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

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.

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The dynamics of a vesicle in simple shear flow

Journal of Fluid Mechanics ◽

10.1017/s0022112011000115 ◽

2011 ◽

Vol 674 ◽

pp. 578-604 ◽

Cited By ~ 85

Author(s):

HONG ZHAO ◽

ERIC S. G. SHAQFEH

Keyword(s):

Shear Flow ◽

Simple Shear ◽

Lipid Membrane ◽

Shear Plane ◽

Plane Motion ◽

Boundary Integral ◽

Boundary Integral Method ◽

Lipid Vesicle ◽

Simple Shear Flow ◽

Motion Patterns

We have performed direct numerical simulation (DNS) of a lipid vesicle under Stokes flow conditions in simple shear flow. The lipid membrane is modelled as a two-dimensional incompressible fluid with Helfrich surface energy in response to bending deformation. A high-fidelity spectral boundary integral method is used to solve the flow and membrane interaction system; the spectral resolution and convergence of the numerical scheme are demonstrated. The critical viscosity ratios for the transition from tank-treading (TT) to ‘trembling’ (TR, also called VB, i.e. vacillating-breathing, or swinging) and eventually ‘tumbling’ (TU) motions are calculated by linear stability analysis based on this spectral method, and are in good agreement with perturbation theories. The effective shear rheology of a dilute suspension of these vesicles is also calculated over a wide parameter regime. Finally, our DNS reveals a family of time-periodic and off-the-shear-plane motion patterns where the vesicle's configuration follows orbits that resemble but are fundamentally different from the classical Jeffery orbits of rigid particles due to the vesicle's deformability.

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Off-plane motion of a prolate capsule in shear flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.62 ◽

2013 ◽

Vol 721 ◽

pp. 180-198 ◽

Cited By ~ 35

Author(s):

C. Dupont ◽

A.-V. Salsac ◽

D. Barthès-Biesel

Keyword(s):

Shear Flow ◽

Capillary Number ◽

Three Dimensional ◽

Shear Plane ◽

Plane Motion ◽

Boundary Integral ◽

Boundary Integral Method ◽

Simple Shear Flow ◽

Flow Vorticity ◽

Wall Deformation

AbstractThe objective of this study is to investigate the motion of an ellipsoidal capsule in a simple shear flow when its revolution axis is initially placed off the shear plane. We consider prolate capsules with an aspect ratio of two or three enclosed by a membrane, which is either strain-hardening or strain-softening. We seek the equilibrium motion of the capsule as we increase the capillary number$\mathit{Ca}$, which measures the ratio between the viscous and elastic forces. The three-dimensional fluid–structure interaction problem is solved numerically by coupling a boundary integral method (for the internal and external flows) with a finite element method (for the wall deformation). For any initial inclination with the flow vorticity axis, a given capsule converges towards a unique equilibrium configuration which depends on$\mathit{Ca}$. At low capillary number, the stable equilibrium motion is the rolling regime: the capsule aligns its long axis with the vorticity axis, while the membrane tank-treads. As$\mathit{Ca}$increases, the capsule takes a complex wobbling motion at equilibrium, precessing around the vorticity axis. As$\mathit{Ca}$is further increased, the capsule long axis oscillates about the shear plane, while the membrane rotates around a capsule cross-section that also oscillates over time (oscillating–swinging regime). The amplitude of the oscillations about the shear plane decreases as$\mathit{Ca}$increases and the capsule finally takes a swinging motion in the shear plane. It is found that the transitions from rolling to wobbling and from wobbling to oscillating–swinging depend on the mean energy stored in the membrane.

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Stable equilibrium configurations of an oblate capsule in simple shear flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.759 ◽

2016 ◽

Vol 791 ◽

pp. 738-757 ◽

Cited By ~ 13

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.

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Controlling rotation and migration of rings in a simple shear flow through geometric modifications

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.20 ◽

2018 ◽

Vol 840 ◽

pp. 379-407 ◽

Cited By ~ 2

Author(s):

Neeraj S. Borker ◽

Abraham D. Stroock ◽

Donald L. Koch

Keyword(s):

Shear Flow ◽

Cross Section ◽

Simple Shear ◽

Rigid Bodies ◽

Boundary Integral ◽

Simple Shear Flow ◽

Cross Sectional ◽

Aspect Ratios ◽

Gradient Direction ◽

Continuous Rotation

A ring with a cross-section that has a blunt inner and sharper outer edge can attain an equilibrium orientation in a Newtonian fluid subject to a low Reynolds number simple shear flow. This may be contrasted with the continuous rotation exhibited by most rigid bodies. Such rings align along an orientation when the rotation due to fluid vorticity balances the counter-rotation due to the extensional component of the simple shear flow. While the viscous stress on the particle tries to rotate it, the pressure can generate a counter-vorticity torque that aligns the particle. Using boundary integral computations, we demonstrate ways to effectively control this pressure by altering the geometry of the ring cross-section, thus leading to alignment at moderate particle aspect ratios. Aligning rings that lack fore–aft symmetry can migrate indefinitely along the gradient direction. This differs from the periodic spatial trajectories of fore–aft asymmetric axisymmetric particles that rotate in periodic orbits. The mechanism for migration of aligned rings along the gradient direction is elucidated in this work. The migration speed can be controlled by varying the cross-sectional shape and size of the ring. Our results provide new insights into controlling motion of individual particles and thereby open new pathways towards manipulating macroscopic properties of a suspension.

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Influence of internal viscosity on the large deformation and buckling of a spherical capsule in a simple shear flow

Journal of Fluid Mechanics ◽

10.1017/s0022112011000280 ◽

2011 ◽

Vol 672 ◽

pp. 477-486 ◽

Cited By ~ 49

Author(s):

É. FOESSEL ◽

J. WALTER ◽

A.-V. SALSAC ◽

D. BARTHÈS-BIESEL

Keyword(s):

Shear Flow ◽

Simple Shear ◽

Viscosity Ratio ◽

Fluid Motion ◽

Boundary Integral ◽

Boundary Integral Method ◽

Simple Shear Flow ◽

Flow Strength ◽

Capsule Deformation ◽

Internal Viscosity

The motion and deformation of a spherical elastic capsule freely suspended in a simple shear flow is studied numerically, focusing on the effect of the internal-to-external viscosity ratio. The three-dimensional fluid–structure interactions are modelled coupling a boundary integral method (for the internal and external fluid motion) with a finite element method (for the membrane deformation). For low viscosity ratios, the internal viscosity affect the capsule deformation. Conversely, for large viscosity ratios, the slowing effect of the internal motion lowers the overall capsule deformation; the deformation is asymptotically independent of the flow strength and membrane behaviour. An important result is that increasing the internal viscosity leads to membrane compression and possibly buckling. Above a critical value of the viscosity ratio, compression zones are found on the capsule membrane for all flow strengths. This shows that very viscous capsules tend to buckle easily.

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Shear flow over a liquid drop adhering to a solid surface

Journal of Fluid Mechanics ◽

10.1017/s0022112096000080 ◽

1996 ◽

Vol 307 ◽

pp. 167-190 ◽

Cited By ~ 39

Author(s):

Xiaofan Li ◽

C. Pozrikidis

Keyword(s):

Contact Angle ◽

Shear Flow ◽

Contact Line ◽

Simple Shear ◽

Liquid Drop ◽

Contact Angle Hysteresis ◽

Boundary Integral ◽

Simple Shear Flow ◽

Contact Lines ◽

Transient Deformation

The hydrostatic shape, transient deformation, and asymptotic shape of a small liquid drop with uniform surface tension adhering to a planar wall subject to an overpassing simple shear flow are studied under conditions of Stokes flow. The effects of gravity are considered to be negligible, and the contact line is assumed to have a stationary circular or elliptical shape. In the absence of shear flow, the drop assumes a hydrostatic shape with constant mean curvature. Families of hydrostatic shapes, parameterized by the drop volume and aspect ratio of the contact line, are computed using an iterative finite-difference method. The results illustrate the effect of the shape of the contact line on the distribution of the contact angle around the base, and are discussed with reference to contact-angle hysteresis and stability of stationary shapes. The transient deformation of a drop whose viscosity is equal to that of the ambient fluid, subject to a suddenly applied simple shear flow, is computed for a range of capillary numbers using a boundary-integral method that incorporates global parameterization of the interface and interfacial regriding at large deformations. Critical capillary numbers above which the drop exhibits continued deformation, or the contact angle increases beyond or decreases below the limits tolerated by contact angle hysteresis are established. It is shown that the geometry of the contact line plays an important role in the transient and asymptotic behaviour at long times, quantified in terms of the critical capillary numbers for continued elongation. Drops with elliptical contact lines are likely to dislodge or break off before drops with circular contact lines. The numerical results validate the assumptions of lubrication theory for flat drops, even in cases where the height of the drop is equal to one fifth the radius of the contact line.

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