Viscous fingering under elastic membranes

Viscous fingering in Hele-Shaw cells in which viscous fluid, contained in the narrow gap between two rigid plates, is displaced by a less viscous fluid is an archetype for front-propagating, pattern-forming phenomena: if the less viscous fluid is injected at a sufficiently fast rate so that viscous forces exceed surface tension forces, the axisymmetric interface between the two fluids is linearly unstable to non-axisymmetric perturbations. The nonlinear growth of this instability causes the development of distinct fingers whose tips become subject to a similar instability, leading to so-called tip-splitting. Repeated tip-splitting combined with the arrest of the interface after the passage of the finger tips ultimately creates a complex dendritic pattern as shown in Movie (a). Elastic deformations of the plates that bound the fluid can have a dramatic and unexpected effect on the onset and nonlinear development of this instability (see Movie (b)). In an elastic-walled Hele-Shaw cell where one of the bounding plates is replaced by a latex membrane, we find that the instability is suppressed and the interface remains axisymmetric (Movie (b)) for values of the injection rate at which the rigid system exhibits strongly nonlinear interfacial growth (Movie (a)). Moreover, the critical injection rate beyond which the axisymmetrically expanding interface becomes unstable to non-axisymmetric perturbations in the elastic-walled system (seeMovie (c)) is 100 times larger than the corresponding predicted by Paterson (JFM 113:513-529, 1981) for the rigid system.

(a) Rigid cell (b) Elastic cell (c) Fingering under elastic membrane at larger flow rate

A. Juel (2012) Flattened fingers. News & Views, Nature Physics, doi: 10.1038/nphys2408
- D. Pihler-Puzovic, P. Illien, M. Heil & A. Juel (2012) Suppression of viscous fingering by elastic membranes. Phys. Rev. Lett. 108, 074502, doi:10.1103/PhysRevLett.108.07450 (Editor's choice)

Bubble motion and displacement flows in complex tube geometries

A simple change in pore geometry can radically alter the behavior of a fluid-displacing air finger. In particular, partial occlusion of a rectangular cross-section can force a transition from a steadily propagating centered finger to a state that exhibits spatial oscillations formed by periodic sideways motion of the interface at a fixed distance behind the moving finger tip. After the passage of the finger, which continues to advance at a constant speed, the interface rapidly approaches a state of quasi-static equilibrium and the periodic deformations form a spatially periodic pattern that remains fixed in the laboratory frame. The existence of these novel propagation modes suggests that models based on over-simplification of the pore geometry will suppress fundamental physical behaviour present in practical applications, where pore geometry often contains many regions of local constriction, e.g., connecting or irregularly shaped pores in carbonate oil reservoirs, and airway collapse or mucus buildup in the lungs. Moreover, these modes offer further potential for geometry-induced manipulation of droplets for lab-on-the-chip applications, in which geometric variations have so far been restricted to the axial direction.

oscillations in constricted channels  

- M. Pailha, A.L. Hazel, P.A. Glendinning & A. Juel (2012) Oscillatory bubbles induced by geometrical constraint. Phys. Fluids 24, 021702; doi:10.1063/1.3682772.  (reprint)
- A. de Lózar, A. Heap, F. Box, A.L. Hazel & A. Juel (2009) Partially-occluded tubes can force switch-like transitions in the behavior of propagating bubbles. Phys. Fluids. 21, 101702; doi: 10.1063/1.3247879. (reprint)
- A.L Hazel, M. Pailha, S.J. Cox & A. Juel (2012) Multiple states of finger propagation in occluded tubes. Submitted to Phys. Fluids. (preprint)


Bubbles and capsules in confined geometries

Multiphase flows of practical interest are characterized by complex vessel geometries ranging from natural porous media to man-made lab-on-a-chip devices. Models based on the over-simplification of the pore geometry often suppress fundamental physical behavior. We study the effect on bubble motion of a sudden streamwise expansion of a square tube. The extent to which a bubble driven by constant flux flow broadens to partially fill the expansion depends on the balance between viscous and surface tension stresses, measured by the capillary number Ca. This broadening is accompanied by the slowing and momentary arrest of the bubble as Ca is reduced towards its critical value for trapping Cac. For Ca < Cac the pressure drag forces on the quasi-arrested bubble are insufficient to force the bubble out of the expansion so it remains trapped. We examine the conditions for trapping by varying bubble volume, flow rate of the carrier fluid, and length of expanded region, and find that Cac depends non-monotonically on the size of the bubble. Using a capillary static model we verify that a bubble is released if the work of the pressure forces over the length of the expansion exceeds the surface energy required for the trapped bubble to reenter the constricted square tube.

- G. Dawson, S. Lee & A. Juel (2012) The trapping and release of bubbles from a linear pore. Submitted to J. Fluid Mech. (preprint)

Benchtop model of airway reopening

Airway reopening is an important physiological event, as exemplified by the first breath of an infant that inflates highly collapsed airways by driving a finger of air through its fluid-filled lungs. Whereas fundamental models of airway reopening predict the steady propagation of only one type of bubble with a characteristic rounded tip, our experiments reveal a surprising selection of novel bubbles with counterintuitive shapes that reopen strongly collapsed, liquid-filled elastic tubes. Our multiple bubbles are associated with a discontinuous relationship between bubble pressure and speed.

Air finger travel from right to left

Top view


Side view

- A. Heap & A. Juel  (2009) Bubble transitions in elastic tubes.  J. Fluid  Mech.  633, 485-507 (reprint)
- A. Heap & A. Juel  (2008) Anomalous bubble propagation in elastic tubes. Phys. Fluids 20, 081702. (reprint)
(selected for the Virtual Journal of Biological Physics,
- A. Juel & A. Heap (2007) The reopening of a fluid-filled, collapsed tube.  J. Fluid Mech. 572, 287-310. (reprint)

Displacement flows in channels: from Bretherton flows to viscous fingering

Top view of steadily propagating air finger.

- A. de Lózar, A. Juel & A.L. Hazel (2008) The propagation of an air finger in a rectangular channel.  J. Fluid Mech. 614, 173-195. (reprint)
- A. de Lózar, A.L. Hazel & A. Juel  (2007) Scaling properties of coating flows in rectangular channels.  Phys. Rev Lett. 99 (23), 234501. (reprint)

Fluctuations in viscous fingering

Our experiments on viscous Saffman-Taylor fingering in very wide Hele-Shaw channels reveal finger width fluctuations that were not observed in previous experiments, which had lower aspect ratios and higher capillary numbers Ca. These fluctuations intermittently narrow the finger from its expected width. The magnitude of these fluctuations is described by a power law, Ca-0.64, which holds for all aspect ratios studied up to the onset of tip istabilities. Further, for large aspect ratios, the mean finger width exhibits a maximum as Ca is decreased instead of the predicted monotonic increase.



- M.G. Moore, A. Juel, J.M. Burgess, and H.L. Swinney (2002) Fluctuations in viscous fingering.  Phys. Rev. E 65, 030601(R).
- M.G. Moore, Anne Juel, J.M. Burgess, and H.L. Swinney (2003) Fluctuations and pinch-offs observed in viscous fingering.  7th Experimental Chaos Conference, AUG 26-29, 2002. Experimental Chaos, 189-194 (pdf preprint).

Oscillatory Kelvin-Helmholtz waves

Under construction...

- S.V. Jalikop & A. Juel (2012) Oscillatory transverse instability of interfacial waves in horizontally oscillating flows.   Phys. Fluids.  24, 044104 (2012); doi: 10.1063/1.4704602 (preprint)
- S.V. Jalikop & A. Juel  (2009) Capillary-gravity waves in oscillating two-layer flows.  J. Fluid  Mech. 640, 131-150.  (reprint)
- E. Talib & A. Juel (2007) Instability of a viscous interface under horizontal oscillation.  Phys. Fluids 19, 092102.   (reprint)
- E. Talib, S.V. Jalikop & A. Juel  (2007) The influence of viscosity on the frozen wave instability: theory and experiment.  J. Fluid Mech. 584, 45-68. (reprint)

Transition to turbulence in the Reynolds pipe

From O. Reynolds, Proc. R. Soc. London 35, 84 (1883).

Osborne Reynolds performed his pioneering experiments at the University of Manchester. His original experiment can be viewed upon request in the School of Engineering.

The puzzle of why the flow of a fluid along a pipe is typically observed to change from laminar to turbulent as the flow rate is increased is a well-known paradox of
hydrodynamic stability theory. The issue is both of deep scientific and engineering interest since most pipe flows are turbulent in practice even at modest flow rates. All
theoretical and numerical work indicates that the flow is linearly stable. It is natural to assume that finite amplitude perturbations are therefore responsible for triggering turbulence and these become more important as the nondimensionalized flow rate, the Reynolds number, Re, increases. We performed experiments to measure the threshold amplitude of turbulence-producing perturbations in pipe flow over approximately an order of magnitude in Re, and uncovered a novel scaling law, which indicates that the amplitude of perturbation required to cause transition scales asRe-1.

- B. Hof, A. Juel & T.  Mullin (2003) Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett.  91, 244502. 

- New experiments set the scale for the onset of turbulence in pipe flow, by R. Fitzgerald. Physics Today, February 2004, p. 21

Marangoni convection

Two superposed liquid layers display a variety of convective phenomena that are inaccessible in the traditional system where the upper layer is a gas. We consider several pairs of immiscible liquids. Once the liquids have been selected, the applied temperature difference and the depths of the layers are the only independent control parameters. Using a perfluorinated hydrocarbon and silicone oil system, we have made the first experimental observation of convection with the top plate hotter than the bottom plate. Since the system is stably stratified, this convective flow is solely due to thermocapillary forces.
Convection patterns heating from above (a) and heating from below (b).

- J.M. Burgess, A. Juel, J.B. Swift, W.D. McCormick, and H.L. Swinney (2001) Suppression of dripping from a ceiling.  Phys. Rev. Lett. 86, 1203-1206. 
- A. Juel, J.M. Burgess, W.D. McCormick, J.B. Swift, and H.L. Swinney (2000) Surface tension-driven convection patterns in two liquid layers.Physica D 143, 169-186. 

Work on the use of thermocapillary forces to suppress dripping from a ceiling received widespread attention with reviews in Nature Science Update,Science News  (vol.159, No. 1,p. 7, 2001), Physics Today  (February 2001) and in the AIP Physics News Update

Magnetohydrodynamic convection in molten gallium

Under construction...

D. Henry, A. Juel, H. Ben Hadid &  S. Kaddeche (2008) Directional effect of a magnetic field on oscillatory low  Prandtl-number-convection.  Phys. Fluids. 20034104. (reprint)
-  B. Hof, A. Juel & T. Mullin (2005) Onset of oscillatory MHD convection in molten gallium.  J. Fluid Mech. 545, 193-201.
- B. Hof, A. Juel, L. Zhao, H. Ben Hadid, D. Henry & T. Mullin (2004) Onset of oscillatory convection in molten gallium.  J. Fluid Mech. 515, 391-413.
- B. Hof, A. Juel & T. Mullin (2003) Magnetohydrodynamic damping of sidewall convection.  J. Fluid Mech. 482, 163-179. 
- A. Juel, T. Mullin, H. Ben Hadid, and D. Henry (2001) Three-dimensional convection in molten gallium.  J. Fluid Mech. 436, 267-281. 
-  A. Juel, T. Mullin, H. Ben Hadid, and D. Henry (1999) Magnetohydrodynamic convection in molten gallium. J. Fluid Mech. 378, 97-118. 
- M.G. Braunsfurth, A.C. Skeldon, A. Juel, T. Mullin, and D.S. Riley (1997) Free convection in liquid gallium. J. Fluid Mech. 342, 295-314. 


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