andong he dissertation

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The Pennsylvania State University

The Graduate School

CONFORMAL MAPPING AND VARIATIONAL METHODS FOR

INTERFACIAL DYNAMICS IN FLUIDS

A Dissertation in

Mathematics

by

Andong He

c2011 Andong He

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

August 2011


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The dissertation of Andong He was reviewed and approved by the following:

Andrew Belmonte

Associate Professor of Mathematics

Dissertation Co-Advisor, Chair of Committee

Mark Levi

Professor of Mathematics

Dissertation Co-Advisor

Qiang Du


Diane Henderson


Tong Qiu

Assistant Professor of Civil Engineering

John Roe


Head of the Department of Mathematics

Signatures are on file in the Graduate School.


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Abstract

In this thesis we will be considering three problems in fluid dynamics whose com-mon features involve free surface dynamics. In a two-dimensional space, theseproblems can be conveniently formulated by conformal mapping methods. Thehistory and status quo of application of conformal mapping methods to fluid dy-namics are briefly discussed in Chapter 1.

In Chapter 2 we study theoretically and experimentally the deformation of afree surface between two fluids in a gravitational field, due to a jet in the lighterfluid impinging at right angles to the surface. A mathematical model is builtusing the method of conformal mapping. The strength of our method lies in itsgeneral applicability to analytically study the interface between two fluids in agravitational field, one of which has an arbitrary potential velocity field, while theother is assumed to be motionless. An asymptotic solution is derived for the cavityshape with the density ratio of fluids as the small expansion parameter.

We present in Chapter 3 an unsteady nonlinear Darcys equation which includesinertial effects for flows in a Hele-Shaw cell, and discuss the conditions under whichit reduces to the classical Darcys law. In the absence of surface tension we derive ageneralized Polubarinova-Galin equation in a circular geometry, using the methodof conformal mapping. The linear stability of the base-flow state is examined byperturbing the corresponding conformal map. We show that inertia always tendsto stabilize the interface, regardless of whether a less viscous fluid is displacing amore viscous fluid or vice versa.

In Chapter 4 a mathematical model of reactive Hele-Shaw flows when twoimmiscible fluids meet, chemically react and form an elastic interface is considered.This reaction brings about significant changes in the interfacial tension, which iscrucial in determining the stability of such a system. We model this by treatingthe interface as an elastic membrane whose bending stiffness depends on the localcurvature. We derive from energy variation a dynamic boundary condition at the

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interface. An analysis of the roles that several parameters play in affecting thestability is performed. We are able to qualitatively account for the anomalous

fingering instabilities that have been seen experimentally.

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Table of Contents

List of Figures viii

List of Tables xii

Acknowledgments xiii

Chapter 1Introduction 11.1 Precursor of complex variables in fluids Joukowski transformation 11.2 Surface deformation due to an impinging jet . . . . . . . . . . . . . 31.3 Hele-Shaw flows and Saffman-Taylor instability . . . . . . . . . . . 3

1.4 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter 2Deformation of a liquid surface due to an impinging gas jet 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Experimental observations . . . . . . . . . . . . . . . . . . . . . . . 102.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Analytic continuation of the force balance equation . . . . . 162.3.2 Inclusion of surface tension . . . . . . . . . . . . . . . . . . . 172.3.3 The dipole jet exact solution . . . . . . . . . . . . . . . . . 182.3.4 Specifying the jet Milne-Thomson jet . . . . . . . . . . . . 20

2.3.5 Approximations and solutions . . . . . . . . . . . . . . . . . 232.4 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . 29

Chapter 3Inertial effects on viscous fingering 323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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3.2 A heuristic interpretation of viscous fingering . . . . . . . . . . . . . 343.3 Derivation of the Unsteady Nonlinear Darcys equation . . . . . . . 37

3.4 Conformal mapping approach . . . . . . . . . . . . . . . . . . . . . 413.4.1 Polubarinova-Galin equation . . . . . . . . . . . . . . . . . . 413.4.2 Exact solutions to the PG equation . . . . . . . . . . . . . . 443.4.3 Generalization of the Polubarinova-Galin equation . . . . . . 47

3.5 Linear stability via conformal mapping . . . . . . . . . . . . . . . . 513.6 Asymptotical stability ofb(t) for small c1 . . . . . . . . . . . . . . . 533.7 Effects of small inertia on the linear stability: WKB approximation 563.8 Conserved quantities in Hele-Shaw flows . . . . . . . . . . . . . . . 58

3.8.1 Conservation of scaled linear momentum . . . . . . . . . . . 583.8.2 Conservation of scaled angular momentum . . . . . . . . . . 61

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 4Fingering instability in a reactive system 644.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Derivation of stability condition . . . . . . . . . . . . . . . . . . . . 664.3 Elastic boundary condition . . . . . . . . . . . . . . . . . . . . . . . 684.4 Bending stiffness function . . . . . . . . . . . . . . . . . . . . . . . 694.5 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5.1 Intrinsic instability . . . . . . . . . . . . . . . . . . . . . . . 734.5.2 Thorough instability (zero flux) . . . . . . . . . . . . . . . . 744.5.3 Instability (zero flux) . . . . . . . . . . . . . . . . . . . . . . 74

4.5.4 Non-zero flux the role of Atwood numberAM . . . . . . . 744.5.4.1 Positive Atwood number . . . . . . . . . . . . . . . 754.5.4.2 Non-positive Atwood number . . . . . . . . . . . . 77

4.5.5 Onset of instability . . . . . . . . . . . . . . . . . . . . . . . 784.6 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.7 Conclusions and Future thoughts . . . . . . . . . . . . . . . . . . . 82

Appendix ADerivation of Polubarinova-Galin equation with surface tension

(3.53) 85

Appendix BVariations of a two-dimensional curve 88

B.0.1 Variations along the normal direction . . . . . . . . . . . . . 88B.0.2 Variations along the tangential direction . . . . . . . . . . . 89

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Bibliography 91

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List of Figures

1.1 Preimages (dashed circles) and images (solid curves) under theJoukowski transformation (1.2). Circles are centered at (, ) withradiusR: (a) = 1, = 1.2, R= 3.3. (b) = 0.4, = 0.9, R=2.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 A cavity in a cup of apple juice is produced by blowing air througha straw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 The basic phenomenon: a gas jet (here nitrogen) originating froma nozzle (height H, diameter ) impinges on the free surface of aliquid (here oil), producing a cavity of depth hand width w. . . . . 9

2.3 The depth h and surface width w of the cavity as functions ofH,at fixed jet velocity. The dashed lines shown are fits to the scaling:h Hc, andw Hc, withc = 0.67; the solid line in (a) representsthe implicitly defined curve h (h+H)2 from [106]. . . . . . . . . 12

2.4 A rough, time-dependent cavity for a high-speed nitrogen jet im-pinging into water. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 (a) Jet impinging onto water exhibits breathing effect: the cavityis in its smallest regime. The liquid is 50% glycerol+50%water, = 6 cP, = 1.13 g/cm3, = 58 dyn/cm; (b) the cavity is inits largest regime, the same fluid as in (a). The oscillation cycleis between (a) and (b). (c) Jet impinging onto a more viscousfluid produces a stable cavity. The liquid is PDMS silicone oil,= 30, 000 cP, = 0.97 g/cm3,= 19 dyn/cm. . . . . . . . . . . . 13

2.6 The steady-state cavity shape is almost unaffected by an interfacebetween two different liquids comprising the bottom layer, nor is

there any evidence of flow deformation of this interface. The threefluid regions shown are: 1 - air; 2 - corn oil (= 0.92 g/cm3, = 65cP); 3 - 80/20 glycerol/water mixture (= 1.21 g/cm3, = 60 cP). 14

2.7 Sketch of the conformal mappingffrom the parametricZ-plane tothe physical W-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Milne-Thomson jet and its image with the solid wall in theZ-plane. 21

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2.9 Representation of the jet in the -plane. . . . . . . . . . . . . . . . 222.10 Free streamlines of Milne-Thomson jet give by (2.24), withd= U=

1. The wall sits at y = 0. . . . . . . . . . . . . . . . . . . . . . . . . 232.11 Approximating the function 1(x, 0): the circles are taken from(2.19), and the solid line is the fitted Gaussian function to a Milne-Thomson jet with U= 30,d= 0.5. . . . . . . . . . . . . . . . . . . 26

2.12 Calculated cavity shapes for the Milne-Thompson jet: (a) fixed jetwidthd= 0.15, forU= 25, 30, 35, 40, 45; (b) fixed jet speedU= 30,for d = 0.1, 0.125, 0.15, 0.175, 0.2. . . . . . . . . . . . . . . . . . . . 27

2.13 (a) A vaselike cavity in silicone oil produced by a jet of nitrogen;(b) deformed liquid surfaces under impingement of dipoles centeredat k= 2, with strengths C= 100, 300, 500(eq. (2.14)). . . . . . . . . 28

2.14 Experimental visualization of the jet exiting the orifice and expand-

ing to a coned region. Dots have been added to guide the eye. . . . 302.15 Cavity in a silicone oil formed by an inclined jet. The jet angle (from

liquid surface) is 60oC. (a) onset of the cavity; (b) a new cavityis formed behind the former one, both traveling to the left; (c) thenew-formed cavity grows, while the old one fades away leaving acusp-like tail on the bottom. . . . . . . . . . . . . . . . . . . . . . . 31

2.16 Superposition of 70 images that are taken by Particle Image Ve-locimetry(PIV) technique. . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Constant injection of air into 50/50 glycerol/water mixture. Fin-gering pattern emerges due to Saffman-Taylor instability. . . . . . . 33

3.2 Illustration of viscous fingering: when air (left) displaces oil (right),Saffman-Taylor instability occurs. . . . . . . . . . . . . . . . . . . . 36

3.3 Illustration of viscous fingering: when oil (left) displaces air (right),no Saffman-Taylor instability occurs. . . . . . . . . . . . . . . . . . 37

3.4 Viscous fingering produced by a periodic injection. An air bubbleis created in a Hele-Shaw cell pre-filled with a 80/20 glycerol/watermixture, then an oscillating injection is performed. Due to disparityin other physical parameters such as net injection rate, a directcomparison with Fig. 3.1 should not be expected. . . . . . . . . . . 39

3.5 Polubarinova-Galins cardioid solution to (3.33). A fluid is ex-

tracted at the origin with the outmost circle as the initial shapeof the boundary. The map fceases to be univalent at T = 2.9,when plotting is stopped. . . . . . . . . . . . . . . . . . . . . . . . . 45

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3.6 Rational solutions to the PG equations (3.24). (a) solution given byf1 in (3.35); (b) solution given by f2 in (3.36). The exterior ellipse

is the initial fluid domain and shaded region is the domain whencusps are formed. (taken from [36]) . . . . . . . . . . . . . . . . . . 463.7 Comparison between constructed solution (3.38), and the experi-

ment. Both images are taken from [47]. . . . . . . . . . . . . . . . . 473.8 Sketch of the conformal map from the parametric -plane to the

physicalZ-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.9 Solution to (3.55) usingode45in MATLAB. Initial conditions are

b(0) = b(t) = 0.01, Q0 = 1, Qp = 0.5, w = 10, a(0) = 0.2, c1 =10, n= 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.10 Plot ofc1D1(t) with injection rate given by (3.20). Two differentsets of parameters are: Qp = 0.8, = 20 (solid curve); Qp = 0.5,

= 5 (dotted curve). Other parameters are the same: a(0) = 0.2,c1 = 0.1, n = 3. See also [39]. . . . . . . . . . . . . . . . . . . . . . 57

3.11 A viscous fluid (t) bounded by closed curves 1 and 2 is ex-panding due to the injection of another fluid at the origin O. Thepressure is equal to 0 on 1 andPe(atmospherical pressure) on 2. . 59

4.1 Plot of bending stiffness function (4.17), with0= = 1, C= 0.5. . 704.2 Stable(5A1 + A2 0), unstable(5A1 +A2 > 0) and intrinsically

unstable(A1 0) regions when flux J = 0. The T-unstable regionand I-unstable region coincide in this case. . . . . . . . . . . . . . . 73

4.3 Instability regions for AM = 0.45, J=230, = 1. Dotted line

encloses the I-stability region. Modes larger than 4 have flat tailingregion near Caxis; while n = 3, 4 modes do not. Dotted curveencloses the I-instability region. . . . . . . . . . . . . . . . . . . . . 75

4.4 Instability regions forAM= 0.8, J=200,= 1. Dotted line enclosesthe I-stability region. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Instability regions forAM = 1, J=5, = 1. Dotted line enclosesthe I-stability region. . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.6 Values of the dimensionless inverse square radius= (a )2 verses the

Atwood number of mobility at the onset of instability, for C= 0.8.The simplest mode n = 2 is the most unstable for AM 0. . . . . 79

4.7 Values of the dimensionless inverse square radius= (

a )

2

verses theAtwood number of mobility at the onset of instability, for C= 0.8.Below AM 0.48, the lowest mode n = 2 is the most unstable. . . . 80

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4.8 Jn (black dashed) and J+n (black solid) as functions of effective

radiusa in the log-log plot. Blue lines are from classical Hele-Shaw

flows (Li et al. [91]) for comparison. The parameters are = 1,C= 0.5 and AM = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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List of Tables

4.1 Signs of several linear combinations of A1(C, ) and A2(C, ) forcertain values ofC and . . . . . . . . . . . . . . . . . . . . . . . . 71

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Acknowledgments

Firstly and most importantly, I am indebted to my dissertation advisor AndrewBelmonte, who led me to the realm of fluid dynamics with his enlightening intelli-gence and patience. He has greatly nurtured my enthusiasm in science.

I also thank my co-advisor Mark Levi, as well as other committee members forvaluable suggestions and discussions in helping me finish this thesis.

A seed needs to receive sunshine from outside for a healthy growth, to me itis the GFD summer program at Woods Hole and many other academic occasions.Among people who inspired me the most are Joseph Keller, Harvey Segur, Ed-ward Spiegel, Darren Crowdy, John Lowengrub, ... Although any list would beincomplete, my thankfulness goes to every of them.

My family is aways charging my energy bath when this research is being carriedon. Special thanks are to my wife, Xiaozhou. Without her constant love andsupport this thesis would be impossible.

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Dedication

TO GOD BE THE GLORY.

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Chapter1

Introduction

1.1 Precursor of complex variables in fluids

Joukowski transformation

The analogy between holomorphic functions and fluid flows reflects the glitter of

human wisdom in understanding the relationship between the mathematical and

physical worlds. If u = (u, v) (where u, v : R2 R) represents the velocity

field of a two-dimensional fluid flow, then the incompressibility and irrotationality

conditions read ux+vy = 0,uy vx = 0. (1.1)Therefore a flow being incompressible and irrotational (I-I) is equivalent to the

quantityu ivbeing holomorphic on the complex plane. Moreover, any harmonic

function remains harmonic under a conformal transformation. These two facts

pave the way for applying complex variable methods in fluid dynamics.

The application of complex-variable methods dates back to Nikolai Joukowski(1847-1921), who was regarded as one of the founding fathers of modern aerody-

namics and hydrodynamics. He was the first scientist to explain the mathematical

origin of aerodynamics, and the first to calculate the lift force generated by a body

immersed in a moving fluid based on his circulation hypothesis (see [3] for his

work in aerodynamics). Among his many great contributions, the conformal map-

ping formulation he built provides a systematic way of studying flows past various


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Figure 1.1. Preimages (dashed circles) and images (solid curves) under the Joukowskitransformation (1.2). Circles are centered at (, ) with radius R: (a) = 1, =1.2, R= 3.3. (b) = 0.4, = 0.9, R= 2.7.

objects in R2. The mapping

z z+b2

z, for z C (1.2)

wherebis a real constant is called the Joukowski transformation.

The efficacy of the Joukowski transformation may be best illustrated with the

help of the Milne-Thomsons circle theorem, which explicitly gives the complex

potential of an I-I flow past a unit disk [1]. To fully characterize how an arbitrary

I-I flow pasts an object ofarbitrary1 shape, one only needs to know the conformal

map from the unit disk to the object domain. The existence of such a map is

guaranteed by the Riemann mapping theorem. Fig. 1.1 shows some circles and

their images under the map (1.2).

1The object in our discussion must be simply-connected. A generalization of I-I flows pastdoubly-connected objects can be found in, for example, [21, 49]


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1.2 Surface deformation due to an impinging jet

Conformal mapping approaches can be used to find exact forms of flows not onlyin infinite domains, but also in those with confined geometry. For instance, a jet

exiting through the orifice on a wall can be explicitly calculated [7]. In this thesis

we particularly focus on a different case: two jets impinge towards each other,

producing another two jets. If the upstream velocity and width of each incoming

jet are specified, then those of the resultant jets can be determined [69]. We can

readily use this to model a jet impinging onto a solid wall: two jets with the same

width but opposite velocity can be treated as the mirror image of each other with

respect to the wall. The pressure on the solid wall is of course not uniform: due to

Bernoullis principle the pressure is larger at the center of the wall, and decreases

monotonically away from the center.

What would happen if the solid wall is replaced by a free surface of a liquid? A

free surface is assumed to meet the pressure balance condition, while the pressure

is given by Bernoullis theorem from the jet side, and by hydrodynamics from the

liquid side. We can write down the equation which is satisfied by the conformal

map from the solid-wall plane to the free-interface plane. By doing so we transform

the generally difficult problem of determining a free interface and the corresponding

flows into a relatively simple one solving a differential equation on a fixed simple

domain. In [38] we derived a formulation for the problem described above and

solved the equation asymptotically, with the densities ratio of the flow and bottom

liquid taken to be the small parameter. Since our formulation is independent of

the choice of air flow, any two dimensional I-I flow acting on a heavy fluid in the

gravitational field can be solved in this way.

1.3 Hele-Shaw flows and Saffman-Taylor insta-bility

Another area where conformal mapping approaches have been successfully applied

is on flows in a small gap sandwiched by two flat glass plates. This experimental

setup is called aHele-Shaw celland was originally used by Hele-Shaw [42] to visu-


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alize streamlines of a flow past obstacles. Due to the large ratio of dimensions of a

Hele-Shaw cell (the vertical dimension is much smaller than the horizontal dimen-

sions), flows within it are often treated as two-dimensional. Using a lubrication

approximation one can show that the flow is irrotational. Therefore incompressible

Hele-Shaw flows are natural objects of study in fluid dynamics for the conformal

mapping method.

Saffman and Tylor [93] discovered that when a less viscous fluid pushes a more

viscous one in a Hele-Shaw cell, the interface is unstable. This phenomenon is ofter

referred to asSaffman-Taylor (ST) instability. In their seminal paper [93] they were

able to find exact solutions of the interface using conformal transformations. In

fact, as early as in 1945 Polubarinova [83] and Galin [32] obtained independentlyan equation for a Hele-Shaw flow:

Re

fft

= Q

2, on || = 1. (1.3)

Heref(, t) is the time-dependent conformal map from the unit circle in the para-

metric plane onto the fluid interface in the physical plane, and Re denotes the

real part function. Eq. (1.3) is now called the Polubarinova-Galin (PG) equation.

Thereupon in principle solving the PG equation solves the Hele-Shaw flow problem

with zero surface tension. The surface tension effects can also be included into the

PG equation [36].

An important aspect in the study of Hele-Shaw flows is how ST instability can

be suppressed or controlled. Li et al. [91] have shown that the emerging dendritic

morphology can be confined to a self-similar pattern with k dominant fingers if the

injection rate is C(k)t1/3 whereC(k) is a function ofk . The equation governing

the flow motion they used is the classical Darcys law

u= P, (1.4)

where u is the lateral fluid velocity and P the pressure. Nevertheless, the roles

that the inertia of the fluids are playing remain unknown. To examine how inertia

may affect the ST instability, one has to re-derive the Darcys law from the 3D

Navier-Stokes equation. This has been done by the author in [39]. The equation

we obtained captures two physical parameters which are otherwise neglected in the


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Darcys law. The conformal mapping method enters to generalize the PG equa-

tion. Instead of solving the resultant equation which is algebraically complicated,

we performed linear stability analysis to the base-flow solution. Linear stability

analysis was done in a mathematically convenient and elegant way due to the fact

that {zk}k=0,1,... form a basis for all holomorphic functions. Then by applying a

WKB approximation we showed that small inertia always tends to stabilize the

interface regardless of whether the system originally favors stability or instability

[39].

1.4 Other applicationsDue to its ability of absorbing all geometrical complexity into a time-dependent

conformal map, the conformal mapping method can be used to formulate many

other interfacial dynamics problems. Such examples include deformation of bub-

bles and drops [19], selective withdrawal [105, 108, 109], and surface waves [27].

It should be noted that viscous flows also enjoy the unique advantages of complex

variable methods. For a Stokes flow both the stream function and stress function

are known to be biharmonic. In order to characterize the flow, two conformal maps

f and g in the form of

f+ zg (1.5)

are needed. A complex variable treatment of free surface deformation for Stokes

flows can be found in [53].

Usage of conformal variable theory in fluid dynamics is not limited to the two

classes discussed above, namely, harmonic and bi-harmonic functions. It can also

be extended to a broader class of problems involving non-harmonic fields [8]. This

provides mathematical tools for new applications of conformal mappings to, e.g.

non-Laplacian fractal growth.

In the course of exploring how complex variable methods can be used to solve

problems arising from fluid dynamics and the like, the author was encouraged by a

discussion with V. E. Zakharov, who commented that ...while many approximate

solutions are available to us, its time to look for exact solutions using conformal


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maps.2 With this inspiration, the author is willing to continue his endeavor to

explore more fundamental phenomena in nature and how they can be understood

from a mathematical prospective. This thesis is not only the end of his old journey,

but also the beginning of a new one.

2At the Eastern Sectional AMS meeting, Penn State University (2009).


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Chapter2

Deformation of a liquid surface due

to an impinging gas jet

2.1 Introduction

The tangential flow of a fluid along the interface with another fluid generally in-

duces motion in the second fluid, a process known asentrainment. Another config-

uration is when the fluid flow is normal to the surface; in this case an indentation,

or cavity, is produced due to the momentum of the flow. In natural settings,both processes are usually present, such as when the wind blows across an ocean

or lake, or when one cools off a hot drink by blowing on it (Fig. 2.1). A better

understanding of such air-fluid interactions will enable us to make improvements

in some industrial applications, such as in the steel industry, where a supersonic

jet of oxygen impinges on molten iron to convert it to steel (known as the basic

oxygen conversion process [96]). In the arc welding process, a high energy plasma

jet impinges on a molten metal pool, creating an indentation which can affect the

stability and efficiency by limiting the welding speed [13]. Since our model allowsfor any incompressible and irrotational air flow, it may also be used to model geo-

physical problems regarding wind-wave interaction, such as wave propagation in

water [27, 37].

While entrainment is a viscous effect, the normal impingement of a flow can

deform a surface even in an inviscid flow. In the simplest situation, an air jet at


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Figure 2.2. The basic phenomenon: a gas jet (here nitrogen) originating from a nozzle(heightH, diameter ) impinges on the free surface of a liquid (here oil), producing acavity of depth h and width w.

[88] observed two types of instabilities of the gas-liquid system: oscillations of the

interface, and dispersion of liquid droplets. As the jet velocity exceeds a critical

value, the interface will start to oscillate; when the velocity increases further, the

oscillation of the interface becomes more vigorous, until the instability leading to

droplet creation is triggered.

It is natural to consider a conformal mapping approach to this problem, al-

though this would impose a two-dimensional restriction. While the use of confor-

mal mapping to study fluid flow is a classic subject, pioneered by Joukowski, it is

still an active area of research which has continued to develop in new directions.

In the 1990s Tanveer [102] suggested the use of conformal mappings for the un-

steady free surface problems of irrotational flows, an idea which has been further

developed by others [27, 89]. Crowdy applied this method to a broader category

of flows, and succeeded in obtaining exact solutions for other complicated systems

[19, 20]. Specific to our problem of an impinging jet on a free surface, Olmstead

& Raynor solved for the cavity shape in the small depression case using conformal

mapping method [76]; the numerical scheme was later improved by Vanden-Broeck

[107]. Here we derive a general condition satisfied on the interface between two

fluids in a gravitational field one of which is assumed to be motionless, without


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requiring the cavity to be small. The force balance equation on the interface can

be analytically continued to the upper half plane, somewhat similar to the ap-

proach taken in [19]. This approach is generalized to include surface tension. An

asymptotic expansion of the solution is also obtained.

The majority of this work has been published in [38].

2.2 Experimental observations

To set the stage for a mathematical approach to this problem, we first present a

simple experimental system in which an air jet produces a steady cavity in a heavier

liquid. A transparent cylindrical tank, made of plexiglas with height 7.5 cm anddiameter 10.2 cm, is placed under a nozzle attached to a nitrogen (density= 1.25

103g/cm3) cylinder (pressurized). The nozzle has an inner diameter = 0.075

cm, and can be moved up and down freely along a fixed metal frame, on which

its position is read from a calibrated scale. The ratio of nozzle distance from the

undisturbed surface Hto nozzle diameter ranges from 20 to 72 (see Fig. 2.2).

The jet speed is typically 100 300 cm/s, controlled by a valve and pressure gauge

connected to the nozzle. A high-speed camera (Phantom v5.0) is placed at the

same level as the undisturbed liquid surface to capture the cavity profile. All dataare taken at room temperature, 25oC.

One of the most often used fluids in our experiments is polydimethylsiloxane

(PDMS) silicone oil, which has viscosity = 30, 000cP, density 0.97g/cm3 and a

nominal surface tension with air = 21.5 dyn/cm. Choosing the characteristic

length to be 1 cm (see Fig. 2.3), then the Weber number is

W e=V2L

103,

and the Bond number is

Bo =gL2

40.

Therefore the surface tension is negligibly small compared to both the inertial and

gravitational effects. The Reynolds number for the air jet near the cavity is about

103.

A simple quantitative characterization of the cavity is made by measuring its


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depth h and width w; these quantities for different fluids are shown in Fig. 2.3

as a function of the height of the nozzle H. We first note that both quantities

are independent of viscosity for the three liquids shown: glycerol (= 1, 500 cP),

silicone oil (= 30, 000 cP) and water (= 0.89 cP). Thus all data sets collapse

onto the same curve, even without normalizing the axes. The data appear to follow

the scaling relations h H2/3 and w H2/3, as shown in the figure. These

relations are consistent with the assumption that the total energy expenditure

of the jet to maintain the cavity is constant: as the nozzle gets closer, the cavity

becomes deeper yet narrower, such that the quantity M ghremains constant, where

Mis the total mass displaced from the cavity. Since M hw2, this implies that

hw constant, as observed. Note however a systematic deviation of the depth datafrom the curve h H2/3, for values ofHexceeding about 40 mm (corresponding

to H 50). In fact, a turbulent eddy viscosity approach due to Turkdogan,

assuming a constant momentum flux in the jet [106], leads to the implicit scaling

relationh (h + H)2 which appears to fit the experimental data better for large

H, as shown in Fig. 2.3a.

In our subsequent mathematical treatment of this problem, we will make a num-

ber of simplifying assumptions, which include neglecting the motion of the bottom

liquid under the impinging jet, assuming that the cavity has a steady shape, and

treating the liquid volume as infinite. Here we briefly record experimentally the

deviations of observations from these assumptions. While our experiments are re-

stricted to the cases where the cavity is observed to be stationary, it is easy to push

the experiment to the point where time-dependent surface disturbances, similar to

wind waves [62], are generated on the surface. At higher jet speeds, a rougher

and time-dependent cavity is observed, from which droplets may be ejected, as

shown in Fig. 2.4. In silicone oil, we find that under certain circumstances (large

H/h and high jet velocity), surface waves are generated near the cavity waist and

propagate upwards at a slow speed. These waves are attenuated and fade away

before reaching the cylinder wall. For a less viscous liquid such as water, this

phenomenon is less regular because the cavity is deeper, and thus apparently more

unstable (Fig. 2.4). Generated waves also propagate further along the surface, and

in this case the whole way to the sides of the cylinder in the experiment.

If the viscosity of the liquid under impingement is small or the jet velocity is


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(a)

(b)

Figure 2.3. The depth hand surface width w of the cavity as functions ofH, at fixedjet velocity. The dashed lines shown are fits to the scaling: h Hc, andw Hc, withc= 0.67; the solid line in (a) represents the implicitly defined curve h (h+ H)2 from

[106].

large, then the cavity is not stable. Various types of cavity oscillation have been

categorized [33, 88]. While all of them have been observed by us, we add a direct

comparison between a breathing cavity and a stable one. With other things

being equal, we perform two experiments which differ only in the bottom fluid:


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Figure 2.4. A rough, time-dependent cavity for a high-speed nitrogen jet impinginginto water.

Figure 2.5. (a) Jet impinging onto water exhibits breathing effect: the cavity is inits smallest regime. The liquid is 50% glycerol+50%water, = 6 cP, = 1.13 g/cm3, = 58dyn/cm; (b) the cavity is in its largest regime, the same fluid as in (a). Theoscillation cycle is between (a) and (b). (c) Jet impinging onto a more viscous fluidproduces a stable cavity. The liquid is PDMS silicone oil,= 30, 000 cP, = 0.97 g/cm3,= 19 dyn/cm.

50%glycerol+50%water and PDMS silicone oil (Fig. 2.5). For the glycerol/watermixture case, the cavity undergoes expansion-contraction-expansion circles, and

the cavity surface is very irregular. Fig. 2.5(a) shows a snapshot of its smallest

regime and Fig. 2.5(b) shows a largest one1. On the other hand, the cavity formed

1The reason we do not use pure water is because then the cavity is too irregular for its unstabletype to be identified. It is likely that for fluids with such low viscosity various types of instabilitycoexist.


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Figure 2.6. The steady-state cavity shape is almost unaffected by an interface betweentwo different liquids comprising the bottom layer, nor is there any evidence of flowdeformation of this interface. The three fluid regions shown are: 1 - air; 2 - corn oil(= 0.92 g/cm3, = 65 cP); 3 - 80/20 glycerol/water mixture (= 1.21 g/cm3, = 60cP).

in silicone oil is much more stable (Fig. 2.5(c)). Both height and width of the

cavity are well-defined, so we can assume that it has a steady and stable shape

(except the generated surface wave, as discussed above). Thus a steady-flow theory

is allowed.

To verify that the motion of the lower liquid has little effect on the form of the

cavity, we use two immiscible liquids with different densities but close viscosities

(corn oil and 80/20 glycerol/water mixture) as the bottom fluids (see Fig. 2.6).By varying the depth of corn oil while fixing H and the exiting speed of the jet,

we observe essentially no change at the interface between the two fluids. Notice

that the maximum depth of the cavity in the first plot is slightly larger than that

in other two due to the density difference.

The simple experimental system we have presented above produces a steady

cavity, thus offers a clear set of questions to be studied mathematically. Motivated

by this, we build a 2D mathematical model for the shape of the cavity and the

steady flow pattern producing it. Although our experiment is clearly 3D, we expectthat a similar phenomenon would occur in 2D; moreover an analytic approach

may be more likely with the use of conformal mapping techniques particular to

2D. We will thus assume that the flow is incompressible and irrotational. This is

a reasonable assumption, since the Reynolds numbers of the jet are not small in

our experiment (Re 103). Moreover in previous work the viscosity of the gas

did not have an apparent effect on the main features of the cavity, as discussed in


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[5]. Note however the full validity of this mathematical approximation is untested.

We consider two types of jets: one in which the imposed velocity field approaches

zero infinitely far away, and one in which it continues to flow along the surface out

to infinity. As we will show, only in the second case is it possible for the jet to

produce a central cavity below the original surface.

2.3 Mathematical Formulation

Let us consider a gas jet flowing towards a wall sitting on the real line in the

complex plane, as shown in Fig. 2.7. Denote byZ(z) and W(w) the parametric

plane and physical plane, respectively. Let f :z w be the conformal mappingfrom the flow region in the Z-plane to the region with deformed surface in the

W-plane, with the real line mapped to . We assume that the interface is smooth

(i.e., with no corners or cusps), so f does not vanish or blow up on y = 0. An

extra condition f(z) 1 as z is imposed to guarantee that flows in two

planes are the same at infinity. Denote by F1(z) the complex potential for the

gas flow in theZ-plane, thenF2(w) :=F1(f1(w)) is the complex potential in the

W-plane. The flow speeds q(z) and Q(w) are related by the mapping f as

Q(w) =dF2dw = dF1dz dwdz

1= q(z)|f(z)| . (2.1)We apply Bernoullis theorem to the gas flow in the W-plane, and obtain

1

21Q

2(w) +pg(w) =1

21U

2 +p, (2.2)

where pg(w) is the pressure in the gas at any point w in the flow region, and p

and Uare the pressure and jet velocity at infinity, respectively.

Under the assumption that the bottom liquid layer is motionless, the pressure

in the liquidpfis only due to the hydrostatic head 2gy. Thus for any pointw ,

we have

pf(w) =p 2gIm(w). (2.3)

The liquid is coupled to the flowing gas by the normal stress boundary condition


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Z-plane W-planew=f(z)

1

2Wall Liquid

Figure 2.7. Sketch of the conformal mapping f from the parametric Z-plane to thephysicalW-plane.

at the free surface. Neglecting the surface tension, this condition is simply

pf(w) =pg(w) for w . (2.4)

Combining equations (2.1)-(2.4), we obtain a real-valued force balance equation,

which is written in Z-plane as

q2(z) = (2gImf(z) +U2)|f(z)|2 on y= 0, (2.5)

where= 1/2 is the ratio of densities.

2.3.1 Analytic continuation of the force balance equation

The force balance equation (2.5) satisfied on the real line uniquely determines the

conformal mappingfin the upper half plane C+. Following Crowdy [19], we extend

(2.5) to C+

using analytic continuation. Define the conjugate analytic functionoff asf :z f(z). As is well-known, on y = 0 we have

Im(f) = 1

2i(f(z) f(z)),

|f(z)|2

=f(z)f(z),


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and

q2(z) = dF1

dz 2

=F1(z)F

1(z). (2.6)

With these identities (2.5) becomes

F1(z)F1(z) = [gi(f(z) f(z)) +U2]f(z)f(z) for z C+. (2.7)

The problem to be solved has now been reduced to finding a univalent conformal

mapping f(z) satisfying (2.7), given some complex potential F1(z) prescribed by

the imposed jet.

2.3.2 Inclusion of surface tension

Although the surface tension can be neglected based on our experiment, it is

straightforward mathematically to include it in our analysis, with (2.4) replaced

by

pf(w) += pg(w), (2.8)

on , where denotes the constant surface tension and is the curvature

=

XxYxx XxxYx(X2x+ Y

2x)

3/2 (2.9)

Let the interface be represented by X(x, y) +iY(x, y) = f(z), where f is from

the z= x+iy plane to W =X+iY plane, then we have

f(z) =Xx+iYx,

f(z) =Xxx+iYxx.(2.10)

Using (2.10) we can write (2.9) as

=Im(f(z))f(z)

|f(z)|3 . (2.11)

If we repeat how we obtained (2.5), with (2.4) replaced by (2.8) the following


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force balance equation is obtained

q2(z) = 2gIm(f) +U2 22

Im(f(z))f(z)|f(z)|3

|f|2 . (2.12)Analogously, this equation can also be extended to C+ as we did in sec. (2.3.1)

F1(z)F1(z) = [gi(f(z) f(z)) +U2]f(z)f(z) + i(f(z)f(z) f(z)f(z))

2(f(z)f(z)) 12 .(2.13)

As we shall see in sec. (2.3.5), the conjugate analytic function in both equations

(2.5) and (2.13) can be replaced by fitself and its derivatives using the symmetry

of the jet-cavity system.

2.3.3 The dipole jet exact solution

If the general impinging jet problem is defined only by the occurrence of a cavity,

then there will be many distinct mathematical cases to be classified within this

problem, corresponding to different kinds of imposed jets. This also implies some

freedom in choosing the prescribed jet. For instance, Banks & Chandrasekhara

considered both a turbulent and a laminar jet, with different mathematical ap-proaches for each [5]. We first consider a simple, explicit jet - a dipole in the

complex plane - which leads to exact solutions. Consider in polar coordinates the

velocity field for a dipole u = (u(r), u()) = (Csin r2 ,Ccos r2 ), where u

(r) and u()

are velocity components in r- and - directions respectively, and the constant C

is the strength of the dipole. Forr = 0 this is a 2D potential flow with complex

potential iz [58]. The complex potential for a dipole with a stiff wall at y = 0 is

obtained by superposing a downward dipole centered at (0, k), of complex poten-

tial G1

= Cizki

, with a upward dipole centered at (0, k), of complex potential

G2 = Ciz+ki

. The combined complex potential G = G1+ G2 = 2Ckz2+k2

gives the

complex velocitydG

dz =

4Ckz

(z2 +k2)2.


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As with equation (2.5), the force balance condition reads

q2

(z) = 2gImf(z)|f

(z)|2

on y= 0,

where we note that no U2 term appears since the flow speed vanishes at infinity.

The analytic continuation of this equation is then

16C2k2z2

(z2 +k2)4 = igf(z)f(z) f(z) f(z) for z C+. (2.14)

Equation (2.14) admits two exact solutions: fa(z) = Az(z+ki)2

and fb(z) = Az(zki)2

where A = (4C2k/g)1

3 . In fact, if f(z) is a solution, so is f(z). However,fa and fb do not approach 1 at infinity. Instead, both fa and fb map the realline to unit circles, centered at (0, 14k ) and (0,

14k ), respectively; thus they are

not physically meaningful solutions. We have not so far succeeded in proving the

general existence or uniqueness of solutions to (2.14), with the condition that f

behaves asymptotically aszat infinity. But the following result shows that such a

solution, if it exists, can not be a rational function.

Theorem 2.3.1. There exists no rational functionf(z) satisfying equation (2.14)

and such thatf(z) zasz .

Proof. Suppose that f(z) has a pole of order j at w / R, then

f(z) = (z)

(z w)j,

whereis meromorphic in C+ and(w) = 0. Notice that

f(z)

f(z) =

(z)(z w)j (z)(z w)j

(z w)j(z w)j ,

whose numerator is nonzero at z=w; therefore the right-hand side of (2.14) will

inherit a singularity at w. So w must be either kior ki (k= 0).

Now we can assume that ki and ki are poles of f(z) with order m and n

respectively, thenf(z) has the form g(z)(z+ki)m(zki)n

, withg meromorphic in C+. The

condition f(z) zas z implies g(z) zm+n+1 as z , so we can write


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g(z) =zm+n+1 +azm+n +O(zm+n1),for some a C. Then

f(z) f(z) = 1(z+ki)m(z ki)m [(a a 2ki)z2m +O(z2m1)], ifm n.Sincef(z),f(z) O(1) and z2(z2+k2)4 O( 1z6 ), it follows by comparing two sides of(2.14) thata a= 2ki. Consideringn mgives usa a= 2ki. Contradiction!

2.3.4 Specifying the jet Milne-Thomson jet

So far the jet in the solid-wall plane has not been chosen. It is in fact inessential

for such a choice, so long as it is an incompressible and irrotational ideal flow.

We choose it to be the one given by Milne-Thomson, and shall call it the Milne-

Thomsonjet [69]. While general Milne-Thomson jets consist of two incoming jets

from arbitrary directions with arbitrary widths, we only consider here a special

case: a jet of width 2d and velocity U from (0, +) and its mirror image with

respect to the x-axis (Fig 2.8). Denote by A and C the far upstreams of the jet

and its image, respectively, and denote by B and B the far downstreams. By

simple symmetry argument the resultant flow has only horizontal components on

the xaxis (all kinematic and dynamic boundary conditions are satisfied there).

Therefore we can identify this configuration with one jet impinging onto a solid

wall sitting on the xaxis. Conservation of linear momentum requiresB and B

to be (, 0) and (+, 0) respectively, and the jet width therein is the same as

upstream: 2d. If F1 is the complex potential on the Z-plane, then the complex

velocity of the flow is given by

(z) =dF1

dz =u iv. (2.15)

On the free streamlines AB, CB, AB and CB, the stream function respectively

takes the value 0, 0, 2dUand 2dU(Fig. 2.8 and Fig. 2.9), and

=U ei. (2.16)


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Figure 2.8. Milne-Thomson jet and its image with the solid wall in the Z-plane.

Next we apply the following well-known result to f() = iF1() =() i()

andR = U, where and are the velocity potential and stream function, respec-

tively.

Lemma 2.3.2. (Schwarz formula) If f() = () +i() is analytic within|| < R and continuous on || R, then

f() = 1

2

20

(ei)Rei +

Rei d+ i(0), for || < R. (2.17)

The result of such application is

z=2d

log

U+

U +i log

U i

U+i+const. (2.18)

Since z= 0 is a stagnation point ( assumes value zero there), the constant in

the above expression must be zero. Therefore on the solid surface y = 0 where

=u Rwe have

x=2d

[log(

U+ u

U u) +i log(

iU+ u

iU u)]. (2.19)


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This equation gives the velocityuon the wall as an implicit function of the position

x.

Figure 2.9. Representation of the jet in the -plane.

Locations of free streamlines for a Milne-Thomson jet can be determined from

(2.18). Take AB

as an example: we substitute = U ei

into (2.18), and uset= cot 2 (1, ) when (0,

2

) to get

z= x+iy=2d

log

1 +ei

1 ei

+i log

1 iei

1 +iei

=

2d

log

i cot

2

+i log

i cot

4

2

=

2d

2+ log(cot

2) +i

2+ log cot

4

2

.

(2.20)

Splitting the real and imaginary parts gives us

x= d+ 2d log cot 2 , (2.21)

y= d+ 2d

log cot4

2

. (2.22)


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From (2.21) we can solve for t to get

t= exp(x d)2d . (2.23)

Plugging (2.23) into (2.22) we obtain the relation between x andy as

y= d+2d

log ctanh

4

xd

1

. (2.24)

Free streamlinesAB and AB are drawn in Fig. 2.10.

4 3 2 1 0 1 2 3 40

1

2

3

4

A

B B

Figure 2.10. Free streamlines of Milne-Thomson jet give by (2.24), with d = U = 1.The wall sits at y = 0.

2.3.5 Approximations and solutions

In order to solve the interface equation, we can take advantage of the left-right

symmetry of the jet (the analogue of an axisymmetric jet in the 3D case), to

further simplify the problem; of course this limits the generality of the results.

Symmetry of the jet implies that of the cavity, hence the conformal mapping


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f(x, y) =(x, y) +i(x, y) must satisfy

(x, y) = (x, y),(x, y) =(x, y),

(2.25)

for all x, y R. Therefore

f(z) = f(z), for y 0,which implies the following identity on y 0:

Im(f) =

1

2i(f(z) +f(z)). (2.26)

Moreover, the complex velocity of the jet flow in the parametric plane dF1dz =u iv

possesses the same symmetry property as f(z), so

dF1dz

= dF1dz

(z),

and

q2(z) = dF1

dz 2

= dF1

dz dF1

dz= F

1(z)F

1(z). (2.27)

Furthermore, it follows from (2.25) that x(x, y) = x(x, y),

x(x, y) = x(x, y).(2.28)

Thereupon,

f(z) =x(x, y) ix(x, y) =x(x, y) +ix(x, y) =f(z). (2.29)

Using eq. (2.26), (2.27) and (2.29) the interface equation (2.5) can be reduced to

F1(z)F

1(z) = [gi(f(z)+f(z))+U2]f(z)f(z) for z C+. (2.30)

The advantage of studying (2.30) rather than (2.7) is that the right-hand side of

(2.30) does not depend on f.


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In general, exact solutions to (2.7) or (2.30) can only be expected when the

prescribed complex potential of the jet F1(z) has certain analytical form. We

therefore seek the approximate solution of the conformal mapping f in terms of

the density ratio, which is typically in the order of 103 in our experiments. Iff

is assumed to possess an asymptotic expansion in of the form

f(z) =f0(z) +f1(z) +2f2(z) +O(

3), (2.31)

then (2.7) becomes

F1F1= [gi(f0f0)+(gi(f1f1)+U2)+O(2)][f0+f

1+O(2)][f0+f1+O(

2)]

TheO(1) term yields:

f0(z) f0(z) = 0.Although this equation does not have a unique solution, we choose the leading-

order solution to bef0(z) =z. Since= 0 corresponds to the case when the density

of the impinging jet is zero, this particular solution has the physical meaning that

an impinging jet with zero momentum will cause no depression on the bottom

liquid at all.

TheO() term yields:

F1F1= gi(f1 f1) +U2, (2.32)

which on the real line reduces to

1(x, 0) =q2(x) U2

2g , (2.33)

where 1(x, y) = Imf1(z). In particular, |1(0, 0)| = U2/2g is the maximum

depth of the cavityhmax, which is independent of the jet width d. In other words,

2ghmax = 1U2/2, which is the same as the standard pressure scaling given in

[5]. We remark that a similar treatment for the equation involving surface tension

(eq. (2.13)) yields atO()

F1F1 = gi(f1 f1) +U2 + i2 (f1 f1 ). (2.34)


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For the profile of a symmetric 2-D jet, we adopt the one given by Milne-

Thomson for a jet impinging on a rigid plate (sec. 2.3.4): The jet velocity u(x)

at any point x on the solid wall is given implicitly by eq. (2.19). Note that the

Milne-Thomson jet is not the only candidate, and in fact our mathematical for-

mulation and solution procedure are independent of the choice of jet profile.

Figure 2.11. Approximating the function 1(x, 0): the circles are taken from (2.19),and the solid line is the fitted Gaussian function to a Milne-Thomson jet with U= 30,d= 0.5.

The harmonic function 1(x, y) on the upper half plane with boundary con-

dition (2.33) is solved numerically by MATLAB using a finite difference method.

The real part off1 at any point Pcan be calculated as

1(P) =

PO

1y

dx 1

xdy,

where O is chosen to be the origin. The domain is chosen to be [L, L] [0, 2L]

for a large L, with boundary condition 1 =y on the three sides of the rectanglefor which y = 0. The boundary value 1(x, 0) in (2.33) is given by the Milne-

Thompson jet via equation (2.19), which we approximate by a Gaussian function

ae(xc)2, as has been done previously [5]. The main reason for this is numerical con-

venience; indeed, the Gaussian provides an excellent approximation, as illustrated

by Fig. 2.11.

Since there are two parameters at our disposal (jet width d and jet speed at


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infinityU), we plot the approximate solution f(z) =f0(z) + f1(z) in Fig. 2.12 for

various d and U. From this we see that, when jet velocity increases with width

fixed, both the effective depth and width of the cavity increase; when jet width

increases with velocity fixed, only the width of the cavity gets larger.

(a)

(b)

Figure 2.12. Calculated cavity shapes for the Milne-Thompson jet: (a) fixed jetwidth d = 0.15, for U = 25, 30, 35, 40, 45; (b) fixed jet speed U = 30, for d =0.1, 0.125, 0.15, 0.175, 0.2.

We also find from numerical solutions that a vaselike cavity is possible when

U/d is large (see the outermost in Fig. 2.12a), i.e., the cavity wall curves inward.

Indeed, such a cavity is observed experimentally (Fig. 2.13a). Even though the

model is 2D whereas the experiments are 3D, it qualitatively matches the cavity

shapes observed under an impinging gas jet, for different exiting velocities and jet

widths.


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Figure 2.13. (a) A vaselike cavity in silicone oil produced by a jet of nitrogen; (b)deformed liquid surfaces under impingement of dipoles centered atk = 2, with strengthsC= 100, 300, 500(eq. (2.14)).

We return briefly to the case of a dipole jet for comparison, and remark that the

liquid interface does not exhibit a depression at the center. Instead, the bottom

liquid is pulled up to form two bumps, as shown in Fig. 2.13b. This is readily

understood from the Bernoulli equation: the speed of the dipole jet vanishes at

infinity, and at any stagnation point in the flow; since the origin is a stagnation

point, it must has the same level as a surface point at infinity. Note that these

approximate solutions are quite different from the exact solutions discussed inSec. 2.3.3, since here f(z) = 1 is an imposed condition at infinity. If we define a

localized jetto be a jet whose velocity vanishes at infinity, then the above argument

can be summarized as follows.

Claim 2.3.3. A localized jet can not depress anywhere of an interface below its

original position; instead, it pulls up the interface. In particular, any stagnation

point has the same height as the infinity.

The deformed surface shapes show in Fig. 2.13b bear some resemblance to thelip formed around the cavity that has been reported in the literature for relatively

large cavities [5, 74]. We have not observed such a lip in our experiment. However,

based on our model we believe that a superposition of a Milne-Thomson jet and a

dipole could provide a realistic description of a lipped cavity. On the other hand,

if the impinging jet has a very high temperature such as in the process of basic

oxygen steelmaking, the effect of heat transfer could possibly be responsible for


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the formation of the lip [84], which lies outside the scope of the present study.

2.4 Conclusion and discussion

By knowing the pressure distribution on the interface one may generalize our model

to consider the two-phase flow problem. However this will require matching two

analytic functions from the upper and lower half plane, which is in general difficult;

exact conformal mapping solutions in such two-phase problems have been found

only in specific physical scenarios [21]. A further generalization would allow for a

time-dependent cavity, although this would necessitate finding a series of conformalmappings, each of which satisfies appropriate dynamic conditions on the interface

at each moment.

In this paper the problem of a gas jet impinging on a liquid is studied. Theo-

retically we treat the jet flow as two-dimensional, incompressible and irrotational

so that the method of conformal mapping can be applied. A complex-valued first

order differential equation (2.7) is obtained from the force balance condition satis-

fied on the interface. The analysis allows for a choice of jet profile, and we consider

two types: one mathematically more convenient and the other more physical. Al-

though an analytic solution for the latter is not feasible, the asymptotic expansion

of the solution based upon small density ratio is found. Experimentally we study

an axisymmetric nitrogen jet impinging onto several different liquids, and find that

surface tension, fluid viscosity and the container size have negligible effects on the

cavity; it is rather the density ratio that plays a role in determining the cavity size.

Although the impinging jet is assumed to be of uniform width, the actual

jet is not. In fact, the jet expands after exiting the orifice and forms a coned

region. Fig. 2.14 shows an picture from the experiments where small particles are

added nearby the orifice exit for visualization purpose. Accordingly the pressure

distribution on the solid wall will be different from Fig. 2.11. To find an exact form

of incompressible and irrotational flow with such a cone region is more involved

than the simple case we considered here. Another possible extension is to use

an infinite two-dimensional jet exiting a hole on the wall, whose exact solution is

available ([7] pp.495). Nonetheless, these choices should make only quantitative


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30

differences since our formulation is applicable for anygiven flow profile.

Figure 2.14. Experimental visualization of the jet exiting the orifice and expanding toa coned region. Dots have been added to guide the eye.

It is obvious that when a jet impinges at a non-zero angle, the cavity will not

be stable and a flow in the bottom is induced. In Fig. 2.15 three snapshots of the

transition are recorded. At the onset of impingement, a tilt cavity is formed which

propagates to the left (Fig. 2.15a); then a new cavity is formed and grows behind

the former one, moving together to the left (Fig. 2.15b); as the former cavity moves

further it starts to diminish leaving a cusp-like tail (Fig. 2.15c). This process seems

to circle repeatedly without entering a different stage.

Even in the case of normal impact, the bottom fluid is not completely motion-

less. Fig. 2.16 is a superposition of 70 images taken by particle image velocimetry

(PIV) technique, which shows clearly that a swirling motion in water under normal

impingement of an air jet. The circling orientation is clockwise for the right-half

image. This is because fluid particles right beneath the surface are dragged out-

ward due to the viscosity of the air, and sink down to the bottom and then climb

along the centerline of the container. Therefore the Kelvin-Helmholtz instability is

effective near the surface, and surface waves (discussed in Sec. 2.2) are generated

as well.


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31

Figure 2.15. Cavity in a silicone oil formed by an inclined jet. The jet angle (fromliquid surface) is 60oC. (a) onset of the cavity; (b) a new cavity is formed behind theformer one, both traveling to the left; (c) the new-formed cavity grows, while the oldone fades away leaving a cusp-like tail on the bottom.

Figure 2.16. Superposition of 70 images that are taken by Particle Image Velocime-try(PIV) technique.


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Chapter3

Inertial effects on viscous fingering

3.1 Introduction

In fluid dynamics an important aspect is to understand the evolution of the in-

terface between two immiscible fluids when one is flowing into the other. In a

fluid held between two closely spaced parallel plates, known as a Hele-Shaw cell

[42], if the displacing fluid has lower viscosity than the displaced fluid the interface

will develop hydrodynamical instability which results in highly ramified patterns

[11, 12, 78, 93]. This phenomenon is known as viscous fingering (see Fig. 3.1).Other effects such as viscoelasticity [60, 72] and chemical reaction [81] may also

influence the instability and produce dendritic morphology in the system. Pattern

formation of a similar type has been observed in a variety of nonequilibrium sys-

tems besides viscous fingering, such as crystal growth [59], electrodeposition [66]

and solidification [50].

The canonical mathematical model for Hele-Shaw flows is Darcys law, where

the flow velocity is proportional to the pressure gradient. Under the assumption

that the flow is incompressible, the pressure field satisfies a Laplaces equation;therefore, such an evolution of the free interface is also called Laplacian growth

process[6]. By averaging the velocity over the direction perpendicular to the cell,

one can reduce the 3D flow problem to 2D, thus allowing the use of complex

variable techniques.

The method of conformal mapping has been used as a powerful tool for both

analysis [11, 48] and numerical computations [2, 23] of Hele-Shaw flows. It has a


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Figure 3.1. Constant injection of air into 50/50 glycerol/water mixture. Fingeringpattern emerges due to Saffman-Taylor instability.

great advantage of transforming the generally difficult task of solving a moving-

free boundary problem into finding solutions to a single differential equation of

an analytic function on a fixed domain, usually the half plane or the interior of

the unit disk. Translating the dynamics implicit in Darcys law to this formalism

leads to an equation of the map for Hele-Shaw flows, derived independently by

Polubarinova-Kochina [83] and Galin [32] in 1945. This equation is now known as

the Polubarinova-Galin(PG) equation [48] whose many exact solutions have now

been obtained (see [36] for a comprehensive overview). The linear stability analysis

for radial fingering done by [78] can also be obtained using the PG equation, as

shown below. In the two-phase Hele-Shaw flow problem, much less progress has

been made since in general it is difficult to find a conformal map which takes a

region and its complement in the parametric plane onto the corresponding regions

occupied by the two phases. Exact solutions are only attainable in some specialcases [21, 49].

Despite the richness of studies in quasistatic Hele-Shaw flows, little effort can

be found in the literature to understand the character of the fluids inertia. The

need to answer this question manifests itself when a fluid is injected in a time-

dependent, especially fast-oscillating manner. In a recent work, Li et al. showed


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34

experimentally and numerically that the interfacial Saffman-Taylor instability in

a circular cell can be suppressed by pumping the fluid at a rate Q(t) t1

3 [91].

Yet, the equation they considered is the Darcys law. Thus, the role inertia of the

fluids plays remains veiled. Gondret & Rabaud [34] and Ruyer-Quil [90] took into

consideration the inertial effects and generalized the Darcys law under slightly

different assumptions. The equations they obtained are of the same type but have

different coefficients. Chevalieret al. [18] examined experimentally that in a linear

Hele-Shaw cell the inertial effects can be significant if the displacing fluid has low

viscosity, or large velocity, or if the cell thickness is large; namely, the modified

Reynolds number is not too small. They found that inertia has similar effects as

the capillary force, in the way that they both tend to slow down and widen thefingers.

A complete treatment of the Hele-Shaw problem, one which could include in-

ertial effects, is still lacking. It is the main goal of this Chapter to investigate how

inertial effects may alter the structure of this mathematical system, and partic-

ularly the stability of the free interface. Our aim is to provide a mathematical

description on the basis of conformal mapping method to enhance our understand-

ings of the full Hele-Shaw flow problem.

The main results of Sec. 3.3-3.7 have been published in the article [39].

3.2 A heuristic interpretation of viscous finger-

ing

Consider Darcys law

u= k

P, (3.1)

which governs the flow motion in a porous medium. Eq. (3.1) was first formulated

by Darcy based on the experimental results of water flow through beds of sand

[25]. Here k is the effective permeability, is the (dynamic) viscosity, u is the

three-dimensional fluid velocity, and Pis the pressure field.

Later on, Saffman & Taylor [93] discovered that flows in Hele-Shaw cells can


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35

also be modeled by the same law,

ui = MiP, (3.2)

with Mi = b2

12ias the fluid mobility, b the thickness of the cell, ui the two-

dimensional velocity in thexyplane, and = ( x

, y

) if the cell is perpendicular

to the zdirection. Here the subscript i= 1, 2 represents two different fluids. It

is obvious that (3.1) can be written in the same form as (3.2) by defining k as the

mobilityof Hele-Shaw flows. Justification of (3.2) for Hele-Shaw flows is shown in

Sec. 3.3: one applies the lubrication approximation to the Navier-Stokes equation,

and average them without nondimensionalization. We remark that the lubrication

approximation can also be used to study the Kelvin-Helmholtz instability in a

Hele-Shaw cell [80].

The fact that a less viscous fluid displacing a more viscous one is unstable

and the converse is stable may seem puzzling at first glance: does this lack of

fluid-fluid symmetry not contradict with our common sense that all motions are

relative? Here we provide a heuristic interpretation of this phenomenon.

We consider a scenario when air displaces oil uniformly from infinity. Since a

fluid with smaller viscosity has a larger mobility, it follows from (3.2) that mod-

erate changes in the velocity will not cause appreciable changes in the pressure.

Therefore we assume that the motion of less viscous fluid can be neglected and

the pressure is constant. We also ignore surface tension effects which only adds a

secondary effect to the interfacial stability. It then follows that the interface is a

level curve of pressure.1

In the absence of disturbances, the interface is a vertical line as in Fig. 3.2.

When disturbances occur, some region (B) on the surface bulges into the oil side

more than others (such as A) so that a finger is formed. Since the interface has

equal pressure we must haveP(A) =P(B). (3.3)

Take another point C on the same horizontal position as A and same vertical

position as B, then C is to first approximation on the same streamline as A. It

1A more complete treatment taking into account the surface tension effects can be found in[67].


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36

follows from Darcys law (3.2) that the pressure gradient is in the opposite direction

as the velocity; therefore

P(A)> P(C). (3.4)

Combining (3.3) and (3.4) we can see that the pressure gradient betweenB(higher)

andC(lower) due to the disturbance will induce a flow from B toC(see Fig. 3.2).

As a consequence, oil near the finger (B) tends to flow toward the unbulged region,

enhancing the intrusion of the fingering.

Air Oil

A

B

C

(induced flow)

Figure 3.2. Illustration of viscous fingering: when air (left) displaces oil (right),Saffman-Taylor instability occurs.

On the other hand, when oil displaces air we can assume a similar disturbance

on the surface as in Fig. 3.3. We can follow the above argument by claiming that

(3.3) still holds and (3.4) is replaced by

P(C)> P(B). (3.5)

Thus a flow in the oil fromC(higher pressure) toA(lower pressure) will be causedwhich slows down the propagation of the finger.


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37

Oil Air

A

B

C

(induced flow)

Figure 3.3. Illustration of viscous fingering: when oil (left) displaces air (right), noSaffman-Taylor instability occurs.

3.3 Derivation of the Unsteady Nonlinear Darcys

equation

We denote by , v = (u,v,w) and P the dimensional fluid density, velocity field

and pressure field in the 3D space, respectively. Let =

be the kinematic

viscosity. Consider in a Hele-Shaw cell the dimensional 3D Navier-Stokes equation

ut+ (v )u= 1Px+(uxx+uyy +uzz), (3.6)

vt+ (v )v= 1Py+(vxx+vyy +vzz), (3.7)

wt+ (v )w= 1

Pz+(wxx+wyy+wzz), (3.8)

and the incompressibility condition

ux+vy+wz = 0. (3.9)

LetUbe the characteristic horizontal flow speed, L the horizontal length scale

and h the thickness of the cell. For a typical Hele-Shaw cell hL 1, so it follows

from (3.9) that w is of order UhL . Thus we assume that

w= 0. (3.10)


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38

It follows from the z-component of (3.8) that Pis a function ofx and y only. The

above argument can be found in many standard textbooks in fluid dynamics, such

as [1].

Dimensionless variables can be defined as follows:

(x, y, z) = 1

L(x,y,

L

hz), t =

t

,

(u, v) = 1

U(u, v), P =

h2P

UL,

(3.11)

with a time scale . Using these dimensionless variables we can rewrite (3.6), (3.7)

and incompressibility condition (3.9) as

U

u t+

U2

L(uux+ v

uy+wuz) =

U

h2Px+

U

h2((

h

L)2uxx+ (

h

L)2uyy+ u

zz),

U

v t+

U2

L(uvx+v

vy+wvz) =

U

h2Py+

U

h2((

h

L)2vxx+ (

h

L)2vyy+ v

zz),

ux+v

y = 0.

(3.12)

It can be seen in the above equations that the 2

x2 and

2

y2 terms are of lower

order than 2

z 2 . Therefore the Laplacian term 2

x2 + 2

y 2 + 2

z 2 can be approximated

by 2

z 2 . In doing that and multiplying first equations in (3.12) by h2

U we obtain

ut+ Re(v )u = Px+u

zz , (3.13)

vt+ Re(v )v = Py+v

zz , (3.14)

wherev= (u, v), = ( x

, y

), = h2

is a dimensionless number, and

Re = Uh2

L = ( h

L)Re

is the modified Reynolds number. Since it can be inferred from equation (3.13)

and (3.14) that when = Re = 0 the velocity v has a parabolic profile, it is

reasonable to assume it remains parabolic when and Re are small enough. In


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39

Figure 3.4. Viscous fingering produced by a periodic injection. An air bubble is createdin a Hele-Shaw cell pre-filled with a 80/20 glycerol/water mixture, then an oscillatinginjection is performed. Due to disparity in other physical parameters such as net injectionrate, a direct comparison with Fig. 3.1 should not be expected.

dimensionless form this reads

u(x, y, z, t) =z(z 1)A1(x, y, t),

v(x, y, z, t) =z(z 1)A2(x, y, t), (3.15)

for some functions A1 andA2 independent ofz. Differentiating (3.15) twice with

respect to zyields

uzz = 2A1,

vzz = 2A2.(3.16)

We can solve (3.16) for (A1, A2) = 1

2

(uzz, v

zz).

We defineu to be the averaged value ofv= (u, v):

u(x, y, t) = (u(1), u

(2))

10

v(x, y, z, t)dz = 1

6(A1, A2),


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40

From (3.15) we can rewrite the nonlinear terms in (3.13) as

(v

)u

= u

u

x+ v

u

y =z2

(z

1)2

(A1A1x+ A2A1y)

= 36z2(z 1)2(u )u(1). (3.17)

Similarly,

(v )v = 36z2(z 1)2(u )u(2). (3.18)

We can then integrate (3.13) and (3.14) along the z-direction from 0 to 1, to obtain

the following equation (with primes dropped)

ut+6

5Re

(u )u + 12u= P. (3.19)

The dimensional version of this equation was to our knowledge first derived by

Gondret&Rabaud [34]. A similar approach was taken by Ruyer-Quil [90], who

obtained an equation in the same form but with different coefficients due to a dis-

crepancy in the way the horizontal flow velocities were averaged. Here we derive

the same equation using slightly looser assumptions than [34]; namely, the hori-

zontal velocities have more general form (as in (3.15)). Throughout this paper,

eq. (3.19) (with possibly different constant coefficients) will be called theunsteady

nonlinear Darcys (UND) equation. We remark that in the circular geometry, even

when the flow is steady (so that the inertial term ut is zero) the convective term

(u )udoes not vanish.

Our derivation assumes that and Re are both small. It is evident from

(3.19) that Darcys law is a limiting case of the UND equation when both and

Re vanish. In most experiments performed in Hele-Shaw cells, and Re are

small. Note that the Reynolds number Re = Uh is not necessarily small. In order

to understand the physical meaning of , let us consider a viscous fluid being

pumped into or removed from a cell through a point at a rate of area change

Q(t) =Q0+Qpsin(t), (Q0, Qp and are constants) (3.20)

Fluid injection will correspond to Q >0 and extraction Q


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41

|Q0| to avoid a scenario which includes both2. Leta(t) be the radius of the radially

growing (or shrinking) interface, then it follows from mass conservation that

rate of change of area = d

dt(a2(t)) =Q(t). (3.21)

It following immediately that

a(t) =

t0

Q

ds+a(0)2, (3.22)

for an initial radius a(0) 0. The time scale can be defined to be the minimal

value of the ratio of velocity of the growing circle to its acceleration within a periodof oscillating injection

mint[t0,t0+

2]

aa= min

t[t0,t0+2]

Qpcos(t)Q Q2a21 . (3.23)

Two distinct cases are worth emphasizing: (i) Qp = 0, implying 2t0+ 2a(0)2

Q0.

Therefore there is no additional timescale. We can choose = LU which identifies

with Re; (ii) Qp = 0 and is large. The dominant term in the absolute value

sign of (3.23) is, socan be chosen to be 1 , and = h2

will not be small. Wecan then infer from the second case that a rapid oscillation in the injection will

invalidate Darcys law. From an experimental point of view, the fingering pattern

produced by an oscillating injection (Fig. 3.4) seems considerably different from

the constant injection case (Fig. 3.1).

3.4 Conformal mapping approach

3.4.1 Polubarinova-Galin equation

The first complex-variable-based formulation of Hele-Shaw flow problem was given

by Polubarinova-Kochina [83] and Galin [32] independently. If surface tension

2It would be interesting to examine the case when Q0 = 0, so that the net injection over aperiod is zero. The system will then sway between stableand unstable.


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42

effects are ignored, then the time-dependent conformal mapping f(, t) from the

parametric plane (-plane) to the physical plane (z-plane) must satisfy the equation

Re

fft

= Q

2, on || = 1. (3.24)

Eq. (3.24) is called thePolubarinova-Galin (or PG) equation. There are typically

two different ways to derive the PG equation. Here we provide the one that is

extendable to later sections (sec. (3.4.3)). The other derivation can be found in

Appendix (A).

Consider a conformal mapping f(, t) from the parametric (-) plane to the

physical (Z-) plane. The complex potentials in two planes G(, t) and F(z, t) are

related through f by

G(, t) =F(f(, t), t) =F(z, t). (3.25)

Since the origin is a point source of the flow, we have for any >0,B(0,)

u nds= (area) mass flux through origin =Q.

Using Greens formula we can rewrite the above expression asB(0,)

dV =

B(0,)

nds=

B(0,)

u nds= Q.

Therefore

= Q(x), (3.26)

for a Dirac delta distribution (x). Since G(, t) is an analytic function in || 1

with vanishing real part on || = 1, it must have the form

G(, t) = Q

2log().

Due to Darcys law (3.2), the velocity potential is equal to the opposite of the

pressure P (up to a positive scalar factor). SoF(z, t) = P+i where is the

stream function. Therefore we have


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P

t = ReFt ,P =

F

z

,

u=

F

z

.

(3.27)

Notice that it is zand t, not and t that are independent variables. We taket of the conformal mapping

z= f(, t), (3.28)

to obtain

t = ftf

. (3.29)

We take t and z of (3.25) and use (3.29) to obtain

F

t =

Qt2

= Qft2 f

.

F

z =

Qz2

= Q

2 f.

(3.30)

If surface tension is ignored, then the pressure is constant on the interface. More-

over, the pressure will remain constant as the interface involves. Therefore the

dynamical condition for the pressure yields (using (3.27) and (3.30))

0 =DP

Dt =

P

t + (u )P

= Re

F

t + |

F

z|2

= Re Qft2 f | Q2 f|2(3.31)

After reorganization we finally obtain the PG equation (3.24) from the above

equation.

It should be remarked that even though in the above arguments the injection

rate Q is constant, we can easily extend it to the case when Q is a function oft


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44

(sec. (3.4.3)).

3.4.2 Exact solutions to the PG equation

The first non-trivial solution to the PG equation was constructed by Polubarinova-

Kochina [82, 83], and Galin [32], which is called the Polubarinova and Galins

cardioid. They have shown that a quadratic function

f(, t) =a1(t)+a2(t)2, (3.32)

solves the PG equation if the real coefficients a1(t) anda2(t) satisfy the system of

algebraic equations

a21(t)a2(t) =a21(0)a2(0),

a21(t) + 2a22(t) =a

21(0) + 2a

22(0) +

Qt

.

(3.33)

Any initial condition with |a2a1 | < 12 will give an univalent function f(, t) for t

[0, T). At some finite time T, f ceases to be univalent and a cusp forms. This

phenomenon of finite-time-cusp emerges before the moving boundary reaches

the sink. Such a solution is plotted in Fig. 3.5.As a matter of fact, the PG equation admits a polynomial solution of arbitrary

order:

f(, t) =nk=1

ak(t)k, ak(t) are complex-valued. (3.34)

By rotatingf(, t) toeif(ei, t) we can assume that an(0) is real. Substituting

(3.34) into the PG equation we obtain a system ofn equations forak(t) which can

be numerically solved. However, it was shown that all these solutions exhibit the

finite-time-cusp phenomenon [44].

Rational solutions to the PG equation were first found found by Kufarev [56,

57], who found two such solutions:

f1(, t) =a(t)(1 b(t))

1 c(t) , (3.35)


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1 0.5 0 0.5 1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Figure 3.5. Polubarinova-Galins cardioid solution to (3.33). A fluid is extracted at theorigin with the outmost circle as the initial shape of the boundary. The map fceases tobe univalent at T = 2.9, when plotting is stopped.

where

a(t) = 2x4 x2 + Qt

2x3 , b(t) =

x3 + xQt

2x4 x2 + Qt

, c(t) = 1

x,

and x(t) is a function oft solving

2x6

5 +

2Qt

x4 +

Qt

2= 0,

and

f2(, t) =a(t)(1 b(t)2)

1 c(t)2 , (3.36)

witha(t), b(t) andc(t) so chosen that the final domain consists of two equal disks

touching at the sink. The fluid domain corresponding to f1 and f2 are plotted inFig. 3.6. Once again, both of them have finite-time-cusp.

Considerable effort is in the literature to construct more physical solutions.

We note that it is not the occurrence of cusps on the interface that make some

solutions unphysical, because it is known that cusps on the interface between

different fluids do exist in certain configuration [105, 108], even in experiments


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Figure 3.6. Rational solutions to the PG equations (3.24). (a) solution given byf1 in(3.35); (b) solution given by f2 in (3.36). The exterior ellipse is the initial fluid domainand shaded region is the domain when cusps are formed. (taken from [36])

[53, 109]. Many researcher have made success in obtaining solutions that resemble

the experiments well [70, 71, 97]. They all assume the form of a finite series of

simple mappings with unknown coefficients, which are determined by a system of

algebraic equations. From these solutions we briefly discuss a typical one: Howison

found that a solution of the form

f(, t) =Nn=1

nlog( cn(t)), (3.37)

always exists, where cn(t) corresponds to the trough between growing fingers [47].

By realizing the general form of (3.37) is too complicated to make a full study, he

constructed a solution of the form

f(, t) =a(t)

+1

N

k=1 k log(c1(t)

k ) +2

N

k=1 k1/2 log(c2(t)

k+1/2 ),

(3.38)

where N = 1, 1 and 2 are positive constants, and real function a(t), c1(t) and


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47

Figure 3.7. Comparison between constructed solution (3.38), and the experiment. Bothimages are taken from [47].

c2(t) satisfy

a2 N a(1c1+2c2) = Qt

+K0,

ac1+1

Nk=1

k log(c21k 1) +2

Nk=1

k1/2 log(c1c2k+1/2 1) =K1,

ac2+2

Nk=1

k log(c22k 1) +1

Nk=1

k1/2 log(c1c2k+1/2 1) =K2.

Here K0, K1 and K2 are constants of integration. A plot of this solution next

to an experimental image is in Fig. 3.7. It should be remarked that despite the

satisfactory resemblance between the two, there is no way from the construction

process to predict the solution shape. For example, the number and positions of

finger troughs are completely arbitrary upon ones choice, and can not be inferred

from other information given in the problem.

3.4.3 Generalization of the Polubarinova-Galin equation

In this section we set up an equation to describe the motion of the free interface

including inertial effects, by using the conformal mapping method. The equation


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andong he dissertation

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