Laplacian Growth: Interfacial Evolution in a Hele-Shaw Cell

Laplacian Growth: Interface Evolution in a Hele-Shaw Cell

A dissertation presented to

the faculty of

the College of Arts and Sciences of Ohio University

In partial fulfillment

of the requirements for the degree

Doctor of Philosophy

Khalid R. Malaikah

August 2013

This dissertation titled

Laplacian Growth: Interface Evolution in a Hele-Shaw Cell

KHALID R. MALAIKAH

has been approved for

the Department of Mathematics

and the College of Arts and Sciences by

Tatiana Savin

Associate Professor of Mathematics

Robert Frank

Dean, College of Arts and Sciences 3

ABSTRACT

MALAIKAH, KHALID R., Ph.D., August 2013, Mathematics

Laplacian Growth: Interface Evolution in a Hele-Shaw Cell (101 pp.)

Director of Dissertation: Tatiana Savin

Laplacian growth is the interface dynamics where the normal component of velocity of a free boundary is proportional to the normal derivative of a harmonic function defined in a moving domain. The interface evolution in a Hele-Shaw cell is described by the Laplacian growth model. In this study we derive governing equations in terms of the Schwarz function of the interface for some specific Hele-Shaw flows in which the interface is not equipotential. This is a generalization of the well-known equation ( ) derived for the free boundary one-phase Hele-Shaw problem.

Here, is the complex potential and is the time derivative of the Schwarz function.

The structure of the thesis is as follows:

o In Chapter 1 we give an introduction to the history of the problem, and discuss the

methods and the state of the art.

o Chapter 2 is devoted to the Schwarz function equation for the two-phase Hele-

Shaw flows. Here we re-derive the equations earlier obtained by D. Crowdy using

a slightly different method. Our derivation is based on an introduction of a single-

valued complex velocity potential.

o In Chapter 3 we derive the Schwarz function equation for a class of generalized

Hele-Shaw flows and apply it to the case of an interior problem in a cell with the 4

time-dependent gap. This generalizes the governing equation of the interfacial motion in a Hele-Shaw cell in the presence of an arbitrary external potential.

DEDICATION

To my parents: Reda and Zakia.

To my family: my wife Marwa and my daughter Puneen.

To all my brothers and sisters.

ACKNOWLEDGMENTS

I would like to express my special gratitude to my advisor Dr. Tatiana Savin, for her excellent guidance, support, and patience throughout the course of this research.

I would like to thank my committee members, Dr. Archil Gulisashvili, Dr. David

Tees, and Dr. Xiaoping A. Shen, for their encouragement.

I would like to acknowledge the Government of Saudi Arabia for providing me financial support and Taibah University for granting me academic leave for higher studies at Ohio University.

TABLE OF CONTENTS

Page

Abstract ...... 3

Dedication ...... 5

Acknowledgments...... 6

List of Figures ...... 9

1 Preliminaries ...... 10 1.1 Introduction ...... 10 1.2 Mathematical background ...... 15 1.2.1 Conformal invariance ...... 15 1.2.2 Schwarz function for an analytic curve ...... 18 1.2.3 Cauchy transform ...... 25 1.2.4 Subharmonic and superharmonic functions ...... 27 1.3 Newtonian fluid ...... 28 1.3.1 Equations of motion ...... 29 1.3.2 Potential flow ...... 36 1.3.3 Reynolds number ...... 42 1.4 Summary of results and future plan ...... 43 1.5 Organization of thesis ...... 49

2 Two-phase displacement in a Hele-Shaw cell ...... 53 2.1 One-phase Hele-Shaw flow ...... 53 2.1.1 Thin film approximation ...... 56 2.1.2 Explicit Solutions ...... 59 2.2 Governing equation for the two-phase Hele-Shaw flow ...... 68 2.3 Mother Body ...... 73 8

2.3.1 Mother body and continuation of the gravitational field ...... 73 2.3.2 A dynamic mother body in a Hele-Shaw flow ...... 74

3 Generalized Hele-Shaw flows ...... 79 3.1 Hele-Shaw flow with external potential field ...... 79 3.2 Governing equation for a class of the generalized Hele-Shaw flow ...... 84 3.3 Governing equation for the time-dependent gap Hele-Shaw flow ...... 90

References ...... 95

LIST OF FIGURES

Page

Figure 1: Laplacian growth process ...... 11

Figure 2: Hele-Shaw cell ...... 12

Figure 3: Rotation of a tangent vector ...... 22

Figure 4: Newton’s parallel plate experiment ...... 29

Figure 5: The system and the fixed control volume ...... 30

Figure 6: Surface force of two dimensional fluid’s medium ...... 34

Figure 7: Geometric interpretation of streamlines ...... 39

Figure 8: Elementary plane flows ...... 41

Figure 9: Typical length scale in Hele-Shaw cell ...... 56

Figure 10: Polubarinova-Galin Equation ...... 60

Figure 11: Saffman-Taylor finger ...... 87

1 PRELIMINARIES

1.1 Introduction

A wide range of physical and mathematical applications are observed in nature as a contour dynamic where an interface moves between two immiscible phases. The interfacial motion is generated due to a driven mechanism such as mass distribution and/or heat fluxes. The present thesis investigates some examples of interface dynamic in which the free boundary moves with a velocity proportional to the gradient of a harmonic field. This pattern is called the Laplacian growth process (see Figure 1), which governs a variety of natural growth phenomena such as viscous fingering and the growth of bacterial colonies [31]. Laplacian growth is a non-linear complex dynamics which has been attracting the attention of mathematicians and physicists for more than a century.

Specifically, we have a simply-connected domain Ω( ) with an interface ( ); inside the region Ω( ) the potential satisfies the Laplace’s equation

. (1.1.1)

The moving boundary is supposed to be equipotential or constant-field, i.e., an interface satisfying the boundary condition

constant. (1.1.2)

Furthermore, the velocity of the point of the moving boundary ( ) is defined as the gradient of a scalar field

( )

, (1.1.3) 11

where is called the velocity potential. Assume the continuity of the velocity potential within the simply-connected domain up to the boundary. Given the initial domain, one wants to study the evolution of the simply-connected domain which occupied by a viscous fluid whose motion is generated by a driven mechanism such as several sources or sinks. In this study we are interested in the interfacial motion where the interface is no longer equipotential so that the velocity potential equals a specific function of the boundary point.

( )

Figure 1: Laplacian growth process

The central theme of the present dissertation is a characterization of an interfacial motion between two immiscible phases in a Hele-Shaw cell which is an apparatus used to investigate plane flow problems (see Figure 2). In particular, two immiscible fluids of different viscosities are confined in a narrow gap between a pair of parallel plates.

Specifically, motion of a sharp interface between a viscous fluid domain such as oil and its complement arises due to injection (suction) of oil through an inlet in the middle of 12

one of the plates. Here, the complement is occupied by less viscous fluid (e.g. air or water) whose physical quantities are constant.

Figure 2: Hele-Shaw cell [67]

The interfacial motion in a Hele-Shaw cell has been extensively studied due to the mathematical analogy with other important free boundary problems, such as ice crystal growth and solidification, and some patterns that are constructed under special restriction in the laboratory. Thus, it has been an important prototype tool that aims to provide research with support that is helpful for understanding interface dynamics. For instance, a free boundary Hele-Shaw problem and flows in a porous medium, which is used in the oil industry, are governed by Darcy’s law. Thus, the Hele-Shaw cell can be used experimentally to realize flows in the porous medium. As such, the present results may be interpreted in terms of interface dynamics in a porous medium. Injection of molten plastic into a mould can be also governed by a free boundary Hele-Shaw problem [47]. Next, prior to presenting a comprehensive detailed overview, we will have a brief summary of topics included throughout the course of this research. 13

In Chapter 1 we discuss preliminary results, including mathematical description and an introduction to fluid mechanics. Indeed, many of the two-dimension flow problems are treated within the frame of complex variable methods. In this study, we are restricted to a Newtonian incompressible flow regime in which the numerical value of a

Reynolds number is low. This pattern is formed due to high fluid viscosity and/or slow flow velocity so that viscous forces predominate over inertial forces.

The dynamics of a Newtonian and incompressible flow is described by the so- called Navier-Stokes equations (neglecting body forces):

( ( ) ) (1.1.4)

(1.1.5)

Here, is density, is velocity, is pressure and is viscosity. Eq. (1.1.4) is called the momentum equation while Eq. (1.1.5) is called the continuity equation. On the left-hand side of (1.1.4), the first term is the unsteady (time-dependent) inertial term while the second term is the nonlinear inertial component. In contrast, the right-hand side consists of the gradient of the pressure and the viscous dissipation (both are linear). Nonlinearity in (1.1.4) impedes the construction of exact solutions. Nevertheless, certain flow regimes allow the use of complex variable methods.

For illustration, consider the Stokes’ flow in which the Reynolds number has a numerically small value. Therefore, we may be able to drop entirely the substantial derivative, i.e., the left-hand side of (1.1.4), reducing the momentum equation to a linear system

. (1.1.6) 14

Here the viscous force is balanced by the pressure (neglecting the body force). Eq. (1.1.6) is called Stokes’ equation which is attributed to Gorge Stokes who first investigated this type of flow. The linearity of (1.1.6) indicates a great simplification as follows. We follow Pozrikidis’ [44] method for the discussion. The following mathematical identities

( ) (1.1.7)

( ) (1.1.8)

( ) (1.1.9) are true in general for any vector A and a scalar function . In addition, the vorticity vector is defined as the curl of the velocity field

. (1.1.10)

Using (1.1.5) and (1.1.10) so that (1.1.7) has the form

(1.1.11) where is replaced by the velocity field. From (1.1.6) it follows that eq. (1.1.11) has the form

(1.1.12)

It is evident from (1.1.12) that the magnitude of the vorticity and the pressure satisfy the

Cauchy-Riemann equations

(1.1.13)

In addition, it follows immediately from (1.1.8) and (1.1.12) that the pressure satisfies the Laplace equation 15

(1.1.14)

Using (1.1.9), the curl of the Stokes’ equation yields that the vorticity satisfies the

Laplace equation

(1.1.15)

Now, introduce a complex function such as

( ) , where is a square root of minus one and . It is evident from equations

(1.1.13) – (1.15) that the function ( ) is analytic. As a consequence, a plane Stokes flow is amenable to the complex variable theory.

1.2 Mathematical background

1.2.1 Conformal invariance

Several natural observations and experiments are described through a planar

Laplace equation. Due to the fact that the harmonic function of two variables is connected to the complex function, complex variable methods are significant for constructing solutions to these applications. Indeed, the real and imaginary parts of an analytic complex function are harmonic and satisfy the Cauchy-Riemann equations.

Consider an analytic function ( ) ( ) ( ), then the complex derivative is written as partial derivatives of the real and imaginary parts

( )

. (1.2.1)

Cauchy-Riemann’s equations follow immediately from equating the real and imaginary parts 16

, (1.2.2)

The transformation ( ) relates the point in a physical domain

( ) to a point ζ η in a parametric domain ( ). In addition, the relation ( ) is conformal provided that it is one to one so that each point

corresponds to a unique point in , and the derivative ( ) is not vanishing so that the inverse function is differentiable

( ) ( ) ( ) at . (1.2.3)

In summary, a harmonic function ( ) defined in a simply-connected domain

Ω is the real part of an analytic function ( ) ( ) ( ) where . In addition, the harmonic function is a conformal invariant so that the conformal transformation plays an elegant role in the investigation of the planar Laplacian growth.

In particular, consider an arbitrary simply-connected domain in which

( ) is harmonic, the conformal transformation maps to a convenient domain, including a unit circle or half-plane, where the explicit solution to Laplace’s equation is known. Thus, a harmonic function in is mapped into another harmonic function in the parametric domain th rough the conformal transformation [8].

We follow Currie [12] to illustrate the method. Let ( ) be a harmonic function in a simply-connected domain where so that

(1.2.4) 17

and ( ) be a conformal transformation that maps the prescribed domain to parametric simply-connected domain in , such as

( ) ( ) ( ) where .

The analyticity of ( ) implies that its real and imaginary parts are harmonic

(1.2.5)

and also satisfy the Cauchy-Riemann equations

(1.2.6)

Now, calculating the partial derivatives of ( ) with respect to and yields

( ) ( )

Therefore,

( ) ( ) (1.2.7) similarly,

( ) ( ) (1.2.8)

It is evident from (1.2.4) that

( ) ( ) ( ) ( )

( )

Equations (1.2.5) and (1.2.6) yield the relations

( )( )

( )( ) , therefore,

The result follows immediately from the condition of conformality, i.e., ( ) for every point of the transformed domain (see Eq. (1.2.3)). Now we introduce the

Riemann’s mapping theorem by which the first exact solution to a classical Hele-Shaw flow was constructed [16], [42] and [43].

Theorem 1.2.1 (Riemann mapping Theorem) [8] Let Ω be a simply-connected domain which is not the whole plane, and . Then there exists a unique analytic function having the properties:

) ( ) and ( ) ,

) is one to one,

) ( ) { | | }

1.2.2 Schwarz function for an analytic curve

A two-dimensional Laplacian growth can be reformulated in terms of the Schwarz function of the interface. The planar Laplacian growth is reduced to a simple dynamic description of the singularities of the Schwarz function of the interface. In other words, 19

the time derivative of the Schwarz function coincides with a multiple-valued function that is analytic in the simply-connected domain with the exception of the driven mechanism.

This observation appeared implicitly in the work of Richardson [48], and was first stated explicitly by Lacey [32] and then by Howsion [23] using a different approach.

Davis [13] has defined the Schwarz function of an analytic curve as follows. Let

be an analytic curve given algebraically by the equation

{( ) | ( ) } (1.2.9) where ( ) is a polynomial with real coefficients, and its partial derivatives and do not vanish simultaneously along points of the boundary. Performing the change of variables

̅ ̅

where (1.2.10) into (1.2.9) yields

̅ ̅ { | ( ) ( ̅) } . (1.2.11)

For every point of the boundary, we have

( ) ̅

then

̅ .

Therefore, by the implicit function theorem we can solve with respect to ̅ in terms of and we obtain

̅ ( ). (1.2.12) 20

Here, ( ) is a unique analytic function in a strip-like shape containing in its interior.

The function ( ) is called the Schwarz function of . For illustration of the definition, let us present some examples of familiar curves.

Example 1

Consider a circle of radius and a center at the origin. Its boundary is given by the equation . Performing the change of variables (1.2.10), the corresponding

( ) ( ) Schwarz function has the form , and thus it has a simple pole at the origin

[13].

Example 2

Let be a boundary of a Neumann’s oval that is described by the equation

( )

From (1.2.10) it follows that

( ) √ ( )

where ( ) is the Schwarz function and . Thus, ( ) has two interior

simple poles , and two exterior branch points [56].

Example 3

A curve of the fourth order (the ovals Cassini) whose boundary given by the equation

( ) ( ) where and are positive. This equation describes a simple closed curve if , then the corresponding Schwarz function is 21

√ ( ) . √

Thus, the Schwarz function has two interior singularities , and two exterior singularities √( ) [56].

We shall list several useful properties of the Schwarz function of an analytic curve. Specifically, the directional derivatives along and normal to the boundary can be expressed in terms of the derivatives with respect to , its conjugate and the Schwarz function. Let be an analytic curve with the Schwarz function ( ), then the following properties hold along the boundary:

(1) The derivative of the Schwarz function does not vanish for any point on the curve,

that is

( ) . (1.2.13)

Indeed, since ̅ ( ) along , we obtain

̅ ( ) , (1.2.14)

hence,

| ( ) |

(2) The tangent vector along the curve can be written in terms of the Schwarz

function as follows

√ (1.2.15) ( )

Consider an element of the curve, the latter equation follows immediately from that 22

̅̅̅ √( )( ) √ =√ ( ) .

In addition, since ̅ ( ) along we obtain

̅ ( ) ( ) √ ( ) (1.2.16)

Equations (1.2.15) and (1.2.16) are the tangent vector and its complex conjugate given in terms of the Schwarz function. Furthermore, the normal vector and its complex conjugate can be simply obtained by rotating the tangent vector and its complex conjugate clockwise with an angle (see Figure 3).

Figure 3: Rotation of a tangent vector

vector

Thus,

̅ ̅ , √ ( ) √ ( )

(3) The derivative of a function ( ̅) along and normal to the boundary

respectively: 23

̅ ,

therefore, from the latter property it follows that

√ ( ) , (1.2.17) √ ( ) ̅

( √ ( )). (1.2.18) √ ( ) ̅

(4) The curve’s curvature can be expressed in terms of the Schwarz function as

follows

. (1.2.19) ( )

For illustration, let be the angle between the tangent to and the real axis, then

(1.2.20) thus,

. (1.2.21)

From (1.2.14) it follows that

(1.2.22)

thus,

(1.2.23) ( )

Substitution of (1.2.22) and (1.2.23) into (1.2.21) yields the relation

. (1.2.24)

Indeed, the curvature is the ratio of the change in the angle of the tangent that moves over a given arc to the length of that arc, i.e., how fast the curve change its direction so that

. (1.1.25)

Substitution of (1.2.15) and (1.2.24) into (1.2.25) yields the relation (1.2.19).

Proposition 1.2.1 [23], [30]. Let ( ) be an interface that has a Schwarz function, then the interface normal velocity in terms of the Schwarz function is

. (1.2.26) √

Proof. Let ( ) be a velocity field in the real plane after the change of variables

̅ and ,̅ the velocity field can be written as ( ̅ ) in the complex plane. Now consider a boundary’s point such that ( ) (( ̅) ( ̅) )

( ) so that the corresponding velocity is

̅ ̅ ( ) (1.2.27) while the outward unit normal vector

̅ ̅ ( ) ( (√ ) ( √ )) (1.2.28) √ √

Note that to prove (1.2.28), we used the second property of the Schwarz function.

Therefore, the normal velocity

( √ ̅ ) (1.2.29) √

Along the boundary we have ̅ ( ) differentiating with respect to time indicates

̅ (1.2.30)

Substitution of (1.2.30) in to (1.2.29) finishes the proof. 25

1.2.3 Cauchy transform

A Cauchy transform of a two-dimensional bounded domain with an analytic boundary is described [9], [38], and [48] as follows,

( ) , . (1.2.31)

Thus, ( ) is an analytic function in , and it may be analytically continued to .

Therefore, all singularities of this continuation lie in . A physical interpretation of the

Cauchy transform of Ω is that it is the gradient of a two-dimensional gravitational potential for a uniform mass distribution occupying the simply-connected domain . In particular,

( ) | | . (1.2.32)

Here, we discuss the relation between the Cauchy transform of a simply- connected domain Ω whose boundary is and the Schwarz function of the boundary itself. Indeed, the Schwarz function is closely linked to the Cauchy transform of a simply- connected domain [9]. Consider a closed, simple and analytic curve , and let and

be the interior and the exterior of the curve respectively. The Schwarz function of can be represented as a sum of two analytic functions (analytic in ) and

(analytic in ), such that

( ) ( ) ( ). (1.2.33)

This decomposition is unique if ( ) vanishes at infinity [38].

Without a loss of generality, we assume the origin belongs to . For every point

the function 26

( ) , (1.2.34) is analytic in and all singularities of its continuation lie inside . In addition,

( ) as . Expanding ( ) as an inverse power series near infinity, we have

( ) ∑ . (1.2.35)

Now, the Richardson’s harmonic moments [47], of are defined as the integral of integer power of over a simply-connected domain

An application of Green’s theorem (complex form) results

( ) (1.2.36)

The formula (1.2.36) indicates that the Laurent coefficients in (1.2.35) are the harmonic moments.

The Cauchy transform of an unbounded domain is handled similarly [9].

Consider

( ) (1.2.37)

where is a circle of radius and a center at the origin. The integral in (1.2.37) is

independent of for while ( ) is analytic in with all singularities of its

continuation lying in . Expanding ( ) as a power series near the origin, we have

( ) ∑ , (1.2.38) 27

In a conclusion, whenever ( ) and ( ) can be analytically continued, their sum is related to the Schwarz function as follows

( ) ( ) ( )

such that ( ) ( ) and ( ) ( ).

As an illustration of this decomposition, we mention two basic examples:

Example 1

The Schwarz function of a circle of radius and center is

( ) ̅ ̅ ̅+ ,

therefore,

( ) , ( ) ̅ ̅ ̅.

Example 2

The Schwarz function of a boundary of an ellipse with the half axis and , where

, is

( ) √ , therefore,

( ) ( √ ) ( ) , .

1.2.4 Subharmonic and superharmonic functions

Definition 1.2.1 [8]. Suppose that Ω is an open set. A function is subharmonic if

, and superharmonic if . 28

The definition has interesting comments: subharmonic or superharmonic function is a function whose Laplacian does not change sign, and a harmonic function is both subharmonic and superharmonic. The following theorem is regarded as the most important result on harmonic functions which also holds for subharmonic functions.

Theorem 1.2.2 (The strong maximum principle) [51]. Let be a subharmonic function on an open set Ω. If attains a global maximum on Ω, then is constant. In particular,

̅ if is compact, is subharmonic on Ω and ( ) for all , then

on Ω .

1.3 Newtonian fluid

A fluid is a substance which does not resist shear stress, irrespective of stress value. In other words, fluid materials move and deform indefinitely over the time due to shear stress, contrasting with solid materials, which do not. A fluid in which the time rate of strain is linearly related to a given shear stress is known as a Newtonian fluid [18]. In particular, shear stress is proportional to the velocity gradient, and the constant of proportionality is the coefficient of viscosity. The viscosity measures the fluid’s resistance to flow. As a practical demonstration of this definition, we consider the

Newton’s parallel plates experiment in which a Newtonian fluid is sandwiched between two parallel plates, the upper plate of which moves at speed while the lower is fixed

[66] (see Figure 4). As a consequence, each fluid particle experiences a rate of strain, because of the applied shear stress, that is in the direction of the x-axis where the fluid motion is heading. Strictly speaking, the angle, increases continuously over time as long as shear stress is maintained. 29

( ) ,

. ( )

( )

Figure 4: Newton’s parallel plate experiment

According to the equation, (1.3.2), the velocity gradient is positively correlated to the given shear stress through the viscosity coefficient,

( )

It is evident from ( ) that the viscosity is independent of the velocity gradient. On the other hand, the viscosity is influenced by temperature and pressure.

1.3.1 Equations of motion

Equations of motion have been derived via two basic physical laws: conservation of mass and Newton’s second law which are applied to fluid’s flow. Here we shall use 30

the control volume approach. The control volume is a finite and fixed region in the fluid’s flow in which the Reynolds’ transport theorem is applied [17] (see Figure 5). The fluid mass moves through the control surface which is the outer boundary of the control volume. The conservation of mass principle yields the so-called continuity equation while

Newton’s second law results ultimately in the momentum equation. These two equations together are called the Navier-Stokes equations. The motion of a fluid is observed from a stationary position in the control volume. Therefore, the control volume represents the

Eulerian viewpoint in which the fluid flow passes through a specific location in the space over time [17].

Figure 5: The system and the fixed control volume

Theorem 1.3.1 (Reynolds’ transport theorem) [66]. Consider a control volume, 퓒V, occupying a fluid mass. Let (extensive property) be a fluid’s property such as velocity 31

or temperature, (intensive property) be the amount of per unit mass such that

퓒 퓒 . , and V is surrounded by surface The extensive property relates to the intensive one through , where is the fluid density. Therefore, the time rate change of the extensive property is

( ) , (1.3.4) where is the velocity vector, and is the outward vector normal to .

Remark that the normal velocity is if the fluid flows outwardly but is

in case of inward flow

We are now in the position to derive the continuity equation, which is based on the conservation of mass, and momentum equation, which is an application of the Newton’s second law, using the Reynolds’ transport theorem. These two equations together are called the Navier-Stokes equations.

(1) Conservation of mass.

Consider a control volume, occupying fluid mass , then the conservation of

mass principle indicates that since the mass never created not destroyed. As an application of Reynolds’ transport theorem, we have Thus,

. (1.3.5)

From Gauss’ divergence theorem , we have

(1.3.6) thus, Eq.(1.3.5) has the form 32

( ) (1.3.7) therefore,

(1.3.8)

In case of an incompressible fluid, in which the density is considered uniform throughout the period of motion, equation (1.3.8) has the form

(1.3.9)

Therefore, the velocity field is divergence-free for the case of incompressible flows.

(2) Conservation of momentum for frictionless flow.

The inviscid flow is a flow of a neglected viscosity so that the shear stress plays no role, it is also called an ideal flow [52]. The conservation of momentum is an application of the Newton’s second law. Specifically, the momentum of fluid mass occupying an arbitrary control volume is and according to the Newton’s second law its change over time is equal to external forces (body forces and surface forces). The surface forces consist of normal stress and shear stress (acts tangentially) while the body forces, such as gravity (acting on fluid mass). The body force acting on fluid mass occupying a control volume is . Due to the absence of viscosity, the only surface force is pressure , acting normally inward to the surface. Thus,

, (1.3.10)

The meaning of this formula is as follows: the time rate change of fluid momentum equals external forces (body forces plus the pressure force) where the minus sign 33

indicates the inward direction of pressure. As an application of the Gauss’ divergence theorem, equation (1.3.10) has the form

( ) ( ) . (1.3.11)

Considering incompressible flows in which the density is constant, we have

( ) , therefore,

(1.3.12)

This equation was derived by Euler, it is applicable only to frictionless fluids. Next, we derive the counterpart of this equation in which the viscous force is taken into account.

(3) Conservation of momentum for viscous flow.

The presence of viscosity imposes burden on the derivation of the momentum equation. For illustration, consider a planar infinitesimal region of fluid, the stress tensor of a two-dimensional fluid flow is defined as

[ ]

Figure 6 illustrates that the surface forces are decomposed as normal stress (pressure) and shear stress which is acting tangentially. Remark that the stress tensor is symmetric so that [4]. We have already mentioned that the fluid medium moves and deforms indefinitely over the time as long as shear stress is sustained. As a result, shear stress is zero if a fluid is at rest so that the flow of fluid is influenced only by pressure.

Hence, the stress component is expressed as 34

(1.3.13) where is Kronecker delta, is pressure and is shear stress [12]. The fluid under consideration is Newtonian so that the shear stress is linearly related to velocity gradient, and viscosity is the coefficient of proportionality

(1.3.14)

( )

where ( ) is velocity vector. We are interested in the gradient of the stress tensor

Figure 6: Surface force of two dimensional fluid’s medium.

[ ( ) ( )]

[ ] .

Newton’s second law yields

on axis,

on axis, where the body forces are donated by and is the surface forces.

Equations ( ), ( ) and (1.3.9) yield the relation

( )

Thus,

( ( ) ) (1.3.15) where ( ) is the velocity field. In the absence of the viscosity, Eq. (1.3.15) reduces to

Euler equation. The left-hand side of (1.3.15) is the acceleration which is the unsteady

(time-dependent) inertial term and the nonlinear inertial components. On the other hand, the right-hand side is linear (the pressure, viscous forces and body forces). Equations

(1.3.15) and (1.3.9) together are called the Navier-Stokes equations for Newtonian incompressible flow. In section 1.3.3 we express these equations in dimensionless variables. As a consequence, an important parameter, the Reynolds number, emerges 36

from the momentum equation while the continuity equation remains unchanged. The value of Reynolds number determines the nature of fluid flow: either laminar or turbulent. In this thesis we are interested in a flow of a numerically low Reynolds numbers so that viscous forces predominate over the inertial force. As such, the left-hand side of (1.3.15) is very small compared to the right-hand side, and thus the nonlinear terms might be omitted. As a result, complex variable methods are feasible to construct exact solutions for the plane flow problems, for instance, for the two-dimensional irrotational ideal flows.

1.3.2 Potential flow

Flow, in which the components of the velocity field are expressed as the gradient of a scalar function, is called a potential flow,

(1.3.16)

Here, is the velocity potential. We follow Currie [12] to illustrate the method. Having found , it is straightforward to calculate the velocity field. The following identity

(1.3.17) is true for every scalar function. Applying (1.3.17) on (1.3.16) yields the condition of irrotationality, i.e., the vorticity vector

(1.3.18)

If the flow is two-dimensional and incompressible then the problem is amenable to complex variable techniques (e.g., conformal transformation) by which a large variety of exact solutions has been constructed. Equations (1.3.9) and (1.3.16) indicate that the velocity potential satisfies the Laplace’s equation 37

(1.3.19)

Next, we introduce a stream function, ψ, which exists for any two-dimensional flow whether is rotational or not, as follow

. (1.3.20)

The stream function satisfies the condition of incompressibility. Consider irrotational flow, and then Eq. (1.3.18) indicates that the stream function satisfies the Laplace’s equation

(1.3.21)

In conclusion, a two-dimensional incompressible and irrotational fluid’s flow implies the existence of the velocity potential and the stream function, and both are harmonic. Moreover, lines of constant stream functions, and constant velocity potentials, are called streamlines and equipotential lines respectively, and they are orthogonal to each other [66] as follows,

( ) , (1.3.22) and

( ) (1.3.23)

Note that to derive (1.3.22 ) and (1.3.23) we use (1.3.20) and (1.3.16). From (1.3.22) and

(1.3.23), we obtain

Therefore,

| | .

Since the product of the slopes is negative, the streamlines and equipotential lines are perpendicular. From a physical viewpoint, the amount of fluid through two streamlines is calculated as the difference of the value of the stream function between these streamlines

(see Figure 7). For illustration, consider volume flow rate, Q, through a control surface,

with a velocity field ( ) then the normal velocity is

( ̅ ( ) ).

Indeed, the inner produce of two complex numbers regarded as vectors is ( ̅ ).

Thus,

( ̅ ( ) )

Here (1.3.22) has been used in the last equality.

The velocity components and can be obtained via either the velocity potential or the stream function. From (1.3.16) and (1.3.20), we obtain

(1.3.24)

(1.3.25)

Let us define a complex function

( ) 39

It is evident from (1.3.24) and (1.3.25) that the Cauchy-Riemann equations are satisfied, and from (1.3.19) and (1.3.21) that the real and imaginary parts are harmonic. Therefore, the complex function ( ) is analytic. In addition, the physical interpretation of its derivative, indicates that the derivative is a single-valued function (the first partial derivatives of and are continuous) that can be expressed as follows

Figure 7: Geometric interpretation of streamlines

̅ (1.3.26)

Here, ( ) is the velocity vector so that the derivative is the complex conjugate of the velocity vector. The function ( ) is called the complex potential while its derivative is called the complex velocity of a potential flow. In general, 40

real and imaginary parts of an analytic function with a single-valued derivative may represent a velocity potential and stream function respectively. The relevant velocity field can be calculated through (1.3.24) and (1.3.25) while the represented flow can be determined via investigation of constant streamlines. Now we are in the position to determine a flow for a given analytic function with a single valued derivative.

Example 1

A complex potential which describes a flow that is driven by a point source of constant strength

( ) ( ),

, , where is the source strength, i.e., volume flow rate. The velocity potential represents radial lines while the stream function represents concentric circles; (see Figure 8(a)).

Thus, velocity components express as

Example 2

Consider a uniform flow parallel to x-axis, the relevant complex potential

( )

From (1.3.26) we have

, obviously, the velocity component in direction is zero; (see Figure 8(b)).

Figure 8: Elementary two-dimensional flows; solid lines are streamlines and dashed lines are equipotentials

In summary, complex variable methods play an elegant role in constructing exact

solutions to plane problems of a potential flow since many of them can be described by

an analytic function with a single-valued derivative. For instance, the conformal

transformation is exploited since the Laplace’s equation is a conformal invariant. Strictly

speaking, harmonic function in the real domain is transformed into another harmonic

function in an auxiliary domain where the solution of this harmonic function is known.

Therefore, the complex potential of the real flow is expressed by a counterpart defined on

the auxiliary domain which are correlated to one another through the relation ( )

Consider a complex potential in plane

( ) ( ) ( ) for

the conformal mapping yields

( ) ( ( )) ( ) ( ) for 42

1.3.3 Reynolds number

A mathematical model of a physical phenomenon is a list of equations by which the relevant physical quantities are correlated. Dimensions of these quantities must adhere strictly to the same standard unit system. Writing the momentum equation for viscous fluid in dimensionless variables yields the Reynolds number by which the nature of the fluid flow is described [17]. The momentum equation for a viscous fluid

has three basic dimensions: mass, distance and time. In addition, the independent variables and can be expressed in dimensionless form by introducing the characteristic length and velocity scales, respectively,

: Length scale, velocity scale, while viscosity and density are regarded as uniform. The pertinent dimensionless variables are given

( ) ( )

The dimensionless form is obtained as follows,

( )

In a similar manner, we obtain

( ) ( ) ( )

Thus, the Navier-Stokes equation in dimensionless variable is 43

(1.3.27) where,

is called the Reynolds number, that is the ratio of the inertial force to the viscous force.

The value of the Reynolds number determines the behavior of the fluid flow: either laminar or turbulent. Specifically, a low Reynolds number ( ) means that the viscous force is important while a high Reynolds number indicates a nearly ideal fluid or high velocity [17]. The smallness of indicates that extremely small length scale, high viscous fluid and/or very slow fluid flow. Eq. (1.3.27) has one parameter, and the dependent variables are pressure and velocity while the independent variables are and

, i.e., position and time. If the boundary and initial conditions of two flows as well as

Reynolds numbers are identical regardless of the comparison among the constant value of

and , their solutions are similar in dimensionless form, i.e., there is the dynamical similarity. In contrast, the condition of incompressibility written in dimensionless variables yields no parameter

(1.3.28)

Eqs. (1.3.27) and (1.3.28) together are the dimensionless form of the Navier-Stokes equations for a Newtonian and incompressible fluid of a constant viscosity.

1.4 Summary of results and future plan

We have derived governing equations in terms of the Schwarz function of the interface for some Hele-Shaw models. The velocity field was proportional to the gradient 44

of a harmonic field (pressure) for both models, but the harmonic field was complemented by different boundary conditions. Our results generalize the well-known equation

( ) (1.4.1) where is the complex potential and is the Schwarz function of the interface. The latter equation describes the free boundary one-phase problem in which the model for the pressure and the normal velocity is

( ) in Ω( ), (1.4.2)

on Ω( ) (1.4.3)

( ) on Ω (1.4.4) where is the Dirac measure. Eq. (1.4.1) indicates that singularities of the complex potential coincide with those of the time derivative of the Schwarz function. In other words, is equal to a derivative with respect to of a complex potential, which is analytic in the fluid domain with the exception of the singularities of . Eq. (1.4.2) is obtained from the continuity equation since the fluid under consideration is incompressible. Eq. (1.4.3) is the Young-Laplace equation for the pressure while Eq.

(1.4.4) indicates that the interface moves with the fluid. The reformulation of the one- phase problem in terms of the Schwarz function of the interface is summarized in the following theorem.

Theorem 1.4.1 [23]. Let be an analytic curve. Then there exists a multiple-valued analytic function ( ) defined in a neighborhood of ( ) satisfying the equation

(1.4.5) 45

whose real part ( ) satisfies the boundary conditions (1.4.3) and (1.4.4).

In the presence of an arbitrary external potential, problem (1.4.2)-(1.4.4) for pressure and the normal velocity has the form

( ) in Ω( ), (1.4.6)

on Ω( ) (1.4.7)

( ) on Ω (1.4.8)

The only difference with the former model is that the interface is not equipotential in this case. Therefore, the flow is driven by the gradient of pressure and the external potential.

Note that both may have singularities in the computational domain. Problem (1.4.6)-

(1.4.8) is reformulated in terms of the Schwarz function by the following theorem.

Theorem 1.4.2 [37]. Let be an analytic curve. Then there exists a multiple-valued analytic function ( ) defined in a neighborhood of ( ) satisfying the equation

( ) (1.4.9) whose real part ( ) satisfies the boundary conditions (1.4.6) and (1.4.8).

Theorem (1.4.2) is a generalization of theorem (1.4.1). In other words, if there is no external potential, , Eq. (1.4.9) reduces to Eq. (1.4.5). Next we summarize free boundary Hele-Shaw problems considered in the present study. Namely two-phase displacement and a class of generalized flows which is motivated by the Hele-Shaw problem with the time-dependent gap.

o problem A

We considered two-phase displacement in a Hele-Shaw cell in which the dynamics of each of which fluids is described by

in ( ) (1.4.10)

on Ω( ), (1.4.11)

( ) on Ω . (1.4.12)

Here, is a time-dependent distribution, whose support has a nonzero co-dimension (a set of points and/or curves) and located strictly within the simply-connected domain for any time. Eq. (1.4.11) is the continuity of the pressure and Eq. (1.4.12) indicates that the interface moves with the fluid. The equation of state for each of which pressures here is equally treated. Problem (1.4.10)-(1.4.12) is reformulated in terms of the Schwarz function in the following theorem.

Theorem 1.4.3. Let be an analytic boundary. Then there exist multiple-valued analytic functions ( ), where i=1,2, defined in the neighborhood of satisfying the equation

( ) , (1.4.13)

whose real parts ( ) satisfies the boundary conditions (1.4.11) and (1.4.12).

This theorem is related to the work of Crowdy [10]. Here we re-derive a simpler equation using a different approach. Our derivation is based on a single-valued velocity potential instead of a multiple-valued complex potential. The present result relates the velocity field on one side of the interface to the velocity field on the other side. In particular, the 47

velocity field of the first phase equals its counterpart of the second phase plus the time derivative of the Schwarz function of the interface. In other words, having found the complex velocity on one side of the interface, it is straightforward to calculate the complex velocity on the other side.

o problem B

A class of the generalized Hele-Shaw flows was also considered. The equation of state for the pressure for this case is as follows

in ( ), (1.4.14)

on Ω( ) (1.4.15)

[ ] ( ) on Ω . (1.4.16)

Here, and are arbitrary functions while is the normal velocity of the interface. The interface evolution is driven by the pressure gradient and the functions and so that all of them may have singularities within the fluid domain. This is a generalization of problem (1.4.6)-(1.4.8). Specifically, the interface normal velocity is given here as a sum of the normal derivative of the pressure and an arbitrary function while Eq. (1.4.8) indicates the absence of the arbitrary function. Problem (1.4.14)-(1.4.16) is reformulated in terms of the Schwarz function in the following theorem.

Theorem 1.4.4. Let ( ) be analytic boundary. Then there exists a multiple-valued analytic function ( ) defined in a neighborhood of ( ) satisfying the equation

( ( )) √ , (1.4.17) whose real part ( ) satisfies the boundary conditions (1.4.15) and (1.4.16). 48

In the case , theorem (1.4.4) reduces to theorem (1.4.2 ). Now we apply the

Theorem 1.4.4 to the Hele-Shaw problem with the time-dependent gap. The equation of state for the pressure and the normal velocity for this case is as follows:

̇ ( ) ( ) ( ) in , (1.4.18)

on Ω( ) (1.4.19)

( ) ( ) on Ω , (1.4.20)

It is evident from (1.4.18) that the pressure satisfies the Poisson equation with density function depending on time only. Problem (1.4.18)-(1.4.20) is reformulated in terms of the Schwarz function in the following theorem.

Theorem 1.4.5. Let ( ) be an analytic boundary. Then there exists a multiple-valued analytic function ( ) defined in a neighborhood of ( ) satisfying the equation

̇ √ ̃ { } (1.4.21) whose real part ( ) satisfies the boundary conditions (1.4.19)-(1.4.20).

Here, { } is the Schwarizain derivative defined as a derivative of the curvature.

o Future plans

In the present study, we have considered the free boundary Hele-Shaw problems with fluid’s permeability is at most a function of time, but not of position. In other words, the permeability is either constant or time-dependent. As a consequence, the corresponding boundary value problem is governed by the Laplacian growth. The generalization of the

Laplacian growth arises when the permeability is non-constant. In particular, the fluid’s permeability may vary with time and position so that the system concerns either variable 49

viscosity and/or non-uniform upper plate. As such, the pressure is no longer harmonic, but satisfies the equation ( ) where is the pressure and is the permeability

. Here, is the gap width and is the fluid’s viscosity. This paradigm is called the elliptic growth which is motivated by the practical applications. The elliptic growth is a natural generalization of the Laplacian growth. It will be interesting to consider problems A and B for the case of variable permeability.

1.5 Organization of thesis

In Chapter 2 we start with some background regarding the derivation of a governing equation for the Hele-Shaw flow and the explicit solutions obtained by complex variable methods. In particular, we discuss solutions for the interior Hele-Shaw problem: a bounded domain is occupied by viscous fluid, and is surrounded by an exterior non-viscous fluid, occupying the rest of the plane, and vice versa. The physical quantities of the frictionless fluid are kept constant while those of the viscous fluid are assumed to be continuous within the bounded region up to the boundary. Thus, the pressure for one-phase flow is constant on the interface (surface tension is neglected and the atmospheric pressure is assumed to be constant),

on Γ, (1.5.1)

while the normal velocity of the boundary is given by the normal derivative of the pressure

on Γ. (1.5.2)

In the present study, we investigate the unsteady two-phase Hele-Shaw problem in which a sharp interface separates two Newtonian incompressible fluids. Due to the continuity of 50

the pressure within the simply-connected domains up to the boundary, the boundary value of the pressure is

on Γ, (1.5.3)

where the subscripts correspond to the first and second phases respectively. Eq. (1.5.3) is complemented by the conservation of mass

(1.5.4)

where is the fluid permeabilities. Eq. (1.5.4) indicates that the boundary moves with the fluid. The constructed Schwarz function equation is essentially related to the work of

Crowdy [10], but we introduce the idea in a different manner. Specifically, this derivation is based upon the single-valued complex velocity potential instead of the multiple-valued complex potential. As a result, a proper choice of branch cuts is not necessary.

In chapter 3 we investigate the generalization of the Hele-Shaw flow. The following generalization relates to the works of Entov et al. [15] and was first stated in terms of the Schwarz function by McDonald [37]. This model considers the presence of an external potential field so that the evolution of the fluid domain is generated by the gradient of the pressure and the external potential. The boundary value of the pressure equals the value of the external potential

( ) on Γ, (1.5.5) so that the interface is no longer equipotential. The conservation of mass remains unchanged

on Γ. (1.5.6) 51

The fluid evolution is driven by the pressure gradient and the external potential field, and both may have singularities in the fluid domain.

In the this study, we generalize the latter model as follows: the boundary condition, (1.5.5), remains unchanged while the normal boundary velocity is given by the normal velocity of the pressure plus an arbitrary function.

( ) (1.5.7)

The evolution of the fluid domain is driven by the gradient of the pressure, external potential and ( ), and all of them may have singularities in the fluid domain. This model is motivated by a Hele-Shaw cell in which the gap width is a function of time, but not of a position. Such flow arises when the upper plate is being lifted (squeezed) slightly and slowly at a uniform rate. As a result, the velocity field is no longer divergence-free, and the modification of the continuity equation is

( ) . (1.5.8)

Here is the gap thickness and is the volume flow rate pumped in through a small hole in the upper plate. The new continuity equation is also applicable to a gap with moving plates, a non-uniform gap thickness and a gap with distributed fluid injection [14]. This model is related to the work of Shelley et al. [61], and to the best of the author’s knowledge, was first mentioned by Entov et al. [14]. Kang et al. formulated this model in terms of a multiple-connected domain [29]. If the gap thickness is uniform, we have

, (1.5.9) thus 52

( ) (1.5.10)

Such flow is called the time-dependent gap Hele-Shaw problem.

2 TWO-PHASE DISPLACEMENT IN A HELE-SHAW CELL

2.1 One-phase Hele-Shaw flow

A Hele-Shaw cell is an apparatus used to study two-dimensional flows; it consists of two parallel, closely separated plates between which two immiscible fluids of different viscosities are sandwiched [20] (see Figure 2). In this section we are interested in a droplet of an incompressible Newtonian fluid that trapped in a Hele-Shaw cell and surrounded by frictionless fluid (e.g. air) occupying the rest of the plane, and vice versa.

The fluid movement is driven due to injection or suction of the material through a small hole in the upper plate.

The mathematical formulation is derived based upon the following assumptions.

First, the cell is placed horizontally, and thus the gravitational force plays no role.

Second, the distance across the gap is much smaller than the length and width of the plates. Therefore, the momentum equation reduces ultimately to the well-known Hele-

Shaw theory by which the pressure is related to the averaged velocity vector in the mid- surface of the gap as follows

( ) . (2.1.1)

Here is the velocity potential, is the fluid mobility, is the gap width, and is the viscosity. Eq. (2.1.1) is called the Darcy law by which the components of the velocity field are exactly the same as that the gradient of a scalar function. The derivation of ( ) is exactly the same as that of the Reynolds equation in lubrication theory. Due to the incompressibility condition, equation (1.3.9), the potential in Eq. 54

(2.1.1) satisfies the Laplace’s equation over the entire simply-connected domain which is occupied by the viscous fluid, except for the singular point of injection as follows.

Surrounding the point source of a constant strength by a circle of a small radius ,

( ) and then applying the Guess divergence theorem we obtain

( ), (2.1.2) ( ) ( ( )) where is the total flow through the interface ( ( )). The left-hand side of

(2.1.2) vanishes since the averaged velocity field is divergence-free (see equation (1.3.9)) so that

( ) ( ) , (2.1.3) ( ) where is the amount of fluid by which the area is created. Applying the Gauss divergence theorem using Eq. (2.1.1), we obtain

( ) in , ( ) where is the Dirac distribution supported at the point . From (2.1.4) it follows that the velocity potential defined by (2.1.1) is harmonic in the fluid domain with the exception of driven singularity. Hence, the Hele-Shaw flow is classified under the so-called Laplacian growth process by which a wide range of free boundary problems are governed.

To specify the boundary conditions, assume that the flow under consideration is not in touch with a rigid boundary. In addition, and the interface is a simple closed curve so that either Ω(t) or ( ) is a simply-connected domain, the former refers to a droplet of fluid while the latter refers to a finite bubble surrounded by infinite fluid. In addition, we stress that the physical quantities of the fluid are continuous within the 55

simply-connected domain up to boundary so that the boundary value for the pressure satisfies

atmospheric pressure surface tension on ( )

Eq. (2.1.5) is the Young-Laplace equation for the pressure. We suppose that the atmospheric pressure is constant and the surface tension is neglected so that the boundary value of pressure is constant (say zero). Generally speaking, addition of a constant to the velocity potential has no influence on the velocity field since only the gradient of the velocity potential is relevant. Thus,

on (2.1.6)

The particles of fluid at the interface must remain at the interface during the period of motion so that the normal boundary velocity is expressed as the normal derivative of the velocity potential.

on , (2.1.7)

Eq. (2.1.7) indicates that the velocity of the point of fluid on the boundary equals the velocity of the boundary point. Rescaling the pressure using the transformation ̃ , and substituting in equations (2.1.4) – (2.1.7), we have

( ) (2.1.8)

( ) in (2.1.9)

on (2.1.10)

, on . (2.1.11) 56

Now we are in the position to derive the equation of state for the pressure, eq. (2.1.1), by which the dynamics of fluid in the simply-connected domain is described. The derivation is identical to that of the lubrication theory or thin film approximation.

2.1.1 Thin film approximation

Consider a droplet of fluid in a thin gap between two parallel fixed plates at a small distance, compared with the lateral dimensions of plates. We follow

Howison [26] to illustrate the derivation.

Figure 9: Typical length scale in Hele-Shaw cell [7]

The dynamic of fluid droplet is described by the Navier-Stokes equations (ignoring body forces)

, (2.1.12)

(2.1.13)

Consider a fluid flow in which the Reynolds number is numerically small so that the fluid 57

velocity is slow and/or the viscosity is high. Therefore, the viscous force predominates over the inertial force. Thus, the acceleration term is much smaller than the viscous force so that can be dropped. Eq. (2.1.12) reduces to a linear system resulting a great simplification. The Hele-Shaw theory is derived by expressing the equation (2.1.12) in dimensionless form as follows. Let be the characteristic length scale in z-axis and is the characteristic length scale in xy-plane such that ( ) since the gap width is small. Each of which length scale corresponds to a separated characteristic velocity scales ( and ) normal to plates and in xy-plane respectively. Having two different scales is a crucial key toward the derivation. Let

( ) ( ), ( ) ( )

be the dimensionless variables for position, velocity and pressure respectively. Thus, the dimensionless form of (2.1.13) is

(2.1.14)

In order for the dimensionless components of velocity be divergence-free, we choose

( ) so that ( ) Thus, the velocity component across the gap (z-axis component) is much smaller than the other components (xy-plane components) that is . On the other hand, the dimensionless form of (2.1.12) in x-axis is

( )

Multiplying this equation by ( ), we have 58

( ) (2.1.15)

In order to balance terms, we set ( ) , which yields .

Canceling out the small terms taking into consideration that the Reynolds number has a numerically low value and the z-derivative predominates due to the edge effect, Eq.

(2.1.15) has the form

(2.1.16) the velocity component in y-axis is handled similarly, so that

(2.1.17)

Note that equations (2.1.16) and (2.1.17) imply that the velocity field achieves its maximum at middle of the cell, and vanishes on the plate-fluid boundary because of the no-slip condition. Thus, the derivatives with respect to and of the components of velocity ( and since is negligible) are much smaller than the corresponding derivative with respect to for the flow under consideration as long as the gap width is small ( ) [19]. In contrast, the momentum equation in z-direction reduces to

( )

Canceling out small terms, we have

(2.1.18)

Thus, the pressure is independent of the z-axis so that ( ). Now, let the lower plate be at height while the upper plate is at ( since the plates are uniform 59

and fixed, constant). Integrating (2.1.16) and (2.1.17) across the gap yields expresions for the components of velocity ( bearing in mind the no-slip condition)

( )

Averaging the velocity components across the gap, we have

Then, the averaged velocity vector is given by

(2.1.19)

The velocity field is defined as the gradient of a scalar field (pressure).

2.1.2 Explicit Solutions

We have already mentioned that two-dimensional flow problems are amenable to complex variable methods. From (2.1.19) it follows that the pressure plays a role of the velocity potential, and due to the incompressibility condition by which the averaged velocity field is divergence-free, the pressure is harmonic

(2.1.20)

From complex variables, every harmonic function is a real part of an analytic function

( ) so that ( ) . For each instance of time, ( ) is a multiple-valued function defined on Ω( ), whose real part satisfies the Dirichlet boundary condition 60

(2.1.10). Here we discuss several exact solutions for the one-phase Hele-Shaw flow that are obtained by complex variable methods:

(1) Polubarinova-Kochina and Galin equation.

The first exact solution to the moving boundary problem of a Hele-Shaw flow dates back to 1945, and was derived independently by Polubarinova [42] ,[43], and Galin

[16]. The method employed is essentially the application of the Riemann mapping theorem due to the conformal invariance of harmonic functions. In particular, a time- dependent conformal transformation ( ) maps a unit disc into a family of domains

{Ω( )}, the physical domain, where | | is mapped to ( ) (see Figure 10).

( )

Auxiliary domain Fluid domain

Figure 10: Polubarinova-Galin Equation

Consider the transformation,

( ): ( ) where, { | | }.

The mapping ( ) is normalized such that ( ) and ( ) so that it is a univalent mapping. The interface of the physical domain is obtained by the boundary of the auxiliary domain Ω( ) ( ). The free boundary problem of a Hele-Shaw flow is reformulated in terms of the transformation ( ) by which the following nonlinear boundary value problem is satisfied

( ) ( ( ) ̅̅ ̅̅̅(̅ ̅̅ ̅ ̅̅)) | | ,

( ) ( ) where the initial domain is given by ( ).

We follow Gustafsson et al. [20] to illustrate the method. Consider a point sink of a constant strength placed at the origin, the corresponding complex potential is

( )

( ) (2.1.21) due to the conformal invariance of the Green’s function. The normal velocity of the interface is obtained as the inner product in a complex form

( ̅ ) (2.1.22)

Here and | |. From the relation ( ) it follows that

Thus, Eq. (2.1.22) has the form

̅̅̅ ̅̅ ̅ ̅ ̅ ̅̅ [( ) ] ( ) | | (2.1.23) | | | |

On the other hand, the normal velocity could be obtained by simply taking the normal derivative of the velocity potential (after suitable rescaling), using Eq. (2.1.19) yields

(2.1.24)

From (2.1.21), and ( ), we have

( ) ( ) | |

Therefore,

. (2.1.25) | |

From (2.1.23) and (2.1.25) it follows that

( ̅̅ ̅̅) | | (2.1.26)

Only constant source strength ( ) has been considered here. A variable strength

( ) can be dealt with a change of variable with respect to time, such that . The physical domain is expanding if while it is contracting if but the problem is time-reversible ( and ). However, the two situations are significantly different: the injection case is well-posed, while the suction case is ill-posed.

Polubarinova-Kochina [42], [43] and Galin [16] have considered a quadratic polynomial as a conformal transformation by which family of domains { ( )} are images of a unit disk | |

( ) ( ) ( ) . (2.1.27) 63

It is evident from (2.1.26) that and satisfy the nonlinear system of algebraic equations

[ ( ) ( )]

(2.1.28)

[ ( ) ( )] .

The transformation (2.1.27) represents a solution of a moving boundary Hele-Shaw

| | problem if which is the sufficient and necessary condition for the conformability (see eq. (1.2.3)). The derivative of the quadratic polynomial vanishes

( ) ( ) | | if and thus the inverse of the transformation is not analytic.

Theorem 2.2.1 [24]. If the initial transformation is a polynomial of second degree or higher then cusp formation is inevitable before the boundary reaches the sink.

The Schwarz function of an interface is closely related to the conformal transformation. Let the moving boundary ( ) has Schwarz function then ̅ ( ) along the boundary, then the Schwarz function of the relation ( ) is given by

( ) ( ) .

Thus, the corresponding Schwarz function of the relation (2.1.27) is

( ) ( ) ( ) | | (2.1.29)

Here ( √ ) with (2.1.29) having a singular point at the origin.

Thus, the Laurent expansions of (2.1.29) near the origin 64

( ) ( ) ( ) ( ) . (2.1.30)

Here, ( ) and ( ) are the zero and first harmonic moments, and their relations to the

Laurent coefficients are defined in section 1.2.3. On the other hand, if the transformation is a polynomial, the relation between harmonic moments and Laurent coefficients can be obtained using a different approach [47],

( ) ̅ ̅ ( ) ( ) | | (2.1.31)

Note that to prove (2.1.3) we use complex form of Green’s theorem. Therefore, equation

(2.1.30) becomes

( ) ( ) ( ) ( ) ( ) ( ) (2.1.32)

Below, we discuss Richardson’s observation regarding harmonic moments that is harmonic moments are time-dependent for while the zero moment grows linearly.

(2) Richardson harmonic moments.

Richardson [47] has defined the harmonic moments as integrals of over a simply-connected domain such that

( ) ( ) (2.1.33)

Richardson has proved that the harmonic moments are conserved in time, except the zero one. In other words, ( ) does not change over time for while ( ) grows linearly at rate that is

( ) { (2.1.34)

Note that harmonic moment when represents the area. 65

For illustration, the physical interpretation of the time change of an injected fluid domain is the derivative of the zero harmonic moment with respect to time

( )

Tian [63] and [64] has demonstrated the equation (2.1.34) for the case of a Hele-Shaw flow with a single source of a constant strength at the origin as follows. As an application of Green’s theorem, we have

, (2.1.35) ( ) ( ( )) where ( ) is a circle of a small radius centered at the origin. The left-hand side of

(2.1.35) vanishes due to the fact that and are harmonic so that

(2.1.36) ( )

The right-hand side of this equation vanishes as well because of that

| | ‖ ‖ | |

( ) and

( ) ( )

Note that to derive the latter equations we used the velocity potential of a single source at the origin that is

| | as .

Thus, (2.1.36) has the form

. (2.1.37) 66

On the other hand, applying the Reynolds transport theorem to the harmonic moments

(2.1.33) for yields the relation

( )

( ) ( ) (2.1.38)

Note that to prove (2.1.38) we used equation (2.1.11). Substitution of (2.1.37) into

(2.1.38), we obtain the relation

( ) ,

Neglecting the surface tension, one obtains the desired formula (2.1.34).

(3) The Schwarz function approach.

A free boundary problem of a one-phase Hele-Shaw flow reduces to a simple dynamic description of singularities of the Schwarz function [23], [32] and [48]. The dynamics of a fluid droplet in a Hele-Shaw cell is described by the Darcy’s law, i.e.,

Due to the incompressibility condition, the pressure is harmonic. Consider a multiple-valued function ( ) defined on a simply-connected domain, whose real part

( ) satisfies the boundary conditions

. (2.1.39)

For what follows, we discuss that singularities of the multiple-valued function coincide with those of the time derivative of the Schwarz function. The normal velocity of the interface is expressed in terms of the Schwarz function as follows (see equation (1.2.26))

(2.1.40) √ 67

Differentiating the function ( ) with respect to and applying Cauchy-Riemann equations in ( ) coordinates, where is a parameterization along the interface, we have

√ ( )

√ ( ) √ ( ) . (2.1.41) √

Note that to prove (2.1.41) we used (1.2.15), (2.1.39) and (2.1.40). Eq. (2.1.41) indicates that the singularities of the complex velocity coincide with those of the time derivative of the Schwarz function. Strictly speaking, singularities that are not related to the pressure are necessarily stationary and they are those of the initial Schwarz function, while the singularities that are related to the pressure must coincide with those of the time derivative of the Schwarz function. The following theorem is implicitly related to the work of Richardson [48], and was first stated in term of the Schwarz function by Lacey

[32], and then by Howison [23] where the idea was introduced using a different approach.

Theorem 2.1.1 [23]. Let be an analytic curve. Then there exists a multiple-valued analytic function ( ) defined in the neighborhood of satisfying the equation

, (2.1.42)

whose real part ( ) satisfies the boundary conditions (2.1.39)and (2.1.40).

In the next section we prove the generalization of theorem (2.1.1) taking into consideration the unsteady two-phase Hele-Shaw flow in which the interface is no longer equipotential. 68

2.2 Governing equation for the two-phase Hele-Shaw flow

This section consists of the paper [36], which has been published in the

International Journal of Applied Mechanics and Engineering.

Here, two-phase displacement in a Hele-Shaw cell is considered. In particular, a free boundary moves between two immiscible viscous fluids, with a velocity proportional to the gradient of a harmonic field. We construct a governing equation in terms of the

Schwarz function of the interface for the unsteady two-phase Hele-Shaw problem. The constructed solution is related to the work of Crowdy [10], but we introduce the idea in a different manner. The derivation is based upon the single-valued complex velocity potential instead of the multiple-valued complex potential. As a result, a proper choice of branch cuts is not necessary in the domain. Furthermore, the interface evolution in the two-phase problem is closely linked to its counterpart in a one-phase problem.

The two-phase displacement in a Hele-Shaw cell is a motion of an interface ( ) between two Newtonian incompressible fluids. For each instant of time, the domain Ω( ) consists of a blob of a viscous fluid ( ), which is imbedded into another infinite

̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ viscous fluid ( ), i.e., ( ) ( ) = ( ), ( ) ( ) ( ) and ( )

( ) . The dynamics of fluid in each of which simply-connected domains is described by the equation of state for the pressure, i.e., Darcy law,:

, (2.2.1)

in ( ) (2.2.2)

on ( ), (2.2.3)

( ) on . (2.3.4) 69

Eq. (2.2.2) is obtained due to the incompressibility condition. Eq. (2.2.3) is the Young-

Laplace equation which indicates the continuity of pressures within the corresponding simple connected domain up to the interface (neglecting the surface tension). Eq. (2.2.4) indicates that a particle of the fluid on the interface remains on the boundary itself throughout the period of motion. Indeed, the interface normal velocity is given by the normal derivative of the velocity potential. The dynamics of each of the two fluids is equally treated.

The first exact solution to the unsteady two-phase Hele-Shaw cell was constructed by Howison [25], but it was restricted in its scope. Specifically, the fluids’ permeabilities

̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ were assumed to be alike in both phases. The fluids domain ( ) ( ) ( ) is occupied by a single fluid with a different color in each subdomain, i.e., a multiply- colored fluid. Indeed, Howison aimed to prove the observation, pointed out by Saffman and Taylor [53], who were considering two-phase displacement in a channel Hele-Shaw cell, that is the pressure on one side of the interface equals the pressure on the other side plus a linear function.

The general form of this construction was developed by Crowdy [10] in terms of the complex potential. Consider multiple-valued functions ( ) and ( ) defined on and respectively, the real part of each of the two complex potentials is given by the corresponding velocity potential as

where is the fluid permeability and is the pressure defined by equations (2.2.1) -

(2.2.4). From (2.2.3) it follows that 70

̅̅̅ ̅ ( ) where . Differentiating this equation with respect to , we obtain

( ) ̅̅ ̅̅̅(̅ ̅ ̅ ̅) ̅ ( ) ̅ ̅ ̅̅̅(̅ ̅ ̅ ̅) ̅ ( ) (2.2.5)

From (2.2.4) it follows that the interface normal velocity is defined as the inner product of two complex numbers, regarded as vectors,

( ̅ ) (2.2.6) where is the unit tangent vector. From (1.3.26), we have that the complex potential is related to the averaged velocity field through the relation

̅ , thus, (2.2.6) has the form

( ) ̅ ̅ ̅̅̅(̅ ̅ ̅ ̅)̅ ̅̅ ( ) ̅ ̅ ̅̅̅(̅ ̅ ̅ ̅)̅ ̅̅

, then,

( ) ̅ ̅ ̅̅̅(̅ ̅ ̅)̅ ̅ ( ) ̅ ̅ ̅̅̅(̅ ̅ ̅)̅ ̅

(2.2.7)

Addition of (2.2.5) and (2.2.7) yields the relation

( ) ( ) ̅ ̅ ̅̅̅(̅ ̅ ̅ ̅)̅ ̅̅ [( ) ( ) ] (2.2.8)

In the following theorem we want to re-derive (2.2.8) in terms of the Schwarz function of the interface using a single-valued complex velocity potential.

Theorem 2.2.1. Let be an analytic curve. Then there exist multiple-valued analytic functions ( ), where i=1,2, defined in the neighborhood of satisfying the equation 71

( ) , (2.2.9)

whose real parts ( ) is defined by (2.2.1) - (2.2.4).

Proof. Let be a parameterization along the interface, the tangent velocity of the interface, , where is the averaged velocity field defined in (2.2.1) and

is the unit tangent vector. Consider a multiple-valued complex potential

( ) defined on , whose real part is given by ( ) where i=1,2. The averaged velocity field is related to the complex velocity potential through the relation

(1.3.26) (see, for example, [24]) as follows

̅ ̅ . (2.2.10)

In addition, we have ̅ ( ) on the interface so that

̅ ̅ . (2.2.11)

From (1.2.15) and (1.2.16), the tangent vector is expressed in terms of the Schwarz function as

, ̅ √ . (2.2.12) √

It is evident from (2.2.1) that the component of the velocity field is defined as the gradient of a scalar velocity potential function. Thus, the directional velocity is obtained by taking the directional derivative of the velocity potential along that direction,

. (2.2.13)

The scalar product of two complex numbers and , regarded as vectors, is ( ̅ ).

Thus, from (2.2.10) it follows that 72

( ) . (2.2.14)

Therefore,

̅̅̅ ̅ ̅̅ ̅̅̅̅ [ ] (2.2.15)

and

̅̅̅ ̅ ̅̅ ̅̅̅̅ [ ] . (2.2.16)

Subtracting (2.2.16) from (2.2.15), and taking into account (2.2.3), we have

̅̅̅ ̅ ̅̅ ̅̅̅̅ ̅̅̅ ̅ ̅̅ ̅̅̅̅ [ ] [ ] .

It follows immediately from (2.2.12) that

̅ ̅̅ ̅̅ ̅ ̅̅̅ ̅ ̅̅ ̅ [ ] [ ] . (2.2.17)

Substitution of (2.2.11) into (2.2.17) finishes the proof.

Eq. (2.2.9) is defined in the neighborhood of the interface, and it might be analytically continued to a wider domain. The result relates the potential on one side of the interface to its counterpart in the other side. In other words, the complex velocity on the infinite fluid domain is given by the complex velocity on its complement plus the time derivative of the Schwarz function, and vice versa. Thus, having the solution of the interior problem, it is straightforward to calculate the solution of the exterior problem.

As an application of the formula (2.2.9), i.e., two-phase displacement in a Hele-

Shaw cell, we consider a dynamical mother body in a Hele-Shaw cell proposed by Savin et al [56]. 73

2.3 Mother Body

Historically the concept of mother body was introduced in the context of continuation of gravitational field. In the next subsection we give its brief description following to [55].

2.3.1 Mother body and continuation of the gravitational field

Let ( ) be a massive body with a nonnegative mass density, ( ) ,

such that ( ) can be continued up to an entire function in the complex plane ( ) and the boundary defined by the equation

{( ) | ( ) } (2.3.1) where ( ) is a polynomial with real coefficients. Savina et al. [55] defined the mother body for a given massive body used throughout this context as follows.

Definition 2.3.1 [55],[58]. A mother body of a given massive body is a tree-like shape comprising a finite set of curves and/or points with a positive mass density producing the same external gravitational field, provided that , contained entirely within and does not bound a two-dimensional subdomain of

In summary, the external gravitational field, which coincides with the mass

density produced by , has a potential ( ) and continues harmonically to ( )

. This harmonic continuation is considered a multiple-valued function so that the mother body is a set of branch cuts selecting a single-valued branch of the multiple- valued function itself. The mother body of a given massive body Ω is a set of singularities of a continuation of the external gravitational potential, taking into 74

consideration that the singularities must be at most of logarithmic type. The system of branch cuts subjected to the following conditions:

(1) The branch cuts are admissible such that the limit values of the gravitational

potential coincide in both sides of cuts,

(2) The mother body is a proper subset of the massive body,

2.3.2 A dynamic mother body in a Hele-Shaw flow

Savina et al. [56] have proposed an application of the Schwarz function equation for a single-phase Hele-Shaw problem derived by Howison [23],

, (2.3.2) where is the complex potential and is the time derivative of the Schwarz function.

Note that to prove equation (2.3.2) the interface was assumed to be equipotential, and the interface normal velocity was given by the normal derivative of a harmonic field. Eq.

(2.3.2) classifies the driven singularities of Laplacian growth into two types: physical singularities which coincide with the potential and non-physical singularities which are those of the initial Schwarz function. A dynamical mother body in a Hele-Shaw flow is described as follows: Given Ω( ) as a one parameter family of domains, we have to determine the driven mechanism by which the simply-connected domain is completely removed without cusp formation.

Definition 2.3.2 [56]. Let ( ) be a one parameter family of domains in the complex plane with algebraic boundaries. The dynamical mother body for a given ( ) in the

Hele-Shaw cell is a union of singularities of ( ) and a system of cuts subject to the following conditions: 75

(1) The cut must be admissible in the sense that the limiting values of the pressure on

either side of the cut must coincide.

(2) All cuts must be contained in the domain ( ).

Proposition 2.3.1 [56]. Let be a singular point of ( ), then

(1) if is a moving singular point, there is only one direction of admissible cut near

this point defined by

(2.3.3)

(2) if is stationary, then the direction of the admissible cuts given

(2.3.4) where is the angle of the singular point with respect to the positive

In order to formulate an algorithm of construction of a mother body, we have to take into account the following conditions:

(1) The boundary has an algebraic representation,

(2) The singularities are at most of logarithmic type.

The suggested algorithm is obtained through the following steps:

(1) Determining singularities of the Schwarz function,

(2) Constructing a set of admissible cuts,

(3) Selecting a set of bounded admissible cuts.

For illustration, we consider examples of the two-phase displacement in a Hele-Shaw cell, where a bounded fluid region embedded into another infinite fluid. Note that the interior behavior of a dynamical mother body was considered by Savin et al. in [56] while here we add the exterior behavior, which includes the far field flow. 76

Example 1

Let Ω( ) be a boundary of circular domain ( ( ) ) occupied by a viscous fluid which is embedded into another infinite viscous fluid. The corresponding Schwarz function is ( ) ( ) so that ( ) has only a simple pole at the origin, and no branch cuts. From (2.2.9) it follows

̇ ( ) (2.3.5)

Therefore, the circular droplet can be totally removed by a point sink located at the center while the far field flow is generated by a point source at infinity.

Example 2

{ } ( ) ( ) Let Ω = ( ) ( ) be an elliptic domain with the half-axis and , where

( ) ( ), embedded in another infinite fluid. The Schwarz function of is

( ) ( ) ( ) ( ) ( ) √ ( ) ( ) ( ) , (2.3.6) where √ is half of the interfocal distance. Thus, ( ) has a square root branch along the segment joining . Differentiating the Schwarz function with respect to time we have

( ) √ ( ) ( ) (2.3.7) √

From (2.2.9) follows that

( ) [ ( ) √ ( ) ( )] (2.3.8) √

First, we determine the bounded flow behavior (within the elliptic blob). From proposition 2.3.1 it follows that the singular point has two schemes: 77

(1) Stationary singular points so that ( ) ( ) for all the time, then equation

(2.3.8) becomes

√ ( ) ( ). (2.3.9)

From (2.3.4) it follows that at the point , the angle so that the only admissible cut is when by which the admissible cut points toward the other singular point , and it is contained in the fluid domain. Otherwise, the admissible cut intersects with the interface.

(2) Moving singular points

From (2.3.3) it follows that at the angle so that

( ), while the angle so that

( )

Thus, the only admissible cut is the interfocal. In contrast, integrating (2.3.9) with respect to we have

√ ( ) [ ( ) ( )

( √ ) ] ,

the complex potential is analytic in the infinite fluid, and the far field flow consists of a stagnation point along with a point source.

In conclusion, we derived the governing equation in terms of the Schwarz function for interface dynamics in the two-phase Hele-Shaw cell. The obtained equation 78

relates the potential on one side of the interface to its counterpart on the other side. In particular, the velocity field in the bounded fluid domain equals the velocity field of the infinite fluid domain plus the time derivative of the Schwarz function. The constructed equation is a re-derivation of an equation earlier obtained by Crowdy [10], but we introduced the idea using a different approach. Our derivation is based upon a single- valued velocity potential instead of a multiple-valued complex potential.

3 GENERALIZED HELE-SHAW FLOWS

This chapter is devoted to a class of the generalized Hele-Shaw flow, which is motivated by the interior problem when the gap width is time-dependent, and based on the paper [35], written jointly with Tatiana Savin, which has been submitted for publication. In the next section we discuss the previously generalized model of a free boundary Hele-Shaw problem prior to present our generalization.

3.1 Hele-Shaw flow with external potential field

The generalized Hele-Shaw flow arises when a droplet of fluid experiences an arbitrary external potential so that the flow of fluid is driven by the pressure gradient and the external potential. In addition, both the pressure and the external potential may have singularities in the simply-connected domain. Due to the continuity of the velocity field within the fluid domain up to the boundary, the boundary value of the pressure given by the atmospheric pressure together with the value of the external potential (the surface tension is neglected). Therefore, the interface is no longer equipotential, that is

constant. We assume a constant atmospheric pressure (say zero since only the pressure gradient is relevant to the fluid flow), then the pressure boundary value equals the value of external potential which may vary in both time and position. The dynamics of fluid within the simply-connected domain is described by

( ) ( ) in Ω (3.1.1)

in Ω( ) (3.1.2)

( ) = , on Ω (3.1.3) 80

on Ω( ) (3.1.4)

Eq. (3.1.1) is the Darcy law, and the incompressibility condition yields Eq. (3.1.2). Eq.

(3.1.3) is the conservation of mass while Eq. (3.1.4) is the continuity of pressure (the

Young-Laplace equation). Therefore, Eq. (3.1.4) is the only difference with the classical free boundary Hele-Shaw problem in which the pressure is constant-field on the boundary. This model was formulated from two perspectives: Entov et al. [15] used the

Richardson’s harmonic moments while McDonald [37] exploited the properties of the

Schwarz function of the interface as follows; the former generalized the usual harmonic moments while the latter generalized Eq. (1.2.41).

Entov and et al. have derived the generalized Richardson’s harmonic moments.

The only difference with the previously used harmonic moments is that the interface is not equipotential. According to Green’s theorem (see, for example, [47]), we have

(3.1.5)

where, is a circle of a small radius and centered at the singular point of the flow. The right-hand side of (3.1.5) reduces to

| | ‖ ‖ | |

( ) and

Thus, equation (3.1.5) has the form 81

(3.1.6)

Note that to prove (3.1.6) we used the velocity potential of a point source given by

| | .

Using the boundary condition (3.1.4), Eq. (3.1.6) can be written in the form

(3.1.7)

Applying Gauss theorem on the left-hand side of this equation, we obtain

( ) ( ) ( ) (3.1.8)

Therefore, Eq. (3.1.7) becomes

( ) (3.1.9)

On the other hand, applying the Reynolds transport theorem to the Richardson’s harmonic moments yields the relation

( ) ( )

( ) . (3.1.10)

We used (3.1.3) in the first equality, and (3.1.4) in the second equality. From (3.1.9) and

(3.1.10), it follows that the generalized Richardson’s harmonics moment has the form

( ) ( ) (3.1.11)

This is the generalization of Richardson harmonics moments which takes into consideration the presence of the external potential field. In other words, the pressure equals a specific function of the boundary point here while the pressure vanishes on the 82

boundary in the usual harmonic moments. Therefore, the formula (3.1.11) reduced to the original harmonic moments in the absence of the external potential .

McDonald has derived a governing equation in terms of the Schwarz function of the interface for the problem defined by (3.1.1) – (3.1.4) as follows. Eq. (3.1.1) yields that the tangent velocity is obtained by taking the tangent derivative of the velocity potential so that

̅ (3.1.12) note that we used (3.1.4) in the second equality. Also, the tangent velocity can be written as inner product of two complex numbers regarded as vectors

( ̅ ) , (3.1.13) where / is the unit tangent vector. Therefore,

̅ ( ̅ ) ̅ (3.1.14) then

̅ ̅̅ ̅ [ ̅ ] ̅ (3.1.15)

Form (1.2.15) and (1.2.16), in which the tangent vector is expressed in terms of the

Schwarz function,

̅ , √ √ it follows that

[ ̅ ] ̅ (3.1.16)

Along the boundary we have ̅ ( ) thus 83

̅ ̅ ̅ . (3.1.17)

Substitution of (3.1.17) into (3.1.16) yields the form

( ) ( ( ) ̅ . (3.1.18)

On the other hand, consider a multiple-valued complex potential function ( ) defined in Ω( ) whose real part ( ) satisfies the boundary condition (3.1.3). The relation between the complex potential and the velocity field is defined through (1.3.26) as

( ) ̅

thus, equation (3.1.18) can be written in the form

( ) ( ) ( ( )

. (3.1.19)

Eq. (3.1.19) is defined along the boundary, and it may analytically continued to a wider domain. This equation is a generalization of (2.1.41) where the external potential is absent ( ). This derivation is summarized by the following theorem.

Theorem 3.1.1 [37]. Let be an analytic curve. Then there exists a multiple-valued analytic function ( ) defined in the neighborhood of satisfying the equation

( ) ( ) ( ( )

whose real part ( ) satisfies the boundary conditions (3.1.3) – (3.1.4).

In the present study we generalize the above theorem taking into consideration that equations (3.1.1) and (3.1.2) are supplemented by the following boundary conditions

( ) = , on Ω (3.1.22)

on Ω( ) (3.1.23) 84

Here, Eq. (3.1.4) remains unchanged while Eq. (3.1.3) is modified, i.e., the interface normal velocity is given as a sum of the normal derivative of the pressure and an arbitrary function while the former, (3.1.1)-(3.1.4), describes a model in the absence of the arbitrary function. Hence, the flow in the simply-connected domain is driven by the pressure gradient, the external potential and the function ; and all of them may have singularities in the fluid domain. In the next section we discuss the motivation by which our model is developed.

3.2 Governing equation for a class of the generalized Hele-Shaw flow

A class of generalized Hele-Shaw flows is motivated by the time-dependent gap

Hele-Shaw flow. Consider a one-phase Hele-Shaw flow in which the gap thickness is a function of time, but not of a position. This case is called a time-dependent gap Hele-

Shaw flow, and arises when the upper plate is being lifted (squeezed) uniformly at a specific rate resulting interfacial motion. Assume that the upper plate is not lifted rapidly, so that the viscosity force still predominates over the inertial force, and the plate is being lifted up to the allowed height in which the two-dimensional approximation is retained. The conservation of mass indicates that

( ) ( ) ( ) ( ) (3.2.1) where is a distance across the gap and is the plane view in the mid-surface of the cell.

The relevant mathematical statement is identical to that of the usual Hele-Shaw flow, except for the continuity equation [61]. For illustration, the three-dimension velocity vector of an incompressible fluid is divergence-free 85

Averaging the latter equation across the gap, we obtain

( ) ( ) ̅̅̅ ̅ ̅̅ ( | | ) ( ) ( ) ( ) (3.2.2)

Indeed, fluid velocity at a liquid-solid boundary has a velocity of that boundary due to the no-slip condition. Thus, a particle of fluid touching the upper plate has the velocity of the upper plate. Let z be the third component in ( ) coordinates, then the corresponding velocity component at the height ( ) across the gap has the form

( ( )) ( )

. (3.2.3)

Due to the no-slip condition, we have | since the lower plate is fixed.

Substitution of (3.2.3) into (3.2.2) yields the modified continuity equation

̇ ( )

( ) (3.2.4)

̇ where is the averaged velocity field in the plane and ( ) ( ) | ( ) is the third velocity component in ( ) coordinates. It is evident from (3.2.4) that the averaged velocity field in the plane is no longer divergence-free. Since the dynamics of fluid in the simply-connected domain is described by the Darcy law,

in ( ) , (3.2.5) and the velocity field is not divergence-free (3.2.4), the pressure is no longer a harmonic function, i.e., the pressure satisfies the Poisson’s equation with a density function depending on time only

̇ ( ) in ( ) (3.2.6) ( ) ( ) 86

The latter equation is complemented by boundary conditions:

( ) ( ) on Ω , (3.2.7)

on Ω( ) (3.2.8)

Here ( ) is the two-dimensional gap-averaged velocity, is the pressure, ( ) is the time-dependent gap thickness and is the viscosity. Eq. (3.2.7) is the conservation of mass, i.e., a particle of fluid on the boundary remains on the boundary throughout the course of motion. Eq. (3.2.8) indicates the pressure is continuous within the simply- connected domain up to the boundary. Remark that equation (3.2.8) called the Young-

Laplace equation for the pressure,

atmospheric pressure + surface tension.

We stress that the atmospheric pressure is constant (say zero since only the pressure gradient is relevant to the flow) and the surface tension is neglected.

In summary, the only difference with the usual Hele-Shaw cell defined by (2.1.8)

- (2.1.11) is that the continuity equation. If the upper plate is lifted, the interface rushes inwardly so that a viscous fluid such as oil, is displaced by a less viscous one (i.g., air).

Thus, Saffman-Taylor fingers may occur [53] (see Figure 11). In fact, the time derivative of ( ) is nonnnegative (nonpositive) when ( ) is increasing (decreasing) so that the fluid domain is contracted (expanded). This observation is consistent with the following lemma.

Lemma 3.2.1 [36]. If the initial value problem has a smooth solution { ( )}, ( ) is contracting when ̇ ( ) , and expanding when ̇ ( ) . 87

Proof. From (3.2.6) it follows that is subharmonic when ̇ ( ) It evident from theorem (1.2.2) together with the boundary condition (3.2.8) that . Therefore, the

( ) pressure achieves its maximum on so that on the boundary where is the outward normal. In view of the relation (3.2.7), the simply-connected domain is contracted. Thus, given that ̇ ( ) , we have a contracted domain. The other case, when ̇ ( ) , is handled similarly.

Figure 11: Saffman-Taylor finger [67].

Now, rescaling the pressure and the surface tension coefficient using the change of variables ̃ , and ̃ , and substituting into equations (3.2.6) - (3.2.8) we have 88

̇ ( ) ̃ ( ) ( ) in , (3.2.9)

̃ ( ) ( ) on Ω , (3.2.10)

̃ ̃ on Ω( ) (3.2.11)

Eq. (3.2.9) indicates that the pressure satisfies the Poisson equation with density function depending on time only. Thus, the difference ̃ ( ̇ ( ) ( )) ( ) satisfies the

Laplace equation. Expressing ̃ as a sum of two functions

̇ ( ) ̃ ̅ ( ) ( ) (3.2.12)

We ultimately have

̅ in ( ), (3.2.13)

̇ ( ) ̅ ̃ ( ) ( ) ( ) on Ω (3.2.14)

̅ ̇ ( ) ( ) [ ( )] ( ) ( ) on Ω . (3.2.15)

The latter can be consider as a special case of a more general form

( ) in ( ), (3.2.16)

( ) on Ω( ) (3.2.17)

( ) [ ( )] ( ) on Ω (3.2.18) with

̇ ( ) ( ) ( ) ( ) (3.2.19) and

̇ ( ) ( ) ( ) ( ) (3.2.20) 89

Here, is a time-dependent distribution, whose support has nonzero co-dimension (a set of points and/or curves) and located strictly within ( ) for any time. In this section we reformulate the problem defined by (3.2.16) - (3.2.18) in terms of the Schwarz function of the interface. The main result is formulated in the following theorem.

Theorem 3.2.1. Let be an analytic curve. Then there exists a multiply-valued analytic function ( ) defined in a neighborhood of ( ) satisfying the equation

( ( )) √ , (3.2.21)

( ) ( ) ( ) [ ( )] whose real part satisfies Ω

Proof. Recall the following properties of the Schwarz function of the interface discussed in section 1.2.2:

̅ √ √ ( ) (3.2.22) ( )

√ ( ) , (3.2.23) √ ( ) ̅

( √ ( )) (3.2.24) √ ( ) ̅

. (3.2.25) √

Eq. (3.2.22) is the unit tangent vector written in terms of the Schwarz function. Equations

(3.2.23) and (3.2.23) are the directional derivatives of a function ( )̅ along and normal to the interface respectively. Eq. (3.2.25) is the interface normal velocity expressed in terms of the Schwarz function. Now consider a multiple-valued function

( ) defined in ( ) whose real part ( ) satisfies the boundary conditions 90

( ) ( ) [ ( )] Ω

Differentiating the function ( ) with respect to and applying Cauchy-Riemann equations in ( ) coordinates, where is a parameterization along the interface, we have

√ ( ) (3.2.26) note that we used (3.2.22). From (3.2.23) it follows that

√ ( ), (3.2.27) √ ( ) ̅ note that we used the first boundary condition. From the second boundary condition it follows that

, (3.1.28) √ note that we used (3.2.24) in the second equality. Substitution of (3.2.27) and (3.2.28) into (3.2.26) finishes the proof.

Now we are in the position to illustrate the work of theorem (3.2.1) on a specific case, i.e., the free boundary Hele-Shaw flow with time-dependent gap.

3.3 Governing equation for the time-dependent gap Hele-Shaw flow

Consider the problem defined by (3.2.6)-(3.2.8), i.e., a time-dependent gap Hele-

Shaw flow. In this case eq. (3.2.21) derived in the previous section can be written as

( ( )) √ ( ) (3.3.1) where

̇ ( ) ̃ (3.3.2) ( ) ( ) 91

and

̇ ( ) ( ) ( ) (3.3.3)

Here, the second term of (3.3.2) is the surface tension given in terms of the Schwarz function (see section 1.2.2). Differentiating equation (3.3.2) with respect to , we have

̇ ( ) ( ) ̃ { } ( ) , (3.3.4) where { } is the Schwarzian derivative which is the derivative of the surface tension.

Using the formula (3.1.24), Eq. (3.1.29) has the form

̇ ( ) ( √ ). (3.3.5) ( ) √

Substitution of (3.3.4) and (3.3.5) into (3.3.1) yields the following theorem.

Theorem 3.3.1. Let be an analytic curve. Then there exists a multiple-valued analytic function ( ) defined in a neighborhood of ( ) satisfying the equation

̇ √ ̃ { } (3.3.6) whose real part ( ) satisfies the boundary conditions (3.2.7)-(3.2.8).

Next, we shall illustrate the work of the above theorem, i.e., the Schwarz function equation for the time-dependent gap Hele-Shaw cell, using different examples.

Example 1

We start with the simplest example when the blob has a circular boundary, with initial radius ( ) and gap width ( ) . The corresponding Schwarz

function is ( ) ( ) , thus ̇ and . The problem (3.2.13)

-(3.2.15) for the pressure and the normal velocity for this case 92

̅ in ( ) (3.3.7)

̇ ̃ ̅ ( ) ( ) ( ) on (3.3.8)

̅ ̇ ( ) ( ) ( ) on (3.3.9)

Since any circle has a constant curvature, the Schwarzian derivative, { } equals zero.

Thus, equation (3.3.6) has the form

̇ ( ̇ ) (3.3.10)

From the conservation of volume (see equation (3.2.1)),

( ) ( ), (3.3.11) it follows that the complex potential is a function of time only so that the boundary condition (3.3.8) defined in the fluid domain. Thus, the real part of the complex potential

̇ ̃ ̅ ( ) ( ) (3.3.12)

Then,

̇ ̇ ̃ ̃ ( ) ( ) ( ) (3.3.13) or in polar coordinates

̇ ̃ ̃ ( ) ( ) (3.3.14)

Thus,

̃ ̇

(3.3.15)

̇ and , i.e., the normal velocity is proportional to the radius as it is the case of the evaporation of a thin circular film considered in [2]. 93

Example 2

{ } ( ) ( ) Let Ω= ( ) ( ) be an elliptic domain with the half-axis and , where

( ) ( ). The problem (3.2.13) - (3.2.15) for the pressure and the normal velocity is as follows

̅ in ( ) (3.3.16)

̇ ̅ ̃ ( ) on (3.3.17)

̅ ̇ ( ) ( ) ( ) on (3.3.18)

For simplicity assume that ̃ so that (3.3.6) has the form

(3.3.19)

From the conservation of volume (see equation (3.2.1)) it follows that

( ) ( ) ( ) (3.3.20) where ( ) ( ) and ( ). Eq. (3.3.19) becomes

( ) ( ) (3.3.21)

Ths Schwarz function of the interface is

( ) ( ) ( ) ( ) ( ) √ ( ) ( ) ( ) , (3.3.22) where √ is half of the interfocal distance. Differentiating the Schwarz function with respect to time we have

( ) √ ( ) ( ) (3.3.23) √

Substitution of and into (3.3.21) yields the relation 94

( ) [ ( ) ( )]

( ) √ [ ( )] ( ) (3.3.24) √

From a physical viewpoint, if there are no sinks/sources in Ω( ), the latter formula indicates that Ω( ) must be a family of co-focal ellipses ( ( ) ( ) ). Then,

( ) ( ) (3.3.25) where ( ( )) is determined by the boundary conditions on ( ) Thus,

̇ ̇ ̅ [( ) ̇] (3.3.26)

Then,

̇ ̇ ̇ ̃ [( ) ̇] ( ) (3.3.27)

The structure of this formula up to the time-dependent coefficients coincides with the formula derived using harmonic moments for the case of evaporation of a thin elliptic film [2].

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