Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Dufresne, Ariel
Master of Science - MSc
University of British Columbia
This thesis aims to explore and understand the Saffman-Taylor instability at the interface of a Hershel-Bulkley fluid and air. We describe the flow in a Hele-Shaw cell theoretically and experimentally. For our theoretical analysis, we follow the usual Hele-Shaw approximations for incompressible flows with a Herschel-Bulkley constitutive law. We extend this model by allowing the fluid to slip on the smooth walls of the cell. We conduct a linear stability analysis for both planar and circular geometries and predict that effective slip stabilizes the interface. We also use numerical computations to examine the non-linear regime of a single finger in the planar geometry. We find that slip generates wider fingers and makes the plugged-up area at the roots of the fingers smaller and appear at later times. We conduct experiments using corn syrup as a control case and aqueous suspensions of Carbopol as a model Herschel-Bulkley fluid. The fluids are pumped into a Hele-Shaw cell through a circular vent, forming an initial disk in the cell. Then, we either pump air into the center of the disk, creating an expanding annulus, or we withdraw the disk through the vent. In both cases, the interface is in an unstable Saffman-Taylor configuration causing fingers to appear. We look at the effect of the type of cell wall, the gap size, the flux and the size of the initial disk. The instability and the observed patterns are very different between a cell with smooth or rough walls, confirming our theoretical prediction on the influence of effective slip. The trends with the other parameter variations are in reasonable agreement with the theoretical predictions.
A study of the Saffman-Taylorinstability in viscoplastic fluidsbyAriel DufresneB. Sc., University of British Columbia, 2020a thesis submitted in partial fulfillmentof the requirements for the degree ofMaster of Scienceinthe faculty of graduate and postdoctoralstudies(Mathematics)The University of British Columbia(Vancouver)August 2022© Ariel Dufresne, 2022The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:A study of the Saffman-Taylor instability in viscoplastic fluidssubmitted by Ariel Dufresne in partial fulfillment of the requirements forthe degree of Master of Science in Mathematics.Examining Committee:Neil Balmforth, Professor, Mathematics, UBCSupervisorMark Martinez, Professor, Chemical and Biological Engineering, UBCSupervisory Committee MemberiiAbstractThis thesis aims to explore and understand the Saffman-Taylor instabilityat the interface of a Hershel-Bulkley fluid and air. We describe the flowin a Hele-Shaw cell theoretically and experimentally. For our theoreticalanalysis, we follow the usual Hele-Shaw approximations for incompressibleflows with a Herschel-Bulkley constitutive law. We extend this model byallowing the fluid to slip on the smooth walls of the cell. We conduct a linearstability analysis for both planar and circular geometries and predict thateffective slip stabilizes the interface. We also use numerical computationsto examine the non-linear regime of a single finger in the planar geometry.We find that slip generates wider fingers and makes the plugged-up areaat the roots of the fingers smaller and appear at later times. We conductexperiments using corn syrup as a control case and aqueous suspensions ofCarbopol as a model Herschel-Bulkley fluid. The fluids are pumped into aHele-Shaw cell through a circular vent, forming an initial disk in the cell.Then, we either pump air into the center of the disk, creating an expandingannulus, or we withdraw the disk through the vent. In both cases, theinterface is in an unstable Saffman-Taylor configuration causing fingers toappear. We look at the effect of the type of cell wall, the gap size, the fluxand the size of the initial disk. The instability and the observed patternsare very different between a cell with smooth or rough walls, confirming ourtheoretical prediction on the influence of effective slip. The trends with theother parameter variations are in reasonable agreement with the theoreticalpredictions.iiiLay SummaryFluid mechanics is the branch of science that studies the movement of liquidsand gases. Fluid mechanics explains the flight of airplanes, the flow of lavaand avalanches, the currents of air and ocean around the earth and allother flows. However, there is still much we do not understand about thebehaviour of fluids. This is where the language of mathematics can helpus. This thesis aims to understand a specific phenomenon that occurs atthe intersection between two different fluids that do not mix. When oneof the fluids is pushing the other, the contact surface between the fluidscan sometimes bend and deform instead of remaining flat, referred to as an“unstable interface”. In this study, we look at the famous Saffman-Taylorinstability and try to understand the mathematics that model it and howdifferent parameters influence it.ivPrefaceThis manuscript follows the work published in two articles, of which I amone of the authors.A version of Chapter 2 excluding Section 2.3 is taken with permissionfrom “Viscoplastic Saffman-Taylor fingers”, by A. P. Dufresne, T. V. Balland N. J. Balmforth, currently under review for publication. Section 2.3is adapted from work that appears in “Viscoplastic fingers and fracturesin a Hele-Shaw cell”, by T. V. Ball, N. J. Balmforth and A. P. Dufresne,published in 2021 in the Journal of Non-Newtonian Fluid Mechanics [5].All of the experiments explored in this manuscript as well as in the twopapers mentioned above were done by Dr. Thomasina Ball and myself, underthe supervision and guidance of Dr. Neil Balmforth in the Complex FluidLaboratory at the University of British Columbia, Point Grey Campus.Portions of the introductory text are also modified from previously writ-ten material in my Honour’s thesis entitled “Instability of the interface be-tween a displacing Herschel-Bulkley fluid and displaced water in a Hele-Shawcell” (2020) completed at the University of British Columbia.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Hele-Shaw Flows . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Characterisation of fluids . . . . . . . . . . . . . . . . . . . . 42 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Reduced model . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 General formulation . . . . . . . . . . . . . . . . . . . 72.1.2 Dimensionless equations . . . . . . . . . . . . . . . . . 92.1.3 The flow pattern . . . . . . . . . . . . . . . . . . . . . 112.1.4 The flux-gradient relation . . . . . . . . . . . . . . . . 142.1.5 Boundary conditions . . . . . . . . . . . . . . . . . . . 172.2 Planar fingers . . . . . . . . . . . . . . . . . . . . . . . . . . . 18vi2.2.1 Linear instability . . . . . . . . . . . . . . . . . . . . . 192.2.2 Nonlinear fingers . . . . . . . . . . . . . . . . . . . . . 202.3 Axisymmetrical flow . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Linear stability . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Expanding viscoplastic annulus . . . . . . . . . . . . . 292.3.3 Extraction of a viscoplastic disk . . . . . . . . . . . . 302.3.4 Limitations of the model . . . . . . . . . . . . . . . . . 313 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Control Experiments . . . . . . . . . . . . . . . . . . . . . . . 343.2.1 Newtonian Case . . . . . . . . . . . . . . . . . . . . . 353.2.2 Miscible Case . . . . . . . . . . . . . . . . . . . . . . . 413.3 Viscoplastic Saffman-Taylor: The impact of slip . . . . . . . . 453.4 Changing gap thickness, flux and size of initial disk . . . . . . 473.5 Comparison with linear theory . . . . . . . . . . . . . . . . . 494 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59viiList of TablesTable 3.1 Viscosity and density of the Newtonian fluids used in ourexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . 34viiiList of FiguresFigure 1.1 Plot of the one-dimensional constitutive laws of differentfluids, showing the relationships between stress and rateof strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 2.1 The geometry of the three displacement problems consid-ered in the present paper. The inset images in (b) and(c) show photographs (from above) of experiments with ablue-coloured Carbopol suspension, as outlined in Chap-ter 3; the vent is visible as the central dark circle (with aradius of 2.75mm for rough plates and 3.8mm for smoothplates), and the unstable interface, as identified thoughimage processing, is shown by the red dotted line . . . . . 8ixFigure 2.2 The flux gradient relation in Equation 2.35 illustrated forthe parameter settings indicated and experiments with(a) expanding or (b) extracted axisymmetrical flow froma source or sink at the centre of the cell. The red andblue curves indicate the cases without (Σ = 0) or with(Σ > 0) wall slip, respectively. The switches at S = Biand S = Bw are indicated. The thicker sections of thecurves indicate the excursion taken by fluid expandingor retreating axisymmetrically; in each case, in the ex-periments an interface is positioned initially at a radiallocation corresponding to Q = 1. On the right, we showthe corresponding plots of the quantity S√Q(S)Q′(S),which controls the basic linear instability (§2.2). . . . . . 16Figure 2.3 Numerical solution for a retreating, planar interface ofNewtonian fluid (n= 1, Bi = Σ = 0; σ= 0.5). In panel (a),we display three snapshots of the flux Q(x,y, t), togetherwith a succession of the equally spaced previous snapshotsof the retreating interface x= Xˆ(y,t) (equally spaced by0.5). The panels on the right show time series of (b) 〈Xˆ2〉(angular brackets denoting a spatial average) and (c) themaximum (blue) and minimum (red) interface positions.In (b), the dashed lines show the expected linear growthand the stars indicate the times of the snapshots of Qshown in (a). . . . . . . . . . . . . . . . . . . . . . . . . . 22xFigure 2.4 Numerical solutions for a retreating interface of non-slippingBingham fluid (n= 1, Bi = 2, Σ = 0; σ = 1). Three snap-shots of Q(x,y, t) are again displayed in (a), along withprevious interface positions (equally spaced by 0.25). Thedotted (green) contours show the yield surfaces. The pan-els on the right show time series of (b) 〈Xˆ2〉 and (c) themaximum (blue) and minimum (red) interface positions.In (b), the dashed lines show the expected linear growthand the stars indicate the times of the snapshots of Qshown in (a). In (c), the shaded region shows the extentof the growing plug along y = 0 and 2pi. . . . . . . . . . . 23Figure 2.5 Numerical solutions for a retreating interface of non-slippingBingham fluid (n = 1, Σ = 0; σ = 1) with varying Bi (asindicated). Snapshots of the interface in the frame of ref-erence of the finger tip are shown in the top row (i.e. inthe frame with origin at (x,y) = (Xtip,pi) where Xtip =Min(X)). The snapshots are roughly equally spaced byintervals equal to a fiftieth of the final time in each compu-tation. The lower panel shows the times series of Min(X)and Max(X) for all the solutions. The lower left insetshows the profiles of the all the finger tips, rescaled by thefinger width ∆, defined as the vertical distance betweenthe final yield points on the interface. The right-handinsets show the final speed of the fingertips vtip, then itsinverse and ∆/pi against Bi. . . . . . . . . . . . . . . . . 24xiFigure 2.6 Numerical solutions for a retreating interface of slippingBingham fluid (n = 1, Bi = 2) with (Σ,σ) = (12 ,25) (top)and (Σ,σ) = (34 ,23) (bottom); Bw =15Bi. In (a), the finalsolution for Q(x,y, t) is plotted along with a succession ofprevious interface positions (equally spaced by 0.25). Thedashed (green) contours show the internal yield surfaceswhere S = Bi; the (red) dotted lines show the contoursQ = 0.1. The panels on the right show time series of (b)〈Xˆ2〉 and (c) the maximum (blue) and minimum (red)interface positions. In (b), the dashed lines show the ex-pected linear growth. In (c), the solution from Figure2.4 is shown by the dash-dotted lines. The stars indicatewhere the pressure gradient S reaches Bi; the small circlesindicate where the cell first becomes blocked (S reachesBw for Σ> 0, or Bi for Σ = 0). . . . . . . . . . . . . . . . 25Figure 3.1 Schematic of the experimental setup. A horizontal Hele-Shaw cell of thickness H, filled with ambient air. A fluidis then pumped into the center of the cell from below andspread radially outwards. The experimental pictures weretaken from above. . . . . . . . . . . . . . . . . . . . . . . 33Figure 3.2 The flow curves for the three aqueous suspension of Car-bopol Ultrez 21, with a concentration C ≈ 0.13% usedin our experiments. The data was collected using a con-trolled shear-rate ramp on a Kinexus Malvern rheometer.The ramp up and down data are denoted by red and bluemarkers respectively. The solid black curves show theHerschel-Bulkley fit for the data on the down ramp. . . . 35Figure 3.3 Pictures of an expanding annulus of corn syrup (yellow),in a cell with gapH = 3.3mm and fluxQ= 59ml/min. Theinterface between the syrup and the air is highlighted inblack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36xiiFigure 3.4 Steps taken to process the images from the experiments,and pick out the interface. (a) is the original photo. (b)is the red color channel minus the blue color channel. (c)is the binary image of (b). (d) is the boundary capturedfrom the binary image. (e) is the edge displayed on topof the original photo. . . . . . . . . . . . . . . . . . . . . . 37Figure 3.5 Snapshots of expanding interfaces for experiments withcorn syrup, where Q= 59ml/min, H= 3.3mm and Vinit =50ml. The interfaces are colour coded in time from redto blue. The dotted black lines show the positions ofsmall local protrusions in gap thickness arranged to excitemodes with m = 3 to 9. The top-left snapshots of theinterface show an experiment with no such protrusions. . 38Figure 3.6 First 20 angular modes of the Fourier transform of theexpanding interface of corn syrup, where Q = 59ml/min,H = 3.3mm and Vinit = 50ml, with no protrusion, colourcoded in time from red to blue. . . . . . . . . . . . . . . . 40Figure 3.7 Two different diagnostics explored to quantified the strengthof the instability of the interface between air and cornsyrup. The black curves show the scaled arclength definedin Equation 3.4 for all eight annulus experiments shownin Figure 3.3. The blue curves correspond to the stan-dard deviation of the scaled radius SR defined in Equation3.3, for the seven corn syrup annuli perturbed with stickytape. The red curve is the one unperturbed experiment. . 42xiiiFigure 3.8 Mode amplitude plots of the syrup experiments shown inFigure 3.5. The thicker line shows the initially excitedmode, and the dashed lines present the results of linearstability analysis, both scaled to align the data for someintermediate volume. The lower right panel shows an ex-periment in which there were no initially excited mode.This panel shows the diagnostic SR(t) in Equation (3.3),again scaled to an intermediate volume. The inset at thebottom right shows the initial mode amplitudes plottedagainst m for the first snapshot of each test; the dashedline indicates m− 32 , the initial distribution that is usedto compute the theoretical version of SR(t) immediatelyabove. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 3.9 Snapshots of expanding interfaces from one immiscibleexperiment with air and corn syrup (top row) and a mis-cible one with water (dyed in blue) and syrup (bottomrow), where Q= 59ml/min, H= 3.3min and Vinit = 50ml.The scale bar of 1.32 cm represents the predicted widthof fully developed non-linear miscible fingers by Paterson[36]. Because the immiscible fingers are less numerous andwider that the miscible ones, we conclude that our gap islarge enough to not fully control the pattern formation. . 44Figure 3.10 Repeated tests for expanding annuli in cells with eitherrough or smooth plates, plotting the standard deviationof the radius SR =√〈(R/Rav−1)2〉 against the volume Vof air in the cell. For the rough cell, the test is repeatedseven times; for the smooth cell, the test is repeated fourtimes. The fainter lines show the individual tests; theaverage, with errorbars based on the standard deviation,are shown with darker lines. For both cells, outlines of theinterface are shown for four sample tests. Q= 50ml/min,H = 1.7mm, Vinit = 50ml. . . . . . . . . . . . . . . . . . . 46xivFigure 3.11 A similar plot to Figure 3.10, but for extractions of a disk,and for the volume V of Carbopol in the cell. The testsin both rough and smooth cells are repeated seven times.Q= 50ml/min, H = 1.7mm, Vinit = 50ml. . . . . . . . . . 46Figure 3.12 Time series of the local, radial maxima and minima (thefinger tips and roots) for the tests of Figures 3.10 and3.11, with the results for rough plates annuli in (a), forsmooth plates annuli in (b), for rough plates extractionin (c) and for smooth plates extraction in (c). The fainterdotted lines and symbols show all the extrema (above athreshold level exceeding the noise in the original images);the lighter solid lines show the global maxima and min-ima. The darker solid lines show the averages over all therepeated tests, and the black dashed line is Rav(t). . . . . 48Figure 3.13 Expanding annuli in roughened cells with different ini-tial volumes Vinit (as indicated), showing the standarddeviation of the radius SR against the volume V of airin the cell, plus sample outlines of the interface. Eachtest is conducted at least three times, except for the testwith the largest initial annulus, which was conducted onlyonce. The inset shows corresponding predictions of thelinear stability theory, assuming an initial spectrum withRˆm(0) = m−32 . The most amplified mode has wavenum-ber m= 8 or 9. Parameter settings: H = 1.7mm, with Qincreasing monotonically from 41ml/min for the smallestVinit up to 68ml/min for the largest initial volume. . . . . 49xvFigure 3.14 Expanding annuli with different (a,c) gap size (Q= 53ml/minin (a), 45ml/min in (c); Vinit/H = 294cm2), and (b,d)flux (H = 1.7mm, Vinit = 50ml), as indicated. For (a,b)the plates are roughened; in (c,d), the plates are smooth.The diagnostic SR(t) is plotted against area, V/H, in (a,c)and volume, V , in (b,d). The insets show the predictionsof linear stability theory for the same gaps and fluxes(holding all other parameters fixed in each case), assum-ing an initial spectrum with Rˆm(0) =m−32 . The numbersindicate the angular wavenumber m of the most amplifiedmode at the end of each time series. . . . . . . . . . . . . 50Figure 3.15 Fourier mode amplitudes observed for the first snapshotsof the tests shown in Figures 3.10 (red) and 3.11 (blue),for both rough and smooth plates. The modes are scaledby the power in modes 5<m< 33, and the dashed linesshows m− 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.16 Linear stability predictions for the amplification factors,|Rˆm|/R0(t), of the angular mode m when the volumewithin the unstable interface is V . Panels (a,b) corre-spond to the annulus experiments of Figure 3.10, withthe tests for roughened cells shown in (a) and for thesmooth cells in (b). Shown are the amplitudes, |am(t)|,of the Fourier modes of R(s, t), normalized by their ini-tial values and plotted as a density on the (V,m)−planewith the colour map indicated. The black lines show tenequally spaced contours of constant |am(t)| from the linearstability analysis; the dotted lines extend these contoursoutside the color scale used for the experiments (with thesame spacing). . . . . . . . . . . . . . . . . . . . . . . . . 52xviFigure 3.17 Observed and predicted mode amplitudes |Rˆm|/R0, forthe angular wavenumbers m indicated (colour coded fromred to blue for increasing m). The experiments corre-spond to those in Figures 3.10 and 3.11, and the panelsare laid out as in Figure 3.16. The solid lines and errorbars show the average, plus or minus the standard devi-ation, of the repeated tests; in (a), the dotted lines showthe results for each test. The dashed lines are the theo-retical predictions, scaled to match the observed averagesat a particular volume. . . . . . . . . . . . . . . . . . . . 53xviiAcknowledgmentsI would like to express my sincere gratitude and appreciation to those whoassisted me during this master’s program.First, Dr. Neil Balmforth for serving as my advisor since 2019. Neiloffered me the opportunity to work on this project, allowing me to learnabout the field of fluid mechanics and the power of applied mathematicaltechniques to understand some of the complexities of our world. Without hisguidance, his willingness to teach me, his faith in me and most importantlyour multiple conversations at the coffee shop, the completion of this thesiswould have been impossible.Dr. Thomasina Ball for her advice, patience and immense help in thelaboratory as well as with the writing of this thesis.To all of the professors throughout my time at UBC for teaching me somuch and inspiring me to want to learn more.I would like to dearly thank my parents, sisters and my partner, Myriam,for their unwavering support and unconditional love. Likewise, to all myfriends for their encouragement and kindness.Finally, financial support from the Natural Sciences and EngineeringResearch Council of Canada through their Canada Graduate Scholarships –Master’s program, as well as Dr. Neil Balmforth through his research grantare gratefully acknowledged.xviiiChapter 1IntroductionThe displacement of a fluid by another fluid occurs in a wide range of sit-uations, from industrial settings to geophysical phenomena. For example,during petroleum recovery, water and polymer solutions are injected intooil reservoirs and push the oil out to extract it. It also occurs when iceshelves displace water, when lava displaces air as it flows, and during cloudformation in the atmosphere. In these flow circumstances when there existsan interface, a line or surface of contact between two different fluids, hy-drodynamic instabilities often occur. These instabilities disrupt the smoothinterface and can create fascinating patterns are are still at the forefront ofresearch in the field of fluid mechanics. The Kelvin-Helmholtz instability,the Rayleigh–Taylor instability and the Saffman-Taylor are some famous ex-amples, to name only a few. This thesis focuses on exploring and gainingsome understanding of the Saffman-Taylor instability in viscoplastic Hele-Shaw flows through both theoretical analysis and experimental examination.For this research, a stable interface means a straight line between the dis-placed and displacing fluid in the planar geometry and an smooth disk inthe axisymmetric geometry.This phenomenon was first described mathematically in Newtonian fluidsby P. G. Saffman and G. I. Taylor in their paper published in 1958 [40]. Itwas explored further in a plethora of other articles by Saffman [39], Paterson[35, 36], Wilson [44] and many others. They established that this instability1occurs in a thin slot, called a Hele-Shaw cell, when a less viscous fluiddisplaces a more viscous one. This disturbance of the interface leads tothe formation of labyrinthian fingering patterns. These patterns have beenshown to also appear in viscoplastic fluids, the yield stress being unable todampen the instability [5, 11, 14, 18, 19, 21–23, 28, 32]. In this thesis, weconcentrate on the latter problem.This problem has been studied previously both experimentally and the-oretically [12, 26]. In these studies, it has always been assumed that vis-coplastic fluid does not slip on the walls of the Hele-Shaw cell however, manycomplex fluids do suffer from effective slip on smooth walls [10]. While con-ducting experiments with Carbopol on both smooth and roughen plates, weobserved a major impact on the resulting instability. We then hypothesisedit was the result of effective slip. The goal of this thesis is then to model theslip of viscoplastic fluids in a Hele-Shaw cell and how exactly it impacts theSaffman-Taylor instability. We follow a recently published stick-slip modelthat has been calibrated experimentally by Daneshi et al. [16].Outline of the thesisThe end of the introductory chapter aims to give some necessary backgroundinformation helpful to understanding the rest of the paper. In Section 1.1,we set the stage for the theoretical section of this thesis by introducingthe governing equations for a flow in a Hele-Shaw cell. In Section 1.2 weprovide a short overview of some models that are used to characterise fluids,in particular the Herschel-Bulkley model for viscoplastic fluids, which waslater used in our theoretical analysis.In Chapter 2, we lay out the theoretical model for a viscoplastic Hele-Shaw flow in detail. We present the relevant scales and the following dimen-sionless equations. Then we provide some details of the behaviour of theflow in the sheared regions, the pseudo-plug region and the position of thefake yield surfaces. We follow by explicitly writing the relationship betweenthe flux and the pressure gradient and we show the impact of adding slip inthe model on the planar linear instability prediction. The planar stability2analysis is done in Section 2.2. We derive the most unstable wavelengthand its growth rate predicted by the linear stability analysis. We then pushour model a little further and use a numerical scheme to predict the non-linear evolution of a single finger in a cell with or without slip. The linearstability analysis is also done in a circular geometry, for both an expandingviscoplastic annulus, where the interface is under tension and an extractedviscoplastic disk, where the interface is now under compression forces.Finally, in Chapter 3, we present our experimental exploration of theSaffman-Taylor instability. We first looked at an expanding annulus of cornsyrup, as a control experiment using a Newtonian fluid. Then we presentmultiple suites of experiments using a Carbopol solution as our model vis-coplastic fluid. We analyse the impact of slip in both the annulus expansionand disk extraction configurations. For the annulus experiment, we alsolooked at the impact of different parameter settings such as the size of theinitial disk, the gap thickness and the flux of the air pumped into the cell. Inthe final section of this chapter, we compare our experimental results withthe linear stability theory presented in Chapter 2.The remainder of the thesis is a detailed summary of the results of thisthesis in Chapter 4.1.1 Hele-Shaw FlowsLet us now understand the equations of motion governing the flow of fluidsbetween two parallel flat plates separated by a small gap of thickness H.Figures 2.1 and 3.1 show sketches of the geometry of the flow and the ex-perimental setup. This configuration is called a Hele-Shaw cell, named afterHenry Selby Hele-Shaw who first used it in 1898 [25].We start with the full Navier-Stokes equations in vector form, assumingthat the fluid is incompressible,∇ ·u= 0, (1.1)∂∂t(ρu) +∇ ·(ρuuT)=−∇p+∇ ·τ +F, (1.2)3where ρ is the density of the fluid, u = (u,v,w) is the velocity field, p isthe pressure, τ is the deviatoric stress tensor and F is the sum of all bodyforces acting on the fluid. Equations (1.1) and (1.2) represent conservationof mass and conservation of momentum respectively. We then ignore anybody forces F such as gravity, and as the flow is slow enough, we can alsoignore all the inertia terms. We also align the Cartesian coordinate systemwith the mid-plane which lies along z = 0, with the z-axis perpendicular tothe mid-plane. Then Equations (1.1) and (1.2) become∇ ·u= 0, (1.3)∇ ·τ =∇p. (1.4)Note that these governing equations for Hele-Shaw flows are identical tothose of inviscid potential flows and to the flow of fluids though a porousmedium, known as Darcy’s Law.1.2 Characterisation of fluidsFor Newtonian fluids τ = µγ˙, where µ is the fluid viscosity and γ˙ is therate of strain. Replacing this into Equation 1.2 would lead to the familiarµ∇2u term in the conservation of momentum Navier-Stokes equation. Thisrelationship between the deviatoric stress and the rate of strain is referredto as the constitutive law of the fluid. For Newtonian fluids it is a linearrelationship, as shown in Figure 1.1, where the slope is defined as the viscos-ity of the fluid. Common examples of Newtonian fluids are air, water andsyrups. In its more general form f(τ , γ˙) = 0, the constitutive law can alsocapture the behavior of non-Newtonian fluids. For example a cornstarchsolution is considered to be a shear thickening fluid, which means that theeffective viscosity of the fluid increases with the shear rate, or deformationrate. Ketchup on the other hand is considered shear thinning, so its ef-fective viscosity decreases with shear rate. The effective viscosity is thenµ = Kγ˙n−1, where K and n are material parameters. These examples arealso shown in Figure 1.14There exist another category of fluids known as yield stress or viscoplasticfluids. These fluids behave as solids and do not flow until the deviatoricstress breaches some threshold value defined as the yield stress. Once thatcritical stress value is breached the fluid flows, and its effective viscositycan then increase linearly or not as shown in Figure 1.1. Some examples ofviscoplastic fluids are toothpaste, mayonnaise and most polymer solutionsincluding Carbopol. The general constitutive law for yield stress fluids, iscalled the Herschel-Bulkley law:γ˙ = 0, if τ < τY , (1.5)τ =(Kγ˙n−1 + τYγ˙)γ˙, otherwise, (1.6)where τ is the second invariant of the stress tensor and τY is the yield stress,both defined explicitly in Equation (2.5).Note that this law also describes Bingham plastics, when n = 1 andτY 6= 0, power law fluids when n 6= 1 and τY = 0 and Newtonian fluids whenn= 1 and τY = 0.There exist multiple other types of fluid such as viscoelastic fluids butthere are not within the scope of this thesis. For our experiments in Chap-ter 3, we used corn syrup as a model Newtonian fluid, and Carbopol as ashear thinning viscoplastic fluid.5Figure 1.1: Plot of the one-dimensional constitutive laws of differ-ent fluids, showing the relationships between stress and rateof strain.6Chapter 2Theory2.1 Reduced model2.1.1 General formulationAs mentioned in Sections 1.1 and 1.2, we consider a two-dimensional, incom-pressible Herschel-Bulkley fluid flowing between two plates with constantseparation, H. We align a Cartesian coordinate system with the mid-plane,so that the z-axis is perpendicular to the midplane, which lies along z = 0(see Figure 2.1). The fluid has velocity u= (u,v,w), pressure p and devia-toric stress tensor τ . Ignoring inertia and any body forces such as gravity,conservation of mass and momentum demand that∇ ·u= 0, (2.1)∇ ·τ =∇p. (2.2)Furthermore, the Herschel-Bulkley constitutive law isγ˙ = 0, if τ < τY , (2.3)τ =(Kγ˙n−1 + τYγ˙)γ˙, otherwise, (2.4)7whereγ˙ij =∂ui∂xj+ ∂uj∂xi, γ˙ =√12∑i,jγ˙2ij , τ =√12∑i,jτ2ij , (2.5)and the material parameters are the consistency K, the power-law index nand the yield stress τY .Figure 2.1: The geometry of the three displacement problems consid-ered in the present paper. The inset images in (b) and (c) showphotographs (from above) of experiments with a blue-colouredCarbopol suspension, as outlined in Chapter 3; the vent is visi-ble as the central dark circle (with a radius of 2.75mm for roughplates and 3.8mm for smooth plates), and the unstable inter-face, as identified though image processing, is shown by the reddotted line8Along the midplane of the slot, symmetry demands thatw = τxz = τyz = 0 on z = 0. (2.6)At the walls, z = ±12H, we allow the fluid to slip. In particular, we adoptthe stick-slip model proposed by Daneshi et al. [16],(uv)=(00), if√τxz2 + τyz2 < τW ,(τxzτyz)=∓(A+ τW√u2 +v2)(uv), otherwise,at z =±12H, (2.7)where A and τW denote material parameters describing the interaction be-tween the fluid and walls.2.1.2 Dimensionless equationsFor a narrow slot, the gap H is much less than the characteristic lengthscale L for variations in the slot’s plane, i.e.: = H2L 1. In this setting, thetransverse velocity w must be smaller than the velocity components alongthe slot, u and v, by a factor of O(). Similarly, the pressure gradientsdriving flow must overcome the resistance stemming from the shear stressesacross the slot, implying that p is larger than τxz and τyz, by a factor ofO(−1). To bring out such features of the problem, we remove dimensionsfrom the equations by setting(x,y) = L(xˆ, yˆ), z = 12Hzˆ, (u,v) = U(uˆ, vˆ),w = U wˆ, t= LU tˆ, p= −1S pˆ, τ = S τˆ ,(2.8)where we connect the characteristics scales for stress and velocity, S and U ,using a characteristic viscosity µ∗:S = 2µ∗UH , µ∗ =K(2UH)n−1. (2.9)9For the moment, we leave open the choice of the scales L and U , and estab-lish the scaling of time t when we consider the boundary conditions (which iswhere the dynamics enters this Stokes-flow-type problem). After droppingthe hat decoration identifying the dimensionless variables, the leading-orderexpression of force balance becomes∂p∂x= ∂τxz∂z+O(),∂p∂y= ∂τyz∂z+O(),∂p∂z=O().(2.10)Evidently, (τxzτyz)= z∇p+O(). (2.11)The yielded part of the constitutive law then becomesτ =(γ˙n−1 + Biγ˙)2ux (uy +vx) uz + 2wx(uy +vx) 2vy vz + 2wyuz + 2wx vz + 2wy −(ux+vy) , for τ ≥ Bi,(2.12)where the dimensionless yield stress, or Bingham number, isBi = τYS =HτY2µ∗U , (2.13)and we have used a shorthand subscript notation on u and v in place ofpartial derivatives. Note that the Bingham number represents the ratiobetween the plastic and viscous components of the flow. At the walls, theslip law becomes(τxzτyz)=∓(UAS +Bw√u2 +v2)(uv), at z =±1, (2.14)10if√τxz2 + τyz2 ≥ Bw, whereBw =τWS ≡τWτYBi. (2.15)We assume that τW < τY , and so Bw < Bi.2.1.3 The flow patternStrongly sheared regionsThe form of the deformation tensor on the right of equation (2.12) sug-gests that the stress tensor τ is dominated by the components τxz and τyz.Therefore, provided that τ2 ≈ τxz2+τyz2 exceeds Bi2, we have γ˙ ≈√u2z +v2zandz∇p=(τxzτyz)=(γ˙n−1 + Biγ˙)(uzvz). (2.16)That is, (uzvz)= (|z|−Y ) 1nS 1n−1∇p sgn(z), (2.17)whereS = |∇p|=√p2x+p2y, Y =BiS. (2.18)Evidently, 1> |z|> Y for such a solution to exist. Provided that one does,the velocity profile over this region can be found by integrating (2.17), sub-ject to the conditions(uv)∣∣∣∣∣z=±12≡(UV)=− Σ(2 + 1n)Max(0,S−Bw)∇pS, (2.19)whereΣ = (2 + 1n)SUA ≡ (2 +1n)2µ∗HA. (2.20)11It further follows that the slot-averaged flux is12∫ 1−1(uv)dz =(UV)−∫ 10z(uzvz)dz (2.21)=−(2 + 1n)−1QS∇p−∫ Y0z(uzvz)dz, (2.22)whereQ(S) = Σ Max(0,S−Bw) +(1 + nYn+ 1)S1n [Max(0,1−Y )] 1n+1. (2.23)The pseudo-plugIn |z|< Y , the preceding solution is no longer valid, and the approximationτ2 ≈ τxz2 + τyz2 motivated by the form of γ˙ in (2.16) seemingly suggeststhat τ <Bi, implying we must take (uz,vz)→ 0. However, this conclusion isfalse because if uz and vz both become small, and in particular O(), thenγ˙ = O() and all components of τ become of similar order in view of thefactor Bi/γ˙ stemming from the yield stress term in (2.12). In other words,once z→±Y and (uz,vz)→O(), we must look at the terms of next orderand take insteadu= u0(x,y, t) + u1(x,y,z, t) + ...v = v0(x,y, t) + v1(x,y,z, t) + ...,(2.24)leading toγ˙ ≈ √Γ2 +u21z +v21z,Γ2 = 2u20x+ 2v20y + 2(u0x+v0y)2 + (u0y +v0x)2(2.25)andτ = Bi√Γ2 +u21z +v21z2u0x u0y +v0x u1zu0y +v0x 2v0y v1zu1z v1z −u0x−v0y . (2.26)12Because the (dimensional) deviatoric stress is now given by τ ≈ τY γ˙/γ˙, theregion −Y < z < Y can be viewed as a perfectly plastic (von Mises) materialthat, despite τxz and τyz continuing to be prescribed by z∇p, actually re-mains yielded because of the other components of the stress tensor. Indeed,one may solvez∇p= Bi√Γ2 +u21z +v21z(u1zv1z), (2.27)for the shear rates, u1z and v1z, in terms of the in-plane deformation rate,Γ [9]. Nevertheless, the flow profile −Y < z < Y is plug-like, in view ofequation (2.24), leading to Walton & Bittleston’s use of the terminology“pseudo-plug.” Only when u0 and v0 turn out to be independent of positiondoes the pseudo-plug develop into a true plug [7, 29]. Consequently, since(uz,vz) =O(), we may discard the final integral in equation (2.22), to arriveat the flux gradient relation defined by the function Q(S) in equation (2.23).The fake yield surfacesThe pseudo-plug is connected to the sheared regions across the “fake” sur-faces, z = ±Y . This connection actually takes place over a thin layer ofthickness δ, wherez =±Y + δζ, u ∼ u0(x,y, t) + δα+1f(x,y,ζ, t),v ∼ v0(x,y, t) + δα+1Υ(x,y,ζ, t).(2.28)To related δ to  and determine the power α, we insert these relations into(2.16) (using the full form of γ˙) and focus on the surface at z = Y . Then,z2S2 =(1 + δ ζY)2Bi2 =[(2Γ2 + δ2α∆2)n/2 + Bi]2 δ2α∆22Γ2 + δ2α∆2 , (2.29)where∆ =√f2ζ + Υ2ζ . (2.30)13Provided δ2α 2, both sides of this equation equal Bi2 to leading order.Equating the three next-order terms impliesδSζ = δnα∆n− 2BiΓ22δ2α∆2 . (2.31)Hence,α= n−1, δ = 2nn+2 and Sζ = ∆n− BiΓ22∆2 , (2.32)which confirm that δ2α ≡  4n+2 2. The final algebraic equation for ∆(ζ)in (2.32) bridges between the limit of the pseudo-plug solution,∆∼ Γ√Y2|ζ| ,following from (2.27), for ζ → −∞, to the limit of the solution for thestrongly sheared region,∆∼ (Sζ) 1ngiven by (2.17), for ζ→∞. Because the bridge arises across a narrow regionof O(2nn+2 ), the contribution to the flux in (2.22) can be ignored.2.1.4 The flux-gradient relationAn integral of the mass conservation equation across the slot implies that∂∂x∫ 1−1u dx+ ∂∂y∫ 1−1v dy = 0. (2.33)Hence, using (2.22), we arrive at∇ ·q = 0, q =(qxqy)=−Q(S)S∇p, (2.34)14where |q|=Q(S) is the flux-gradient relation given by (2.23), which we nowrepeat, but in the slightly different form,Q(S) = Σ Max(0,S−Bw) + 1S2(S+ nBin+ 1)[Max(0,S−Bi)] 1n+1. (2.35)This relation repeats the one derived in [5, 26] when Σ→ 0, which corre-sponds to eliminating any slip over the walls. If we further take Bi = 0, weemerge with the flux-gradient relation for a power-law fluid, and when, inaddition, n = 1, we arrive at the Newtonian law Q(S) = S. The relationswith and without slip are illustrated in Figure 2.2 for parameter settings typ-ical of the experiments we report later. In the circular geometry for whichwe conduct those experiments, the imposition of a constant flux throughthe slot from a source or sink at the centre actually implies that Q varieswith radial position. The fluid therefore progresses along an entire sectionof the flux-gradient relation in its passage through the cell; excursions forexpanding or extracted axisymmetrical flow are indicated in Figure 2.2.The full flux-gradient relation contains two switches: if S < Bw < Bi,equivalent to τxz2 + τyz2 < Bw2 < Bi2 at z =±1, the fluid is not sufficientlystressed to either slip over the walls or deform between them. Instead, astagnant plug bridges across the gap. On the other hand, if S > Bi > Bw,then Bi2 < τxz2+τyz2 at z =±1, and the shear stresses are sufficient to forcethe fluid to both slide over walls and shear internally against those bound-aries, with a pseudo-plug occupying a section surrounding the midplane. Thesituation Bi > S > Bw corresponds to a stress state with Bi2 < τxz2 + τyz2at z = ±1. In this case, sliding must occur, but the shear stresses are notsufficient to generate any internal shear adjacent to the walls. The interpre-tation is that the pseudo-plug now fills the entire slot with (U,V )≡ (u0,v0).Unless the sliding velocity is independent of position, the fluid does not slideas a rigid block, and the in-plane stress components contribute to holdingthe fluid just above the yield stress, as in §2.1.3. The flux-gradient rela-ton Q = Σ Max(0,S−Bw) that applies for Bi > S > Bw is, coincidentally,equivalent to the simplifed forms for flow with no-slip adopted by Coussot[14] and Alexandrou & Entov [1]. Figure 2.2 highlights how the fluid flow15Figure 2.2: The flux gradient relation in Equation 2.35 illustrated forthe parameter settings indicated and experiments with (a) ex-panding or (b) extracted axisymmetrical flow from a source orsink at the centre of the cell. The red and blue curves indicatethe cases without (Σ = 0) or with (Σ > 0) wall slip, respec-tively. The switches at S = Bi and S = Bw are indicated. Thethicker sections of the curves indicate the excursion taken byfluid expanding or retreating axisymmetrically; in each case, inthe experiments an interface is positioned initially at a radiallocation corresponding to Q= 1. On the right, we show the cor-responding plots of the quantity S√Q(S)Q′(S), which controlsthe basic linear instability (§2.2).through the Hele-Shaw cells of the experiments can take place entirely dueto sliding where the fluxes are lowest, and partly by sliding where the fluxesare highest, unless slip is removed.Note that the form of the flux-gradient relation relies on the assumptionthat the sliding parameter Σ is O(1). Had this parameter instead beenO(−2), then the magnitude of the slip velocity would have violated our16asymptotic scalings, and the implied in-plane stresses could no longer bediscarded in the leading-order force balance equations (2.10). A revision ofthe asymptotic reduction would then be demanded, more like that followedto describe free viscous or viscoplastic films [4, 8, 15, 30, 31, 34].2.1.5 Boundary conditionsIn the Saffman-Taylor problem equation (2.34) must be supplemented withthe conditions holding at the fluid interface, x = X(y,t). Assuming thatthis interface maintains a constant curvature across the slot (with associateddynamic contact lines), we have an approximate kinematic condition,Xt+ qyXy = qx, (2.36)and Laplace-Young relation,p(X,y,t) = Pamb−σK (2.37)where Pamb denotes the pressure beyond the fluid, K denotes the interfacecurvature and σ is a parameter that is related to the dimensional surfacetension σ∗ byσ = σ∗H2L2S . (2.38)To arrive at (2.36), the dimensional time must be scaled by (2 + 1n)L/U ,given that our non-dimensionalization of the problem implies that the aver-age velocity across the slot is (2 + 1n)−1U(qx, qy).Finally, the fluid must be driven through the slot by imposing a flux con-dition. In the planar problem, if the net dimensional flow speed down theslot is V , then we choose U = (2+ 1n)−1V (implying that the net dimension-less flux is unity). We further select L to be a typical finger scale, adoptingperiodic boundary conditions in y. The interfacial curvature is given byK = Xyy(1 +X2y )32. (2.39)17For problems with circular geometry, we may force fluid through the cellby imposing a flux Q through a vent of radius rv. Transforming to thecurvilinear coordinates (r,θ), we set q = (qr, qϑ) and write (2.34) and (2.36)in the alternative forms,1r∂∂r(rqr) +1r∂qϑ∂θ= 0 and Rt+ qϑRθ = qr at r =R(θ, t). (2.40)We may then take L = ro, the initial radial position of the interface,and impose the net radial velocity qr = 1 at r = 1. This corresponds tofixing U = (2 + 1n)Q/[2piroH]. In this case, the interface, at r =R(θ, t), hascurvature,K = R2 + 2R2θ−RRθθ(R2 +R2θ)32. (2.41)For an expanding interface, ro & rv, but for a retreating interface, rv ro,suggesting that any lengthscale based on rv should be irrelevant (motivatingthe selection L = ro).For the circular problem, we may also supplement an inner, retreatinginterface with a second. outer interface to model the expanding viscoplasticannulus shown in Figure 2.1(c). This problem was considered explicitly in[5], and partly reproduced in Section 2.3. We use this analysis to providepredictions from linear stability theory for expanding annuli in Chapter 3.2.2 Planar fingersWith the reduced model in hand, we may explore the dynamics of viscoplas-tic Saffman-Taylor fingers. We first consider this problem for a planar inter-face, before turning to circular geometry. In the planar problem, we choosethe domain length in y to 2pi (so that L is selected as the wavelength of thegravest finger), and consider the moving domain ςt ≤ x ≤X = ςt+ Xˆ(y,t),where ς =±1, for a problem with an advancing or retreating interface. Thekinematic condition (2.36) then translates to Xˆt+qyXˆy = qx− ς, whereas weimpose the flux condition qx = 1 at x= ςt.182.2.1 Linear instabilityThe solution for a steadily advancing planar interface is given byX =X0(t) =X0(0) + ςt, (qx, qy) = (ς,0),Q= 1, p= ςS0(X0−x), S = S0,(2.42)where S0 is the pressure gradient required to drive the background flux,Q0 = Q(S0). For the planar problem, we may choose characteristic scalesso that Q0 = 1, as stated above. However, in the circular problem, thebackground flux varies with radial position. For a comparison with thatproblem, we therefore avoid this particular choice.Considering small perturbations about this state, we setX =X0(t) + Xˆ`(t)ei`y, p= ςS0(X0−x) +p`(x,t)ei`y, (2.43)where `= 1,2, ..., is the wavenumber, and then discard terms in the equationsof the reduced model that are nonlinear in the amplitudes Xˆ` and p`. Wefirst arrive at the relations,S = S0− ς ∂p`∂xei`y, (2.44)(qx, qy) = (ςQ0,0)−(Q′(S0)∂p`∂x,i`Q0S0p`)ei`y, (2.45)which then imply∂2p`∂x2= `2Q0p`S0Q′(S0), (2.46)orp` =A(t)exp[`(x−X0)√Q0S0Q′(S0)], (2.47)for some A(t), if we assume that the viscoplasic fluid extends to x→−∞.From the developments of (2.36) and (2.37), we then finddXˆ`dt =−`A√Q0Q′(S0)S0and A− ςS0Xˆ` = σ`2Xˆ`. (2.48)19Hence,Xˆ`(t) = Xˆ`(0)exp−`√Q0Q′(S0)S0(ςS0 +σ`2)t . (2.49)As long as the flux-gradient relation is always increasing (Q′(S)> 0, whichis the case for the constitutive laws considered here, as illustrated in Fig-ure 2.2), the advancing interface is therefore always stable; the retreatinginterface is unstable to the `th mode if S0 > σ`2.Equation (2.49) predicts a most unstable wavelength given by 2pi√3σ/S0,with a growth rate equal to 23S0√Q0Q(S0)′/3σ. The factor, S0√Q0Q′(S0),indicates how the rheology enters the linear problem to impact stability.Notably, as highlighted by Figure 2.2, the addition of slip leads to a reduc-tion in this factor, suggestive of a corresponding reduction in the degreeof instability for our experiments (although the geometry is not planar).Also, in the limit that Q0→ 0, we must have S0→ Bw or S0→ Bi (with orwithout slip), implying that the dimensional, most unstable wavelength isproportional to√σ∗H/τW or√σ∗H/τY (cf. [14]).2.2.2 Nonlinear fingersNumerical schemeTo look at the initial development of unstable viscoplastic Saffman-Taylorfingering, we solve (2.34), (2.36) and (2.37) numerically for Bingham fluid(n= 1) beginning from the initial condition, Xˆ(y,0) = 20 + 10−2 cosy.To solve (2.34) at each instant of time, and given the current position ofthe interface, we first map the domain into a fixed rectangle by introducingthe variable ξ = (x− ςt)/(X− ςt) = (x− ςt)/Xˆ. We then adopt a fixed gridin y and use the fast Fourier transform to evaluate spatial derivatives inthat direction. This turns (2.34) into a set of coupled ODEs in ξ, that canbe solved using MATLAB’s in-built solver bvp5c. The grid in y has 64 or128 points; bvp5c introduces an adaptive grid with a variable number ofpoints for ξ. To accelerate the computations we also exploit the reflectionsymmetry about y = pi to further halve the grid in y.20In solving (2.34), we also smooth the switch in the flux-pressure-gradientrelation at Q = 0 by replacing Q(S) in (2.34) by√Q2 +εS2, where ε is asmall regularization parameter. Practically, ε is chosen to be 10−2 or smaller,and the effect of this regularization is gauged by varying the precise value.The regularization also leads to a prescribed pressure solution over any trueplugs, which would otherwise be indeterminate.With the solution for the pressure in hand, we construct the flux at theinterface, and then advance the interface position x=X(y,t) in time, using(2.36). For this task, we employ MATLAB’s implicit integrator, ode15s.Note that the mapping ξ = (x− ςt)/Xˆ implicitly limits the numericalstrategy to fingers that do not thin behind the advancing finger tip. i.e. tointerfacial shapes for which x=X(y,t) is single valued. For the Newtonianproblem, it is known that fingers can develop such thinned sections at latetimes [17, 43]. Such shapes are inaccessible for the current numerical schemeand the computation is necessarily stopped when Xy becomes too large.Solutions without slipFigures 2.3 and 2.4 compare numerical solutions for retreating interfaces ofNewtonian (Bi = Σ = 0) and non-slipping Bingham (Bi = 2, Σ = 0) fluid.For the two solutions, the surface tension parameter is chosen so that onlythe first (`= 1) Fourier mode in y is unstable (σ = 0.5 for Bi = 0, and σ = 1for Bi = 2). The development of Saffman-Taylor fingers on the interfaceis similar between the two solutions with the interface deforming into agrowing finger. The main difference between the two solutions is that, for theBingham case, the flux falls significantly adjacent to the rearward portionsof the interface, with the stress decreasing to the yield stress there. Theprogress of the lagging parts of the interface thereby becomes arrested, andfluid retreats only along the front of the finger. For the Newtonian case,instead, the interface slows at the back, but does not halt.A main feature of the numerical solutions in Figures 2.3 and 2.4 is thelengthening of the interface x=X(y,t) along the sides of the finger, whichbecome relatively flat. The mapping used to solve (2.34) on a rectangular21Figure 2.3: Numerical solution for a retreating, planar interface ofNewtonian fluid (n = 1, Bi = Σ = 0; σ = 0.5). In panel (a),we display three snapshots of the flux Q(x,y, t), together witha succession of the equally spaced previous snapshots of theretreating interface x = Xˆ(y,t) (equally spaced by 0.5). Thepanels on the right show time series of (b) 〈Xˆ2〉 (angular brack-ets denoting a spatial average) and (c) the maximum (blue) andminimum (red) interface positions. In (b), the dashed lines showthe expected linear growth and the stars indicate the times ofthe snapshots of Q shown in (a).domain becomes problematic in this situation, with the limited spatial reso-lution prompting the development of short-wavelength oscillations in y. Weterminate the computations before these spurious features become too se-vere, but they are visible in some of the final solutions shown in the figures,particularly in the yield surface of the Bingham cases.Alexandrou & Entov [1] have previously given solutions for the advanceof a finger of inviscid fluid at constant speed into a cell filled with a (misci-ble) yield-stress fluid. As mentioned earlier, however, these authors exploitan approximation of the flux-gradient relation that is more similar to thepurely sliding law we have derived here (i.e. (2.35) with Bi→∞). Never-theless, their solution possesses plugs on either side of the advancing finger,somewhat like those seen in Figure 2.4(a). The development of the blockedsection of the cell along the lines y = 0 and 2pi is shown in more detail in22Figure 2.4: Numerical solutions for a retreating interface of non-slipping Bingham fluid (n = 1, Bi = 2, Σ = 0; σ = 1). Threesnapshots of Q(x,y, t) are again displayed in (a), along withprevious interface positions (equally spaced by 0.25). The dot-ted (green) contours show the yield surfaces. The panels on theright show time series of (b) 〈Xˆ2〉 and (c) the maximum (blue)and minimum (red) interface positions. In (b), the dashed linesshow the expected linear growth and the stars indicate the timesof the snapshots of Q shown in (a). In (c), the shaded regionshows the extent of the growing plug along y = 0 and 2pi.Figure 2.4(c). The retreat of the yield surface here appears to match that ofthe interface at late times, suggesting that the solution may indeed convergeto a steadily translating form.Further evidence for such convergence is shown in Figure 2.5, whichpresents solutions with varying Bingham numbers, holding all other pa-rameters fixed. Plotted are snapshots of the interface in the frame of thefingertips; i.e. after shifting the origin to (x,y) = (Xtip,pi) where Xtip(t) =Min(X). This shift increasingly collapses the profile of the fingertips nearx=Xtip as time progresses, suggesting the convergence to a steadily propa-gating form. The profiles also highlight how the finger widths ∆ and speed230 1 2 3 4 5 6 7 8 9 108101214161820-20 2 0 1 2 3-1010 10 2005100 1010.20.40.60.81Figure 2.5: Numerical solutions for a retreating interface of non-slipping Bingham fluid (n = 1, Σ = 0; σ = 1) with varying Bi(as indicated). Snapshots of the interface in the frame of ref-erence of the finger tip are shown in the top row (i.e. in theframe with origin at (x,y) = (Xtip,pi) where Xtip = Min(X)).The snapshots are roughly equally spaced by intervals equal toa fiftieth of the final time in each computation. The lower panelshows the times series of Min(X) and Max(X) for all the so-lutions. The lower left inset shows the profiles of the all thefinger tips, rescaled by the finger width ∆, defined as the verti-cal distance between the final yield points on the interface. Theright-hand insets show the final speed of the fingertips vtip, thenits inverse and ∆/pi against Bi.of advance vtip depend on Bingham number, with the finger narrowing andbecoming faster as the yield stress increases. In Figure 2.5, the width isestimated by measuring where the yield surface intersects the interface foreach final solution.The limiting tip speed and finger width are connected since in anysteadily translating final state, mass conservation demands that ∆vtip =pi. As the plug is still in expansion at the end of the computations and24Figure 2.6: Numerical solutions for a retreating interface of slippingBingham fluid (n = 1, Bi = 2) with (Σ,σ) = (12 ,25) (top) and(Σ,σ) = (34 ,23) (bottom); Bw =15Bi. In (a), the final solution forQ(x,y, t) is plotted along with a succession of previous interfacepositions (equally spaced by 0.25). The dashed (green) contoursshow the internal yield surfaces where S = Bi; the (red) dottedlines show the contours Q= 0.1. The panels on the right showtime series of (b) 〈Xˆ2〉 and (c) the maximum (blue) and min-imum (red) interface positions. In (b), the dashed lines showthe expected linear growth. In (c), the solution from Figure 2.4is shown by the dash-dotted lines. The stars indicate where thepressure gradient S reaches Bi; the small circles indicate wherethe cell first becomes blocked (S reaches Bw for Σ> 0, or Bi forΣ = 0).the fingertip is accelerating, the measurements of v−1tip and ∆/pi plausi-bly bracket the limiting relative finger width (the viscoplastic analogue ofSaffman & Taylor’s λ parameter). Evidently, with surface tension, a partic-ular finger width is selected, as found previously in the Newtonian problem[17, 33, 39, 43]. Here, the finger width is further controlled by the yieldstress, which constricts the fingers and lowers the relative width ∆/pi below12 , the bound encountered in the Newtonian problem for σ→ 0.25Figure 2.5 also highlights another feature of the finger solutions, namelythat scaling the interface position by the finger width ∆ leads to some col-lapse of all the profiles. This scaling is equivalent to that noted by Pitts[38] in the Newtonian Saffman-Taylor problem. In the current viscoplasticproblem, we may rationalize this observation by noting that when the fin-ger becomes relatively thin and surrounded by a plug, the domain adoptedin y must become secondary in setting the finger profile, raising the ques-tion of what lengthscale sets the width. The key points are that the pres-sure gradient S must remain O(Bi) in order to drive the finger forward, sop ∼ S∆ ∼ Bi∆, whereas the capillary stress demands that the pressure isp ∼ σ∆−1. Thus, ∆ ∼√σ/Bi, and since this prescribes the only relevantlengthscale, a collapse is implied by the scaling of the profile by ∆. Thescaling of ∆ with Bi− 12 is indeed observed for the solutions shown in Figure2.5 (see the inset panel on the right). In dimensional terms, because oursurface tension parameter is given by (2.38), we conclude that the fingerwidth is set by the lengthscale√σ∗H/τY , which also corresponds to thewavelength of the most unstable mode in the plastic limit (see §2.2.1; [14]).Slippy solutionsNumerical solutions including slip are shown in Figure 2.6; again, the surfacetension parameter is chosen so that only the ` = 1 mode is unstable. Forthe solution shown at the top of Figure 2.6(a), the slip parameter Σ isset to a value so that the viscoplastic fluid is internally yielded when it isuniformly withdrawn from the cell (i.e. S0 > Bi). However, although thefluid is consequently internally yielded to begin with, once an unstable fingerdevelops, the stress declines sufficiently to either side of the finger that Sfalls below Bi there (at the time indicated by the blue star in Figure 2.6(c)).The fluid over these portions of the cell then deforms entirely as a slidingpseudo-plug.The second case, shown at the bottom of Figure 2.6(a), has a differentvalue for Σ, such that the fluid purely slides in uniform withdrawal (i.e. Bw<S0 < Bi). The development of an unstable finger, however, now raises the26stress near the finger tip such that the fluid eventually fully yields internallythere. The red star in Figure 2.6(c) indicates the moment that internalyielding first arises.For the solutions shown in Figure 2.6, the solution for the pressure gra-dient over the nearly blocked parts of the cell becomes particularly sensitiveto the spatial resolution and the regularization parameter ε. We cannot,therefore, reliably identify any rigid regions, or the detailed shape of anyyield surfaces where S = Bw. Nevertheless, the computations suggest thatthe yield stress is reached at the leftmost point of the interface close tothe end of both computations in Figure 2.6) (at the times indicated by thefilled circles). Before this happens, the computations do reliably indicaterelatively low fluxes to the left of the cell, as highlighted by the contourswhere Q = 0.1 that are also indicated in Figure 2.6(a). The shape of thesecontours is more like the yield surfaces presented by Alexandrou & Entov,which might be due to the similarity between the two flux-gradient relationswhen S < Bi.Although the solutions with slip in Figure 2.6 suggest that the fluid mayblock the cell at late times, the plugs appearing in the solution withoutsliding (Figure 2.4(a)) are rather more pronounced. In particular, reliablydetected plugs and yield surfaces arise without slip, with the stress certainlyfalling below the yield point well before the computation in Figure 2.4 termi-nates. By contrast, with slip the pressure gradient declines more gradually(see Figure 2.6(c)), but a better numerical scheme is called for to determinethe late-time behaviour.2.3 Axisymmetrical flowTo complement our experiments in the next chapter, we also solve the linearstability problem in a circular geometry. Because in this setting the basestate is time-dependant, a normal-mode analysis as done in Section 2.1 isnot valid. Instead we need to solve the linear equations as an initial-valueproblem after breaking it down into each angular Fourier modes. For theexpanding annulus problem, we consider the paired inner and outer inter-27faces separately, and for the extracted disk we only need to consider the oneouter interface.2.3.1 Linear stabilityWhen the net dimensionless flux is constant and equal to 2pi, an axisymmet-rical flow is possible, driven by a local radial pressure gradient, S→ S(r, t),satisfying the algebraic relation,1r=Q(S ;n,Bi), (2.50)Where the flux-gradient relation Q(S;n,Bi) is still given by Equation (2.35).For an expanding annulus, the flux is directed outwards and the pressuregradient is negative. The corresponding circular fluid edges areRI (t) =√1 + 2t≡RI (t),RO(t) =√R2A+ 2t≡RO(t),(2.51)where RA denotes the initial outer radius. For an extracted disk, the flux isdirected inwards and the background pressure gradient is positive; the singleinterface is atr =R(t) =√R2A−2t, (2.52)where RA denotes its initial position. If we neglect the pressure of theambient fluid beyond r = RO(θ, t) or r = R(θ, t) then the base pressuredistribution isP(r, t) =∫ ROrS(r¯)dr¯ or −∫ RrS(r¯)dr¯. (2.53)We add linear non-axisymmetrical perturbations to this stable base statesolutions, with angular dependence eimθ and a pressure distinguished by ahat decoration. These perturbations must then satisfy(rQ′pˆr)r =m2pˆrS Q(S;n,Bi)−→m2pˆr2S , (2.54)28whereQ′ = Σ + 1r[n+ 1(n+ 1)S+nBi +n+ 1n(S−Bi) −2S]. (2.55)Note that when the fluid does not slip on the surface (Σ = 0), one recoversthe equations presented in [5].2.3.2 Expanding viscoplastic annulusWhen air is pumped into an annulus of viscoplastic fluid, the pressure withinthe inner interface is almost uniform in space but adjusts in time so thatthe net flux through the cell is fixed. To see this in detail, we first solveLaplace’s equation for the pressure perturbation in 1< r <RI , applying afixed influx condition at the vent r = 1 (using the radius of that orifice inthe scaling of the problem): pˆr(1, t) = 0. To first order in the perturbations,this gives the pressure distribution,p(r, t) = P(RI , t) + µµ∗log(RIr)+ rm+ r−mRmI +R−mIpˆin(t)eimθ (2.56)if µ viscosity of air, and given that the scaling of the problem implies thelocal flux-pressure gradient relation S = (µ/µ∗)Q. Here, pˆin(t) is the ampli-tude of the pressure perturbation at r = R¯I . Matching pressures and theradial velocity at the inner interface (but expanding all quantities about theunperturbed position) now demands thatpˆ(RI , t) =[S(RI , t)− µµ∗R−1I]RˆI(t)eimθ + pˆineimθ, (2.57)and[Q′pˆr]r=RI =µ∗µm(R2mI −1)RI(R2mI + 1)pˆineimθ, (2.58)where RˆIeimθ denotes the perturbation to the edge position. Eliminatingpˆin(t) leads to the boundary condition,pˆ(RI , t)[S(RI , t)− µµ∗R−1]RˆI(t)eimθ + [Q′pˆr]r=RIµµ∗RI(R2mI + 1)m(R2mI −1).(2.59)29As long as m 6= 0, we may therefore impose pˆ= SRˆI(t)eimθ at r = R¯I whenµ µ∗.Likewise, at the outer edge, we assume that the pressure is held at thatof the ambient fluid (which we again neglect): p = 0 at r = RO. The lin-earization again implies that pˆ−SRˆOeimθ = 0 at r =RO, where RˆOeimθ isthe shift of the outer edge.We may therefore find two solutions to Equation (2.54), pˆI(r, t)eimθ andpˆO(r, t)eimθ, satisfying the conditions,pˆI =S at r =RI0 at r =RO , pˆO =0 at r =RIS at r =RO . (2.60)Writing the edge perturbations as RˆIeimθ and RˆOeimθ, the desired linearsolution is then,pˆ= [RˆI(t)pˆI(r, t) + RˆO(t)pˆO(r, t)]eimθ. (2.61)The linearizations of the kinematic conditions now furnish the systemdRˆIdt=−[(Q′pˆIr + r−2)RˆI +Q′pˆOr RˆO]r=RI, (2.62)dRˆOdt=−[Q′pˆIrRˆI + (Q′pˆOr + r−2)RˆO]r=RO. (2.63)2.3.3 Extraction of a viscoplastic diskFor the extraction problem, the theoretical description is simpler: we writer = R(θ, t) =R(t) + Rˆ(t)eimθ and pˆ = pˆA(r, t)Rˆ(t)eimθ for the disturbed in-terface position and pressure. The linear perturbations satisfy Equation(2.54) subject to the boundary conditions,pˆAr (1) = 0 and pˆA(R, t) =−S(R), (2.64)which correspond to specifying the extracted flux at the vent r= 1 and fixingthe ambient pressure. The linearized kinematic condition then provides the30instantaneous growth rate,Rˆ−1dRˆdt=−[Q′pˆAr ]r=R−1R2 . (2.65)2.3.4 Limitations of the modelThe Hele-Shaw formulation for displacement problems in a cell is a wellstudied simplified model to describe our experiments in Chapter 3. It ade-quately predicts the linear instability of the problem but it has some limi-tations. Firstly, the size of the gap must be significantly smaller than thehorizontal length scale. The flux and pressure must also be small enough tosupport our omission of the inertial terms in the Navier-Stokes equations.Near the inlet region this is not a valid assumption. We can not assumethe flow to be laminar and purely horizontal. This problem is addressedby taking the inner boundary to be at the vent radius, when the fluid hasalready formed a small disk. Furthermore, we use the Herschel-Bulkley con-stitutive law to characterize Carbopol, the fluid used in our experiments.However, Carbopol is more complex than we assume in this chapter. It dis-plays thixotropic behavior, as can be seen in Figure 3.2. Finally, in both theplanar and axysymmetric geometries, we treated the interface between thetwo fluids as a flat vertical moving front. The front has a parabolic flow pro-file but our crude simplification of the interface does not have a significantimpact on the linear stability analysis.31Chapter 3ExperimentsIn this chapter, we discuss experiments conducted to observe and analysethe Saffman-Taylor instability. Our investigation is done by pumping vari-ous fluids into a Hele-Shaw cell, made of two horizontal plates of plexiglass,and separated by spacers of thickness H = 1.7, 3.3 and 4.3mm. The platesare either left in their original smooth state or roughened with 60-grit sand-paper to try to eliminate effective slip. Figure 3.1 shows a sketch of theexperimental apparatus. The vent through which the fluids were pumpedinto the cell was located in the center of the bottom plate, and had a radiusof rv ≈ 2.75mm for the roughened cell and rv ≈ 3.8mm for the smooth cell.The fluid was pumped at various fluxes Q between 5ml/min to 100ml/min.To get some quantitative measurements of the interfaces in our exper-iments, photographs and videos are taken from above, to image the fluidsin plan view, as seen in Figures 3.1 and 3.3. The images are binarized onMATLAB to clearly pick out the interface between the displaced and thedisplacing fluids. Figure 3.4 shows the steps followed to do so.For this study, corn syrup and air are used as model Newtonian fluids andan aqueous suspension of Carpobol as a model viscoplastic fluid. Carbopolis a well studied complex fluid that can be characterised using the Herschel-Bulkley law, as discussed later in §3.1. For the disk extraction problem, oneof the fluids mentioned above, other than air, is pumped into the empty cell,displacing the ambient air as it expands. Because all of these fluids have32Figure 3.1: Schematic of the experimental setup. A horizontal Hele-Shaw cell of thickness H, filled with ambient air. A fluid isthen pumped into the center of the cell from below and spreadradially outwards. The experimental pictures were taken fromabove.a higher effective viscosity than the air, the expanding disk is in a stableconfiguration and should expand asymmetrically. After the disk has reachedsome given radius the pump is reversed and the fluid is extracted out of thecell, displaced by the ambient air.For the expanding annulus problem, we pump a first fluid into the cellas before. After the disk has reached a given initial radius, we changed thetube at the vent of the cell, and pump in a second fluid, usually air unlessmentioned otherwise. The air pushes the resulting annulus outwards and wetake pictures or a video from above to capture the interface again.333.1 MaterialAs previously mentioned, we used a variety of fluids for this study. Air andcorn syrup were used as model Newtonian fluids. Their respective viscosityand density are reported in Table 3.1.Fluid Viscosity (Pa· s) Density (g/cm3)Air 10−5 0.0012Corn syrup 8.2 1.38Table 3.1: Viscosity and density of the Newtonian fluids used in ourexperiments.As a model yield stress fluid, we used an aqueous suspension of Carbopol(Ultrez 21) with a concentration C ≈ 0.13% by weight, and a density of ρ≈1g/cm3. Figure 3.2 shows the flow curves of the three batches of Carbopolused in our experiments, obtained from a Kinexus Malvern rheometer fittedwith roughened parallel plates, and using an up-down, controlled shear-rate ramp. The Herschel-Bulkley fit obtained from the flow curve gives thefollowing average material parameters:τY = 7.0±0.3 Pa, K = 5.3±0.1 Pa, n= 0.42±0.01, (3.1)where τY is the yield stress, K is the consistency index and n is the power-law index.3.2 Control ExperimentsHaving in mind the goal of understanding the viscoplastic Saffman-Taylorinstability, we begin by discussing a few control experiments. We first con-ducted a brief suite of test with an expanding annuli of corn syrup, displacedby air. We also explored the miscible version of the problem, by replacingthe air by water to displace the syrup. This allowed us to investigate theimportance of surface tension in this setting, and to make sure the fingerpatterns were not driven by the chosen gap size.34Figure 3.2: The flow curves for the three aqueous suspension of Car-bopol Ultrez 21, with a concentration C ≈ 0.13% used in ourexperiments. The data was collected using a controlled shear-rate ramp on a Kinexus Malvern rheometer. The ramp up anddown data are denoted by red and blue markers respectively.The solid black curves show the Herschel-Bulkley fit for thedata on the down ramp.3.2.1 Newtonian CaseThe Newtonian version of the Saffman-Taylor instability is very well studiedand easier to model than it’s viscoplastic counterpart. The problem actuallyreduces to solving Laplace’s equation in both fluids. We therefore conducta few experiments of expanding annuli of corn syrup in a Hele-Shaw cellwith gap H = 3.3mm and flux Q= 59ml/min. These experiments are doneon smooth plexiglass plates as the corn syrup satisfies a no-slip boundarycondition on the plates. Figure 3.3 shows snapshots of one of those exper-35iments at successive times. To make sure no residual syrup was left in thenozzle attached on the plate after switching the syrup pipe and the air pipe,we had to flip the cell upside down. The tube is therefore visible in thepictures taken from above. To remove it from the edge highlighted in black,we fit a smoothing spline connecting the interface on both sides of the pipe.The edge detection algorithm was done in MATLAB, and is shown in 3.4.In panel (b), we split the original image into its red, blue and green colorchannels and subtracted the blue from the red one. In (c), we chose anintensity threshold to binarize the image in (b). The yellow pixels are onesand the blue are zeros. Finally to retrieve the interface in (d), we found theboundary in the binary image and filled the gap where the pipe was usinga smoothing spline.Figure 3.3: Pictures of an expanding annulus of corn syrup (yellow),in a cell with gap H = 3.3mm and flux Q = 59ml/min. Theinterface between the syrup and the air is highlighted in black.The final pattern and the evolution on fingers is triggered by small ran-dom perturbations to the interface near the vent. To try and control morecarefully these perturbation, we add small pieces of tape to the bottom plate,close to the vent. These bumps create small changes in the gap thickness,36and placing them at regular angular spacing allows to kick one particularFourier mode. Figure 3.5 shows the placement of the pieces of tape, trig-gering modes m= 3 to 9 and one experiment with no tape on the surface ofthe cell, and the resulting final patterns.Figure 3.4: Steps taken to process the images from the experiments,and pick out the interface. (a) is the original photo. (b) isthe red color channel minus the blue color channel. (c) is thebinary image of (b). (d) is the boundary captured from thebinary image. (e) is the edge displayed on top of the originalphoto.The number of fingers for the experiments kicking modes m = 6,7 and8 remain constant until the end of each test. However, for the experimentswhen modesm= 3 to 5 are kicked, as the interface expands, more fingers ap-pear and the pattern loses angular symmetry. The experiment with 9 piecesof tape also loses symmetry and becomes distorted as lower wavenumbersgrow. Thus, for these particular experimental parameters, it appears thatwavenumbers over the range 5≤m≤ 8 are preferred, in line with the shapeof the interface in the test without tape perturbations.Counting fingers is one method to quantify the instability. It gives anidea of the shape of the pattern, as well as the preferred modes. Furthermore,the most unstable mode number can be predicted by the linear stabilityanalysis presented in Chapter 2, the number of fingers we count in theexperiments allow us to test these theoretical predictions. However, thatmeasure can be somewhat ambiguous and open to some bias. It also does notgive any information on the strength and growth rate of each mode, anotherpredicted measure from the stability analysis. As a result, it is importantto supplement our examination with other diagnostics. During the analysis37Figure 3.5: Snapshots of expanding interfaces for experiments withcorn syrup, where Q= 59ml/min, H= 3.3mm and Vinit = 50ml.The interfaces are colour coded in time from red to blue. Thedotted black lines show the positions of small local protrusionsin gap thickness arranged to excite modes with m= 3 to 9. Thetop-left snapshots of the interface show an experiment with nosuch protrusions.of the photos taken with the camera, we measure the instantaneous radiusR(ϑ,t), expressed in terms of the angular variable ϑ= 2pis/st, based on theratio between the local arclength s and the total arclength st. We thenbreak down this periodic variable into its discrete Fourier seriesr =R(ϑ,t) =∑mRˆm(t)eimϑ (3.2)where m is the mode number. Note that ϑ is chosen not to be the polarangle θ, to avoid the problem that arise once the fingers become nonlinearand render the radius a multi-valued function of θ. Moreover, in the linearregime, ϑ and θ are equivalent.Figure 3.6 shows the Fourier spectrum of the corn syrup annulus exper-iment with no protrusion, the same experiment in the first panel of Figure383.5. The peak in the spectrum is at m = 7, which is consistent with whatone would get by counting the fingers in the experiment. Additionally, wenow can also see the final amplitudes of all the other modes, as well astheir growth with time. The mode amplitude |Rˆm|/R0, for each angularwavenumber m can now be compared with its theoretically predicted value.which is done in Figure 3.8. However, we notice a lot of noise at early times,when the interface is nearly circular, and at higher wave numbers.In an attempt to further clean up this diagnostic and make it easierto compare multiple modes between experiments and with the theory, weconsider summing up each angular mode of the Fourier series of the scaledradius R(ϑ,t)/Rav. This is also equivalent to the scaled standard deviation,SR =√〈(R/Rav−1)2〉. (3.3)The angular brackets denote the average over ϑ. The blue curves in Figure3.7 show SR against volume for the seven corn syrup annuli experiments,perturbed with sticky tape. The red curve shows SR for the unperturbedexperiments.Another possible diagnostic we considered to quantify the instability inour experiments is the scaled arclength of the edgeAc= L2piRav−1, (3.4)where L is the instantaneous arclength of the interface and Rav is theaverage radius. Figure 3.7 also shows Ac against volume for all eight syrupexperiments. We note that both diagnostic look very similar. Their shapeis almost exactly the same, only scaled by a factor of 10. This shows thatare diagnostics are somewhat robust. If one experiments appears as moreunstable with the SR diagnostic, Ac will display the same feature. Onebenefit of using SR over Ac is that it is less noisy at early times in theexperiments. Ac suffers from the pixelation of the image when the annulusis small. The averaging over R for the standard deviation removes a lotof that noise. Furthermore, the relationship between SR and the angular39Figure 3.6: First 20 angular modes of the Fourier transform of theexpanding interface of corn syrup, where Q = 59ml/min, H =3.3mm and Vinit = 50ml, with no protrusion, colour coded intime from red to blue.modes of the Fourier series of the interface is much more transparent thanthe one between Ac and the modes. This makes it easier to compare theexperiments with the linear stability theory. For these reasons, we chose toonly use SR in the rest of the analysis in this thesis.The plots in Figure 3.8 show the growth of the lower Fourier modes3≤m≤ 9 for the eight syrup experiments. Even though each of the tests iskicked differently by the protrusions, these individual modes grow at largelythe same rate and differ mainly in initial amplitude (as highlighted in theplots, where the mode amplitudes are scaled to align the data at an interme-diate volume), except when they reach appreciable amplitude, or obscuredby noise. Consequently, the diagnostic SR in Equation 3.3 also differs largelyby a purely multiplicative factor. Notable exceptions to this rule are modesthat are excited initially with relatively low amplitudes (and become im-pacted by noise), and the m = 3 and 4 modes of the first test, which rapidly40becomes overtaken by the m = 6 mode and thereafter decay, presumably asa result of nonlinear effects. Therefore, we conclude that the modes growmostly in the linear regime, and the detail manner in which the interface isperturbed is significant only in selecting the initial amplitudes. The insetshows the initial mode amplitudes plotted against m for the first snapshotof each experiments. This initial spectrum is used to initially perturb thelinear stability analysis.Also included in the mode amplitude plots are the predictions of linearstability analysis. These predictions imply that the most amplified modesare m= 6,7 and 8 (using Rˆm ∝m− 32 ), and match well with the observationsfor the lower modes (m= 3 to 5), but then increasingly overestimate growthat higher wavenumbers. That feature, and how it again contaminates theprediction for SR(t), mirror the results reported later in this chapter forCarbopol.3.2.2 Miscible CaseFor miscible flows in an unstable Saffman-Taylor configuration, it is knownthat the finger pattern is dictated entirely by the gap thickness of the Hele-Shaw cell. The dominant wavelength for well-developed non-linear fingers isabout three (in microgravity experiments [3]), four (in radial displacements[36]) or five (in a vertical Hele-Shaw cell with density differences [27]) timesthe gap of the Hele-Shaw cell. In the immiscible Saffman-Taylor instability,surface tension and therefore the capillary number plays an important rolein stabilising the instability. It reduces the number of fingers and increasestheir width. However, it can also happen with immiscible flows that thegap controls the size and number of fingers when the Hele-Shaw cell is thinenough. If this is the case in our experiments, it would render our analysiserroneous. All of our parameter variations, the use of a viscoplastic fluid andthe introduction of slip would all be irrelevant to the dominant linear insta-bility. To ensure that this is not the case, we conducted one miscible annulusexperiment in our cell of thickness H = 3.3mm, using tap water instead ofair to displaced the corn syrup. As water and corn syrup are miscible, the41Figure 3.7: Two different diagnostics explored to quantified thestrength of the instability of the interface between air and cornsyrup. The black curves show the scaled arclength defined inEquation 3.4 for all eight annulus experiments shown in Figure3.3. The blue curves correspond to the standard deviation ofthe scaled radius SR defined in Equation 3.3, for the seven cornsyrup annuli perturbed with sticky tape. The red curve is theone unperturbed experiment.width of the fingers should be about 4∗H = 1.32cm, as predicted by Pater-son [36]. Figure 3.9 shows a comparison between our experiment with airwithout sticky tape and the miscible experiments. The emerging patternsare very different in these two cases. There are 15 fingers in the misciblecase and only 7 in the immiscible one. The water fingers are also thinnerthan their air counterparts. However, the average width of the water fingersseem to be slightly larger than the predicted 1.32 cm shown in Figure 3.9,but the fingers are not yet fully developed and one could imagine the fingersgetting thinner as some of them split. We therefore conclude that our gap is42Figure 3.8: Mode amplitude plots of the syrup experiments shown inFigure 3.5. The thicker line shows the initially excited mode,and the dashed lines present the results of linear stability analy-sis, both scaled to align the data for some intermediate volume.The lower right panel shows an experiment in which there wereno initially excited mode. This panel shows the diagnostic SR(t)in Equation (3.3), again scaled to an intermediate volume. Theinset at the bottom right shows the initial mode amplitudesplotted against m for the first snapshot of each test; the dashedline indicates m− 32 , the initial distribution that is used to com-pute the theoretical version of SR(t) immediately above.43large enough to ensure that it does not fully control the number and shapeof the fingers in our experiments.Figure 3.9: Snapshots of expanding interfaces from one immiscibleexperiment with air and corn syrup (top row) and a miscibleone with water (dyed in blue) and syrup (bottom row), whereQ= 59ml/min, H = 3.3min and Vinit = 50ml. The scale bar of1.32 cm represents the predicted width of fully developed non-linear miscible fingers by Paterson [36]. Because the immisciblefingers are less numerous and wider that the miscible ones, weconclude that our gap is large enough to not fully control thepattern formation.443.3 Viscoplastic Saffman-Taylor: The impact ofslipWe now explore the Saffman-Taylor instability for a flow of Herschel-Bulkleyfluid, using Carbopol. We conducted a suite of experiments in cells of thick-ness H = 1.7 mm with an initial disk of volume Vinit = 50 ml. For the ex-panding annuli, air was pumped into the roughened cell at Q= 53 ml/minand into the smooth cell at 45ml/min. For the extracted disk, the pump wasreversed and the Carbopol was pulled at 45 ml/min out of the rough celland at 23ml/min out of the smooth cell. Sample results for fingers formingon the interface of an expanding annulus and extracted disk are presentedin Figures 3.10 and 3.11, respectively. In both cases, time series of the di-agnostic SR defined in Equation 3.3 are shown, along with sample series ofoutlines of the unstable interface, for both smooth and roughened cells. Theinterface loses its initial axisymmetry after a short period, developing thedistinctive fingers of the Saffman-Taylor instability. The tests are haltedwhen the longest finger approaches either the outer air-Carbopol interface(for the annulus), or the vent (for the extraction). For the former, the outerinterface largely maintains its axisymmetry until shortly before the longestfinger breaks through it.A main conclusion that one immediately draws from Figures 3.10 and3.11 is that the development of the instability is very different in cells withroughened plates and cells with smooth plates, a feature that must be dueto effective slip. In particular, the fingers develop significantly more rapidlyin the rough cells. Once formed, the fingers also lengthen faster in theroughened cells, with the rearward sections of the interface barely movingat late times; by contrast, in the smooth cells, the entire interface looksto remain in motion for the duration of the experiment. These featuresare illustrated further in Figure 3.12, which shows the advance of all thetips and roots of the fingers (i.e. the local radial maxima and minima ofthe interface) detected in the annulus and extraction experiments shown inFigures 3.10 and 3.11.45Figure 3.10: Repeated tests for expanding annuli in cells with eitherrough or smooth plates, plotting the standard deviation of theradius SR =√〈(R/Rav−1)2〉 against the volume V of air inthe cell. For the rough cell, the test is repeated seven times; forthe smooth cell, the test is repeated four times. The fainterlines show the individual tests; the average, with errorbarsbased on the standard deviation, are shown with darker lines.For both cells, outlines of the interface are shown for four sam-ple tests. Q= 50ml/min, H = 1.7mm, Vinit = 50ml.Figure 3.11: A similar plot to Figure 3.10, but for extractions of adisk, and for the volume V of Carbopol in the cell. The testsin both rough and smooth cells are repeated seven times. Q=50ml/min, H = 1.7mm, Vinit = 50ml.46The finger pattern arising in the experiments is never the same if the testis repeated, as illustrated by the sample interface outlines shown in Figures3.10 and 3.11. For example, for the experimental conditions used in thetests of Figure 3.10, between 4 and 7 fingers appear on the interface of theexpanding annulus in the rough cells; between 7 and 9 fingers appear on theinterface in the smooth cells. The fingers typically have different strengthsand angular positions relative to one another between tests (confirming theuniformity of the gap). We attribute the variability in the finger patternsto the impact of minor differences in the initial conditions, which promptsthe growth of a different selection of the unstable angular Fourier modes. Inorder to reduce the impact of this variability on the diagnostics we employto characterize the finger patterns, tests are repeated a number of times(typically between three and seven times), and the results are averaged overthese repetitions (see Figures 3.10 and 3.11).3.4 Changing gap thickness, flux and size ofinitial diskThe effect of the initial emplaced volume Vinit is illustrated for annulus ex-periments in roughened cells in Figure 3.13. Relatively small initial disks(Vinit = 5 or 10ml) develop fingers that amplify sharply and quickly breakthrough to the outer edge. However, increasing the initial volume aboveabout 25ml leads to finger patterns that develop in much the same man-ner, except near the final moments when the longest finger breaks through.Indeed, for V > 25ml, the variation in the time series of the diagnostic 3.3with V init observed in Figure 3.13 is less than the variability seen betweenthe seven repetitions of the V = 50ml tests in Figure 3.3 (as indicated bythe error bars for that case). There is also no obvious trend in Figure 3.13for Vinit > 25ml. Such observations therefore led us to take Vinit = 50mlfor the remainder of the experiments (corresponding to an initial radius ofabout 10cm in the cell with a gap of H = 1.7mm) and otherwise ignore thisparameter.47Figure 3.12: Time series of the local, radial maxima and minima (thefinger tips and roots) for the tests of Figures 3.10 and 3.11,with the results for rough plates annuli in (a), for smooth platesannuli in (b), for rough plates extraction in (c) and for smoothplates extraction in (c). The fainter dotted lines and symbolsshow all the extrema (above a threshold level exceeding thenoise in the original images); the lighter solid lines show theglobal maxima and minima. The darker solid lines show theaverages over all the repeated tests, and the black dashed lineis Rav(t).Trends with varying gap size and flux are illustrated for annulus exper-iments with rough and smooth cells in Figure 3.14. The most significantvariation is with cell size, with any increase in H significantly reducing thenumber of the fingers and their strength (Figure 3.14(a)). The effect of vary-ing flux is less dramatic for rough plates in Figure 3.14(b), with the most48obvious trend being that a single finger dominates more quickly as the flux islowered. The highest flux with smooth plates displays a stronger instability,but the lower fluxes again remain comparable.Figure 3.13: Expanding annuli in roughened cells with different initialvolumes Vinit (as indicated), showing the standard deviationof the radius SR against the volume V of air in the cell, plussample outlines of the interface. Each test is conducted at leastthree times, except for the test with the largest initial annulus,which was conducted only once. The inset shows correspond-ing predictions of the linear stability theory, assuming an ini-tial spectrum with Rˆm(0) = m−32 . The most amplified modehas wavenumber m= 8 or 9. Parameter settings: H= 1.7mm,with Q increasing monotonically from 41ml/min for the small-est Vinit up to 68ml/min for the largest initial volume.3.5 Comparison with linear theoryTo complement the experiments we solve the linear stability problem in cir-cular geometry, as described in Section 2.3.2. We solve the linear equationsas an initial-value problem after breaking down the problem into each angu-lar Fourier mode with wavenumber m, and considering separately the pairedinterfaces of the expanding annulus or the single interface of the extracteddisk.49Figure 3.14: Expanding annuli with different (a,c) gap size (Q =53ml/min in (a), 45ml/min in (c); Vinit/H = 294cm2), and(b,d) flux (H = 1.7mm, Vinit = 50ml), as indicated. For (a,b)the plates are roughened; in (c,d), the plates are smooth. Thediagnostic SR(t) is plotted against area, V/H, in (a,c) andvolume, V , in (b,d). The insets show the predictions of lin-ear stability theory for the same gaps and fluxes (holding allother parameters fixed in each case), assuming an initial spec-trum with Rˆm(0) = m−32 . The numbers indicate the angularwavenumber m of the most amplified mode at the end of eachtime series.Given an initial spectrum for the linear, angular Fourier modes, onecould also formulate suitable superpositions for a direct comparison of the-ory and experiment. However, as the initial states in the experiments projectrandomly onto these modes, formulating such an initial spectrum is chal-lenging, particularly since the experimental Fourier series are contaminatedby noise and image-processing artefacts. The theoretical model also features505 10 15 20 25 3010 -210 -1100101Figure 3.15: Fourier mode amplitudes observed for the first snap-shots of the tests shown in Figures 3.10 (red) and 3.11 (blue),for both rough and smooth plates. The modes are scaled bythe power in modes 5 < m < 33, and the dashed lines showsm−32 .a relatively crude treatment of the interface that likely introduced order-onediscrepancies (cf. [38–40]) and renders suspicious any detailed quantitativecomparison with experiment. Consequently, we opt for a more qualitativecomparison.More specifically, we first observe that the Fourier series of R(ϑ,t) forthe first snapshot of the interface is approximately given by a power-lawm−32 over the range 1 < m < 33 (the best fit exponent is −1.41) for boththe annulus and extraction experiments (see Figure 3.15). We then usethis spectrum to initialize the modes in the linear stability analysis, adoptthe Herschel-Bulkley fit in Equation (3.1), match the average radius of thefirst snapshot ro = L and the experimental parameters, (Q,H,Vinit), andassume that the surface tension is that of water (cf. [13, 24]). For thecomparison with tests in roughened Hele-Shaw cells, we assume that theroughening the plates fully removes an effective slip. For the tests withsmooth cells, we adopt the the wall-slip model outlined in Chapter 2 withparameters A = 1.56× 104 Pa sm−1 and τW = 0.18τY , guided by the cali-brations of [16] (although those calibrations do not correspond precisely tothe experimental conditions or fluids that we use here). All this translates51to dimensionless parameter settings of (Bi,σ,Σ,Bw) = (0.31,0.014,0,0.057)and (0.29,0.028,0.21,0.051), for the rough and smooth plates of Figure 3.10(respectively), or (Bi,σ,Σ,Bw) = (0.75,7.1× 10−4,0,0.135) and (0.9,1.3×10−3,1,0.16), for those of Figure 3.11.Figure 3.16 presents amplification factors, |Rˆm|/R0(t) as densities overthe (V,m)−plane for the experiments of Figures 3.10 and 3.11. Here, gen-eralizing §2.2.1, we set R = R0(t) + Rˆm(t)eimθ for the perturbed unstableinterface in the linear stability analysis, where r = Ro(t) =√1±2t is the(dimensionless) unperturbed position. When a single mode is excited, thisamplification factor corresponds to SR(t)/SR(0).Figure 3.16: Linear stability predictions for the amplification factors,|Rˆm|/R0(t), of the angular mode m when the volume withinthe unstable interface is V . Panels (a,b) correspond to the an-nulus experiments of Figure 3.10, with the tests for roughenedcells shown in (a) and for the smooth cells in (b). Shown arethe amplitudes, |am(t)|, of the Fourier modes of R(s, t), nor-malized by their initial values and plotted as a density on the(V,m)−plane with the colour map indicated. The black linesshow ten equally spaced contours of constant |am(t)| from thelinear stability analysis; the dotted lines extend these contoursoutside the color scale used for the experiments (with the samespacing).For the annulus experiments in Figure 3.16(a,b), the linear theory pre-dicts that the most amplified modes are similar to those seen experimentally.Roughened cells are also predicted to be rather more unstable than smoothcells in line with the arguments in §2.2.1, numerical observations [20] andexperimental observations of displacements of viscoplastic foams [28]. De-521 2 3 4 5 6 710 -210 -11005 10 15 20 25 3010 -210 -110035404510 -110015202510 -210 -1Figure 3.17: Observed and predicted mode amplitudes |Rˆm|/R0, forthe angular wavenumbers m indicated (colour coded from redto blue for increasingm). The experiments correspond to thosein Figures 3.10 and 3.11, and the panels are laid out as inFigure 3.16. The solid lines and error bars show the average,plus or minus the standard deviation, of the repeated tests;in (a), the dotted lines show the results for each test. Thedashed lines are the theoretical predictions, scaled to matchthe observed averages at a particular volume.spite this qualitative agreement, the amplifications predicted theoreticallyare rather higher than those seen experimentally (by a factor of ten or so),as found previously for Newtonian Saffman-Taylor fingering [37]. For thesmooth cells, the most amplified modes are predicted to be higher than seenexperimentally, and for rough cells the amplifications are again about tentimes too high.One possible explanation for the discrepancy in amplification factor isthat the finger patterns do not remain in the linear regime for the duration ofthe experimental time series (cf. [37]). However, a more likely explanationis that the theory systematically overestimates the degree of growth in thehigher wavenumbers, as we highlight in Figure 3.17. This figure compares aselection of experimental Fourier mode amplitudes with theoretical predic-tions for the experiments of Figures 3.10 and 3.11. Although comparisonsof this kind are often obscured by experimental variability, some of the testsshow satisfying repeatability (e.g. the mode m = 3 in Figure 3.17(a), with53relatively low error bars). Moreover, one sees clearly in Figure 3.17 thatthe theoretical amplification factors are consistent with observations for thelower modes. Once the angular wavenumber reaches m= 6 or higher, how-ever, the theory overestimates growth, a feature that becomes steadily worseuntil surface tension cuts off the range of growing wavenumbers. This over-estimation is not only a property of the viscoplastic fingering problem: aswe show in the Appendix, the experiments that we conducted with cornsyrup also demonstrate the same discrepancy. Therefore, the issue may wellpoint to a limitation of the Hele-Shaw approximation for our experiments(which is not altogether surprising in view of our simplistic treatment of theinterface).Note that the third case in Figure 3.17, for extractions in roughenedcells, has amplification factors that are almost exponential (on this semi-logarithmic plot, the dashed lines are almost straight), because the radius ofthe interface does not vary by much throughout the tests. The planar linearstability analysis of §2.2.1 therefore applies approximately, with `≈m. Forthe other experiments, the curvature of the amplification factors emphasizesthe time-dependent axisymmetric state.Because the theory probably overestimates the growth of high wavenum-bers, diagnostics of the kind in Equation (3.3) are inevitably also overesti-mated. Nevertheless, we continue with this diagnostic to explore whetherthe trends observed experimentally in §3.4 can be broadly reproduced bythe model. The predictions of this analysis for the annulus tests (vary-ing one of (Vinit,H,Q) whilst fixing the others) are included as insets inFigures 3.13 and 3.14. The agreement is qualitative, if not quantitative,barring the sharp upturns in SR(t) at the end of the experiments where thelongest finger is about to break through the outer interface. The trends ininitial volume are consistent with weakly-nonlinear analysis and nonlinearsimulations for displacement of a Newtonian annulus [2, 45]. However thetrends with varying flux are less well captured. Despite this, the most am-plified modes predicted by the linear stability analysis are, once more, infairly good agreement the trends in the observed numbers of fingers. This isconsistent with observations from separating plate experiments [19, 23] and54planar Saffman-Taylor experiments [28, 32] that show good agreement withthe theoretical wavelength of the most unstable mode in §2.2.1 (cf. [14]).Note that model results for smooth plates in Figures 3.14, 3.16 and 3.17predict that the fluid deforms mostly by both sliding and internal shear.Only in the test with the lowest flux in Figure 3.14(d) does a significantfraction of the fluid eventually fall below the internal yield criterion givenby S = Bi (cf. Figure 2.2). In particular, this arises for radii that are aboutfour times the initial position of the inner interface.55Chapter 4ConclusionsThis thesis has presented an in-depth theoretical and experimental analysisof the Saffman-Taylor instability in a Hele-Shaw cell. This phenomenon findsapplications in many industrial and geophysical flows, and this research areaof fluid mechanics is currently very active and relevant.Our main motivation for studying the Saffman-Taylor instability in aHele-Shaw cell was to understand the impact of fluid slip on the walls of thecell on the strength of the instability. Previous work on this topic alwaysassumed Carbopol did not slip and the surfaces of the cell did not makea difference. However, while conducted experiment on both smooth androughened plexiglass plates we observed drastic differences. We hypothesisedthis to be caused by slip. This was indeed confirmed by our theoretical andexperimental analysis.In Chapter 1, after introducing the focus of this manuscript, we gave abrief overview of the relevant literature. We then covered some importantpreliminary background knowledge, setting the stage for the next chapters.We introduced the governing equations for the flow of a fluid in a Hele-Shaw cell, first derived by Henry Selby Hele-Shaw [25]. Next, we discussedconstitutive laws of Newtonian, power-law and viscoplastic fluids, and inparticular, the Herschel-Bulkley model, which we used to describe the Car-bopol solution used in our experiments.56In Chapter 2, we laid out in detail the full theoretical model for a vis-coplastic Hele-Shaw flow. We gave the relevant scales and the dimension-less equations for the problem. We then described the sheared regions, thepseudo-plugs and the fake yield surfaces. The main point of this chapter wasto analyse the impact of introducing some slip to the problem. By allowingthe fluid to slip at the walls of the cell, following the stick-slip model, weshowed that the basic linear instability was damped, for both the expand-ing annulus and retracting disk. Thus, we concluded that the roughenedcell was more prone to instability than the smooth one. We then looked atthe impact of slip on the nonlinear instability of a single finger on a pla-nar interface. This investigation suggested that adding slip may plug upand completely block the cell at late times, but the plug regions are muchsmaller than the ones in the solution without slip. Finally, we conducted asimilar linear stability analysis for the expanding annulus and the retract-ing disk, deriving the formulae for the instantaneous growth rates for bothcases.In Chapter 3, we explored the Saffman-Taylor instability under an exper-imental lens, in the circular geometries. First, we looked at the Newtonianannulus problem and explained some of the diagnostics we used to quantifythe instability of the interface. We also looked at a miscible Newtonianexperiment to ensure that the gap of our Hele-Shaw cell was not too smalland that it was not fully controlling the instability. Then, we delve intothe viscoplastic case. We used an aqueous suspension of Carbopol, with ayield stress of about 7 Pa. We conducted repeated tests on smooth andrough plates and, for both expanding annuli and disk extraction, with thesame gap thickness, initial volume and flux. We were able to quantify thedifference in the instability of the interface between the rough and smoothcases using our diagnostic SR. We conclude that the experiments on roughplates are much more unstable as the fingers grow much quicker, in accor-dance with our theoretical prediction. Furthermore, we notice that the rootsof the fingers in the rough cell become immobile at late times, in contrastwith the smooth cell experiments. Next, we completed suites of experimentschanging the gap thickness, flux and initial volume in the annulus configura-57tion, on rough and smooth plates. The main variation was with the cell size,with an increase in h significantly reducing the number of fingers and theirstrength. Varying flux was less impactful on rough plates, with the maintrend observed being that a single finger dominates more quickly as the fluxis decreased. The highest flux on smooth plates displays a stronger insta-bility, but the lower fluxes again remain comparable. Finally, we comparedthe amplification rates of the lower angular Fourier modes with the theoryand observed fairly good agreement. However, the higher wavenumbers areoverpredicted theoretically.Further possible workFollowing the results shown in this thesis, there are many possible avenues formore research. First, one could consider developing an improved numericalscheme, aiming to gain further knowledge on the impact of slip on the non-linear instability. It would be interesting to see what differences emerge whenconsidering more than one finger in the slot and the circular geometry.We established in this research that slip can have a drastic impact on theSaffman-Taylor instability. One might consider experiments where the slipis even more extreme than the ones we showed. Using hydrophobic platesof simply a more “slippy” fluid are two possible options.Another possible path would be to look at fully sliding free surface flows,where another interesting instability can occur. In work published in 2019,Sayag & Worster [41, 42] describe the existence of linear instability in atwo-dimensional extensional flow of power-law fluids, further confirmed andgeneralized to include flows of yield stress fluids by Ball & Balmforth [6].However, this instability has not yet been observed experimentally. The pat-terns in the experiments in [41, 42] are caused by a fracturing phenomenon.Therefore, one might try to observe the extensional flow instability andalso see what patterns would emerge when the flow is both in an unstableSaffman-Taylor and unstable Sayag-Worster configuration.58Bibliography[1] A. N. Alexandrou and V. Entov. 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Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Balmforth, Neil J.
A study of the Saffman-Taylor instability in viscoplastic fluids
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