developing a framework for their behavior. The general rotational behavior of spheroids resembles that of a kayak paddle, hence the term “kayaking.” The two extreme behaviors are termed “logrolling motion,” where the particle rotates about its vorticity axis, and “tumbling,” where the particle rotates in the flow gradient plane. However, when inertia is considered (Re > 0), the infinite set of orbits is reduced to one^73,74. Lashgari et al. ⁷⁵ used IBM to conduct the first systematic numerical study on ellipsoidal particles in microchannels, while Nizkaya et al. ⁷⁶ used LBM and Hafemann et al. ⁷⁷ used IBM to explore the inertial focusing and migration of oblate spheroidal particles. Further studies by Hu et al. ⁷⁸ investigated the self-organization of rectangular and elliptical particles in a channel flow using LBM, by exploring the effect of Re, confinement ratio, and particle concentration on the formation of particle trains (Fig. 4c).

Most computational work on non-spherical particles in microfluidic channels has predominantly utilized the LBM-IBM approach. While SPH has been extensively applied in various fluid dynamics problems, its use in inertial microfluidics has been limited. In 2022, our group pioneered the application of SPH to model non-spherical particles in microfluidic flow³⁷. Unlike mesh-based methods like FEM and LBM, SPH offers advantages in modeling the inertial migration dynamics of non-spherical particles within complex flow fields. SPH implicitly considers boundaries by incorporating them into the interaction forces between particles, leading to a more accurate representation of fluid-solid interactions. Additionally, as a mesh-free method, SPH is well-suited for modeling complex geometries and non-uniform particle distributions, providing a more natural handling of non-spherical particles compared to mesh-based methods.

Lauricella et al. ³⁷ employed SPH to numerically map the migration dynamics of a prolate particle in a straight rectangular microchannel at moderate Re. While the SPH model was previously validated for spherical particles⁵¹, Lauricella et al. included validation against Jeffery’s theory and existing computational work by Lashgari et al.^37,72,75. Through a parametric investigation of various particle aspect ratios and confinement ratios, they identified a threshold of the particle confinement ratio beyond which the particle reaches a stable logrolling motion, in contrast to the commonly reported tumbling motion. Experimental validation using aggregates of cancer cells with the same shape and aspect ratio as the simulated prolate ellipsoid confirmed these findings under the same flow conditions and channel geometry (Fig. 4d). The study demonstrated that the rotation period of prolate particles depends on both size and aspect ratio, challenging previous results that attributed the rotation period solely to particle size or aspect ratio.

In a separate experimental study, we observed similar logrolling motion in triplets of cancer cells as they moved closer to the channel center. Lauricella et al. also applied SPH to explore how ellipsoidal particles behave in inertial shear flow within a simplified system featuring two parallel walls, confining the flow along the z-axis, and applying periodic boundary conditions (PBCs) in the x and y directions⁷⁹. Expanding on prior research⁸⁰ that uncovered a supercritical pitchfork bifurcation for spherical particles in shear flow using LBM, they employed SPH to verify this phenomenon. The use of SPH allowed modeling at higher Re, showing that, unlike spherical particles, ellipsoidal particles returned to the central position from off-center locations at this new Re range. FEM was used for flow visualization to unravel the underlying mechanism of this bifurcation reversal, attributing it to altered streamwise vorticity and symmetry breaking of pressure.

In addition, surrogate methods have been used to rapidly predict the inertial focusing patterns in microfluidic devices. Notably, Mashhadian and Shamloo⁸¹ proposed a method for fast prediction of particles within the channel cross-section using a velocity profile matching technique. This method approximates the velocity profile of a non-rectangular channel by combining segments of velocity profiles from various rectangular channels and predicts the focusing behavior of particles based on three types of equilibrium positions relative to the channel walls.

Curved channels and other geometries

In applications involving spiral and serpentine channels, a secondary flow arises due to curvature, and is characterized by vortices formed within the channel cross-section due to centrifugal acceleration in curved channels^82,83. As fluid traverses the curved channel, these vortices create regions of varying velocity, leading to a pressure gradient orthogonal to the main flow, resulting in the Dean drag force (F_D). Understanding the governing equations and dominant forces in curved geometries can be challenging due to complex flow characteristics. Most available data are from experimental investigations, introducing ambiguities in particle interactions and migration across the channel cross-section.

Several computational studies have explored inertial focusing in curved and spiral channels, primarily using FEM and IBM. For example, Do et al. ⁸⁴ implemented an IBM algorithm in OpenFOAM to study inertial focusing in spiral microchannels with rectangular and trapezoidal cross-sections. They introduced a unique periodic boundary condition using a shadow particle, similar to the SPH formulation⁸⁵, allowing for periodicity in the surrounding domain. This approach allowed the investigation of inertial migration in a spiral channel, revealing stable equilibrium locations near the inner wall in rectangular cross-sections and near the outer wall in trapezoidal channels.

In another study, Ince et al. ⁸⁶ examined how the curvature angle of a curved channel influences inertial focusing, employing experiments and a Volume of Fluid (VOF) multiphase approach implemented in Ansys Fluent. They demonstrated that higher channel curvature improves focusing quality. Valani et al. ⁸⁷ used the FEM to exploit bifurcations in the particle equilibria regarding duct bend radius for particle separation in spiral channels. They extended their study to non-neutrally buoyant spheres, exploring how inertial and gravitational forces affect particle migration with duct bend radius. They found that non-neutral buoyancy may hinder size-based separation, suggesting that density-based separation is a more promising principle in curved channels.

While these studies focused on spherical particles, Hafemann et al. ⁸⁸ investigated the behavior of prolate and oblate particles in spiral ducts, comparing them with spherical particles. They used a particle-resolving Euler–Lagrange approach coupled with IBM⁸⁹, highlighting differences in focusing position and migration velocity between spheres and ellipsoids (Fig. 5a). For spheres with confinement ratio a/H = 0.2, a single stable equilibrium position near the inner wall was observed. In contrast, ellipsoidal particles exhibited two stable equilibria positioned further away from the wall (Fig. 5b). Additionally, non-spherical particles had lower migration velocities, requiring longer times and lengths to achieve focusing.

Fig. 5: Modeling strategies and results for curved and other non-straight channel geometries.

a Schematics of the Euler–Lagrange approach coupled with IBM developed for curved channels; (i) coordinate transformation between the cylindrical and toroidal coordinate systems. (ii) relation between the basis vectors of the particle and the Eulerian grid cells used in the volume integral; Reproduced with permission from ref. ⁸⁹. Copyright © AIP Publishing. b Simulated trajectories of prolate and oblate particles of different sizes projected on the cross-section of the channel; Reproduced from ref. ⁸⁸, licensed under CC BY 4.0. c Schematics and setup of the DNS-PT method, reproduced with permission from ref. ⁹⁰; (i) The sperm is modeled with manually disabled rotation due to the tail effect. (ii) The inertial force field is projected onto the spiral channel to simulate particle trajectories. d The migration trajectories of the spherical particles and sperm cells within the channel cross-section. Solid circles indicate the initial positions, while hollow circles denote the final position; reproduced with permission from ref. ⁹⁰. e Schematics and the hexahedral mesh configuration of the helical channel simulated in Ansys Fluent; Reproduced from ref. ⁹³, licensed under CC BY 4.0. f Schematics of the cross-slot geometry and top view of the spiral vortex at Re = 120. In this work, LBM-IBM was used to study flow and particle dynamics. Reproduced from ref. ⁹⁵, licensed under CC BY 4.0

Recently, Naderi et al. ⁹⁰ utilized the DNS-PT approach to model the unique focusing behavior of sperm cells. The novelty of their work lies in considering the oval-shaped geometry of the sperm head and incorporating the effects of the sperm tail through an appropriate boundary condition on the particle surface (Fig. 5c). These simple modifications to traditional DNS settings enabled accurate prediction of sperm focusing near the outer wall of a spiral channel. They deciphered the detailed migration pathways of the sperm cells within the channel cross-section (Fig. 5d) and examined the effects of the inertial lift and the Dean drag forces through a side-by-side comparison with the spherical particles.

In another study, Ebrahimi et al. ⁹¹ used IBM in conjunction with FEM to study deformable particles in curved channels with square and rectangular cross-sections. The simulations covered entire channels, utilizing high curvature to induce faster migration and reduce simulation time. The study observed a single focusing position near the inner wall, transitioning to two focusing locations near the center of the Dean vortices with increasing Re. These equilibrium locations resembled those in microchannels with circular cross-sections and comparable hydraulic diameters. As Re increased, the secondary flow strengthened, causing Dean vortices to move toward the center, shifting equilibrium positions toward the outer wall, independent of the Capillary number.

In addition to spiral and curvilinear channels, recent years have seen studies of more intricate channel geometries using numerical modeling. Kovalčíková et al. ⁹² used LBM to investigate the flow field in toroidal and helical channels with circular cross-sections. A more detailed parametric study of helical microchannels was conducted by Palumbo et al. ⁹³ using a Lagrangian model (Fig. 5e). This model, however, neglected particle-particle interactions and the effect of the fluid domain on particle dynamics, limiting its accuracy in predicting inertial migration dynamics. Ni et al. ⁹⁴ used LBM-IBM-FEM to elucidate focusing mechanisms in a high-aspect-ratio asymmetric serpentine (HARAS) channel, enabling single-line focusing at the center. Their findings revealed that the periodic turn of the Dean flow induces particle movement in waves, facilitating 3-D single-line focusing, adjustable by flow rate.

Kechagidis et al. ⁹⁵ expanded work on cross-slot junctions for deformability cytometry. While previous research focused on flow dynamics and particle trapping, Kechagidis et al. used an LBM-IBM code to investigate interactions between particles and vortices at moderate inertia (Fig. 5f). Their study on rigid neutrally buoyant spherical particles provided insights into the correlation between particle and flow properties with particle trajectory and residence time in the junction. They emphasized the need to extend this study to deformable particles for practical guidelines in designing devices involving clusters of particles⁹⁶ and cells⁹⁷.

De Marinis et al. ⁹⁸ recently developed an incompressible LBM-IBM-FEM solver that is capable of modeling the migration of deformable capsules through curved geometries. Their work numerically compared the migration of capsules through straight and curved geometries, reproducing the known effect that curved channels can separate particles by deformability through the migration to different equilibrium positions. A comprehensive review of complex geometry modeling using the LBM-IBM-FEM can be found in a recent tutorial review by Owen et al. ³³. In this review, the authors state that there is a current need to focus LBM use in inertial microfluidics towards complex geometry modeling.

Surrogate methods have also been used to model the inertial focusing behavior in curved channel geometries. Cruz and Hjort⁹⁹ utilized a semi-analytical force balance model to simulate particle separation in high-aspect-ratio curved microchannels. Harding et al. ¹⁰⁰ explored the inertial focusing dynamics in curved microchannels with trapezoidal cross-section using the perturbation expansion method. They identified an optimal skew level for the trapezoidal cross-section to enhance particle separation and provided a detailed explanation of how particle focusing dynamics vary with the degree of skew.

Modeling focusing on non-Newtonian fluids

The application of non-Newtonian, viscoelastic fluids in microfluidics has been explored for over two decades^8,101. These fluids, often polymeric solutions, have unique properties that enhance particle focusing into a single stream. Viscoelastic fluids exhibit elastic behavior under deformation due to the stretching of polymer chains under applied stresses. This elastic behavior can induce cross-stream migration of particles. The dimensionless Wi number, defined as the product of the fluid relaxation time (λ) and the shear rate ($\dot{\gamma }$), characterizes the magnitude of the viscoelastic force relative to the viscous force. Additionally, the elastic number (El), defined as the ratio of the elastic force to the inertial force, is given by El = Wi/Re. In purely inertial flows, four distinct focusing positions arise from the equilibrium of inertial forces, typically observed for Re > 1 and Wi = 0. In viscoelastic flows, Re << 1 and Wi > 0, and the presence of the elastic forces leads to five focusing positions. Elasto-inertial flows, characterized by both significant inertia and elasticity (Re > 0.01 and Wi > 0), result in the elimination of corner positions, leading to a single focusing position in rectangular channels (Fig. 6a). These flows were first observed in the early 1960s by Karnis & Mason¹⁰² as particle migration toward the center of the channel. The combination of elastic and inertial effects in elasto-inertial focusing has gained attention due to its superior ability to control microparticles^103,104,105, particularly in the sub-micron range, compared to inertial focusing⁸.

Fig. 6: Elasto-inertial focusing mechanisms and recent advances in modeling particle migration in non-Newtonian fluids.

a Schematic of the forces and equilibrium position in inertial only, elasticity dominant, and Elasto-inertial regimes; Reproduced from ref. ⁸, licensed under CC BY 4.0. b A Complex channel design was proposed through DNS simulations and a particle tracer solver using the Oldroyd-B fluid. Numerical results agree with experimental observations for Yeast cells; Reproduced with permission from ref. ²⁷. Copyright © AIP Publishing. c Schematic of the effects of shear-thinning properties near the wall for 7.32 µm particles proposed by DNS simulations using FEM with Giesekus fluid. Reproduced from ref. ²⁶, licensed under CC BY 4.0. d LBM was used to study spheroid particles migration in power-law fluids. (i) Schematic of rotational modes of tumbling (TU) and logrolling (LR) for a prolate and an oblate particle, respectively. (ii) Trajectories of a prolate in the channel cross-section of shear-thinning fluid. (iii) The final rate-of-strain tensor of an oblate spheroid in the channel cross-section of shear-thickening fluid; Reproduced with permission from ref. ¹¹¹, Copyright © 2023, Elsevier. e FEM with a particle tracer model was used to simulate inertial and viscoelastic particle migration in a microchannel with asymmetrical triangular expansions. Particles are collected from the top and the middle outlets, respectively. Reproduced from ref. ¹¹⁸, licensed under CC BY 4.0. f Flow field simulations using FEM showing magnitude and direction of secondary flow in the cross-section of a spiral channel. (i) Asymmetry of the secondary flow is due to the combined effects of the Dean flow and the second normal stress difference (N₂). (ii) Results are used to explain the elasto-inertial focusing mechanics in spiral channels. In this study, DNS simulations failed to converge due to the high Wi number problem. Reproduced from ref. ¹⁵⁷, licensed under CC BY 4.0

Numerical modeling of the behavior of non-Newtonian fluids is crucial to the design of viscoelastic microfluidic systems. Continuity and momentum equations are solved, and an additional stress tensor is computed to account for viscoelasticity¹¹. Various models describe the flow of these fluids, including the Carreau-Yasuda model for shear-thinning fluids^106,107,108, the Oldroyd-B model for linear viscoelastic response¹⁰⁹, and the Giesekus model for capturing nonlinear behavior at high deformation rates¹¹⁰. The Giesekus model includes parameters to describe the nonlinear relationship between the stress tensor and fluid deformation. The polymer stress in the Giesekus model is expressed as:

$$\boldsymbol{{\tau}}^{p}+\lambda \left(\frac{\partial \boldsymbol{{\tau}}^{p}}{\partial t}+\boldsymbol{u}\cdot \nabla \boldsymbol{{\tau}}^{p}-\boldsymbol{{\tau}}^{p}\cdot \nabla u-{\left(\nabla \boldsymbol{u}\right)}^{T}\cdot \boldsymbol{{\tau}}^{p}+\frac{\alpha }{{\mu }_{p}}\boldsymbol{{\tau}}^{p}\cdot \boldsymbol{{\tau}}^{p}\right)={\mu }_{p}\,(\nabla \boldsymbol{u} + {\left(\nabla \boldsymbol{u}\right)}^{T})$$

(9)

where $\lambda$ is the polymer relaxation time and $\alpha$ is the mobility factor¹¹¹. As reported by Alves et al. ¹¹², $\alpha$ is generally kept below 0.5 to avoid unphysical solutions, and when $\alpha$ is set to zero, the model reduces to the Oldroyd-B model.

Wall-bounded flows and straight channels

Recent studies have extensively investigated the impact of elasticity on elasto-inertial focusing mechanisms in straight channels, primarily due to their geometric simplicity, for both spherical and non-spherical particles. Raoufi et al. ²⁷ used DNS with the Oldroyd-B model to calculate elasto-inertial forces. They proposed a two-stage focusing model in straight channels, where particles initially migrate to regions of the lowest elastic force before moving diagonally to their equilibrium positions. They also examined the effect of corner angles on viscoelastic forces, finding that increasing the corner angle enhances and directs elastic forces toward the channel center, even near the corners. Based on these insights, they designed a complex straight channel to achieve a more compact particle focusing band for smaller microparticles across a wider range of flow rates (Fig. 6b).

Jiang et al. ¹¹³ used the Oldroyd-B model in an LBM-IBM formulation to study particle focusing. They adjusted parameters such as particle size, Re, and Wi to identify optimal conditions for efficient focusing in a 2-D Poiseuille flow. Their results indicated that larger particles at higher Re migrate faster toward the centerline, and an optimal elastic number (El = 0.3–0.4) is crucial for achieving a single center position. Charjouei et al. ¹¹⁴ conducted a parametric study of elasto-inertial focusing in straight channels using COMSOL Multiphysics and the Oldroyd-B model. They found that increasing viscoelasticity, particle diameter, flow velocity, and channel length improved focusing, but were inversely correlated with channel hydraulic diameter. Naderi et al. ²⁶ combined experiments and DNS simulations in rectangular microchannels, considering both elasticity-induced secondary flow and shear-thinning effects using the Giesekus constitutive equation. They observed that at low flow rates, elastic force and secondary flow drove particles toward the channel center, while at high flow rates, the elastic force balanced the shear-gradient force, resulting in two focusing positions on the horizontal midline. They also highlighted how shear-thinning effects can amplify the shear-gradient force for larger particles, directing the net inertial lift force toward the channel walls (Fig. 6c). Ghomsheh et al. ²⁵ studied lift forces of a xanthan gum fluid in straight rectangular microchannels. Their 3-D DNS simulations using the Carreau-Yasuda and power-law models showed that a strong shear-thinning effect pushed equilibrium positions toward the channel walls, a behavior observed for larger particles, while smaller particles still migrated toward the center due to the wall repulsive force.

Most research on elasto-inertial focusing has focused on spherical particles, with limited studies on non-spherical particles in viscoelastic channel flows. Computational studies on ellipsoidal particles in non-Newtonian fluids are relatively scarce^49,104,115. Hu et al. ¹¹¹ used LBM to study ellipsoidal particles in shear-thinning and shear-thickening fluids (Fig. 6d), finding that ellipsoidal particles migrate faster in shear-thinning fluids. They also observed that prolate spheroids can exhibit a ‘conditionally stable’ logrolling motion, dependent on initial orientation, with tumbling being the preferred rotational mode. Logrolling prolate ellipsoids focused further from the channel wall, while tumbling prolate particles showed oscillatory angular velocity. More recently, the same group applied the direct forcing/fictitious domain (DF/FD) method to investigate the motion of ellipsoidal particles in a square channel flow of Oldroyd-B viscoelastic fluid within the elasto-inertial regime¹¹⁶. They examined the effects of fluid elasticity and particles’ initial state on the focusing and rotation dynamics and reported four unique equilibrium positions and six rotational patterns for prolate spheroids (five for oblate spheroids).

Curved channels and other geometries

The numerical studies of viscoelastic flows in curved and non-planar structures are relatively limited due to the complexity of the flow field, lift forces, and intricate channel geometries. These challenges make numerical simulations computationally expensive and difficult compared to experiments¹¹⁷. While computational work on curved geometries in the viscoelastic regime is scarce, some studies have combined viscoelastic fluids with non-straight channel geometries primarily for validation and optimization rather than investigating fundamental mechanisms.

Wang et al. used 2-D FEM to study the viscoelastic migration of particles in a channel with triangular cavities¹¹⁸. Their particle tracing results showed that in viscoelastic flow, microparticles were deflected to the top side of the microchannel cavity and collected from one outlet, whereas in Newtonian flow, the deflection was less, and the microparticles were collected from a different outlet (Fig. 6e). While suitable for design optimization, this method has limitations related to potential elastic turbulence in cavities induced by high Wi, leading to numerical instability. Yuan et al. ¹¹⁹ used 2-D DNS to study the oscillatory performance of a T-shaped microchannel. They investigated the use of viscoelastic fluids to generate oscillatory flow for mixing and particle focusing applications and found that specific values of fluid viscosity and elasticity are needed to create regular oscillations in the flow. Nouri et al. ¹²⁰ used DNS to explore the synergetic effects of the elastic and inertial forces in straight and serpentine channels with different corner angles. They showed how a corner angle of 75° can reduce defocusing near the channel center (compared to that of a 90° angle), enabling particle focusing with a smaller focusing width.

Jang et al. ¹²¹ also studied a T-shaped microchannel, using a finite-volume-based solver in Ansys Fluent, and found that cross-sectional geometry and viscoelasticity combined to impact the pattern of particle migration. The study suggested that manipulating the aspect ratio of the T-shaped microchannel and the flow rate could potentially impact particle separation based on size. Jeyasountharan et al. ¹¹⁵ investigated the self-assembly dynamics of particle trains in a microchannel consisting of trapezoidal elements followed by a cylindrical channel using the ALE method. They studied particle motion in a Giesekus fluid and observed that small particles require more considerable distances to achieve self-ordering, and higher particle concentrations prevent the formation of trains. They suggested that the response of fully formed particle trains to flow disturbances should be investigated using numerical simulations and experiments^115,122.

Finally, an emerging direction explores stretchable microchannels, where tunable geometries enable dynamic control over particle focusing. Such systems are of particular interest for designing adaptive devices for cell separation¹²³ and droplet generation¹²⁴. While this concept remains nascent, with seminal studies demonstrating its potential^125,126, a critical challenge lies in numerical modeling, as the time-dependent deformation of these channels leads to continuously evolving inertial lift distributions, complicating predictions of particle trajectories, or even leading to unsteady lift forces in these systems. The development of ad hoc models is still in its infancy¹²⁷, with a few pioneering works which developed fluid-structure interaction models for similar systems, such as the work by Inamdar et al., who proposed a one-dimensional mesh-based lubrication model for the unsteady fluid-structure interaction in a soft-walled microchannel¹²⁸. However, large deformations may affect fluid-particle interactions and inertial lift, as shown in related studies involving shock waves¹²⁹. This suggests that mesh-free methods are better suited for such problems and can provide more accurate results, as demonstrated in a previous 2-D model of a microchannel with upper wall contractions¹³⁰, or the SPH-based approach developed for fluid-structure interaction in flexible channels within biomedical flows¹³¹. We believe that further development and adaptation of mesh-free methods will lead to more effective techniques for studying how flexible channel walls influence particle dynamics and focusing positions, with SPH proving to be a valuable tool for tackling emerging challenges in inertial microfluidics.

Machine learning in inertial microfluidics

In the past decade, artificial intelligence (AI) has experienced significant advancements, leading to its widespread application across various disciplines. In the biological sciences, machine learning (ML) has accelerated research and facilitated discoveries in areas such as drug development¹³², protein folding¹³³, and disease diagnosis¹³⁴. ML algorithms enable robust data modeling, allowing for the extraction of valuable insights from large datasets. Microfluidics, known for its high-throughput data generation capacity^135,136, presents an attractive domain for ML techniques. Consequently, ML has the potential to reduce both the costs and time required for experimentation and optimization. The integration of computational methods has further increased the volume of available data within a short time frame¹², promising extended applications of ML in microfluidics.

ML in microfluidics can be categorized into three main paradigms: supervised learning for labeled data, unsupervised learning for unlabeled datasets, and reinforcement learning for trial-and-error-based optimization schemes¹³⁷. These approaches are well-suited to the varied datasets in size and dimensionality from microfluidic experiments and simulations, offering numerous application possibilities. Additionally, artificial neural networks (ANNs), particularly deep learning networks (DLNs) with multiple hidden layers, have driven revolutionary breakthroughs in fields such as bioinformatics, image processing, and text processing¹³⁸.

Numerous efforts are underway to extend the applications of ML into microfluidics, as evidenced by recent attempts in particle manipulation¹³⁹, droplet microfluidics^140,141, and device design and control^142,143. ANNs can accurately predict critical microfluidic performance parameters like droplet size and generation rate without requiring complex simulations. ML approaches enable both direct design automation through reverse modeling and iterative optimization techniques that efficiently navigate parameter spaces. For device operation, reinforcement learning and Bayesian optimization allow real-time adaptation of parameters, enabling autonomous error correction¹⁴⁴. Data-driven approaches have recently demonstrated remarkable potential in transforming both microfluidic device design and operational control workflows. For example, Lashkaripour et al. ¹⁴⁵ demonstrated this potential by developing the DAFD web-based tool, which uses neural networks to predict flow-focusing droplet generator performance and automate designs based on user specifications. In addition, physics-informed machine learning (PIML) represents a significant advancement in fluid mechanics modeling by integrating domain knowledge with ML algorithms, achieving higher data efficiency and more stable predictions. This approach proves particularly valuable for complex flows where traditional simulations demand substantial computational resources. By embedding physics through data features, model architecture, or loss functions, PIML demonstrates superior generalizability compared to traditional ML approaches, especially for unseen parameter spaces, making it an essential tool for advancing computational microfluidics beyond current limitations¹⁴⁶. Furthermore, recent studies have demonstrated the potential of large language models such as ChatGPT to enhance microfluidic device development, from assisting with computer-aided design to the fabrication of functional components¹⁴⁷.

However, advancements have not yet been fully realized in the inertial microfluidics domain. In inertial microfluidics, focusing of particles in channels is influenced by channel geometry, fluid properties, flow conditions, particle confinement ratio, and particle material properties^1,3,7. A model that establishes a connection between system inputs and the focusing position of particles would greatly benefit the design and control of inertial microfluidic devices. However, the high level of mathematical complexity poses challenges for theoretical analyses, and numerical solvers can become prohibitively expensive, especially when solving time is critical¹¹. A potential solution lies in robust heuristic methods capable of predicting the focusing position of particles given the system’s intricate properties. In this context, deep neural networks offer an effective and feasible approach.

A notable contribution to AI-enhanced inertial microfluidics comes from Su et al. ¹⁴⁸, who used a combination of DNS and ANNs to investigate inertial lift forces and predict particle trajectories in microchannels. Their work involved generating large numerical datasets, considering three distinct channel geometries: rectangular, triangular, and semicircular. For each case, a range of parameters was simulated using DNS, including variables such as channel aspect ratio, Re, and confinement ratio (Fig. 7a). The resulting dataset of 14,160 cases was divided into three parts: a training set (70% of the data), a validation set (15%), and a testing set (15%). During the training process, Su et al. employed a backpropagation neural network architecture with one input layer, two hidden layers, and one output layer (Fig. 7b). These parameters were iteratively updated using the Levenberg–Marquardt optimization algorithm¹⁴⁹ to minimize error. Once trained, the neural network could recognize patterns and predict coefficients, which were subsequently used to calculate particle trajectories in COMSOL Multiphysics. Su et al. then compared these predictions with experimental data from Kim et al. ¹⁵⁰, which included rectangular, triangular, and semicircular cross-sectional channels, and data from Li et al. ¹⁵¹, which involved a multistage channel with a curved section. The results demonstrated a high degree of agreement between the predicted and experimental particle trajectories in both cases (Fig. 7d).

Fig. 7: Machine learning-assisted prediction of inertial focusing of particles.

a Example of a force field (to be used as training datasets for the ML model) calculated using the DNS approach with FEM. Reproduced with permission from ref. ¹⁴⁸; Copyright © Royal Society of Chemistry. b The BP neural network architecture used to train the model; Reproduced with permission from ref. ¹⁴⁸. Copyright © Royal Society of Chemistry. c Schematic diagram of the multistage channel and the velocity field in the rectangular segment. Reproduced with permission from ref. ¹⁴⁸. Copyright © Royal Society of Chemistry. d Comparison of the predicted particle trajectories from the ML model with experiments in the rectangular segment of the multistage channel¹⁵⁰; Reproduced with permission from ref. ¹⁴⁸ and ref. ¹⁵⁰. Copyright © Royal Society of Chemistry and licensed under CC BY 4.0, respectively

Ultimately, Su et al. ¹⁴⁸ highlight the potential of combining ML techniques with numerical simulations to advance the understanding and predictive capabilities of inertial microfluidics, particularly for complex geometries and highly nonlinear scenarios. Their research provides valuable insights into the future development of AI-driven approaches for optimizing microfluidic devices and understanding particle migration in intricate systems.

Overall, the application of ML to inertial microfluidics is limited, with the only notable work being the paper reviewed in this section. However, as ongoing efforts continue to apply ML techniques to computational fluid dynamics (CFD) in general, we anticipate that ML will play an increasingly important role in inertial microfluidics in the near future. Recent advances in ML have already shown promise in enhancing the speed and accuracy of fluid simulations^152,153. As the field progresses, it is likely that similar methodologies will be adapted to optimize simulations in inertial microfluidics, offering the potential for more efficient and accurate computational tools.

Concluding remarks and prospective

The continuous progress in inertial microfluidics underscores the pivotal role of computational modeling in expediting the design and optimization of microfluidic devices. This advancement not only accelerates the development process but also enhances our comprehensive understanding of the fundamental physics underpinning inertial and viscoelastic microfluidics. Despite notable improvements in computational techniques, several critical aspects remain insufficiently explored. In recent years, limited progress has been made, particularly in addressing nonlinear phenomena of non-Newtonian fluids and particles with non-spherical and deformable shapes. To effectively advance the field, including self-alignment, self-orientation, or rotational stability within shear flows, there is a growing need to harness high-performance computing and modern simulation approaches. Table 2 summarizes the commonly used numerical methods discussed in this review. These advanced tools hold the potential to unlock new frontiers in inertial microfluidics research and pave the way for groundbreaking discoveries.

Table 2 Summary of the numerical solvers used in inertial microfluidics

Mesh-based numerical methods (e.g., FEM) have been pivotal in analyzing inertial lift forces and optimizing microfluidic device designs. These methods are particularly effective in scenarios with well-defined fluid flows and relatively simple geometries. However, they face limitations when addressing complex fluid dynamics, including free surfaces, deformable boundaries, moving interfaces, extensive deformations, and fluid-structure interactions. In such cases, LBM coupled with IBM has emerged as an effective solution to overcome the limitations of conventional mesh-based methods. This technique offers flexibility, accuracy, and computational efficiency, making it suitable for capturing fluid-particle interactions in microchannels. Combining these methods with traditional mesh-based approaches like FEM can leverage their complementary strengths and further mitigate individual limitations. However, the choice of coupling algorithms and implementation can be challenging and may vary depending on the specific application. Using IBM and LBM methods, particularly in combination with FEM, may involve increased computational costs, as they can be more demanding than traditional FEM for specific problems. Therefore, selecting the most appropriate numerical approach depends on the nature of the problem and the available computational resources.

In recent years, mesh-free particle-based methods such as SPH have gained extensive applications in fluid mechanics. SPH is widely used in simulating fluid and solid mechanics due to its ability to handle complex deformations and maintain excellent conservation properties. SPH approximates governing fluid equations on particles, making it well-suited for simulating particle behavior in complex microfluidic environments. It has been used to analyze how spherical and ellipsoidal particles migrate and rotate under the influence of inertial forces in microchannels. SPH simulations have been validated against numerical simulations, analytic solutions, and experimental observations, establishing SPH as a valuable alternative for simulating particle behavior in inertial microfluidics.

The integration of ML techniques with inertial microfluidics shows significant potential to transform the design and operation of microfluidic devices. This integration encompasses several crucial aspects. First, ML can streamline data processing and compression tasks, allowing for predictive modeling of particle behavior in intricate microchannel geometries. Its ability to efficiently manage larger datasets generated from numerical simulations, as opposed to experimental data, is particularly advantageous. ML can also function as a complementary bridge between numerical and experimental data, enhancing the overall generality of predictions. Second, ML techniques demonstrate the capability to generalize effectively over biased data and adapt to its inherent high variance. This attribute is particularly valuable in mitigating errors arising from experimental data when combined with relatively accurate numerical simulations. Given that inertial microfluidics operate at microscopic scales, where even minor errors can result in substantial deviations from the desired outcome, the ability of ML algorithms to construct robust models with microfluidic data is paramount. Third, ongoing efforts focus on crafting ML techniques tailored to the distinctive requirements of inertial microfluidics. Continued research is essential to identify modeling demands within the field and delve into the expansive realm of ML mathematics for appropriate solutions. This exploration aims to create ML models capable of precisely predicting particle dynamics in various microfluidic setups, particularly as these systems become progressively more complex.

The overarching aspiration of integrating ML with inertial microfluidics is automation, spanning from the design phase to control and operation. This involves developing ML models capable of predicting particle inertial dynamics for various applications, including particle sorting, separation, and focusing. Such automation holds the potential to significantly accelerate the advancement of state-of-the-art microfluidic devices. To fully realize this potential, future research should delve into multifaceted systems that consider additional factors like viscoelastic fluids, particle shape, and deformability effects. Incorporating parameters such as Wi for elastic fluids can provide finer control over fluid properties. As microfluidic applications become more intricate, the implementation of deeper neural networks with an increased number of hidden layers may be necessary, especially as datasets expand and variables accumulate, though effective management of associated computational complexities will be crucial.