Scholarly article on topic 'Extreme Deformation of Capsules and Bubbles Flowing through a Localised Constriction'

Extreme Deformation of Capsules and Bubbles Flowing through a Localised Constriction Academic research paper on "Materials engineering"

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{"biological fluid dynamics" / capsule / bubble / "fluid-structure interaction" / wrinkling / "folding tip-streaming"}

Abstract of research paper on Materials engineering, author of scientific article — Geoffrey Dawson, Edgar Häner, Anne Juel

Abstract We explore experimentally the motion of deformable objects – bubbles and capsules – through narrow localised constrictions, with widths between 16 and 32% of the diameter of the object. Under constant volume-flux flow, both bubbles and capsules extend to a maximum length as their front passes through the constriction. Rapid contraction occurs as their rear accelerates towards the constriction, followed by relaxation upon exiting the constriction. We find that the large deformations imposed by narrowing constrictions and increasing flow rates highlight distinguishing features between bubbles and capsules, which reflect their distinct mechanics. These include tip-streaming of the rear of the bubble, buckling of the capsule upon contraction and wrinkling of the capsule membrane through excessive compressive strains. Finally, we present evidence of distinct modes of rupture for bubbles and capsule as the flow rate is further increased.

Academic research paper on topic "Extreme Deformation of Capsules and Bubbles Flowing through a Localised Constriction"

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Procedía

I UTA M

ELSEVIER

Procedía IUTAM 16 (2015) 22 - 32

www.elsevier.com/locate/procedia

IUTAM Symposium on Dynamics of Capsules, Vesicles and Cells in Flow

Extreme deformation of capsules and bubbles flowing through a

localised constriction

Geoffrey Dawson, Edgar Haner, Anne Juel*

Manchester Centre for Nonlinear Dynamics and School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK.

Abstract

We explore experimentally the motion of deformable objects - bubbles and capsules - through narrow localised constrictions, with widths between 16 and 32% of the diameter of the object. Under constant volume-flux flow, both bubbles and capsules extend to a maximum length as their front passes through the constriction. Rapid contraction occurs as their rear accelerates towards the constriction, followed by relaxation upon exiting the constriction. We find that the large deformations imposed by narrowing constrictions and increasing flow rates highlight distinguishing features between bubbles and capsules, which reflect their distinct mechanics. These include tip-streaming of the rear of the bubble, buckling of the capsule upon contraction and wrinkling of the capsule membrane through excessive compressive strains. Finally, we present evidence of distinct modes of rupture for bubbles and capsule as the flow rate is further increased.

©2015The Authors.Publishedby ElsevierB.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under the responsibility of the organizing committee of DYNACAPS 2014 (Dynamics of Capsules, Vesicles and Cells in Flow). Keywords: biological fluid dynamics, capsule, bubble, fluid-structure interaction, wrinkling, folding, tip-streaming

1. Introduction

The transportation of bubbles, drops and capsules (a thin elastic membrane enclosing a viscous fluid) by means of a flow in complex vessel geometries spans applications from lab-on-chip devices for biological analysis1,2 to cell and drug encapsulation for targeted delivery3,4. Capsules are fascinating objects in their own right, whose motion and deformation result from fluid-structure interaction. When a capsule is subjected to an external force its deformation depends on its shape, the mechanical properties of the membrane, internal fluid viscosity and the level of pre-inflation5. The capsule alters its shape when it is subject to hydrodynamic forces, and its altered shape in-turn modifies the flow field. Moreover, if the change in geometry is significant, the elasticity problem may become both geometrically and constitutively non-linear. By contrast, bubbles and droplets are simpler deformable objects, whose interface is characterised by a constant surface tension, and cannot support bending6.

* Corresponding author. Tel.: +44-161-2754053 ; fax: +44-161-2755819. E-mail address: anne.juel@manchester.ac.uk

2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under the responsibility of the organizing committee of DYNACAPS 2014 (Dynamics of Capsules, Vesicles and Cells in Flow). doi: 10.1016/j .piutam.2015.03.004

Fig. 1. Top-view of the experimental channel geometry.

The motion of bubbles and capsules in viscous channel flows is primarily governed by a capillary number, Ca, corresponding to the ratio of viscous to surface tension and elastic forces, respectively. As Ca increases, the liquid films between the channel wall and either the bubble interface or the capsule membrane, has been shown to increase monotonically in tubes of circular and square cross-section for bubbles7,8 and for capsules9,10,11,12. However, capsules also exhibit behaviour that distinguishes them from bubbles, such as the development of a steady concave parachute shape in tubes with a circular or rectangular cross-section13,14. Moreover, the presence of an elastic membrane also allows for solid mechanics instabilities, such as wrinkling to occur in shear flows15 or due to indentation16. The geometry of the tube has a dominant influence on the shape and motion of capsules. Variations of the tube geometry in the form of smooth or sudden constrictions may be used to determine the mechanical properties of capsules17, sort red blood cells by size18, or facilitate capsule rupture17. Bubbles passing though a constriction may break up if the film between the tube wall and the bubble is sufficiently thick or non-uniform so that surface-tension-driven instabilities can develop on the time scale of transit through the constriction. This typically occurs for high flow rates and small gap widths of the constriction19,20,21.

In this paper, we study the motion of capsules and bubbles through a localised constriction, and compare their behaviour qualitatively when they undergo large deformation. The study of capsules passing through narrow passages has been previously limited to constrictions longer than the capsule22,17. Recent boundary integral calculations by Park and Dimitrakopoulos 23 have compared the motion of capsules and viscous drops passing through a rectangular constriction, whose length and width are set between one and 2.5 times the capsule radius. They found that the response time for the capsules to react to changes in the flow increases with the ratio of the inner to the outer fluid viscosities. We investigate capsules passing through a localised constriction that is approximately 16% - 32% the equilibrium diameter of the capsule, so that the capsules undergo much larger amplitude deformations than previously addressed. We find that the overall deformation of bubbles and capsules in constrictions is similar and dominated by viscous forces, despite the distinct natures of these deformable objects. We highlight distinguishing features of capsules that arise under extreme deformation by reference to bubble deformation. A description of the experimental methods is given in section 2. The motion of capsules and bubbles through narrow constrictions is quantified in section 3, while distinguishing features are discussed in section 4. Conclusions are given in section 5.

2. Experimental methods

A top-view of the flow channel is shown in Fig. 1. The channel consisted of two float glass plates (width x

length = 100 x 600 mm), which were separated by two 70 x 600 mm steel gauge plates to form a rectangular tube

of height h = 3.04 mm and width W = 8.16 mm. A local axial variation in width was imposed by introducing two

identical pieces in mirror symmetry, which were attached to the gauge plates with double-sided adhesive tape. Each

piece consisted of a rectangular block extended by a triangular tooth with rounded ends (with a radius of curvature of

0.52 ± 0.07 mm, and a maximum length lb = 3.0 ± 0.2 mm) as shown in Fig. 1. The width of the constriction was

varied in the range 0.6 mm < wc < 1.2 mm by modifying the width of the rectangular block, wb. The teeth were

manufactured from epoxy putty Kneadite R (Polymeric Systems Inc). Four constrictions were positioned within the

length of the tube each separated by a spacing of 120 mm, resulting in a length of tube to width ratio of 15 between obstacles.

Both ends of the tube were connected to reservoirs filled with silicone oil (polydimethylsiloxane fluids from Basildon Chemicals, with dynamic viscosity p = 1.045 kg m-1 s-1, density p = 975 kg m-3, and surface tension < = 21.1 mN m-1, measured at the laboratory temperature of 21 ± 1°C). One reservoir was open to the atmosphere, while at the opposite end, the reservoir was sealed and connected to a glass/teflon syringe (gastight series, Hamilton) driven by a syringe pump (KDS210, KD scientific). This allowed the withdrawal of fluid at a constant volumetric flow rate, Q. Top-view visualisation was achieved with a CMOS camera (Dalsa Genie HM1400, 1400 x 1280 pixels, at frame rates of up to 64 fps), with optics that yielded a resolution of 68 pixels/mm. Edge detection of the outer boundary of the bubble/capsule (see Fig. 1) was performed with MATLAB routines to a resolution of ±0.5 pixels. However, edge detection of top-view images failed in situations where the capsule buckled inwards, and the location of the interface was then determined manually with an accuracy of ±3 pixels.

Capsules were manufactured using the method developed by Levy and Edwards-Levy24. Dropwise addition of a solution containing 0.4 g polyglycolic acid (PGA), 1.6 g chicken egg white (ovalbumin, 62-88%, agarose gel electrophoresis, Sigma), 0.3 g sodium alginate (Sigma) and 20 ml distilled water into a 10% solution of calcium chloride (anhydrous granular, < 7.0 mm, > 93.0%, Sigma) was followed by five minutes of agitation by stirring to create gelled beads. It was found through trial and error that a drop height of 4 cm led to the most spherical beads. Once formed, the beads were rinsed three times in a 9 g/l saline solution (NaCl) and then agitated in a 5 ml solution of 0.0045M NaOH (Sigma), to start a transacylation reaction to create a covalently linked membrane of ovalbumin and alginate through an amide linkage. After 5 minutes, the solution was neutralised by adding 0.0045M HCl (Sigma) and agitated for a further 5 minutes. The beads were then again rinsed three times in a 9 g/l saline solution. In order to re-liquify the core, the beads were added to a 10% solution of sodium citrate (> 99% FG, Sigma), and agitated for 10 minutes. The capsules were again rinsed three times in a 9 g/l saline solution, and stored in a fresh 9 g/l saline solution. As the process of liquefaction led to the partial deflation of the capsules, they were left in the solution for 12 hours to re-inflate before use. Sherwood et al25 showed that the volume of inner liquid is set by a Donnan equilibrium, which implies that the capsules can be overinflated but not underinflated. Internal liquid can be forced through the membrane during extreme deformation and this is regarded as failure of the capsule. Millimetric alginate capsules26 have been shown to be best described by Evans and Skalak's27 constitutive law through compression tests. Finally, the internal viscosity of the capsules is less than that of the surrounding fluid. Numerical simulations suggest that a maximum deformation is reached for ratios of internal to external viscosity n < 0.2, but this deformation is only 12 % larger than for n = 1 at a given stress 28.

A batch of approximately 50 capsules was produced for the experiments. The capsules used in the experiments had a diameter in the range 3.56 < D < 3.74 mm and a membrane thickness of 0.16 ± 0.02 mm (4.4% of D ). Hence, they were slightly compressed in the channel of depth h = 3.04 mm. A single air bubble was formed in the channel by injecting air with a syringe pump through a teflon tube, placed inside the main channel where a constant-flux flow was imposed. A large range of bubble sizes could be generated by varying the flow rate of injection relative to the background flow29. The use of air bubbles instead of drops allowed the experiments to be performed in the absence of surfactants in contrast to most microfluidic studies.

In all the experiments, the Reynolds number - the ratio of inertial to viscous forces - was small with Re = pUb/p < 0.02, where U is the speed of the bubble/capsule in a carrier fluid with dynamic viscosity p, and density p. Our bubbles exhibited buoyancy which is quantified by the Bond number - the ratio of buoyancy to surface tension forces - with a value of Bo = Apgh2/4< = 1.06, where Ap is the density difference between silicone oil and air and g the acceleration due to gravity. For this value, the bubble lifts off the bottom boundary of the channel30, so that the bubble may expand both vertically and in the plane of visualisation when subjected to viscous stresses. However, the density of the solution that was used to manufacture the capsule, p = 987 kg m-3, was similar to that of the silicone oil. Hence, the Bond number of the capsule was very small, Bo = 2.2 x 10-4, so that buoyancy effects were negligible. Hence, we compared capsules and bubbles that had the same diameter in top-view when flowing in the unconstricted part of the channel.

The motion of bubbles is governed primarily by the capillary number, Ca = pQ/(A<), which measures the ratio of viscous to surface tension forces, where A = Wh is the cross-section of the channel. For capsules, an alternative capillary number is defined as a measure of the ratio between viscous and elastic forces, Ca = pQ/(AGs), where Gs

1.6 1.4 1.2 1

0.8 0.6

L /L max 0

Increasing Q

Increasing 1 Q }

Fig. 2. Deformation of a bubble as it moves through a localised constriction of width w*c = 0.32 under constant volume-flux flow at Q = 40 cm3/s. The streamwise length of the bubble L/Lq is plotted as a function of the front tip position yC = yt/lb, where yC = 0 indicates the centre of the constriction. Top-view snapshots of the deformed bubble are shown for points (a) to (f) on the graph.

Fig. 3. Deformation of a capsule as it moves through a localised constriction of width wC = 0.32 under constant volume-flux flow at Q = 40 mm3/s. The streamwise length of the capsule L/Lq is plotted as a function of the front tip position yC = yt/lb, where yC = 0 indicates the centre of the constriction. Top-view snapshots of the deformed capsule are shown for points (a) to (f) on the graph.

is the surface shear modulus. The measurement of the shear modulus is a substantial undertaking, which we did not focus on in this study. Hence, we present all results in terms of the dimensional flow rate Q, which gives a measure of the viscous stresses exerted by the mean flow in the channel.

3. Motion and deformation of bubbles and capsules through a localised constriction

The motion and deformation of bubbles and capsules through a localised constriction is shown in Fig. 2 and 3, respectively. Prior to reaching the constriction, the bubble/capsule propagates steadily with a constant length L0 = 4.32 mm. We will refer hereafter to the scaled width of the constriction wC = wc/L0. As it passes through the constriction, the bubble/capsule experiences strong viscous shear forces that induce deformation. The shapes of both capsule and bubble are imposed primarily by the non-uniform pressure distribution in the carrier fluid, but the

(a) 2.5

! Front tip Back tip

at y=0 ! at y=0

(b) 2.5

Front tip} Back tip

at y=0 ! I at y=0

Fig. 4. Speed of front tip vf (black) and back tip vb (red) as a function of front tip position y* for Q = 6.67 mm3/s and w*c = 0.32: (a) bubble; (b) capsule. The speeds are rescaled with the average flow speed through the centre of the constriction Qwch, and every fourth data point is labelled with a cross +. The vertical dashed lines indicate where the tip and tail of the bubble/capsule pass through the narrowest part of the constriction. The maximum rescaled speeds of the tip and tail (at the same flow rate) are also shown for constriction widths of wC = 0.24 (o), wC = 0.22 (a) and wCC = 0.16(D).

mechanics of deformation of the thin shell of the capsule and the free surface of the bubble are distinct. In particular, the capsule membrane is subject to viscous stresses imposed by both the external and internal fluid, whereas the bubble has a uniform internal pressure resulting in a stress-free surface. Despite these differences, the plots of the bubble centreline length, L/L0, as a function of tip position shown in Fig. 2 and 3 indicate that the overall deformations of capsule and bubble are similar. Our measured deformation curves are also consistent with the recent boundary integral calculations of Park & Dimitrakopoulos23 in wide constrictions.

As it approaches the constriction, the bubble/capsule starts to extend due to the strong viscous shear forces acting on its tip, which arise due to the rapid narrowing of the tube, and leads to increased tip speed and curvature. By contrast, the tail of the bubble, which is further away from the constriction, retains an approximately constant curvature until the tip reaches the constriction. Once the front tip has passed the constriction, the bubble/capsulefront expands into the undeformed tube allowing it to reach a maximum length, Lmax/L0, before it starts to contract as its rear tip accelerates. In the range of flow rates studied, the position of the bubble tip when its maximum length is achieved, remains approximately constant (see Fig. 2), whereas the tip position of capsules under maximum extension migrates closer to the constriction as Q increases (see Fig. 3), which is linked to the distinct mechanics underlying deformation. A further feature that distinguishes the capsule from the bubble is the wrinkling and folding of its membrane as it squeezes through the localised constriction, which is discussed in section 4.3.

Also, while the bubble extends considerably more than the capsule, the capsule undergoes a more severe contraction as its rear passes through the constriction. Plots of the velocity of the front and back tips, calculated by finite-differencing time series of the tip positions, are shown in Fig. 4 as a function of front tip position. The maximum front tip speed associated with extension is lower than the maximum back tip speed that corresponds to contraction, and the difference between them increases with flow rate. However these maximum values do not vary significantly with constriction width, suggesting that they are governed by the large difference between the mean flow and local flow conditions in the constriction. The most significant qualitative differences between bubble and capsule deformation occur in the contraction phase. As flow rate is increased, the rear tip velocity of the bubble can diverge, as discussed in section 4.1, resulting in the rupture of the bubble. In cases where the rear tip velocity of the bubble remains finite, its decay is always monotonic by contrast with the non-monotonic decay of the rear tip speed of the capsule. Moreover, the local minimum of the rear tip speed of the capsule, which occurs at approximately yC = yt/lb = 1.5, is associated with a local maximum of the front tip speed. These features are indicative of the buckling of the capsule during contraction, which cannot be supported in the bubble, as discussed further in section 4.2.

The maximum streamwise length Lmax/L0 is shown in Fig. 5. It increases linearly with flow rate as shown for both bubbles (Fig. 5 (a)) and capsules (Fig. 5(b)). The data indicate that in the limit vanishing flow rate, the bubble/capsule is deformed significantly and its length Lmax/L0(Q = 0) decreases with increasing constriction width (Fig. 5(e)). When

0.1 0.2 0.3 w

Q (mm3/s)

Q (mm3/s)

Fig. 5. Variation of the maximum extension Lmax/L0 with flow rate Q for constriction widths wC = 0.32 (+), 0.24 (o), 0.22 (A) and 0.16 (□): (a) bubble; (b) capsule. The solid lines correspond to linear least-squares fits, and their slopes are plotted as a function of constriction width in the insets. Snapshots of a bubble and a capsule at the point of maximum extension are shown in (c) and (d) for a constriction width wC = 0.22 and flow rate Q = 40 mm3/s. (e) Maximum length Lmax/L0 at Q = 0 versus constriction width for bubbles (o) and capsules (★). The solid lines correspond to a linear least-squares fit.

the constriction width is reduced, the bubble/capsule has to reduce its curvature through the constriction and hence elongate in order to achieve a static equilibrium that conserves volume. The bubble deforms more notably because it can also expand vertically due to greater initial confinement imposed by buoyancy forces. In the static limit, however, the bubble is expected to be unstable because any difference in curvature on either side of the constriction will lead to a pressure difference, which will drive the bubble out of the constriction. For capsules, in the limit of Q = 0, the absence of a lubricating film would make the capsule stick to the tube wall.

For both bubble and capsule, the elongation beyond Lmax/L0(Q = 0) due to flow is proportional to Q and only weakly dependent on wC as indicated by the near parallel datasets for different values of wC shown in Figs. 5(a) and (b). This suggests that the elongation is dominated by the mean flow beyond the constriction, which stretches the protruding part of the deformable object, while its rear approaches the other side of the constriction. The weak linear decrease of the proportionality coefficient with wC, shown in the insets of Figs. 5(a) and (b), also indicates a small but measurable dependence on wC.

The larger extension of bubbles compared to capsules and the presence of thicker films between the bubble and the tube wall is illustrated in Fig. 5(c),(d). However, a quantitative comparison between bubble and capsule deformation is not possible because buoyancy forces lift the bubble off the bottom of the undeformed channel30, so that it can expand downwards as well as in the plane of view, when it travels through the constriction.

4. Distinguishing features

4.1. Bubble tail thinning

At the start of the contraction phase, both bubble and capsule tails accelerate as they approach the constriction. Bubbles differ from capsules during this phase, as bubbles exhibit a divergence in tail speed for sufficiently large flow

Fig. 6. (a) Variation of the rear tip speed with flow rate for w'c = 0.32. The solid line is a two-parameter hyperbolic fit of the form vb = a/(Q - Qc )2, with A = 1.02 ± 0.02 and Qc = (29.3 ± 0.1) mm3/s. (b) Top view images of thin tail formation for Q = 27.5 mm3/s with a time interval between images of 7.4 x 10-3 s (the cells on the tail are due to lighting effects). The snapshots illustrate the thinning of the tail and its absorption back into the bulk.

rates, which is associated with an apparently unbound thinning of the tail. By contrast, the tail of the capsule only narrowed to approximately 4 times the membrane thickness. This is because the elastic tensions in the membrane become very large when the membrane bends on the order of the membrane thickness.

The thinning of the bubble tail was recorded using a high frame rate (250 fps) camera for a bubble of size L0/W = 0.72. The speed of the rear tip is shown in Fig. 6(a) as a function of flow rate for a constriction of width w* = 0.22. At low flow rates, the tail curvature and speed remained relatively low. For Q > 19 mm3/s, a cusp formed through the rapid narrowing of the tail as it approached the constriction (see successive snapshots in Fig. 6(b)) . As Q increased further, a thin thread of fluid formed, and the tail speed increased sharply. A two-parameter hyperbolic fit of the form vb = a/(Q - Qc)2, where a and Qc are fitted parameters, applied to the experimental data, yields a critical flow rate Qc = (29.3 ± 0.1) mm3/s. Above Qc, the bubble broke up before the thin thread of fluid at the tail was formed. Similar behaviour was observed by Tsai and Miksis20 in boundary integral simulations of the breakup of drops driven by a pressure gradient through a constriction in a circular capillary. The observed tip formation also bears resemblance to tip-streaming in extensional flows31, although tip-streaming is dominated by surfactant dynamics, not present here, and occurs over longer time-scales.

4.2. Capsule buckling during contraction

The contraction phase of bubble and capsules differ in that the capsule buckles. When the tail of the bubble/capsule reached the constriction, the tail moved faster than the tip resulting in the rear of both capsule and bubble adopting a concave shape. The capsule contracted inwards resulting in the folding of the membrane, as illustrated in Fig. 7. This occurred because capsules cannot significantly reduce their surface area, by contrast with bubbles, and also because membrane thickness and resistance to stretching impose a minimum local curvature of the membrane. By contrast with the bubble, the capsule tail speed relaxed non-monotonically back to its mean flow value, as shown in Fig. 4. This indicates that the buckling of the capsule plays an important role in both the contraction and relaxation mechanics of the capsule. Time scale measurements of relaxation after the contraction phase were recently used to determine the mechanical properties of capsules17. However, the relaxation time scales could not be determined with confidence in this experiment, as the complex constriction geometry allowed insufficient control on the starting shape and position of the bubble/capsule as it began to relax.

Fig. 7. Top-view successive snapshots (with a time interval of 0.156 s, left to right and top to bottom) of the contraction phase for a capsule moving through a constriction of width wC = 0.22 with imposed flow rate Q = 23.3 mm3/s.

Fig. 8. Successive top-view snapshots (left to right) of a capsule wrinkling as it passes through a narrow constriction of width wC = 0.22 at Q = 40 mm3/s, taken with a time interval of 0.078 s.

4.3. Capsule wrinkling

Localised wrinkling and even folding of the central portion of the capsule was observed for Q > 20 mm3/s in one of the four constriction widths, w*c = 0.22. Buckling of thin-walled elastic sheets is typically associated with the presence of compressive in-plane stresses. Once these stresses exceed a certain threshold, it becomes energetically favourable for the membrane to adopt a buckled configuration because the increase in strain energy associated with bending becomes less than that associated with any further in-plane compression32.

Wrinkling only occurred fleetingly as the centre of mass of the capsule approximately reached the narrowest part of the constriction. Leyrat-Maurin & Barthes-Biesel33 calculated longitudinal and hoop extension ratios numerically for an infinitely thin hyperelastic capsule membrane with negligible bending stiffness, enclosing a Newtonian fluid, in a cylindrical tube with a constriction. The authors found that maximum negative hoop and positive longitudinal tensions occurred when the centre of mass of the capsule was in the narrowest part of the constriction, which is consistent with our observations.

In Fig. 8, a series of snapshots illustrate the wrinkling of the capsule as it passes through the constriction at Q = 40 mm3/s. Although it was difficult to quantify the observed wrinkling, the wrinkles became increasingly marked for increasing Q, suggesting an increase in amplitude, while their wavelength remained approximately constant, at A/L0 = 0.13, as shown in Fig. 9. Cerda and Mahadevan32 showed that the most stable wavelength results from a balance between increasing bending energy at short wavelengths and increasing the stretching energy at long wavelengths. They found that the wavelength is governed by the material properties, while the amplitude also depends on the imposed compressive strain. The snapshots also indicate that the capsule membrane exhibits localisation effects as the wrinkling amplitude increases, with a transition to localised folds34. Hence, capsules only wrinkled for one constriction width because for wC > 0.22, the compressive strain was not great enough. For wC = 0.16, the narrow gap width made it difficult to observe wrinkling, and moreover, only one wrinkle would form with the wavelength observed in the wider constriction.

Fig. 9. Top-view images of capsule wrinkling in the constriction of width w* = 0.22 for: (a) Q = 10 mm3/s; (b) Q = 20 mm3/s; (c) Q = 30 mm3/s; (d) Q = 40 mm3/s.

Fig. 10. Last snapshot before bubble breakup in a constriction of width w* = 0.16: (a) Q = 23.3 mm3/s; (b) Q = 30 mm3/s; (c) Q = 40 mm3/s.

Fig. 11. Top-view of ruptured capsules: (a) internal liquid has leaked through the elastic shell at the tip of the capsule; (b) catastrophic rupture.

4.4. Rupture

Bubble and capsule breakup occurs through distinct mechanics. Bubble breakup was observed as the bubble extended and narrowed when passing through the constriction for widths of w* = 0.22 at Q > 40 mm3/s and w* = 0.16 for Q > 20 mm3/s, as illustrated in Fig. 10. Bubbles remained intact for lower values of the flow rate, and within our range of flow rates when passing through wider constrictions. Breakup occurred nearer to the tip, the higher the flow rate and the smaller the gap width. Gauglitz and Radke19 studied bubble breakup through a cylindrical constriction experimentally in the limit of small Q and found that as the bubble stretches, the film between the bubble and the tube wall thickens. Breakup occurs when surface-tension-driven instabilities of the film grow on time scales comparable with the transit of the bubble through the constriction. Hence, snap-off events occur at relatively high Q and small gap widths, when the film is sufficiently thick and deformed20.

By contrast there was no precursor sign of rupture for capsules. Their deformation through the constriction did not deviate from that described in section 3 to within experimental resolution until the capsule visibly ruptured. We observed two modes of capsule rupture: (i) the capsule was deformed through the constriction to the extent that the rise in internal pressure of the fluid forced a small volume of fluid through the semi-permeable membrane at the tip of the capsule (Fig. 11(a)); (ii) the capsule ruptured catastrophically at relatively small deformations, suggesting that the capsule had a manufacturing defect (Fig. 11(b)).

5. Conclusion

We have explored the propagation in a constant volume-flux flow of deformable objects (bubbles and capsules) through narrow constrictions, with widths 0.16 < wc/L0 < 0.32. The dynamics are dominated by strong viscous shear forces that squeeze the bubble/capsule through the constriction, and hence, overall deformation is similar for both bubbles and capsules. However, the large deformations required for transit through the constriction are associated

with distinguishing features, which highlight differences in the mechanics of bubbles and capsules. As the flow rate increases, a divergence of the speed of the rear tip of the bubble, once its front has passed the constriction, is accompanied by unconstrained thinning of the bubble tail, similar to tip-streaming. By contrast, the rear curvature of the capsule always remains finite. The contraction phase that occurs once the front of the bubble/capsule has passed the constriction is more severe for capsules, which buckle inwards into a parachute shape, while the rear bubble interface adopts a less deformed concave configuration. Moreover, novel observations are reported of wrinkling and even localised folding of the membrane as the capsule passes through the constriction, which is promoted by the large compressive strains imposed by the geometry. Finally, at high flow rate, bubbles and capsules are shown to rupture through distinct mechanisms. A limitation of our study lies in its qualitative nature and work is currently underway to quantify the elastic properties of the capsules, which will allow deeper insight into the mechanisms of extreme capsule deformation.

Acknowledgements

The authors thank A-V. Salsac for helpful discussions.

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