Dynamics of a non-spherical microcapsule with incompressible interface in shear flow

2011 ◽  
Vol 678 ◽  
pp. 221-247 ◽  
Author(s):  
P. M. VLAHOVSKA ◽  
Y.-N. YOUNG ◽  
G. DANKER ◽  
C. MISBAH
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Viscosity Ratio ◽  
Perturbation Expansion ◽  
Shear Plane ◽  
Creeping Flow ◽  
Regular Perturbation ◽  
Cell Dynamics ◽  
Initial Shape ◽  
Flow Equations

We study the motion and deformation of a liquid capsule enclosed by a surface-incompressible membrane as a model of red blood cell dynamics in shear flow. Considering a slightly ellipsoidal initial shape, an analytical solution to the creeping-flow equations is obtained as a regular perturbation expansion in the excess area. The analysis takes into account the membrane fluidity, area-incompressibility and resistance to bending. The theory captures the observed transition from tumbling to swinging as the shear rate increases and clarifies the effect of capsule deformability. Near the transition, intermittent behaviour (swinging periodically interrupted by a tumble) is found only if the capsule deforms in the shear plane and does not undergo stretching or compression along the vorticity direction; the intermittency disappears if deformation along the vorticity direction occurs, i.e. if the capsule ‘breathes’. We report the phase diagram of capsule motions as a function of viscosity ratio, non-sphericity and dimensionless shear rate.

Stable equilibrium configurations of an oblate capsule in simple shear flow

2016 ◽  
Vol 791 ◽  
pp. 738-757 ◽  
Author(s):  
C. Dupont ◽  
F. Delahaye ◽  
D. Barthès-Biesel ◽  
A.-V. Salsac
Keyword(s):  
Aspect Ratio ◽  
Shear Flow ◽  
Simple Shear ◽  
Capillary Number ◽  
Stable Equilibrium ◽  
Viscosity Ratio ◽  
Shear Plane ◽  
Initial Orientation ◽  
Simple Shear Flow ◽  
Mechanical Equilibrium

The objective of the paper is to determine the stable mechanical equilibrium states of an oblate capsule subjected to a simple shear flow, by positioning its revolution axis initially off the shear plane. We consider an oblate capsule with a strain-hardening membrane and investigate the influence of the initial orientation, capsule aspect ratio$a/b$, viscosity ratio${\it\lambda}$between the internal and external fluids and the capillary number$Ca$which compares the viscous to the elastic forces. A numerical model coupling the finite element and boundary integral methods is used to solve the three-dimensional fluid–structure interaction problem. For any initial orientation, the capsule converges towards the same mechanical equilibrium state, which is only a function of the capillary number and viscosity ratio. For$a/b=0.5$, only four regimes are stable when${\it\lambda}=1$: tumbling and swinging in the low and medium$Ca$range ($Ca\lesssim 1$), regimes for which the capsule revolution axis is contained within the shear plane; then wobbling during which the capsule experiences precession around the vorticity axis; and finally rolling along the vorticity axis at high capillary numbers. When${\it\lambda}$is increased, the tumbling-to-swinging transition occurs for higher$Ca$; the wobbling regime takes place at lower$Ca$values and within a narrower$Ca$range. For${\it\lambda}\gtrsim 3$, the swinging regime completely disappears, which indicates that the stable equilibrium states are mainly the tumbling and rolling regimes at higher viscosity ratios. We finally show that the$Ca$–${\it\lambda}$phase diagram is qualitatively similar for higher aspect ratio. Only the$Ca$-range over which wobbling is stable increases with$a/b$, restricting the stability ranges of in- and out-of-plane motions, although this phenomenon is mainly visible for viscosity ratios larger than 1.

Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities

1998 ◽  
Vol 361 ◽  
pp. 117-143 ◽  
Author(s):  
S. RAMANUJAN ◽  
C. POZRIKIDIS
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Simple Shear ◽  
Large Deformations ◽  
Viscosity Ratio ◽  
Curvilinear Coordinates ◽  
Triangular Element ◽  
Elastic Membrane ◽  
Simple Shear Flow ◽  
Liquid Drops

The deformation of a liquid capsule enclosed by an elastic membrane in an infinite simple shear flow is studied numerically at vanishing Reynolds numbers using a boundary-element method. The surface of the capsule is discretized into quadratic triangular elements that form an evolving unstructured grid. The elastic membrane tensions are expressed in terms of the surface deformation gradient, which is evaluated from the position of the grid points. Compared to an earlier formulation that uses global curvilinear coordinates, the triangular-element formulation suppresses numerical instabilities due to uneven discretization and thus enables the study of large deformations and the investigation of the effect of fluid viscosities. Computations are performed for capsules with spherical, spheroidal, and discoidal unstressed shapes over an extended range of the dimensionless shear rate and for a broad range of the ratio of the internal to surrounding fluid viscosities. Results for small deformations of spherical capsules are in quantitative agreement with the predictions of perturbation theories. Results for large deformations of spherical capsules and deformations of non-spherical capsules are in qualitative agreement with experimental observations of synthetic capsules and red blood cells. We find that initially spherical capsules deform into steady elongated shapes whose aspect ratios increase with the magnitude of the shear rate. A critical shear rate above which capsules exhibit continuous elongation is not observed for any value of the viscosity ratio. This behaviour contrasts with that of liquid drops with uniform surface tension and with that of axisymmetric capsules subject to a stagnation-point flow. When the shear rate is sufficiently high and the viscosity ratio is sufficiently low, liquid drops exhibit continuous elongation leading to breakup. Axisymmetric capsules deform into thinning needles at sufficiently high rates of elongation, independent of the fluid viscosities. In the case of capsules in shear flow, large elastic tensions develop at large deformations and prevent continued elongation, stressing the importance of the vorticity of the incident flow. The long-time behaviour of deformed capsules depends strongly on the unstressed shape. Oblate capsules exhibit unsteady motions including oscillation about a mean configuration at low viscosity ratios and continuous rotation accompanied by periodic deformation at high viscosity ratios. The viscosity ratio at which the transition from oscillations to tumbling occurs decreases with the sphericity of the unstressed shape. Results on the effective rheological properties of dilute suspensions confirm a non-Newtonian shear-thinning behaviour.

Motion of Single Drops in Linear Shear Flows

2007 ◽  
Author(s):  
Win Myint ◽  
Shigeo Hosokawa ◽  
Akio Tomiyama
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Drag Coefficient ◽  
Shear Flows ◽  
Viscosity Ratio ◽  
Continuous Phase ◽  
Glycerol Water ◽  
Silicon Oil ◽  
Drag Forces ◽  
Linear Shear

Motions of single silicon oil drops rising in linear shear flows of glycerol-water solutions are measured to investigate the effects of the viscosity ratio κ and dimensionless shear rate Sr on drag forces under the conditions of −6.3 ≤ logM ≤ −4.8, 0.94 ≤ κ7.04, 1 < Re ≤ 23 and 3.2 ≤ ω ≤ 6.0 s−1, where M is the Morton number and ω the magnitude of the velocity gradient of the continuous phase. As a result, we confirmed that (1) the drag coefficient CD of a drop in a linear shear flow takes a higher value than the drag coefficient CD0 of the same-size drop in a stagnant liquid, (2) CD/CD0 increases with Sr, and (3) the augmentation of CD becomes large as κ increases.

Lift on a sphere moving near a wall in a parabolic flow

2010 ◽  
Vol 662 ◽  
pp. 447-474 ◽  
Author(s):  
SAMIR YAHIAOUI ◽  
FRANÇOIS FEUILLEBOIS
Keyword(s):  
Shear Flow ◽  
Solid Sphere ◽  
Migration Velocity ◽  
Creeping Flow ◽  
Regular Perturbation ◽  
Translation Velocity ◽  
Perturbation Problem ◽  
Sphere Radius ◽  
Parabolic Flow ◽  
And Migration

The lift on a solid sphere moving along a wall in a parabolic shear flow is obtained as a regular perturbation problem for low Reynolds number when the sphere is in the inner region of expansion. Comprehensive results are given for the 10 terms of the lift, which involve the sphere translation and rotation, the linear and quadratic parts of the shear flow and all binary couplings. Based on very accurate earlier results of a creeping flow in bispherical coordinates, precise results for these lift terms are obtained for a large range of sphere-to-wall distances, including the lubrication region for sphere-to-wall gaps down to 0.01 of a sphere radius. Fitting formulae are also provided in view of applications. The migration velocity of an inertialess spherical particle is given explicitly, for a non-rotating sphere with a prescribed translation velocity and for a freely moving sphere in a parabolic shear flow. Values of the lift and migration velocity are in good agreement with earlier results whenever available.

Large deformations of a cylindrical liquid-filled membrane by a viscous shear flow

1987 ◽  
Vol 179 ◽  
pp. 283-305 ◽  
Author(s):  
George I. Zahalak ◽  
Peddada R. Rao ◽  
Salvatore P. Sutera
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Viscous Fluid ◽  
Series Solution ◽  
Large Deformations ◽  
Boundary Value ◽  
Creeping Flow ◽  
Symbol Manipulation ◽  
Viscous Shear ◽  
Fifth Order

This paper treats the steady flow fields generated inside and outside an initially circular, inextensible, cylindrical membrane filled with an incompressible viscous fluid when the membrane is placed in a two-dimensional shear flow of another viscous fluid. The Reynolds numbers of both the interior and exterior flows were assumed to be zero (‘creeping flow’), but no further approximations were made in the formulation. A series solution of the resulting free boundary-value problem in powers of a dimensionless shear rate parameter was constructed through fifth order. When combined with a conformal coordinate transformation this series gave accurate results for large deformations of the membrane (up to an aspect ratio of 2.5). The rather tedious algebraic manipulations required to obtain the series solution were done by computer with a symbol-manipulation program (reduce), which both formulated the boundary-value problems for each successive order and solved them. Results are presented which show how the shear rate and fluid viscosities influence the internal and external velocity and pressure fields, the membrane deformation and its ‘tank-treading’ frequency, and the membrane tension.This work demonstrates that classical perturbation techniques combined with computer algebra offer a useful alternative to purely numerical methods for problems of this type.

Ellipsoidal capsules in simple shear flow: prolate versus oblate initial shapes

2011 ◽  
Vol 676 ◽  
pp. 318-347 ◽  
Author(s):  
J. WALTER ◽  
A.-V. SALSAC ◽  
D. BARTHÈS-BIESEL
Keyword(s):  
Shear Flow ◽  
Simple Shear ◽  
Capillary Number ◽  
Shear Plane ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Simple Shear Flow ◽  
Initial State ◽  
Initial Shape ◽  
Aspect Ratios

The large deformations of an initially-ellipsoidal capsule in a simple shear flow are studied by coupling a boundary integral method for the internal and external flows and a finite-element method for the capsule wall motion. Oblate and prolate spheroids are considered (initial aspect ratios: 0.5 and 2) in the case where the internal and external fluids have the same viscosity and the revolution axis of the initial spheroid lies in the shear plane. The influence of the membrane mechanical properties (mechanical law and ratio of shear to area dilatation moduli) on the capsule behaviour is investigated. Two regimes are found depending on the value of a capillary number comparing viscous and elastic forces. At low capillary numbers, the capsule tumbles, behaving mostly like a solid particle. At higher capillary numbers, the capsule has a fluid-like behaviour and oscillates in the shear flow while its membrane continuously rotates around its deformed shape. During the tumbling-to-swinging transition, the capsule transits through an almost circular profile in the shear plane for which a long axis can no longer be defined. The critical transition capillary number is found to depend mainly on the initial shape of the capsule and on its shear modulus, and weakly on the area dilatation modulus. Qualitatively, oblate and prolate capsules are found to behave similarly, particularly at large capillary numbers when the influence of the initial state fades out. However, the capillary number at which the transition occurs is significantly lower for oblate spheroids.

Dynamics of red blood cells in oscillating shear flow

2016 ◽  
Vol 800 ◽  
pp. 484-516 ◽  
Author(s):  
Daniel Cordasco ◽  
Prosenjit Bagchi
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Red Blood Cells ◽  
Periodic Motion ◽  
Blood Cells ◽  
Large Amplitude ◽  
Chaotic Dynamics ◽  
Viscosity Ratio ◽  
Reduced Order Models ◽  
Reduced Order

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.

Heat/mass transport in shear flow over a heterogeneous surface with first-order surface-reactive domains

2015 ◽  
Vol 782 ◽  
pp. 260-299 ◽  
Author(s):  
Preyas N. Shah ◽  
Eric S. G. Shaqfeh
Keyword(s):  
Mass Transfer ◽  
Shear Rate ◽  
Shear Flow ◽  
Reaction Rate ◽  
Patch Size ◽  
Patch Area ◽  
First Order ◽  
Effective Boundary ◽  
Order Surface ◽  
Slip Distance

Surfaces that include heterogeneous mass transfer at the microscale are ubiquitous in nature and engineering. Many such media are modelled via an effective surface reaction rate or mass transfer coefficient employing the conventional ansatz of kinetically limited transport at the microscale. However, this assumption is not always valid, particularly when there is strong flow. We are interested in modelling reactive and/or porous surfaces that occur in systems where the effective Damköhler number at the microscale can be $O(1)$ and the local Péclet number may be large. In order to expand the range of the effective mass transfer surface coefficient, we study transport from a uniform bath of species in an unbounded shear flow over a flat surface. This surface has a heterogeneous distribution of first-order surface-reactive circular patches (or pores). To understand the physics at the length scale of the patch size, we first analyse the flux to a single reactive patch. We use both analytic and boundary element simulations for this purpose. The shear flow induces a 3-D concentration wake structure downstream of the patch. When two patches are aligned in the shear direction, the wakes interact to reduce the per patch flux compared with the single-patch case. Having determined the length scale of the interaction between two patches, we study the transport to a periodic and disordered distribution of patches again using analytic and boundary integral techniques. We obtain, up to non-dilute patch area fraction, an effective boundary condition for the transport to the patches that depends on the local mass transfer coefficient (or reaction rate) and shear rate. We demonstrate that this boundary condition replaces the details of the heterogeneous surfaces at a wall-normal effective slip distance also determined for non-dilute patch area fractions. The slip distance again depends on the shear rate, and weakly on the reaction rate, and scales with the patch size. These effective boundary conditions can be used directly in large-scale physics simulations as long as the local shear rate, reaction rate and patch area fraction are known.

Small Reynolds Number Electro-Hydrodynamic Flow Around Drops and the Resulting Deformation

1979 ◽  
Vol 46 (3) ◽  
pp. 510-512 ◽  
Author(s):  
M. B. Stewart ◽  
F. A. Morrison
Keyword(s):  
Reynolds Number ◽  
Flow Field ◽  
Order Approximation ◽  
Creeping Flow ◽  
Small Reynolds Number ◽  
Zeroth Order ◽  
Regular Perturbation ◽  
Low Reynolds Number Flow ◽  
Reynolds Number Flow ◽  
Number Flow

Low Reynolds number flow in and about a droplet is generated by an electric field. Because the creeping flow solution is a uniformly valid zeroth-order approximation, a regular perturbation in Reynolds number is used to account for the effects of convective acceleration. The flow field and resulting deformation are predicted.

Effects of shear rate on rouleau formation in simple shear flow

Biorheology ◽  
1988 ◽  
Vol 25 (1-2) ◽  
pp. 113-122 ◽  
Author(s):  
T. Murata ◽  
T.W. Secomb
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Simple Shear ◽  
Simple Shear Flow ◽  
Rouleau Formation
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