1. Introduction

Figure 1. A schematic overview of applications of computational methods in FSI for micro elastofluidics.

Figure 1. A schematic overview of applications of computational methods in FSI for micro elastofluidics.

2. Fundamentals of Fluid-Structure Interaction in Micro Elastofluidics

2.1. Fundamentals of Fluid-Structure Interaction

In micro elastofluidics, the study of FSI is crucial for understanding the behaviour of fluids confined in flexible microchannels and the impact of this behaviour on the surrounding structures. As described by Duprat and Stone [32], FSI denotes mechanical problems where the flow field affects the orientation, shape, and location of an interacting object, leading to reciprocal modifications in the flow pattern. The flexible and elastic nature of the microchannel does not allow the use of already established rules of rigid microfluidics and necessitates the incorporation of fluid-structure interaction to obtain suitable engineering solutions.

The focus of FSI is on the coupling between fluid dynamics and structural mechanics. Fluid flow exerts forces on a structure, potentially causing it to deform. The magnitude of these deformations depends on fluid pressure, velocity, and material properties of the structure. Minor deformations may not significantly affect the fluid flow. However larger deformations create a feedback loop where the altered structure modifies the behaviour of the fluid. FSI with negligible influence of deformation on fluid flow is considered as one-way fluid-structure interaction. Problems with one-way FSI are relatively easy to manage and comprehend. Two-way or fully coupled FSI induces deformation large enough to affect the fluid flow, which in turn significantly changes the flow-induced deformation of the structure, Figure 2. These complex interactions are important for biomedical problems such as blood flow in flexible vessels [33], and designing medical devices such as micropumps [27,34].

Figure 2. Fluid structure interaction: (A) One-way FSI, (B) Two-way FSI.

Figure 2. Fluid structure interaction: (A) One-way FSI, (B) Two-way FSI.

Considering the interface between fluid and a deformable structure, FSI can also be categorised further into two types (i) fluid-wall interface and (ii) fluid-particle interface, Figure 3. In FSI with the fluid-wall interface, deformable structures are walls of microchannels, Figure 3A. These walls are elastic, flexible, and fixed. For example, in continuous micro elastofluidics, walls of microchannel can be bent and stretched but cannot move. Hence, analysing continuous micro elastofluidic problems should consider FSI with fluid-wall interface. In FSI with the fluid-particle interface, deformable structures are moving particles, Figure 3B. These particles are elastic and deformable but are not fixed at a point. Fluid forces cause the cell to deform as well as change its position, and these changes in shape and position alter the pattern of fluid flow. For instance, problems related to digital micro elastofluidics are analysed through FSI with the fluid-particle interface. Methods have been developed to study this FSI at the particle boundary, with applications in cell sorting and biomechanics of cells.

Figure 3. (A) Fluid-wall FSI, (B) Fluid particle FSI.

Figure 3. (A) Fluid-wall FSI, (B) Fluid particle FSI.

Understanding the relationship between volumetric flow rate $Q$ and pressure drop $∆P$ is crucial for the design and operation of fluidic systems. In deformable microchannels, this $∆P-Q$ relationship is nonlinear, which is a major difference to flow in rigid microchannels [35]. Deformation of channel walls reduces flow resistance, causing lower pressure drop [19,36]. To elucidate this relationship, numerous experimental works have been conducted, and researchers have developed empirical models for pressure-flow characteristics. For instance, Gervais et al. [35] investigated the behaviour of fluid flow in stretchable microchannels using confocal microscopy. A model was created to describe the observed pressure-flow characteristics. The results indicate that flexible microchannels could enable higher flow rates compared to rigid microchannels of identical size driven by the same pressure difference. Subsequently, Hardy et al. [37] investigated the behaviour of PDMS microchannels with flexible walls and reported significant differences in pressure drop compared to rigid channels. The team demonstrated that pressure drop in flexible microchannels decreases down to 35% of that in identical channels with rigid walls.

Another FSI problem is compressible flow in deformable channels. The mass flow rate is a function of, undeformed microchannel dimensions, differential pressure across a microchannel, and characteristics of the channel’s surface such as elastic modulus, thickness, and Poisson’s ratio [19,36,38,39,40,41,42,43,44]. Although efforts have been made toward the development of empirical models for the $∆P-Q$ relationship, these models only apply to specific conditions with small domain characteristics. To comprehend the generalised and complete FSI phenomena, numerical analysis is necessary. The next section discusses fluid dynamics and solid mechanics of FSI and its governing equations.

2.4. Coupling Approaches

The selection of a suitable coupling approach is the cornerstone of successful FSI simulation. This determines the mechanism by which the fluid and structure domains exchange displacement and force information at the interface during iterative solutions or time steps. The selected coupling strategy can significantly impact the numerical stability, convergence rate, and the ability to capture the true physics of the coupled system.

Two primary approaches exist: monolithic and partitioned, Figure 5. In monolithic approaches, the governing equations for fluid and solid domains are integrated into a single, unified framework and solved simultaneously. Interfacial boundary conditions (e.g., no-slip, stress continuity) are implicitly embedded within this framework. Monolithic schemes offer superior accuracy for tightly coupled FSI problems but often demand the development of highly specialised solvers, potentially increasing implementation complexity and computational expense [52,53,54]. In contrast, the partitioned approach considers the fluid and structure as separate computational domains. This approach enables the independent discretization of each domain and the application of numerical solution techniques optimised for the governing physics within each domain. The approach allows for the use of specialised, potentially pre-existing "legacy" solvers optimised for each respective physical domain (e.g., fluid vs. structural mechanics). Interfacial data exchange (displacements, tractions) occurs iteratively, potentially simplifying development time and enhancing flexibility. However, careful attention must be paid to interface tracking and the stable, accurate transfer of data across this dynamic interface. Specialised algorithms designed for moving boundary problems are often required to ensure numerical stability and prevent unphysical solution behaviour.

Figure 5. FSI coupling approaches, (A) Monolithic, (B) Partitioned.

Figure 5. FSI coupling approaches, (A) Monolithic, (B) Partitioned.

An additional classification scheme for the FSI solution is based on the discretization of mesh, which dictates how interfacial boundary conditions are imposed within the discretised system of equations, Figure 6. One is the Conforming Meshing Method, where the fluid-structure interface is considered as a physical boundary. The fluid and solid domain meshes must perfectly match. This often leads to re-meshing (or mesh-updating) as the structure deforms throughout the simulation, hence adding computational cost. The other is the Non-Conforming Meshing Method, where the fluid-structure interface is not treated as a strict alignment boundary. Instead, the interface location and its conditions (e.g., stress continuity) are considered constraints within the governing equations. This approach enables independent meshing for each domain, streamlining simulation setup and avoiding re-meshing.

Figure 6. Mesh discretization approaches, (A) Conforming meshing, (B) Non-conforming meshing.

Figure 6. Mesh discretization approaches, (A) Conforming meshing, (B) Non-conforming meshing.

3. Computational Methods for Studying Fluid-Structure Interactions

Figure 7. Space and Time scale of different computational methods application.

Figure 7. Space and Time scale of different computational methods application.

3.1. Finite Element Method

The Finite Element Method (FEM) is an indispensable computational tool for tackling the complex challenges presented by FSI in micro elastofluidics. This method divides the domain of interest into smaller elements, allowing for the accurate modelling of complex geometries and boundary conditions, Figure 8A. FEM has been widely applied in various aspects of microfluidics. Hung et al. [63] modelled mass transfer using FEM in a high-aspect ratio microfluidic device, which provides a stable and uniform microenvironment for cell growth in a high-throughput mammalian cell culture array. Erickson et al. [64] investigated the role of surface heterogeneity on electrokinetically driven microfluidic with 3D FEM, aiming to enhance mixing in a T-shaped micromixer. Bianchi et al. [65] implemented FEM models to simulate electroosmotic-driven flow division at a T-junction.

FEM has also been applied in various contexts to investigate fluid-structure interactions in microfluidics. For example, Zhang et al. [66] reported a groundbreaking application of the Cell-based Smoothed Finite Element Method (CS-FEM) to computational fluid dynamics (CFD) and FSI simulations. CS-FEM belongs to the broader family of Smoothed Finite Element Methods (S-FEM), which aim to enhance accuracy and robustness by applying strain smoothing techniques. Specifically, CS-FEM simplifies calculations by not requiring explicit shape functions and demonstrates better tolerance to distorted meshes. This study demonstrated that CS-FEM offers greater accuracy and stability in handling deformable interfaces and flow fields that typify FSI problems, making it well-suited for simulations relevant to micro elastofluidics. Erickson et al. [67] applied the same approach to investigate how smart one-way micro-valves behave when FSI is taken into account. Hence, FEM has been proven vital in comprehending FSI in micro elastofluidics.

A typical FEM approach consists of five steps: (i) Discretization: The continuum domain is discretised into a finite number of smaller elements that are easier to manage; (ii) Element Equation Formulation: For each element, the governing physical equations are formulated. These equations typically stem from fundamental conservation laws (such as mass, momentum, and energy conservation) and are expressed in terms of local element variables; (iii) Assembly: The local element equations are assembled into a global system of equations that models the entire problem domain. This step involves integrating the contributions of individual elements to the overall behaviour of the system; (iv) Solution: The global system of equations is solved numerically to find the unknowns, such as displacements and pressures. This solution process may incorporate iterative methods and solvers that handle non-linearities and complex boundary conditions; (v) Post-processing: The solution obtained is interpreted in terms of physical quantities of interest, such as stress distributions, fluid velocities, and pressures. This helps in evaluating the performance and safety of the micro elastofluidic devices. FEM for simulating FSI in micro elastofluidics consists of modelling the fluid domain, modelling the deformable structure domain, and then fluid-structure interface coupling. In every phase, specific mesh strategy and physics are used to simulate and then coupled them. Here we discuss physics and governing equations of these domains.

Figure 8. Computational methods for FSI. (A) Discretisation of fluid and solid domain in FEM; (B) Discretisation of domain in BEM; (C) Molecules interaction in MD method; (D) 2D and 3D configurations of LBM; (E) (i) Bounce Back Method, (ii) Extrapolated Bounce Back Method illustration; (F) IBM as Lagrangian and Eulerian description.

Figure 8. Computational methods for FSI. (A) Discretisation of fluid and solid domain in FEM; (B) Discretisation of domain in BEM; (C) Molecules interaction in MD method; (D) 2D and 3D configurations of LBM; (E) (i) Bounce Back Method, (ii) Extrapolated Bounce Back Method illustration; (F) IBM as Lagrangian and Eulerian description.

3.1.1. Modelling of Fluid Domain

FEM begins by discretizing the fluid domain (e.g., a microfluidic channel) into a mesh of smaller, interconnected elements. These elements are often triangles or quadrilaterals in 2D and tetrahedra or hexahedra in 3D. In this domain, FEM describes the mesh as Eulerian space. The behaviour of the fluid within this mesh is governed by the fundamental Navier-Stokes equations. These equations express the conservation of mass i.e. equation (1) and momentum i.e. equation (2). To solve the Navier-Stokes equations numerically, FEM employs a technique called the Galerkin method. Essentially, the equations are transformed into their 'weak form' and approximated using shape functions. These shape functions describe how the fluid velocity and pressure vary within each element of the mesh. The introduction of shape function in simulations is defined as:

$$u\left(x\right)=\sum N_i\left(x\right)u_i$$

(7)

$$p\left(x\right)=\sum M_j{p}_j$$

(8)

where; $N_i\left(x\right)$ is the shape function and ${\mathit{u}}_i$ is the nodal velocity. Similarly, in equation (8), $M_j$ is the shape function and $p_j$ is the nodal pressure.

3.1.2. Modelling of Deformable Structure

The versatility of FEM extends to handling solid mechanics of deformable structures interacting with the flow. FEM divides the structural domain in the same way as the fluid domain, i. e. into small finite elements. However, the structural domain uses Lagrangian description. The governing equations of solid mechanics i.e. equation (9) describe the balance of forces within the structure and the relationship between stress and strain (deformation). The choice of a constitutive model (e.g., linear elastic, hyperelastic, viscoelastic) dictates how the material responds to these stresses. Similar to the fluid domain, FEM discretises deformable structures with elements and uses shape functions to approximate the displacement field.

$$\nabla .\sigma +f_s=0$$

(9)

$$d\left(x\right)=\sum X_i{d}_i$$

(10)

where $\mathit{\sigma }$ is the stress tensor and ${\mathit{f}}_s$is body force on the structure. More importantly, this body force comes from fluid pressure on the structure in FSI problems. In equation (10), $\mathit{d}\left(x\right)$ is the displacement vector and $X_i$ is the shape function and ${\mathit{d}}_i$ is the nodal displacement vector. For micro elastofluidics, selecting the appropriate shape functions is crucial and directly depends on the material models used to represent the deformable structures. For materials exhibiting linear elastic behaviour, where deformations are small, linear or quadratic shape functions often provide sufficient accuracy. However, microfluidic systems often involve significant deformations, demanding the use of hyperelastic material models. These models describe nonlinear stress-strain relationships, thus necessitating higher-order shape functions within FEM to accurately represent the complex deformation patterns that can occur.

3.5. Immersed Body Method in FSI

Originally intended for deformable membranes inside a flow field, the Immersed Boundary Method (IBM) was created by Peskin [112]. An integral relationship can be used to calculate the membrane force from membrane deformation. Membrane forces are transferred to the local fluid as part of fluid-membrane interactions and the membrane configuration was updated in response to the local flow velocity. This method has the advantage of avoiding problems related to shifting boundaries by allowing numerical techniques to solve fluid flow on a fixed, regular Eulerian mesh. In microfluidics, the immersed body approach has been effectively combined with a solver for both soft and rigid particles [117,118,119,120]. This technique has also been used to investigate the dynamics of RBCs in microcirculation [121,122].

When a fundamental relationship of solid boundaries is not available, alternative relationships must be developed between the desired boundary velocity and the boundary force to stimulate solid particles and moving boundaries in a flow. Many approaches have been proposed. For example, Feng et al. [88] propose modelling solid particles as deformable, using a spring force to represent interactions between each particle and a virtual reference point. This approach allows the simulation of how particles elastically respond to displacement, aiding in understanding their behaviour under various stress conditions. Niu et al. [123] determined the boundary force through the momentum interchange of particle distributions at the boundary. Dupuis et al. [124] used LBM with IBM for modelling the Navier-Stokes equation (2). The boundary force was calculated by comparing the desired boundary velocity with that computed without the boundary force to account for the no-slip boundary condition.

IBM uses the Lagrangian description for the immersed structures and the Eulerian description for the fluid flow. This method takes into account the movement and distortion of immersed bodies while accurately representing fluid behaviour. In order to take into consideration the impact of immersed structures on surrounding fluids, this approach utilises forcing functions. These features allow forces to be transferred from the fluid to the immersed bodies, enabling the fluid to act on the structures and vice versa. A collection of Lagrangian points or markers positioned within the Eulerian grid serves as the representation of immersed boundaries. Interpolation techniques compute the influence of these markers on the surrounding fluid grid as shown in Figure 8F.

Methods such as interpolation and projection are employed to transfer information between the Eulerian and Lagrangian domains, ensuring the accurate exchange of forces, velocities, and displacements between the fluid and structures. The typical IBM process involves several key steps.

The first step is interpolating the velocity of fluid $u\left(X\right)$ at position $x_i$ of every particle mesh vertex $i$ in Figure 8F. The interpolated velocity ${\dot{x}}_i$​ is given by [49]:

$${\dot{x}}_i={∆x}^d{\sum }_{X}u\left(X\right)\delta (X-x_i)$$

(36)

where $d$ is the number of spatial dimensions, $∆x$ is the lattice spacing, $\delta$ and is a discrete delta distribution. After interpolating the fluid velocity, the force $f_i$ acting on each vertex $i$ is calculated. For soft particles, the forces resulting from mesh deformation caused by moving vertices, leading to the changing distance $\left|x_i-x_j\right|$ over time. Then spread each vertex force $f_i$ on the fluid, treating them as body forces $\mathit{F}\left(X\right)$ according to [49]:

$$F\left(X\right)={\sum }_i{f}_i\delta \left(X-x_i\right)$$

(37)

After spreading the force as per equation (37), the method updates the position and orientation of each particle based on the forces acting on it. The position of the vertex can be updated as:

$$x_i\left(t+∆t\right)=x_i\left(t\right)+\dot{x_i}\left(t\right)∆t$$

(38)

The shape of the discrete delta distribution $\delta \left(X-x_i\right)$ is crucial, and a common simplification is using factorised 2D or 3D kernel functions.

For soft particles, the vertex velocity is determined by interpolating fluid velocity. Particle deformation is caused by moving vertices. For rigid particles, challenges involve satisfying rigidity conditions and no-slip conditions simultaneously. Various IBM algorithms for rigid particles exist, such as direct-forcing, implicit IB, and multi-direct-forcing methods.

When a system features symmetric or periodic spatial characteristics, symmetric or periodic boundary conditions can be effectively applied to reduce the computational domain and enhance efficiency. These conditions are particularly well-suited to simulations using IBM due to their adaptability with particle-based models. For periodic boundaries, particles that exit the domain on one side are reintroduced from the opposite side, effectively simulating an infinitely extended horizontal domain with identical repeating units. Additionally, to implement a pressure gradient in a channel, the periodic boundary condition can be modified to account for a pressure (density) variation between the domain’s inlet and outlet [109].

The two main approaches for flow in geometries with streamwise periodic boundary conditions are: (i) imposing pressure drop $∆p$ on the periodic boundary condition and (ii) a body force $\mathit{F}$. The previous approach, which can be substituted by a constant body force $\mathit{F}=∆\mathit{p}$ works well in straight channels with an effective constant pressure gradient. It is often preferable to apply an overall pressure drop $∆\mathit{p}$ between the inlet and outlet for other geometries. Then taking into consideration any pressure variation brought on the particles, the LB algorithm adjusts for the proper pressure field inside the domain. Periodic boundaries are conceptually simple, but they have some difficulties as well. Controlling long-range particle-particle interactions across boundaries is necessary to simulate an infinite number of particles. This frequently involves sensitivity tests of selected channel length [125,126].

Creating more appropriate boundary conditions for complex geometries remains a difficult task. Distortion in the flow field can spread downstream in inertial flows. Any upstream influence is ignored when using a fully formed velocity profile at the inlet, which is not suitable for obtaining accurate results. Further research is required to facilitate practical and accurate LB simulations of microfluidic devices that cannot be approximated with periodically repeating boundary conditions.

Table 1. Summary of Computational Methods and Applications.

Table 1. Summary of Computational Methods and Applications.

Feature		FEM	BEM	MD	LBM	IBM
Basic Principle		Divides problem domain into discrete elements. Solves for field variables.	Reduces dimensionality by focusing on boundaries only. Solves integral equations.	Simulates behaviour of atoms and molecules using Newton's laws of motion.	Simulates fluid dynamics using discrete particle distribution functions on a lattice.	Simulates fluid and structure interaction by embedding the structure in a fluid mesh.
Primary Applications		Structural analysis, micropumps and microvalves.	Acoustic streaming in microfluidics, fluid flow in infinite domains	Biophysics, chemistry, biosensors, and drug delivery systems.	Droplet Generation, fluid flow in complex geometries, multiphase and multicomponent flows	Complex fluid-structure interactions, cardiovascular simulations, biological flows.
Computational Domain		Volumetric domain requires discretization of the entire region.	Surface-based, requires discretization of the boundary only.	Atomic or molecular scale simulates each particle individually.	Volumetric, based on a fixed grid of discrete points.	Hybrid approach combines a fluid mesh with structures that do not conform to the mesh.
Strengths		Flexible, can handle complex geometries and multiphysics problems.	Less discretization is needed, faster for problems with small boundaries.	Provides detailed molecular-level information.	Efficient for complex boundary conditions and scalable for large systems.	Efficiently handles interaction between fluid and immersed structures without requiring mesh conformity.
Weaknesses	Can be computationally intensive, especially for large or complex domains.		Limited to problems where boundary definition is clear and sufficient.	Computationally intensive, limited to relatively small system sizes or short time scales.	Accuracy depends on lattice resolution, and handling complex physics can be challenging.		Can suffer from accuracy issues near the boundary between the fluid and the structure.
Mesh Dependency	Highly mesh-dependent, solution accuracy increases with finer mesh.		Only boundary needs meshing, reducing overall mesh dependency.	No traditional mesh, but particle density and interaction range are crucial.	Mesh (lattice) dependent, though generally less sensitive than FEM.		Mesh of the fluid needs to be fine enough to accurately capture boundary layer phenomena.
Typical Solvers	Direct solvers, iterative solvers.		Boundary integral methods, direct solvers.	Verlet integration, velocity Verlet, leapfrog methods.	Collision and streaming operators, often using BGK approximation.		Direct forcing methods, Lagrangian-Eulerian solvers.
Software Examples	ANSYS, Abaqus, COMSOL Multiphysics.		ANSYS, BEASY, Altair AcuSolve.	LAMMPS, GROMACS, NAMD.	Palabos, OpenLB, LBMflow.).		IBAMR, immersed boundary (MATLAB)

4. Applications

4.1. Microvalves and Micropumps

Microvalves and micropumps are the typical microfluidic components with strong FSI. FSI dictates how flexible membranes and channels within these devices respond to fluid pressure, ultimately shaping their ability to regulate and deliver tiny volumes of liquid in lab-on-a-chip and drug-delivery devices. Much work has been done in analysing the intricate interplay between fluid forces and deformations of flexible structures. Researchers have proposed designs of microvalves and micropumps that achieve unparalleled precision in flow control.

Among the different types of microvalves, elastomeric membrane [127] microvalves are a prime example of where FSI plays a critical role [128]. The core principle behind these valves is the deformation of a flexible membrane in response to fluid pressure [129]. Active valves often utilise external actuation through FSI for precise flow control. Various actuation mechanisms have been proposed [130,131,132,133,134]. Passive valves harness flow forces through FSI to achieve remarkable self-regulation [135,136,137]. In a passive microvalve, the fluid pressure deforms the membrane, which in turn alters the flow resistance. This dynamic interplay between the fluid and the membrane allows the valve to maintain a constant flow rate over a specific pressure range.

Numerous models for passive check valves and passive regulating valves have been proposed. For instance, Nguyen et al. [138] utilised FSI to study passive valves and proposed models for ortho-planar micro check valves for incorporation in polymeric microdevices, Figure 9A. These check valves efficiently prevent backflow and require an inlet pressure of less than 1 kPa to open. Ortho-planar designs provide enhanced sealing performance, thanks to their parallel out-of-plane motion. Later Kartalov et al. [139] proposed a PDMS push-up valve utilising the pressure drop along the channel length and performed FSI simulation to investigate the flow rate and threshold pressure, Figure 9B. This model maintained a constant flow rate of 0.033 mL/min with a threshold pressure of 103 kPa. Yang et al. [140] designed a planar check valve and did an FSI study to model self-adaptive variable resistors to use in microfluidics models, Figure 9C. This valve design achieved a relatively high flow rate of 1.2 mL/min with a threshold pressure of 100 kPa. In addition, Doh et al. [141] developed a passive parallel membrane valve designed for low-threshold pressure operation using the concept of FSI, Figure 9D. The valve consists of two control channels, two vertically oriented membranes, and a single fluidic channel. The autonomous deflection of the membranes within the microchannel enables the valve to achieve flow regulation at a pressure as low as 15 kPa. Moreover, Zhang et al. [142] utilised FSI on flow regulation in microfluidic environments and developed a unique parallel membrane valve featuring a stacked five-layer architecture, Figure 9E. This design, with two horizontal membranes enclosing a fluidic channel and sandwiched between control channels, achieved a remarkable flow rate of 2.79 mL/min with a low 10 kPa threshold pressure. Zhang el at. [137] achieved low threshold pressure in microfluidic high throughput delivery systems and designed a passive valve for stable flow control. The valve utilises an ellipsoid control chamber and a dual micro-hole elastic membrane, Figure 9F. Membrane deflection in response to pressurised flow through the micro-holes dynamically modifies the control chamber's resistance. This self-regulating mechanism maintains a constant flow rate regardless of inlet pressure changes.

Figure 9. Microvalves and micropumps. Microvalves and micropumps. (A) Ortho-planer micro check valves with different stiffness; (B) Poiseuille Law pressure drop self-regulating valve; (C) Self-adaptive planer check valve with flexible cantilever flap; (D) Parallel membrane with low threshold pressure, self-regulating valve; (E) Stacked parallel membrane with low threshold pressure, regulating valve; (F) Ellipsoid control chamber auto-regulating valve.

Figure 9. Microvalves and micropumps. Microvalves and micropumps. (A) Ortho-planer micro check valves with different stiffness; (B) Poiseuille Law pressure drop self-regulating valve; (C) Self-adaptive planer check valve with flexible cantilever flap; (D) Parallel membrane with low threshold pressure, self-regulating valve; (E) Stacked parallel membrane with low threshold pressure, regulating valve; (F) Ellipsoid control chamber auto-regulating valve.

Micropump is another crucial component of many microfluidic systems. Since their invention, micropumps have seen significant advancements, offering advantages such as compact size, portability, energy efficiency, wide range of flow rate, affordability, and potential for integration with other microfluidic components. Micropumps are typically constructed using Micro-electromechanical Systems (MEMS) techniques on biocompatible materials such as silicon, glass, or various polymers (such as Polymethyl Methacrylate (PMMA), PDMS, or SU-8 photoresist) [143,144]. Micropumps fall into two broad categories: (i) mechanical, using moving parts like diaphragms and valves, and (ii) non-mechanical which manipulates fluid flow through hydrodynamic [145], electroosmotic [146], or electrowetting [147] forces. In mechanical micropumps, fluid flow is pressurised by an external force i.e. piezoelectric (PZT), electromagnetic (EM), electrostatic, and thermo-pneumatic, applied to either fixed flexible membranes or moving structures. This force transfer to the fluid occurs through fluid-structure interaction. Much work has been done in harnessing and manipulating the pumping function on the micro-scale through FSI. Wang et al. [148] proposed a piezoelectric micropump utilising fixed-end PDMS valves with integrated compressible space. This micropump utilises a resonantly driven membrane actuator, two fixed-end PDMS check valves for stability and reduced leakage, and strategically placed compressible spaces, Figure 10A.

Figure 10. Pumping schemes. (A) Piezoelectric micropump utilizing fixed-end PDMS valves with integrated compressible space; (B) Magnetically actuated membrane micropump with in-plane check valves; (C) Electrostatically actuated micropump utilizing four electrodes to induce peristaltic motion; (D) Thermo-pneumatic micropump with a thin polyimide membrane actuator.

Figure 10. Pumping schemes. (A) Piezoelectric micropump utilizing fixed-end PDMS valves with integrated compressible space; (B) Magnetically actuated membrane micropump with in-plane check valves; (C) Electrostatically actuated micropump utilizing four electrodes to induce peristaltic motion; (D) Thermo-pneumatic micropump with a thin polyimide membrane actuator.

Piezoelectric actuator deforms the pump membrane This deformation of the membrane affects the fluid inside the pump by changing the space it occupies, which increases or decreases the pressure. Essentially, as the membrane changes shape, it pushes on the fluid, helping to move it through the system. The micropump design relies on two key interactions: (i) electromechanical interaction, where the piezoelectric sheet converts electric signals into movement of the beam, and (ii) fluid-solid interaction, where the pump diaphragm interacts with the working fluid. An alternating voltage causes the beam to deform, driving the membrane and thus the fluid flow. Simultaneously, the fluid resists the movement of the membrane. The reported micropump delivers a maximum flow rate of 105 mL/min and a maximum back pressure of 23 kPa under a 400 V sinusoidal voltage at 490 Hz. Maximum power consumption at zero back pressure is approximately 42 mW.

Ni et al. [149] introduced a magnetic micropump utilising FSI for easy fabrication and seamless integration into other microfluidic systems. The device features in-plane check valves for flow control and a magnetically actuated membrane, Figure 10B. The deformable elastic membrane then interacts with the flowing fluid and in turn, pressurises the fluid. Since actuation is controlled directly by an external magnetic field, enabling efficient wireless operation is ideal for various applications. Experimental results indicate that the micropump can deliver 0.15 μL/min at 2 Hz, offering 1 nL per stroke resolution, and works against 550 Pa backpressure.

Moreover, Lee et al. [150] fabricated an electrostatically actuated micropump incorporating FSI, utilising four electrodes to induce peristaltic motion, Figure 10C. In this study, the micropump makes use of electrostatic force to create bidirectional peristaltic motion. Its unique design features a single deformable membrane with four movable polyimide electrodes hence eliminating the need for valves. Actuation signals cause the membrane to bend in small, sequential steps, which increases the pressure of the fluid in stages. This action splits a single chamber into two, three, or four separate sections, allowing for controlled movement of the fluid within the device. Experiments showed, optimising the actuation signal dramatically increased the flow rate. With a basic signal, the pump achieved 38 μl/min, but an optimised signal boosted this to 136 μl/min (both at 90 V and 15 Hz). This represents a 3.6-fold improvement. Hamid et al. [151] modelled a cost-effective thermo-pneumatic micropump with a thin polyimide membrane actuator, Figure 10D. The model includes a microheater, thermal cavity, and planar valve. This thermo-pneumatic micropump utilises thermal air expansion within a chamber to actuate a thin polyimide membrane. The membrane movement then interacts with the fluid and hence creates pressure fluctuations accordingly. This device effectively controls fluid on the picolitre to nanolitre scale, making it suitable for applications such as artificial kidneys and drug delivery systems, and is both simple and economical to fabricate.

4.2. Cell and Particle Manipulation

Cell sorting is a laboratory technique for isolating a specific cell type from a mixed population. Isolation criteria include physical parameters (size, morphology), cell viability, and the presence of specific intracellular or extracellular proteins [152]. Purified cells obtained through sorting are essential tools for research, diagnostic procedures, and cell-based therapies. Cell sorting encompasses a broad range of established techniques, employing both active and passive mechanisms [153]. Active sorting utilises external fields (electric, acoustic, magnetic, or optical) to manipulate cell trajectories. Passive systems primarily leverage inertial forces, filtration, and cell-surface adhesion for purification. Understanding FSI in both cell-fluid and fluid-channel interactions is crucial for optimizing and designing cell sorting devices. Extensive experimental works have been conducted to utilise FSI phenomena for cell sorting and manipulation. However, numerical simulations are vital to fully comprehend these FSI phenomena. Numerical modelling of cell dynamics complements experimental approaches, enabling the in-depth study of fluid-particle interactions. Accurately simulating these dynamic processes presents challenges due to the complexity of FSI coupling, cell mechanics, and the computational cost of simulating cell-cell interactions at scale.

Sun et al. [154] developed an LBM model to simulate blood flow in realistic microvascular networks and the separation of different types of cells, Figure 11A. This approach allows for the detailed analysis of FSI between blood cells and the vessel wall. This model treated blood as a suspension of particles (red blood cells [RBCs] and white blood cells [WBCs]) within the plasma, explicitly incorporating cell-cell and cell-wall interactions. The LBM approach allows for simulating RBC and WBC interactions as the cells flow through a microvascular network. This approach enables (i) quantification of forces exerted between RBCs and WBCs, tracking of trajectories of individual cells, (ii) analysis of pressure variations within the network due to cellular traffic, and (iii) the evaluation of forces experienced by the vessel walls at any location. Simulations demonstrate that vessel curvature and junctions increase the apparent viscosity and induce stress perturbations near stagnation points. The results suggested a potential link to atherogenesis at stagnation points and may also significantly influence our understanding of endothelial biology and its role in atherosclerosis formation.

Mao et al. [155] conducted computational modelling of particle sorting, specifically addressing FSI, for high-throughput hydrodynamic size-based sorting of solid microparticles in microchannels, Figure 11C(i). With this model, high-resolution separation was achieved by combining cross-stream inertial migration of particles with circulatory flows induced by periodic diagonal ridges on opposite channel walls. A hybrid approach was employed to model the multi-component system of a fluid-filled ridged microchannel and various-sized solid particles. This approach integrates LBM for fluid dynamics with a lattice spring model (LSM) for modelling solids. FSI is captured through appropriate boundary conditions at the solid-fluid interface. Simulations proved to be crucial for designing the ridged microchannel. Optimization for separating neutrally buoyant microparticles by size relied heavily on understanding the complex FSI within a microchannel. Figure 11C(ii) shows the results of this study. The geometry of the microchannel induces unique fluid flow patterns, and the resulting FSI between these patterns and the particles is the key to high-resolution separation.

Figure 11. Cell separation and micromixers. (A) Illustration of considering RBC and WBC as suspended particles in plasma to capture fluid-particle and fluid-wall interactions; (B) RBC discretisation to investigate the change in deformability of cells caused by malaria parasites. reproduced with permission from Hosseini et al. [156]; (C)(i) Setup for particle sorting in microchannels, (ii) Particle separation. reproduced with permission from Mao et al. [155]; (D) DLD device for separating CTCs within blood stream; (E) Mixing with magnetic actuated artificial cilia; (F) Passive mixing with flexible baffles.

Figure 11. Cell separation and micromixers. (A) Illustration of considering RBC and WBC as suspended particles in plasma to capture fluid-particle and fluid-wall interactions; (B) RBC discretisation to investigate the change in deformability of cells caused by malaria parasites. reproduced with permission from Hosseini et al. [156]; (C)(i) Setup for particle sorting in microchannels, (ii) Particle separation. reproduced with permission from Mao et al. [155]; (D) DLD device for separating CTCs within blood stream; (E) Mixing with magnetic actuated artificial cilia; (F) Passive mixing with flexible baffles.

Khodaee et al. [157] carried out a numerical FSI study of fluid-particle interaction in a deterministic lateral displacement (DLD) microfluidic device for effective separation of circulating tumor cells (CTC ) in bloodstreams. Numerical simulations, incorporating FSI using FEM to model deformable cell behaviour, guide the design of a typical DLD array for separating CTCs and leukocytes (white blood cells) under various flow conditions. Figure 11D illustrates the discretization of this method. This study focused on how coupled FSI phenomena, specifically related to flow conditions, stress, and cell deformability impact cell separation in DLD devices. This model provides essential data for optimising DLD devices, enhancing efficiency, and protecting cell viability. The model quantifies the cellular stress experienced during separation and maps the distribution of effective stress at peak deformation.

FSI also plays a fundamental role in accurately simulating cell deformations within a complex environment. Cells are not rigid bodies and respond dynamically to the fluid forces surrounding them. These forces can cause cells to stretch, compress, and change shape. In turn, deformed cells alter the flow field around them. FSI models are essential for capturing this intricate interplay. FSI simulation provides realistic predictions of how cells deform under various conditions. This has far-reaching implications in biomedical research, understanding disease processes where cell deformation plays a role. For instance, Hosseini et al. [156] carried out FSI analysis on RBCs and investigated the change in deformability of cells caused by malaria parasites. Numerical simulations incorporating FSI were employed to investigate this phenomenon. The cell membrane was represented as a collection of interconnected, elastic particles, Figure 11B. The cytosol is modelled as a Newtonian fluid using smoothed particle hydrodynamic techniques (SPH). More importantly, the malaria parasite was treated as a rigid structure, capturing its influence on the overall behaviour of the cells. Healthy RBCs are remarkably flexible, but the presence of the rigid malaria parasite within the cell significantly increases its stiffness. In response to the fluid forces present in the bloodstream, the deformability of infected cells declines. These findings underscore the potential of using cell deformability as a diagnostic marker and highlight the value of FSI simulation in understanding disease-induced cellular changes.

4.4. Modelling Cardiovascular Systems

In the realm of micro elastofluidics, computational methods for FSI unlock a deeper understanding of the complex interplay between biological fluids and the flexible tissues they encounter. The complex dynamics of blood flow, coupled with the flexible nature of heart valves and arterial walls, necessitate FSI simulations for a comprehensive analysis of the cardiovascular system. FSI models provide critical insights into heart valve function, including leaflet deformation, flow patterns, and stress distribution, leading to better diagnostics for heart diseases and improved designs for prosthetic replacements [174,175,176,177,178]. Similarly, FSI models applied to artery flow provide insights into the development of arterial diseases like atherosclerosis, uncovering potential risks for aneurysm formation, and contribute to the optimization of medical devices such as stents [179,180,181,182,183].

Heart valves play a critical role in maintaining blood flow direction. Artificial valves are treatment options, but their design and performance require careful analysis. Laha et al. [174] investigated bi-leaflet mechanical heart valve dynamics through FSI modelling with Smoothed Particle Hydrodynamics (SPH). Figure 12 illustrates the schematic model. The team explored a method for simulating a bi-leaflet heart valve using an SPH open-source code. By incorporating FSI, the SPH technique analysed hemodynamic abnormalities associated with valve dysfunction. The study considered normal and abnormal flow behaviour, valve movement under blockage scenarios, and potential risks associated with blockages. The findings demonstrate the effectiveness of this SPH/FSI approach for capturing the dynamic behaviour of bi-leaflet valves. The versatility of this computational model suggests its potential application to more complex cardiovascular problems.

Figure 12. Bi-leaflet mechanical heart valve dynamics through FSI Modelling with Smoothed Particle Hydrodynamics (SPH); (A) Illustration of mechanical heart valve; (B) Opening and closing position of valve; (C) Illustration of smoothed particles for simulation; (D) Inlet velocity profile to mimic the real pulse; (E) Simulation results. reproduced with permission from Laha et al. [174].

Figure 12. Bi-leaflet mechanical heart valve dynamics through FSI Modelling with Smoothed Particle Hydrodynamics (SPH); (A) Illustration of mechanical heart valve; (B) Opening and closing position of valve; (C) Illustration of smoothed particles for simulation; (D) Inlet velocity profile to mimic the real pulse; (E) Simulation results. reproduced with permission from Laha et al. [174].

Sodhani et al. [175] carried out an FSI study on an artificial aortic heart valve that was reinforced with textile. In this study, an in-silico FSI model was developed using the immersed boundary method to mimic the in-vitro experiment. The model assessed the geometric orifice area and flow rate over a single cycle, while also incorporating the material properties of the implant. The model employed fixed boundary conditions for the structural part. This involved fixing the bottom and stitched regions of the device in all directions, preventing movement in those areas. For the fluid domain, transvalvular pressure is incorporated as a boundary condition at the inlet and a zero-pressure condition at the outlet. Transvalvular pressure, measured in the corresponding in-vitro test. Figure 13 shows the model of the heart valve. The FSI simulation provided valuable insights into device performance i.e. pressure distribution, velocity field, recirculation zones, vortices, and potential leakage points. This work demonstrated the effectiveness of FSI simulations for validating material property determination techniques and predicting the kinematics and flow behaviours of the device.

Figure 13. An artificial aortic heart valve. (A) Effect of asymmetry on valve closure (right) and comparison with a similar tex-valve (left); (B) Illustration of test setup for FSI simulations; (C) Simulation results. reproduced with permission from Sodhani et al. [175].

Figure 13. An artificial aortic heart valve. (A) Effect of asymmetry on valve closure (right) and comparison with a similar tex-valve (left); (B) Illustration of test setup for FSI simulations; (C) Simulation results. reproduced with permission from Sodhani et al. [175].

Arterial diseases, including atherosclerosis and aneurysms, pose significant health risks due to their potential to cause serious cardiovascular events such as heart attacks and strokes. By accurately simulating the dynamic interactions between blood flow and arterial walls, FSI models provide invaluable insights into mechanical forces that contribute to disease progression. For example, Valente et al. [184] carried out the numerical investigation of Ascending Thoracic Aortic Aneurysm (ATAA) through FSI simulations using the open-source software package SimVascular. Figure 14A and 14B illustrate the mesh model and results, respectively. The simulations are based on patient-specific geometric models reconstructed from Computed Tomography (CT) scans. The analysis incorporates specific outlet conditions and temporal flow variations at the model inlet. By assigning prestress, the aorta model accurately reflects the in vivo stress state during the cardiac cycle. The process begins with a CFD analysis on the fluid domain, followed by structural analysis in the solid domain, using the pressures from the CFD phase as boundary conditions. The results from both CFD and Computational Structural Mechanics (CSM) were used as initial conditions for further analysis. The hemodynamic and structural behaviour of ATAA was studied, focusing on the velocity, displacement magnitudes, and wall shear stress distribution during the first cardiac cycle. The results confirm the effectiveness of the simulation in capturing the complex dynamics of ATAA, highlighting its potential for enhanced precision in biomechanical assessments.

Figure 14. Numerical investigation of Ascending Thoracic Aortic Aneurysm (ATAA) through FSI simulations. (A) Mesh discretisation of patient-specific geometric ascending thoracic aorta model reconstructed from CT scans; (B) Simulations results reproduced with permission from Valente et al. [184].

Figure 14. Numerical investigation of Ascending Thoracic Aortic Aneurysm (ATAA) through FSI simulations. (A) Mesh discretisation of patient-specific geometric ascending thoracic aorta model reconstructed from CT scans; (B) Simulations results reproduced with permission from Valente et al. [184].

5. Discussion and Future Perspectives

Abbreviations

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