Evolving Bubbles in a Hele-Shaw Cell

Lachlan TyrrellSupervisors: A/Prof Scott McCue & Dr Chris Green

Queensland University of Technology

February 27, 2016

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Abstract

There is a well-studied problem in applied mathematics of determining the shape ofa bubble as it moves through a device called a Hele-Shaw cell. The problem is two-dimensional, and involves solving Laplace’s equation in an unknown domain. Thus,complex variable techniques can be applied. In particular, the goal is to determinea conformal mapping from a canonical domain (like the unit disc) to the fluid area.One of the relevant boundary conditions includes the effects of surface tension on thebubble boundary. In this project, the so-called “selection problem” of determiningthe shape and speed of bubbles shall be revisited.

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Contents

1 Introduction and Background 4

2 Mathematical Model 62.1 Initial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Deriving the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 The Zero Surface Tension Problem . . . . . . . . . . . . . . . . . . . 102.5 The Non-Zero Surface Tension Problem . . . . . . . . . . . . . . . . . 10

3 Sample Solutions 113.1 Zero Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Non-Zero Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Conclusion 154.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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1 Introduction and Background

Fluid mechanics is a well studied discipline of mathematics, used to describe thebehaviour of liquids and gases under various conditions. Specifically of interest tofields like geology and earth science (hydrology) is the behaviour of fluids in porousmedia; namely fluids in the soil. This fluid mechanics problem can be described byDarcy’s Law, named for Henry Darcy who first derived the equation. Conductingexperiments on the movement of fluids in soil is difficult to complete, as the threedimensionality makes it challenging to monitor and record valid results. However,there is a two dimensional analogue to this three dimensional problem, which canbe much more easily studied. The experimental apparatus used to conduct this twodimensional replication is called a Hele-Shaw Cell.

A Hele-Shaw Cell consists of a pair of glass plates, with a thin gap between them,which is then filled with a viscous fluid. A hole drilled into the top plate allowsfor a second, and less viscous, fluid to be pumped into the cell. This simple ex-periment permits the interface between the two fluids to be observed and studied.The interface describes the shape of the growing bubble of less viscous fluid, andcan be modelled using partial differential equations for fluid dynamics. Prior studiesby Saffman and Taylor [2], show that when a less viscous fluid displaces a more vis-cous one, the interface between them is unstable and will produce a fingering pattern,so called because the shapes made by the pattern seem to resemble elongating fingers.

The difficulties with using fluid dynamics to describe this behaviour arise fromthe complexity of solving moving boundary problems. Thankfully, this area of inter-est allows us to average the three dimensional fluid equations over a small gap, inorder to create a system of two dimensional equations.

When these flows are averaged into two dimensions, the following system is de-rived: v = − d2

12µ∇p, ∇ · v = 0, ⇒ ∇2p = 0 where v is the velocity, p is the

pressure, d is the distance between the two plates of the Hele-Shaw cell, and µ isthe fluid viscosity [3]. These give rise to v = ∇φ, and from the assumption of theincompressibility of fluids, this must follow Laplace’s Equation of φ,

∇2φ = 0 (1)

Laplace’s equation implies that the use of complex variable theory is applicable tothe problem, and is indeed the technique used herein to find solutions to the shape ofthe bubble. Therefore the complexity of the moving boundary issue can be bypassed

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by translating the problem into a nicer domain. Indeed, complex variable theoryallows for exact, and nontrivial, solutions to be found for the problem in its simplestcase: without physical effects, such as surface tension. However, these solutions allcontain the so-called Saffman-Taylor instability which can cause unrealistic shapesto form in the solution, such as cusps with infinite slope. When surface tension is in-troduced to the problem, the Saffman-Taylor instability can be limited, which stopsthe creation of these unrealistic solutions. While this surface tension term makes theproblem more difficult, mathematically, it is able to change the problem from beingill-posed, to being well-posed instead [3].

Consequently, the aim of this report is to describe solutions for the shape of agrowing bubble over time in a Hele-Shaw Cell. This is done firstly for the zero surfacetension case, and secondly for the non-zero surface tension case. The former wassolved analytically, giving exact solutions, while the latter was numerically solved.In both cases, solutions are presented.

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2 Mathematical Model

2.1 Initial Model

Let Ω be a fluid region of viscous fluid surrounding a bubble of less viscous fluid.The boundary between these two regions is δΩ. We also have that, as the distancefrom the source increases, the velocity potential approaches a valueφ ∼ Q ln r as r →∞.

In order to complete the model, we need two boundary conditions: a kinematiccondition and a dynamic condition. The first is given by δφ

δn= vn, which relates the

movement of the interface to the velocity of the fluid (moving in the same, normal,direction). The dynamic condition determines the boundary data on the interfaceallowing for the incorporation of physical effects, such as surface tension. This dy-namic condition is described by φ = σκ, where σ is the surface tension, and κ thecurvature of the bubble. Figure 1 displays the conditions and the state of the cell.From these, the model is built and examined below.

φ = σκ

vn =∂φ

∂n

φ ∼ Q ln r

∇2φ = 0

Figure 1: The experimental setup, with boundary conditions

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∇2φ = 0 in R2\Ω

vn =∂φ

∂non ∂Ω

φ = σκ on ∂Ω

φ ∼ Q ln r as r →∞

In this report, two cases are examined for the dynamic condition. The simplecase without surface tension will be explored first, which simply allows the dynamiccondition to become φ = 0. The non-zero surface tension case is investigated second,where a non-zero value for surface tension is used.

2.2 Conformal Mapping

Due to Laplace’s Equation, complex variable theory is the framework used to developsolutions of this problem. In particular, we will use conformal mapping to transformthe problem into a new domain, where the problem no longer has a moving bound-ary. The idea is to map the interface between fluids onto the surface of a unit circlein the new domain. The viscous fluid maps to the inside of the unit circle, while theless viscous fluid maps to the outside. Finally, the far field infinity is mapped ontothe origin. This mapping can be examined in Figure 2, with shaded sections beingmapped to each other.

On a unit circle, Laplace’s equation is simple to solve, and so the problem becomesa different one. Instead of trying to find a solution to the moving boundary prob-lem, we are only trying to find the equation which maps from the original domainto the new domain. The conformal mapping can be represented by the complexfunction z = f(ζ, t), where the function tends toward a(t)ζ−1 as ζ goes to zero.The coefficient function, a(t), is a real valued function. ζ can be represented asζ = eiν , where |ζ| = 1 is the boundary of the unit circle. Finally, it is knownthat because φ satisfies Laplace’s equation, it is the real component of a functionW (z, t) = φ(x, y, t) + iψ(x, y, t) [3].

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Physical z-plane

φ = σκ

vn =∂φ

∂n

φ ∼ Q log |z|

∇2φ = 0

Auxiliary ζ-plane

1

z = f(ζ, t)

Figure 2: The intention: to map between unit circle and the original domain.

2.3 Deriving the Model

The model must be expressed in terms of the complex function of the mapping inorder to be solved. On the unit circle, δφ, where |ζ| = 1, the normal can be foundby taking the unit tangent to the movement of the interface, and rotating it by π

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radians. This gives the unit normal:

n̂ =−izν|zν |

(2)

This is under the condition that ζ = eiν .

The velocity at any point on the surface of the unit circle can be simply rep-resented as ft, the time derivative of the mapping function. The velocity on theinterface is given by the complex inner product of the velocity with the unit normal.

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vn =Re(vn̂) =Re(ftzν(−i)|zν |

(3)

vn =Re(ft ¯ζfζ)

|ζfζ |(4)

The right hand side of the kinematic condition may be written as

δφ

δn= ∇φ · ¯̂n = Re

( ¯δWδz· ¯̂n)

(5)

Using the definitions of inner product and the unit normal, along with Re(¯δWδz

) =φx + iφy, it can be shown that the right hand side becomes:

δφ

δn= Re(ζwζ) on ζ = |1| (6)

The curvature term involves a more in-depth derivation, but it can be shownstraightforwardly that it is given by:

κ =dt̂

dS· n̂, where (7)

dt̂

dS=

zνν|zν |2

(8)

=−ζ(ζfζ)ζ|zν |2

, substituting in to the equation for κ (9)

κ = Re( dt̂dS· ¯̂n)

(10)

= Re(ζ(ζfζ)ζ ¯ζfζ|ζfζ |3

), where ζ = eiν (11)

Then the kinematic condition may be written as:

Re(ft ¯ζfζ) = Re(ζwζ) (12)

Where w is the complex potential function.

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2.4 The Zero Surface Tension Problem

When there is no surface tension, the dynamic condition is simply zero. This comefrom φ = σκ where, in this case, the surface tension term is zero. This means thatthe complex potential simplies down to

w = Q log(ζ) (13)

This allows the Polubarinova-Galin equation [3] to be produced, combining ourboundary conditions,

Re(ft ¯ζfζ) = Q (14)

Assuming a form of the solution for f allows for many analytical, explicit solutionsto be recovered from this equation. Subsequently, its simplicity permits exact solu-tions to be found using equation 12 and solving the ordinary differential equationsfor the coefficients in the assumed form.

2.5 The Non-Zero Surface Tension Problem

For the case where physical effects are not ignored, the complex potential is signif-icantly more complicated. For non-zero surface tension, the complex potential isgiven by

w = Q log(ζ)− σK (ζ, t) (15)

K is an analytic function in our domain whose real component can be found usingthe curvature term as follows:

Re(K ) = Re(ζ(ζfζ)ζ ¯ζfζ|ζfζ |3

)(16)

So this may be taken and substituted into the boundary condition, finding [3]:

Re(ft ¯ζfζ) = Q− σRe(ζKζ) (17)

Unlike in the zero surface tension case, this cannot be solved for analytically forexplicit solutions of f . However, solutions can be obtained numerically using theassumption of the form of f as a power series. The numerical scheme is able to solvefor the coefficients, so solutions to this case may still be found.

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3 Sample Solutions

3.1 Zero Surface Tension

The zero surface tension case is useful as it allows for explicit solutions to be foundanalytically. One of the possible behaviours which can be exhibited by the interfacefrom these exact solutions is cusp formation. The following function from Howisonexhibit this behaviour [1]:

f(ζ, t) = a(t)ζ + b(t)ζN (18)

Where a and b are real coefficient functions, and N > 1 is an integer. In this case,the solution will be N+1-fold symmetric.

Taking f and substituting it into the model recovers the following pair of equa-tions:

−aȧ+Nbḃ = Q, Nȧb− aḃ = 0 (19)

From the second of these equations, we find that b = �aN , where � is b(0). Usingthis relationship, it is possible to plot the solution up until the point where cuspsform, see Figure 3. At this point the solution ceases to exist, or has blown up, due tothe instability of the problem. This is obviously not representative of reality, wherecusps do not form in the Hele-Shaw cell.

3.2 Non-Zero Surface Tension

For the non-zero surface tension case, the system of equations being solved is signifi-cantly more difficult than the zero surface tension case.In this instance, the functionbeing used is an infinite power series, where the intention is to solve for the coeffi-cients to produce a plot of the solution. As mentioned in section 2.5, the governingequation being solved is:

Re(ft ¯ζfζ) = Q− σRe(ζKζ), with (20)

Re(K ) = κ = Re(ζ(ζfζ)ζ ¯ζfζ|ζfζ |3

)(21)

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−2 0 2

−2

0

2

x

y

Cusps (z-plane)

Figure 3: The exact solution for f(ζ, t) is plotted, showing that it evolves to cuspformation over time.

The power series representation of the mapping function is, from [3]:

f(ζ, t) =∞∑n=0

an(t)ζ2n−1 (22)

, From this, the derivatives needed are recovered to be

ft =∞∑n=0

ȧnζ2n−1 (23)

ζfζ =∞∑n=0

(2n− 1)an(t)ζ2n−1 (24)

ζ(ζfζ)ζ =∞∑n=0

(2n− 1)2an(t)ζ2n−1 (25)

Where an(t) are real coefficients, and we assume symmetry between the x and y axes.

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Figure 4: A six-fold symmetric solution to the non-zero surface tension problem usingsurface tension σ = 0.02

In order to solve this system, we truncate the series at N terms. We then treat ζas our time-like variable and implicitly solve the system at N points according to

ζ = eiπj4N , for j = 0, ..., N − 1 (26)

Using the power series and its derivatives, it is simple to find the values for κ ateach value of j. From these, K can be found and ζKζ evaluated. Once this is done,ode15i in MATLAB can be used to solve implicitly for the coefficients.

Of note in this solution for the interface are the behaviours displayed, which canbe seen in Figure 4. In this solution, both fingering and fjords are present. Fjords arethe thin inlets or troughs in the fingering pattern, which remain filled by the viscousfluid over time. They do not behave exactly as in experimental Hele-Shaw cells, butthey are evidenced here. The fjords, as in the real Hele-Shaw cell, steadily moveoutwards from the source at the original. While the fingering is present, it does

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not contain tip splitting. This means that the fingers remain whole as they growoutwards, and never split into two thinner fingers. Using a power series solutiondoes not produce tip splitting and branching, but the fingers are at least realistic upto that point, unlike some exact solutions where they widen dramatically or formcusps.

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4 Conclusion

The evolution of a bubble in a Hele-Shaw cell can be determined using complex vari-able theory. While more susceptible to instability, the problem can be solved exactlyif physical effects including surface tension are ignored. The solutions produced whensurface tension is introduced are much better behaved, though mathematically muchmore complicated to find. However, these nicer solutions are also not exact solutionsand can only be found numerically. Both of these methods have the capability toproduce characteristic behaviours for a Hele-Shaw cell, though the exact solutionsare not as well behaved and are much closer to blowing up.

This model and the functions examined are not original, and have been studiedbefore. For further work, more advanced applications such as having the bubble movethrough the domain instead of remaining fixed (and expanding), and examining theeffect of surface tension in this case could be studied and may yield more interestingresults if a selection problem was undertaken on this case. One other extension isthe inclusion of other physical effects such as kinetic undercooling.

4.1 Acknowledgements

I would like to personally thank AMSI for running the Big Day In conference, aswell as providing the summer research scholarship. I would also like to thank mysupervisors Associate Professor Scott McCue and Doctor Chris Green for giving methe chance to study Hele-Shaw cells as well as for all of the help and guidance whileI was taking part in the project. Finally, I’d like to also thank Doctor MichaelDallaston for helping to clarify part of the model which was of particular confusion.

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References

[1] S. D. Howison. 1986. “Fingering in Hele-Shaw cells.” Journal ofFluid Mechanics 167: pp 439-453. Accessed February 26th, 2016.doi:10.1017/S0022112086002902

[2] G. Taylor and P. G. Saffman. 1959.“A Note on the Motion of Bubbles in aHele-Shaw Cell and Porous Medium.” The Quarterly Journal of Mechanicsand Applied Mathematics 12 (3): pp 265-279. Accessed February 26th,2016. doi:10.1093/qjmam/12.3.265

[3] Dallaston, M.C. 2013. “Mathematical models of bubble evolution in a Hele-Shaw Cell.” Queensland University of Technology

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Introduction and BackgroundMathematical ModelInitial ModelConformal MappingDeriving the ModelThe Zero Surface Tension ProblemThe Non-Zero Surface Tension Problem

Sample SolutionsZero Surface TensionNon-Zero Surface Tension

ConclusionAcknowledgements