The red blood cell (RBC) is considered as a deformable body, which is immersed in the fluid plasma Ω_f, as shown in Fig. 1(a). The RBC consists of two components: (i) the cellular membrane (Γ_RBC) and (ii) the cytosol fluid (Ω_c). The outer surface of the RBC’s membrane (Γ_RBC) is the interface between the RBC and the fluid plasma, which is considered as a Newtonian incompressible fluid. Inside the RBC, the cytosol fluid occupies the inner part of the domain Ω_c. The viscosity of the cytosol fluid is different from the one of the plasma fluid (viscosity contrast- νcνf≠1). The RBC deforms under the loading from the fluid domain Ω_f onto the interface Γ_RBC. In our hybrid continuum–particle approach, different numerical methods are used to model the dynamics of Γ_RBC, Ω_c, and Ω_f, as explained in the following sections.

The idealized shape of the RBC membrane is given by a set of points (i) with coordinates (x_i, y_i, z_i) in a 3D space with the analytical equation,³⁰ as shown in Fig. 1(b),zi=±D01−4(xi2+yi2)D02a0+a1xi2+yi2D02+a2(xi2+yi2)2D04;(1)the parameters are chosen in this work as D₀ = 7.82 µm (the equilibrium diameter), a₀ = 0.005 18, a₁ = 2.0026, and a₂ = −4.491. Note that the idealized shape will be used as the initial shape of the RBC membrane for all simulation cases. The membrane mechanics that govern the cellular deformation under loadings will be described in the following sections.

As the idealized surface of the RBC membrane is known precisely according to Eq. (1), a triangulation procedure is carried out to mimic the distribution of the spectrin links on the membrane as edges of each triangular element (links),³⁰ as shown in Fig. 1(b). A network of non-linear springs is generated for each edge to model the dynamics of the spectrin links,^29–31 as shown in Fig. 1. At each vertex i, the dynamics of the links are determined by the membrane force Fimembrane, which is linked to Helmholtz’s free energy V_i at the same vertex i through the following relationship:Fimembrane=−∂Vi∂ri,(2)with r_i being the position vector of vertex i. The potential V({r_i}) incorporates the following physical properties of the lipid bilayer: (a) in-plane stretching; (b) bending stiffness; and (c) area and volume conservation,V({ri})=Vin−plane+Vbending+Varea+Vvolume.(3)

In this work, the model for membrane mechanics follows closely standard dissipative particle dynamics (DPD) methodologies.^30–33 The details of the respective potentials are listed in Appendix A 1.

The interaction between the membrane and the cytosol fluid leads to the emergence of the overall RBC dynamics.³⁸ Differentiating between membrane and cytosol particles is essential for accurately capturing RBC mechanics, as it enables realistic modeling of membrane elasticity, internal viscosity contrast, and deformation behaviors under flow.³⁹ This interaction is modeled using the dissipative particles dynamics (DPD method). DPD is a microscopic simulation technique widely used to model the flow of complex fluids. Here, the flow is described as a group of clustered interacting particles according to the Lagrangian approach.³⁰ In this work, the cytosol within the RBC is modeled using a set of randomly distributed DPD particles (N_f) that fill the internal volume of the cell,^30,31 as shown in Fig. 1(a). Therefore, both the membrane and the cytosol fluid are considered collections of DPD particles.

Due to the different nature of the interactions, the components of the force of each particle depend on the nature of the particle i (either membrane or cytosol particle). In general, each DPD particle i interact with surrounding particles j (either a membrane or cytosol fluid particle) within a cutoff radius r_c through three pairwise additive forces: (a) the conservative force FijC; (b) the dissipative force FijD; (c) the random force FijR, and the penalty force FijP. The relative position vector between the particles i and j and related terms are given by r_ij = r_i − r_j, the distance r_ij = |r_ij|, and the unit vector r^ij=rijrij. In addition, v_i,j = v_i − v_j is the relative velocity between the particles i and j with velocities v_i and v_j.

For a DPD particle i of the cytosol fluid, the total internal force Fiinternal isFiinternal=∑j≠iFijC+FijD+FijR+FijP.(5)

For membrane particles, the total internal force Fiinternal acting on each membrane particle is given by the sum of the membrane force Fimembrane, the interaction forces from the surrounding DPD fluid particles (the cytosol), and the bilayer–cytoskeletal interactions force F^E,Fiinternal=Fimembrane+FiE+∑j≠iFijC+FijD+FijR+FijP.(6)

The mathematical formulation of the conservative force FijC, the dissipative force FijD, the random force FijR, and the penalty force FijP for the membrane and the cytosol fluid particles are explained in Appendix A 2.

Physically, the cytosol particles must be contained within the space bounded by the Γ_RBC surface, as shown in Fig. 1(a), at all times. However, the geometrical shape of Γ_RBC does not appear explicitly in the formulation of F^C, F^D, or F^R [Eqs. (A10)–(A18), Appendix A 2 a]. To ensure the containment of cytosol particles, it is required that a penalty force⁴⁰ is used to model a bounce-back reflection of cytosol fluid particles whenever they impact the membrane surface (collision force). This algorithm ensures that the cytosol particles remain within the RBC volume. This impact condition is checked using the dot product between the normal vector at the membrane vertex i (n_i) and the relative position vector (r_ij = r_j − r_i) of the cytosol fluid particle j asni⋅rij=<0InsideΩc>0OutsideΩc.(7)Note that the penalty force is only activated when the cytosol particles impact on Γ_RBC, as illustrated in Fig. 1(b) if |r_ij| < r_c. The normal vector n_i is defined as the average of the normal vectors of supporting elements n_e, which share the same vertex i. If the cytosol particle is confirmed to impact Γ_RBC, the penalty force F^P(r_ij) is activated,FijP=−βeni,(8)where β_e = 500 is the collision force coefficient.

We implemented the modified velocity Verlet algorithm,⁴¹ which consists of two primary steps. The first step involves determining the new position of the particle i (r_i) while predicting the velocity (vi~), and the second step involves correcting the velocity by utilizing the computed force (F_i) based on the predicted velocity and the new position as follows:ri(t+dt)=ri(t)+dtvi(t)+12dt2Fi(t),vi~(t+dt)=vi(t)+ΛdtFi(t),Fi(t+dt)=Fi(ri(t+dt),vi~(t+dt)),vi(t+dt)=vi(t)+12dt(Fi(t)+Fi(t+dt)),(9)where vi~(t+dt) is the predictive velocity at time t + dt and Λ is the variable which accounts for the effects of the stochastic processes. The value of Λ is chosen to be the optimal value⁴¹ Λ = 0.65. Note that the stochastic processes are related to the thermal fluctuations and thus dt must be chosen in the order of microseconds so that these processes can be captured accurately by the simulation.

Here, the computed force is the sum of the external and internal forces as Fi=Fiinternal+Fiexternal. The external force Fiexternal is either known (applied directly on the cell membrane) or computed from the extracellular fluid flows (fluid-structure interaction), as explained in the following.

The blood plasma is considered as an incompressible Newtonian fluid modeled using the incompressible three-dimensional unsteady Navier–Stockes equations, with density ρ and kinematic viscosity ν=μplasmaρ. The governing equations (continuity and momentum) for the domain Ω_f are read in Cartesian tensor notation as follows (i = 1, 2, 3 and repeated indices imply summation):∂ui∂xi=0,∂ui∂t+∂(uiuj)∂xj=−∂p∂xi+ν∂2ui∂xj∂xj.(10)In the above-mentioned equations, u_i is the ith component of the velocity vector u; t is time; x_i is the ith spatial coordinate; and p is the pressure divided by ρ. The characteristic velocity scale is chosen as U₀. The length scale L_s is set to equal 8 µm for all cases. Note that this length scale is chosen to reflect the diameter of the RBC under the equilibrium condition (L_s ≈ D₀). While the problems in the current study consider the dynamics of a RBC in microchannels, our numerical method is generally applicable for a wider range of applications including those with larger shear rates since the all terms of the Navier–Stokes equations are included.

The fluid solver is based on the sharp-interface curvilinear-immersed boundary (CURVIB) method (https://www.osti.gov/biblio/1312901) in a background structured-grid that contains the RBC model.²⁹ The computational grid is formulated using the hybrid staggered/non-staggered methodology.⁴² The Navier–Stokes equations are discretized using second-order accurate differencing for all spatial derivatives. The discrete equations are integrated in time via a second-order accurate fractional step method. The momentum equation is solved with an implicit Runge–Kutta solver. The total mass flux of the entire domain is enforced strictly to machine zero to satisfy the incompressibility constraint of the blood plasma. The flexible generalized Minimal residual (FGMRES) method with multigrid pre-conditioner is used to solve the Poisson equation, which results from the discrete continuity equation.⁴³

The complex surface of Γ_RBC is handled by the sharp-interface immersed boundary method, which requires only the velocity reconstruction at the immersed surface along the local normal to the body.⁴³ The presence of Γ_RBC is represented by a layer of computational grid points (immersed surface—IB nodes), which is distributed around the surface Γ_RBC. The boundary conditions reconstructed at the IB nodes are the three velocity components, which are reconstructed by interpolation.⁴⁴ In this work, a linear velocity profile along the normal direction of the immersed surface is assumed near RBC’s boundary.⁴⁵ Therefore, the presence of the IB nodes in our algorithm can be considered as an interface where the fluid and the cellular structure exchanges. The CURVIB method used here has been applied and validated in various FSI problems across different biological engineering areas.^46–49

To enable the continuum–particle coupling of the blood plasma and the cellular structures in our FSI algorithm, the flow imparted loads (Fiexternal) must be calculated on Γ_RBC as required by Eq. (9).

The fluid stress tensor components σijf are evaluated locally at every blood plasma points using the second-order differencing schemes for the velocity components using the continuum approach,⁴⁴σijf=−pI+μ∂ui∂xj+∂uj∂xi,(11)where I is the unit tensor. The traction vector t_e = σ^fn_e is calculated at the center of the triangular element e [see Fig. 1(b)]. The traction vector at the vertex is interpolated from t_I using the inverse distance weighting method⁴⁴ from elements in the region of support. Finally, the external force (F^external) at the vertex can be determined asFexternal=tIAs.(12)Here, A_s is the total area of the region of support for the vertex, as seen in Fig. 1(b).

In the current work, we employ the loose coupling approach⁵⁰ for the fluid–structure interaction. First, the Navier–Stokes equations are solved (the fluid solver F) to compute the flow field (pressure and velocities in Ω_f) at time step n. The loading (traction vector) on the cell membrane is computed using Eq. (12). Using the continuity of the stresses, the external loading on Γ_RBC is then computed to supply for Eq. (9) (the DPD solver—S) to find the deformation and velocity of Γ_RBC. Via the continuity of the velocity on the immersed boundary, (ri,vi)n+1 are used as boundary conditions on Γ_RBC in the next time step n + 1. In shorthand notation,(ri,vi)n+1=S(F((ri,vi)n)).(13)

Fluid–structure interaction simulations are performed to determine the dynamics of RBC in a confined micro-channel.⁴ The physical properties of the RBC are described in Table I. The corresponding DPD parameters are shown for the corresponding number of vertices N_v in Table II. The computation domain is defined as a rectangular channel containing a single RBC, as illustrated in Fig. 3(a). The dimensions of the domain along the x, y, and z are L_x (the length), L_y (the width), and L_z (the height), respectively. The computational domain is discretized as a structured grid of size N_i × N_j × N_k with the spatial resolution in three directions (i, j, k) are Δx ×Δy ×Δz, respectively. The no-slip condition is applied to all channel walls. The details of the channels used in the simulations are listed in Table III.

The RBC is located initially at t = 0 in an axial distance of x₀ from the inlet. The transverse location of the RBC is placed along the bisector of the first quadrant with a radial shift (r). Thus, the transverse coordinates of the RBC are y₀ = r and z₀ = r, respectively, as shown in Fig. 3(b) (r is the radial shift). With this configuration, the RBC confinement is defined as the ratio between the effective RBC diameter Dr=A0totπ (Table I) and the domain height L_z,χ=DrLz.(14)

The initial shape of the RBC is first set to be the idealized shape [Eq. (1)] for all simulation cases at the initial time t = 0. A short period of relaxation t_relax is allowed for the RBC under no external load (no flows) so that the internal forces of the RBC membrane balance. A uniform flow velocity U(t) is then applied at the channel inlet at t > t_relax to induce RBC’s deformation depending on the controlling strategy. The average shear rate across the channel height is defined as the ratio between the bulk velocity U(t) and the domain’s height,γ(t)˙¯=U(t)Lz.(15)At the channel’s outlet, a fully developed flow (Neumann) condition is imposed at the outlet of the channel.

A stretching test is carried out and aimed to replicate the experimental test of Ref. 51. In our simulations, the parameters used to describe the physical characteristics of the RBC are listed in Table I. Following the coarse-graining procedure, the model parameters for the cell membrane such as the equilibrium length, the persistence length, the spring stiffness, and the spontaneous angle are computed for each value of N_v, as in Table II. The cytosol fluid is modeled by a set of particles N_f = 100, placed within the interior volume of the cell membrane, as shown in Fig. 6(b).

In this experiment, two external forces F_stretch and −F_stretch with opposite directions are applied on both sides of the RBC. The magnitude of the force F_stretch is increased in a stepwise manner from 0 to 200 pN (a total of 16 steps). The axial diameter (D_a) and transverse diameter (Dt) are measured for every step. The definitions of D_a and D_t are shown in Fig. 6(a). D_a is calculated as |x_max − x_min| representing the axial distance between the maximum x position, denoted as x_max, and the minimum x position, denoted as x_min of the membrane vertices. D_t is calculated using the formula 2×maxi=1…Nv(yi−cy)2+(zi−cz)2, where c_y and c_z represent the y and z coordinates of the RBC’s center of mass, respectively. The simulations are performed systematically with different RBC surface mesh resolutions by changing the number of vertices (N_v) (also see Table II).

The current RBC model accurately replicates the elastic response of the RBC under stretching forces, as revealed by the results of D_a and D_t in Fig. 6. During stretching, the dynamic response of cytosol particles is visible, indicating the coupling between the membrane and the cytosol fluid. The shapes of the RBC under loading conditions agree with ones from experimental data of Ref. 51. The computed values of the axial (D_a) and transverse (D_t) diameters agree well with the experimental values, as seen in Fig. 6(a). In particular, the values of D_a and D_t are consistent across the different values of N_v, which indicates a robust performance of the coarse-graining procedure. There is a disagreement between the simulated results and the experimental value of D_t. Examining the shapes of the RBC in the simulations [Fig. 6(b)], it is revealed that the RBC tends to rotate around the stretching direction. This rotation leads to the difference between the experimental and numerical results of D_t. In brief, the mechanics of RBC is well-replicated by the computational model across different levels of coarse-graining. Thus, the value of N_v = 1000 is chosen to report the dynamics of the RBC.

Under constant shear rate conditions (I₀), as described in Sec. II G 1, two districts of the RBC shape are observed: (i) the croissant shape (I₀U₃r₁χ₁‐γ0˙¯=200s−1) and (ii) the slipper shape (I₀U₄r₃χ₁—γ0˙¯=600s−1), as shown in Fig. 7.

Under low shear rate (I₀U₃r₁χ₁), the RBC is initially placed along the centerline of the microchannel (discocyte shape). As the RBC interacts with the incoming flow, it deforms and eventually transitions to a croissant shape, as shown in Fig. 7(a). The terminal shape (croissant) is attained as the RBC continues to propagate along the channel’s symmetry axis. Note that the croissant shape, in this case, is not fully axis-symmetric. In addition, the croissant shape observed in this study is comparable with the parachute shape from our previous work,²⁹ although not identical due to differences in channel geometry and flow conditions.

Under high shear rate (I₀U₄r₃χ₁), the RBC transitions to the slipper shape, as shown in Fig. 7(b), which exhibits a bistability mode with tank-treading behavior. Note that the RBC is placed at a radial shift r₃ = 0.7 µm. Thus, the initial location of the RBC is not at the channel’s symmetry axis. The tank-treading effect is a complex dynamic in which the RBC membrane propagates axially along the channel while it rotates around its own center of mass, as seen in Fig. 7(b). A counterclockwise rotation is observed as indicated by the locations of two membrane particles (Lagrangian points—V₁ and V₂) at different time instances (t₁ = 22 ms and t₂ = 25 ms).

In both the croissant or slipper shapes, the transition from the initial shape (discocyte) to the terminal shape (either croissant or slipper) occurs within around 30 ms, as seen in Fig. 7. Our simulation results agree well with the corresponding experimental data of Ref. 4 as well as those described in recent experiments on RBC transient dynamics.^52,53 Furthermore, these transitions are in good agreement with the shape diagram⁵⁴ for different capillary numbers and confinements. In conclusion, our simulations are able to replicate the dynamics of the croissant and slipper shapes excellently.

The extracellular patterns of the croissant and slipper shapes also agree well with the experimental data.^4,26 The extracellular flow pattern can be visualized by reconstructing the relative flow velocity field²⁶ (co-moving frame). The relative velocity is defined as the difference between the flow velocity and the RBC’s centroid velocity, as shown in Fig. 8. In the croissant shape (I₀U₃r₁χ₁), the velocity streamlines closely resemble an axi-symmetrical flow pattern [Fig. 8(a)]. The downstream side of the RBC membrane deforms significantly, whereas the upstream side barely changes, as depicted in Fig. 8(b). In the slipper shape (I₀U₄r₃χ₁), there exists an asymmetrical vortical structure in the vicinity of the RBC membrane. As the slipper shape emerges, a fully closed vortex ring is created by a separated flow region, which is close to the channel wall. In short, the emergence of the RBC shape dictates the extracellular flow pattern.

Under stepwise flow waveform (I_s), our simulation results agree well with the propulsion step map (Δx_c, Ca^f), which was developed¹³ for both channels χ₂ = 0.5 and χ₃ = 0.38. In both cases, the propulsion step (Δx_c) is observed to monotonically increase with the values of Ca^f. However, Δx_c is higher in the lower confinement channel (χ₃), which indicates the importance of channel confinement.

In all simulation cases (I_sψ₁r₁χ₃, I_sψ₂r₁χ₃, I_sψ₃r₁χ₃, I_sψ₄r₁χ₃, I_sψ₅r₁χ₃, and I_sψ₆r₁χ₃), the RBC transitions from the idealized discocyte to the biconcave shape during the forward phase (0<t<T5) with all values of the peak forward flow (ψ_f = 1.05 mm s⁻¹ to ψ_f = 4.34 mm s⁻¹), as shown in Fig. 7(c). Strikingly, the complex multilobe shape emerges during the backward phase T_b. The elastic response of the RBC membrane to the oscillatory flow during the cycle T is depicted for the case I_sψ₆r₁χ₃ in Fig. 7(c). The reversal of the flow direction during T_b results in membrane buckling and stretching, which give rise to the multilobe shape. Note that these cases are initially placed at the channel center (r = r₁ = 0). Therefore, it is possible to induce complex dynamics of RBCs in a confined channel by changing the inflow waveform only.

Recent studies^4,26 in rectangular microchannels, which are identical to our channels as shown in Fig. 3 and Table III, suggest that the emergence of the RBC shape is dependent on the radial shift (r—see Fig. 3 for its definition). In previous studies,^4,26 the croissant shape emerged at γ0˙¯<300s−1 if the RBC is placed exactly at the channel’s center (r = 0). On the other hand, the slipper shape emerged whenever the RBC was not placed exactly at the centerline (r > 0). The RBC was found to exhibit a (tank-treading) slipper shape at sufficiently high shear rate (γ0˙¯≈500s−1) and off-centered placement (r > 0).^4,26 In cylindrical microchannels,¹⁷ similar observations were confirmed, albeit at lower shear rates (0<γ0˙¯<80s−1). These studies point to the importance of the radial shift in regulating RBC dynamics. Our results shown in Fig. 7 agree with these observations (also see Tables III and IV). Our results show the appearance of croissant-to-slipper transition as the capillary number (and thus γ0˙¯) increases from 0.1 to 0.37 for a confinement of χ = 0.65. The croissant shape emerges when the initial position of the RBC is placed exactly at the channel centerline at a sufficiently low shear rate (Ca = 0.1). When the shear rate is increased to Ca = 0.37, the slipper shape emerges. Furthermore, our model is able to capture the intricate dynamics of the tank-treading motion, which is characterized by the rotation of the membrane at the shear rate of 600 s⁻¹, as illustrated in Fig. 7. Therefore, our results further confirm the importance of the radial shift in the croissant-to-slipper transition.

When the inflow varies in a stepwise manner, as seen in Fig. 4, the shear rate (γ˙¯) changes as a function of time γ˙¯(t) with distinct forward (T_f) and backward (T_b) time phases. In all cases (I_sψ₁r₁χ₂, I_sψ₂r₁χ₂, I_sψ₃r₁χ₂, I_sψ₄r₁χ₃, I_sψ₅r₁χ₃, and I_sψ₆r₁χ₃), the RBC is placed exactly at the channel axis (r = r₁ = 0). The shape transitions are accomplished through consistent transient stretching and compression of the membrane. This occurs as the RBC experiences forward and backward flow phases during the flow cycles. The RBC transitions from a discocyte shape toward a croissant shape during its forward propulsion, as shown in Fig. 7. Although the backward phase induces the buckling of the cellular membrane, the RBC shape remains relatively symmetrical with respect to the channel axis (multilobes) as shown in Fig. 7 at the end of T_b. This is remarkable given that the maximum shear rate during the backward phase can be sufficiently large (γ˙¯f=200s−1) but this symmetry is still maintained. Comparing the case I₀U₃r₁χ₁ and I₀U₄r₃χ₁ in Table IV, our results suggest that the complete break of symmetry⁵² (slipper shape) is observed only when the radial shift exists (r > 0).

When applying different sinusoidal waveforms (I₁, I₂, I₃, and I₄) shown in Fig. 5, our results show the ubiquitous presence of croissant and slipper shapes across all shear rates (γ˙¯f=100,150, and 200 s⁻¹). The slipper shape appears at (t ≈ 0.3T) whenever the RBC is placed off the channel’s axis (r > 0), as shown in Tables VIII–XI. Note that these waveforms are different in terms of the forward (T_f) and backward (T_b) phases, with the backward phase being the shortest in I₄. As shown in Table VIII, the slipper shape is observed even when the waveform is completely symmetric (I₁) given that r > 0, as in (a) I₁U₁r₂χ₁, (b) I₁U₁r₃χ₁, (c) I₁U₂r₂χ₁, (d) I₁U₂r₃χ₁, (e) I₁U₃r₂χ₁, and (f) I₁U₃r₃χ₁. Hence, our results indicate that the flow waveform does not affect the emergence of the RBC shape during the forward phase. Instead, the radial shift plays an essential role in this process.

It has been demonstrated¹¹ that the presence of the discocyte shape is correlated with weak shear rates γ0¯. Under γ˙¯0<10s−1, the RBC typically maintains its discocyte shape with an 80% probability.¹¹ However, as the shear rate gradually rises (10s−1≤γ˙¯0≤400s−1), the likelihood of a discocyte shape decreases to 30%. This observation has been validated even when different viscosity ratios (λ)¹² are considered. In our work, the discocyte shape is ubiquitously observed during the second flow cycle across all applied waveforms (I₁, I₂, I₃, and I₄) and shear rates (γ˙¯f=100, 150, and 200 s⁻¹) with (r > 0) as shown in Tables VIII–XI. Therefore, our results further confirm that discocyte is the most common shape with the range shear rate less than 200 s⁻¹.

In our study, complex shapes evolve from simpler shapes under the influence of time-dependent waveforms. In the experimental work¹¹ under constant shear rates, stomatocytes shape was observed to dominate the RBC population (65%) when the shear rate was between 10s−1<γ0˙¯<400s−1. In our study, the elliptical-rim-shaped stomatocytes only emerge under symmetric waveform (I₁) in the case I₁U₁r₁χ₁ under the peak shear rate of γ˙¯f=100s−1. At high constant shear rates 400s−1<γ0˙¯<2000s−1, experimental data¹¹ showed that polylobes shape could emerge. This polylobes shape is characterized by a large number of lobes on the RBC surface, known as multilobes, trilobes, and hexalobes. In the current study, polylobes are also observed across all applied waveforms, given that the cell is placed initially at the channel axis (r = r₁ = 0) even at weak shear rates (γ˙¯f≤200s−1) as in Fig. 9 and Tables VIII–XI. For example, the multilobes are observed with all waveforms. The trilobes are both observed in the symmetric waveform (I₁U₂r₁χ₁ and I₁U₃r₁χ₁—Table VIII) or the asymmetric waveform (I₃U₂r₁χ₁—Table X). The hexalobes shape only appears under the most asymmetric waveform with (I4U2r1χ1)−γf˙¯=150s−1, as shown in Table XI. In the previous work,⁵⁵ the RBC shape has been reported to deform further into elongated shapes as the shear rate increases. Our results support this observation shown in Fig. 12(g) as the elongated multilobes appear during the backward flow phase. Examining Tables VIII–XI, our results suggest that this elongated shape is generally present regardless of the applied waveform but it only manifests under higher shear rates of at least γ˙¯f=150s−1. In particular, the elongated multilobes are observed under I₁U₂r₁χ₁, I₁U₃r₁χ₁, I₂U₃r₂χ₁, I₂U₃r₃χ₁, I₃U₃r₂χ₁, and I₃U₃r₃χ₁. In essence, our results strongly suggest that the complex shape can appear in a micro-channel even at weak shear rates given that the inflow is time-dependent.

While the radial shift is important for the emergence of the RBC shape, it is less important in controlling the cell displacement (the center of mass), as shown in Fig. 11. The background flow velocities vary largely over the cycle in the forward and reverse phases, depending on the flow waveform, as shown in Fig. 5. This variation dominates the plasma flow surrounding the RBC. Therefore, small radial shifts (0–0.7 µm) play a minimal role in regulating the overall displacement of the RBC’s center of mass.

Microfluidic devices are typically used to isolate and separate cells.⁵⁶ While these devices are promising for many cell-sorting applications,^57,58 the main challenge is the difficulty in obtaining high-throughputs due to the required length of the microfluidic channels. Changing the geometrical design of channels⁵⁹ has been proposed as one efficient way. Recent studies have shown that varying the shear rates in time^13,14 can reduce the required length based on the concept of velocity lift,⁶⁰ which is the factor that drives the RBC’s migration toward the center of the channel.

As the inertial effect is negligible at a very low Reynolds number (R_e ≈ 0.01), the flow is reversible for a rigid body. Thus, a rigid body will return to its initial position if the inflow conditions in the backward phase are reversed in the exact opposite way of its own during the forward phase. However, the RBC is not a rigid body and its membrane is highly flexible. Experiments¹⁴ have shown that the responses of RBCs are different from the ones of stiff beads in oscillatory flows. The RBC migration is observed to have a net actuation in oscillating flows whereas a stiff bead does not.

Our results in Figs. 10 and 11 for the symmetric waveform (I₁) show that the RBC does not go back to its initial position at the end of the cycle. There is an axial shift of the RBC from its original position (Δx_c ≠ 0) at the end of the cycle. Moreover, the RBC migrates significantly in the lateral cross section (Δy_c ≫ 0 and (Δz_c ≫ 0), as shown in Fig. 10. Our results in Figs. 11(b) and 11(e) indicate that the RBC undergoes lateral migration. This migration is particularly evident in the cases I₁U₁r₃χ₁ and I₄U₁r₃χ₁, where the cell migrates in the lateral direction for a distance of ∼0.25L_s and 0.13L_s, respectively. Comparing the shapes of the waveform I₁, I₂, I₃, and I₄ in Fig. 5, our results suggest that significant lateral migration can be induced by adjusting the forward time interval (T_f). With our sinusoidal waveforms shown in Fig. 5, T_f should be less than three times the backward flow phase T_b (TfTb<3) to be able to induce significant lateral migration. This lateral migration in oscillatory flows might be used to separate cells selectively based on their mechanical properties.

Our findings in Figs. 8 and 12 show that the extracellular flow patterns are directly influenced by the dynamics of the RBC. Under a stationary condition in Fig. 8, the extracellular flow dynamics in the croissant shape are distinctively different from the ones of the slipper shape. In particular, the flow around the steady croissant shape was similar to that of a rigid sphere,⁶¹ in which the flow streamlines move nearly symmetrically inward and outward from the cell in the upstream and downstream sides, respectively. In contrast, a fully closed vortex ring (“bolus”) was observed downstream the cell for the slipper shape.⁴ Our results in Fig. 12 suggest that the extracellular flow patterns are far more complicated and highly dependent on the type of inflow waveform in time-dependent shear rates. It is important to note that the extracellular flow has been found to play an important role in drug delivery strategies²⁶ due to its potential use of particle trapping. Therefore, our results suggest that controlling the inflow waveform either by adjusting the peak flow ψ_f or the shape of the waveform (specifically T_f) might lead to the desired effects in delivering small particles (e.g., therapeutic nano-particles) to the RBCs.

While the current model has successfully simulated various RBC morphologies under different flow conditions, it is crucial to acknowledge its limitations. First, the current model assumes that the impact of the collision force of the cytosol particles [Eq. (8)] is negligible in determining the shape of the cellular membrane, as shown in many validations of Sec. III A with high shear rates. However, it is important to recognize that this collision force might have impacts on the dynamics of the membrane and cytosol under very low shear rates. Therefore, future work will need to develop new algorithms for the collision scheme.^30,40 Second, analyzing the distribution of cytosol fluid particles and their conformational temperature could offer valuable insights into cytosol dynamics. In this work, the number of cytosol particles (N_f) is set to be 100, which is much less than the number of membrane particles (N_v). The coarse resolution of the cytosol fluid may lead to significant under-representation of hydrodynamic interactions and spatial gradients in the cytoplasm. Therefore, the intracellular flow cannot be reliably reproduced. In future studies, a larger number of N_f should be implemented to capture the intracellular flow patterns accurately. Third, previous work⁶² has shown that Eq. (A11) does not satisfy angular momentum conservation. This violation might induce inaccurate behaviors in simulating the dynamics of fluids such as the cytosol fluid. This issue needs to be investigated in future work. Fourth, the intracellular pressure coefficient p_bs = 4.12 Pa is chosen empirically, which is based on the range of RBC intracellular pressure.⁶³ Further investigations might be needed to identify the optimal value of p_bs. In addition, clustering of red blood cells is an important phenomenon, which determines the overall rheology of blood flow. Clusters of RBCs are formed by the adherence of individual cells, which have been shown to be dependent not only on the characteristics of the RBC membrane but also on the hydrodynamic condition of the plasma solution.⁶⁴ While this clustering process is beyond the scope of this work, our methodology can be extended to study this phenomenon by incorporating the interaction forces between adjacent cell membranes. This issue will be addressed in our future work. Finally, our coupling methodology utilizes the sharp-interface immersed boundary method,⁴⁵ which requires only the velocities on the cellular membrane as boundary conditions. In this work, the physical time step is chosen in the order of microseconds. Therefore, the thermal fluctuations of the velocity fields on the cellular membrane are fully resolved via the implicit momentum solver of the fractional step method (Sec. II E). Future studies need to be carried out to accelerate the convergence of the fractional step method for the Navier–Stokes equations and allow larger physical time steps.