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Inertial and non-inertial focusing of a deformable capsule in a curved microchannel
- Saman Ebrahimi, Prosenjit Bagchi
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- Journal:
- Journal of Fluid Mechanics / Volume 929 / 25 December 2021
- Published online by Cambridge University Press:
- 27 October 2021, A30
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A computational study is presented on cross-stream migration and focusing of deformable capsules in curved microchannels of square and rectangular sections under inertial and non-inertial regimes. The numerical methodology is based on immersed boundary methods for fluid–structure coupling, a finite-volume-based flow solver and finite-element method for capsule deformation. Different focusing behaviours in the two regimes are predicted that arise due to the interplay of inertia, deformation, altered shear gradient, streamline curvature effect and secondary flow. In the non-inertial regime, a single-point focusing occurs on the central plane, and at a radial location between the interior face (i.e. face with highest curvature) of the channel and the location of zero shear. The focusing position is nearly independent of capsule deformability (represented by the capillary number, $Ca$). A two-step migration is observed that is comprised of a faster radial migration, followed by a slower migration toward the centre plane. The focusing location progressively moves further toward the interior face with increasing curvature and width, but decreasing height. In the inertial regime, single-point focusing is observed near the interior face for channel Reynolds number $Re_{C}\sim {O}(1)$, that is also highly sensitive to $Re_{C}$ and $Ca$, and moves progressively toward the exterior face with increasing $Re_{C}$ but decreasing $Ca$. As $Re_{C}$ increases by an order, secondary flow becomes stronger, and two focusing locations appear close to the centres of the Dean vortices. This location becomes practically independent of $Ca$ at even higher inertia. The inertial focusing positions move progressively toward the exterior face with increasing channel width and decreasing height. For wider channels, the equilibrium location is further toward the exterior face than the vortex centre.
Motion of a capsule in a curved tube
- Saman Ebrahimi, Peter Balogh, Prosenjit Bagchi
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- Journal:
- Journal of Fluid Mechanics / Volume 907 / 25 January 2021
- Published online by Cambridge University Press:
- 25 November 2020, A28
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Cross-streamline migration of deformable capsules is studied in three-dimensional curved vessels using a numerical model. Two geometries are chosen: torus vessels to study the effect of constant vessel curvature, and U-shaped vessels to study the effect of change in curvature. A wide range of inertia is considered to include a broad range of applications, from the microcirculation to inertial microfluidics. Vessel curvature and change in curvature are shown to affect capsule migration because of the way they affect the undisturbed flow. In toroidal vessels at negligible inertia, no secondary flow exists and the axial fluid velocity is shifted toward the inner surface of the vessel. In this limit, capsules settle at a location that is away from the vessel centreline, and between the location of the maximum fluid velocity and the innermost surface. Increasing the vessel curvature results in the equilibrium position being increasingly closer to the inner surface. The results suggest the presence of a curvature-induced migration that drives the capsule continuously toward the region of higher streamline curvature. The equilibrium locations are on the symmetry plane of the torus, and are stable equilibria when inertia is negligible. At finite inertia, capsules released on the symmetry plane first settle at an equilibrium located on this plane, which may either be stable or unstable. This equilibrium location depends on both capsule deformability (characterised by capillary number, $Ca$) and the tube Reynolds number $R{e_t}$. It is closer to the inner edge of the torus for smaller $R{e_t}$ and larger $Ca$, and toward the outer edge for larger $R{e_t}$ and smaller $Ca$. This dependence of the equilibria arises due to the secondary flow (or Dean's vortex), which opposes the curvature-induced inward migration. The equilibrium locations on the symmetry plane are unstable if the secondary flow is strong, in which case capsules depart the symmetry plane, and set on to a spiralling motion, eventually settling near the centres of the Dean's vortices. For the U-shaped vessels, the curvature change is shown to create a radial velocity component, even at small inertia, that can be of comparable magnitude to the axial velocity, and that increases with increasing curvature. This radial velocity causes a large and abrupt change in capsule trajectory toward the inner side of the vessel where the curvature increases, but toward the outer side where the curvature decreases.
The cell-free layer in simulated microvascular networks
- Peter Balogh, Prosenjit Bagchi
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- Journal:
- Journal of Fluid Mechanics / Volume 864 / 10 April 2019
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- 11 February 2019, pp. 768-806
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In the microcirculation, a plasma layer forms near the vessel walls that is free of red blood cells (RBCs). This region, often termed as the cell-free layer (CFL), plays important haemorheological and biophysical roles, and has been the subject of extensive research. Many previous studies have considered the CFL development in single, isolated vessels that are straight tubes or channels, as well as in isolated bifurcations and mergers. In the body, blood vessels are typically winding and sequentially bifurcate into smaller vessels or merge to form larger vessels. Because of this geometric complexity, the CFL in vivo is three-dimensional (3D) and asymmetric, unlike in fully developed flow in straight tubes. The three-dimensionality of the CFL as it develops in a vascular network, and the underlying hydrodynamic mechanisms, are not well understood. Using a high-fidelity model of cellular-scale blood flow in microvascular networks with in vivo-like topologies, we present a detailed analysis of the fully 3D and asymmetric nature of the CFL in such networks. We show that the CFL significantly varies over different aspects of the networks. Along the vessel lengths, such variations are predominantly non-monotonic, which indicates that the CFL profiles do not simply become more symmetric over the length as they would in straight vessels. We show that vessel tortuosity causes the CFL to become more asymmetric along the length. We specifically identify a curvature-induced migration of the RBCs as the underlying mechanism of increased asymmetry in curved vessels. The vascular bifurcations and mergers are also seen to change the CFL profiles, and in the majority of them the CFL becomes more asymmetric. For most bifurcations, this is generally observed to occur such that the CFL downstream narrows on the side of the vessel nearest the upstream bifurcation, and widens on the other side. The 3D aspects of such behaviour are elucidated. For many bifurcations, a discrepancy exists between the CFL in the daughter vessels, which arises from a disproportionate partitioning between the flow rate and RBC flux. For most mergers, the downstream CFL narrows in the plane of the merger, but widens away from this plane. The dominant mechanism by which such changes occur is identified as the geometric focusing of the two merging streams. To our knowledge, this work provides the first simulation-based analysis of the 3D CFL structure in complex in vivo-like microvascular networks, including the hydrodynamic origins of the observed behaviour.
Dynamics of red blood cells in oscillating shear flow
- Daniel Cordasco, Prosenjit Bagchi
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- Journal:
- Journal of Fluid Mechanics / Volume 800 / 10 August 2016
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- 08 July 2016, pp. 484-516
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We present a three-dimensional computational study of fully deformable red blood cells of the biconcave resting shape subject to sinusoidally oscillating shear flow. A comprehensive analysis of the cell dynamics and deformation response is considered over a wide range of flow frequency, shear rate amplitude and viscosity ratio. We observe that the cell exhibits either a periodic motion or a chaotic motion. In the periodic motion, the cell reverses its orientation either by passing through the flow direction (horizontal axis) or by passing through the flow gradient (vertical axis). The chaotic dynamics is characterized by a non-periodic sequence of horizontal and vertical reversals. The study provides the first conclusive evidence of the chaotic dynamics of fully deformable cells in oscillating flow using a deterministic numerical model without the introduction of any stochastic noise. In certain regimes of the periodic motion, the initial conditions are completely forgotten and the cells become entrained in the same sequence of horizontal reversals. We show that chaos is only possible in certain frequency bands when the cell membrane can rotate by a certain amount, allowing the cells to swing near the maximum shear rate. As such, the bifurcation between the horizontal and vertical attractors in phase space always occurs via a swinging inflection. While the reversal sequence evolves in an unpredictable way in the chaotic regime, we find a novel result that there exists a critical inclination angle at the instant of flow reversal which determines whether a vertical or horizontal reversal takes place, and is independent of the flow frequency. The chaotic dynamics, however, occurs at a viscosity ratio less than the physiological values. We further show that the cell shape in oscillatory shear at large amplitude exhibits a remarkable departure from the biconcave shape, and that the deformation is significantly greater than that in steady shear flow. A large compression of the cells occurs during the reversals which leads to over/undershoots in the deformation parameter. We show that due to the large deformation experienced by the cells, the regions of chaos in parameter space diminish and eventually disappear at high shear rate, in contradiction to the prediction of reduced-order models. While the findings bolster support for reduced-order models at low shear rate, they also underscore the important role that the cell deformation plays in large-amplitude oscillatory flows.
Intermittency and synchronized motion of red blood cell dynamics in shear flow
- Daniel Cordasco, Prosenjit Bagchi
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- Journal of Fluid Mechanics / Volume 759 / 25 November 2014
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- 24 October 2014, pp. 472-488
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We present the first full-scale computational evidence of intermittent and synchronized dynamics of red blood cells in shear flow. These dynamics are characterized by the coexistence of a tumbling motion in which the cell behaves like a rigid body and a tank-treading motion in which the cell behaves like a liquid drop. In the intermittent dynamics, we observe sequences of tumbling interrupted by swinging, as well as sequences of swinging interrupted by tumbling. In the synchronized dynamics, the tumbling and membrane rotation are observed to occur simultaneously with integer ratios of the rotational frequencies. These dynamics are shown to be dependent on the stress-free state of the cytoskeleton, and are explained based on the cell membrane energy landscape.
Influence of membrane viscosity on capsule dynamics in shear flow
- Alireza Yazdani, Prosenjit Bagchi
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- Journal:
- Journal of Fluid Mechanics / Volume 718 / 10 March 2013
- Published online by Cambridge University Press:
- 08 February 2013, pp. 569-595
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Most previous numerical studies on capsule dynamics in shear flow have ignored the role of membrane viscosity. Here we present a numerical method for large deformation of capsules using a Kelvin–Voigt viscoelastic model for the membrane. After introducing the model and the related numerical implementation, we present a comprehensive analysis of the influence of the membrane viscosity on buckling, deformation and dynamics. We observe that the membrane viscosity leads to buckling in the range of shear rate in which no buckling is observed for capsules with purely elastic membrane. For moderate to large shear rates, the wrinkles on the capsule surface appear in the same range of the membrane viscosity that was reported earlier for artificial capsules and red blood cells based on experimental measurements. In order to obtain stable shapes, it is necessary to introduce the bending stiffness. It is observed that the range of the bending stiffness required is also in the same range as that reported for the red blood cells, but considerably higher than that estimated for artificial capsules. Using the stable shapes obtained in the presence of bending stiffness, we analyse the influence of membrane viscosity on deformation, inclination and tank-treading frequency of initially spherical capsules. Membrane viscosity is observed to reduce the capsule deformation, and introduce a damped oscillation in time-dependent deformation and inclination. The time-averaged inclination angle shows a non-monotonic trend with an initial decrease reaching a minimum and a subsequent increase with increasing membrane viscosity. A similar non-monotonic trend is also observed in the tank-treading frequency. We then consider the influence of the membrane viscosity on the unsteady dynamics of an initially oblate capsule. The dynamics is observed to change from a swinging motion to a tumbling motion with increasing membrane viscosity. Further, a transient dynamics is also observed in which a capsule starts with one type of dynamics, but settles with a different dynamics over a long time.
Dynamic rheology of a dilute suspension of elastic capsules: effect of capsule tank-treading, swinging and tumbling
- PROSENJIT BAGCHI, R. MURTHY KALLURI
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- Journal:
- Journal of Fluid Mechanics / Volume 669 / 25 February 2011
- Published online by Cambridge University Press:
- 13 January 2011, pp. 498-526
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Three-dimensional numerical simulations are used to study the effect of unsteady swinging and tumbling motion on the rheology of a dilute suspension of oblate-shaped elastic capsules. Unlike a suspension of initially spherical capsules undergoing the steady tank-treading motion for which the rheology is constant in time, the suspension of non-spherical capsules is time-dependent due to the unsteady capsule motion. In a simple shear flow, the non-spherical capsules undergo a transition from the tank-treading/swinging to the tumbling motion with a reduction in the shear rate or an increase in the ratio of the internal to external fluid viscosities. We find that the time-averaged rheology obtained for the non-spherical capsules undergoing the unsteady motion is qualitatively similar to that obtained for the spherical capsules undergoing the steady tank-treading motion, and that the tank-treading-to-tumbling transition has only a marginal effect. The time-averaged rheology exhibits a shear viscosity minimum when the capsules are in a swinging motion at high shear rates but not at low shear rates. This is a remarkable departure from the behaviour of a vesicle suspension which exhibits a shear viscosity minimum at the point of transition. We find that the shear viscosity in a capsule suspension can decrease as well as increase with increasing viscosity ratio during both tank-treading and tumbling motions, while that of a vesicle suspension always decreases in tank-treading motion and increases in tumbling motion. We then seek to connect the time-dependent rheology with the time-dependent membrane tension, capsule orientation, deformation and tank-treading velocity. At low shear rates, the numerical results exhibit a similar trend to that predicted by analytical theory for rigid ellipsoids undergoing tumbling motion. The trend differs during swinging motion due to the periodic deformation and time-dependent variation of the membrane stress. The elastic component of the shear stress is minimum when the capsules are maximally compressed, and is maximum when the capsules are maximally elongated. In contrast, the viscous component is related to the periodic variation of the tank-treading velocity synchronized with the swinging motion, and the rate of capsule elongation or compression. The swinging or tumbling velocity makes no contribution to the time-dependent rheology.
Response of the wake of an isolated particle to an isotropic turbulent flow
- PROSENJIT BAGCHI, S. BALACHANDAR
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- Journal of Fluid Mechanics / Volume 518 / 10 November 2004
- Published online by Cambridge University Press:
- 20 October 2004, pp. 95-123
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The interaction of an isolated spherical particle with an isotropic turbulent flow is considered using direct numerical simulations (DNS). The particle Reynolds number is varied from about 50 to 600 and the particle diameter is varied from about 1.5 to 10 times the Kolmogorov scale. The Reynolds number based on the Taylor microscale of the free-stream turbulent field considered here is 164. The DNS technique employed here is the first of its kind to address particle–turbulence interaction and it resolves the smallest scales in the free-stream turbulent flow and the complex vortical structures in the particle wake. The primary objective of this paper is to present new results on the effect of the free-stream turbulence on the particle wake and vortex shedding, and the modulation of free-stream turbulence in the particle wake. The parameters of the present simulations are comparable to those of the experimental study by Wu & Faeth (1994$a$, $b$), and agreement between the present computational results and the experimental measurement is demonstrated.
The effect of free-stream turbulence on the mean and instantaneous wake structure is studied. The time-averaged mean wake in a turbulent ambient flow shows a lower velocity deficit and a flatter profile. However, in agreement with the experimental results of Wu & Faeth the mean wake in a turbulent flow behaves like a self-preserving laminar wake. At Reynolds numbers below about 210 the effect of free-stream turbulence is to introduce wake oscillations. For Reynolds numbers in the range 210 to 280, free-stream turbulence is observed to promote early onset of vortex shedding. The nature of the shed vortices is somewhat different from that in a uniform flow. Increasing the free-stream turbulence intensity suppresses the process of vortex shedding, and only marginally increases the wake oscillation. The modulation of free-stream turbulence in the wake is studied in terms of the distribution of kinetic energy and RMS velocity fluctuation. The free-stream energy lost in the wake is recovered faster in a turbulent ambient flow than in a uniform ambient flow. The energy of the velocity fluctuation is enhanced in the wake at low free-stream intensities, and is damped or marginally increased at higher intensities. The fluctuation energy is not equi-partitioned among the streamwise and cross-stream components. The RMS streamwise fluctuation is always enhanced, whereas the RMS cross-stream fluctuation is enhanced only at low free-stream intensities, and damped at higher intensities.
Inertial and viscous forces on a rigid sphere in straining flows at moderate Reynolds numbers
- PROSENJIT BAGCHI, S. BALACHANDAR
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- Journal:
- Journal of Fluid Mechanics / Volume 481 / 25 April 2003
- Published online by Cambridge University Press:
- 28 April 2003, pp. 105-148
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The focus of this paper is the effect of spatial non-uniformity in the ambient flow on the forces acting on a rigid sphere when the sphere Reynolds number, $Re$, is in the range 10 to 300. Direct numerical simulations (DNS) based on a pseudospectral methodology are carried out to solve for the unsteady three-dimensional flow field around a sphere which is either held stationary or allowed to translate freely under the hydrodynamic forces. The various components of the total force, namely the inertial, steady viscous, and history forces, are systematically estimated in the context of linearly varying straining flows. The inertial forces are isolated by computing the rapid changes in the drag and lift forces in response to a rapid acceleration of the ambient flow. It is shown that the inertial forces arising due to convective acceleration at moderate Reynolds numbers follow the inviscid flow result. While the effect of temporal acceleration depends only on the sign and magnitude of the acceleration, the effect of convective acceleration is shown to depend also on the initial state of the ambient flow. A simple theoretical argument is presented to support the numerical observations. It is also shown that the effect of convective acceleration on the steady viscous force can be realized on a slower time scale. The results show that the history kernels currently available in the literature are not adequate to represent the effect of non-uniformity on the history force.