The Motion of Bubbles and Drops in Reduced Gravity

, by R. Shankar Subramanian , R. Balasubramaniam

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ISBN: 9780521496056 | 0521496055
Cover: Hardcover
Copyright: 4/9/2001

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Orbiting spacecraft provide a valuable laboratory for experiments on physical and biological systems in a reduced gravity environment. Materials processing experiments have commonly involved the growth of crystals from the melt or solution, and the processing of alloys and composites. Biological experiments have been performed on a variety of subjects, including protein crystal growth, bio-reactors, and the adaptation of humans to extended periods of weightlessness. In these studies, fluid masses containing bubbles and drops are encountered routinely. This book is the first to provide a clear, thorough review of the motion of bubbles and drops in reduced gravity, particularly motion caused by variations in interfacial tension arising from temperature gradients on their surfaces. The emphasis is on theoretical analysis from first principles; experimental results are discussed and compared with predictions where appropriate. Students and researchers interested in fluid mechanics in reduced gravity will welcome this state-of-the-art reference.

Preface

PART ONE. INTRODUCTION

The Role of Gravity and Interfacial Tension in the Motion of Bubbles and Drops

(19)

Bubbles and Drops and the Influence of Gravity

(2)

The Reduced Gravity Environment Aboard Spacecraft

(2)

Mechanisms That Can Cause the Motion of Bubbles and Drops in the Absence of Gravity

(3)

Interfacial Tension

(6)

Elementary Concepts

(4)

The Dependence of Interfacial Tension on Adsorbed Surfactant Concentration

(1)

The Dependence of Interfacial Tension on Temperature

(1)

Description of Interfacial Tension Driven Motion

(4)

Scope of the Book

(2)

The Governing Equations

(25)

Introduction

(2)

Conservation Equations

(5)

Consevation of Total Mass

(1)

Conservation of Momentum

(3)

Conservation of Energy

(1)

Conservation of Species

(1)

Boundary Conditions

(6)

General Comments

(1)

Fluid-Solid Interface

(1)

Fluid-Fluid Interface

(4)

Physical Assumptions

(2)

Important Dimensionless Parameters

(3)

Conservation Equations in Common Coordinate Systems

(7)

PART TWO. THE MOTION OF ISOLATED BUBBLES AND DROPS

Motion Driven by a Body Force

(36)

Motion When Inertial Effects Are Negligible

(12)

Introduction

(1)

Analysis

(6)

Results

(5)

Motion Accounting for Small to Moderate Inertial Effects

(7)

Introduction

(1)

Asymptotic Expansions

(2)

The Role of Small Inertial Effects

(3)

Numerical Solutions for the Case of Intermediate Values of the Reynolds Number

(2)

Motion of a Spherical Bubble at Large Reynolds Number-Asymptotic Analysis

(6)

Introduction

(1)

Analysis at Leading Order

(3)

Higher-Order Analysis

(1)

Hydrodynamic Force on the Bubble

(1)

Motion of a Spherical Liquid Drop at Large Reynolds Number-Asymptotic Analysis

(8)

Introduction

(1)

Inviscid Solutions for the Velocity Field

(1)

Outer Velocity and Pressure Fields

(1)

Inner Velocity and Pressure Fields

(3)

Solution for the Tangential Velocity Fields in the Boundary Layer

(2)

Motion of Highly Deformed Bubbles and Drops

(3)

Introduction

(1)

Determination of the Rise Velocity

(2)

Thermocapillary Motion

(146)

Governing Equations

(5)

Introduction

(1)

Problem Statement, Physical Assumptions, and Scales

(1)

Governing Equations and Boundary Conditions

(4)

Motion When Convective Transport Is Negligible

(11)

Introduction

(1)

The Case of Zero Gravity

(6)

Inclusion of the Gravitational Effect

(5)

Motion When Inertia Is Important, but Convective Transport of Energy Is Negligible

(6)

Introduction

(1)

Calculation of Small Departures of the Shape from a Sphere Due to Inertia

(4)

Correction to the Migration Velocity Due to Deformation

103

(2)

The Case of an Arbitrary Temperature Gradient

105

(3)

Introduction

105

(1)

Analysis

105

(3)

A Useful Special Case

108

(1)

The Case of an Axisymmetric Interfacial Tension Gradient

108

(3)

Introduction

108

(1)

Solution for the Fields

109

(1)

The Hydrodynamic Force and the Quasi-Steady Velocity

110

(1)

Motion in the Presence of Surfactants

111

(4)

Introduction

111

(1)

The Way in which Surfactants Affect the Motion of Drops

111

(2)

Modeling the Effect of Surfactants on Thermocapillary Migration

113

(2)

Effect of Insoluble Surfactant When Convective Transport Dominates - Stagnant Cap Limit

115

(6)

Solution for the Drag on a Drop and Its Migration Velocity

115

(2)

Surfactant Distribution along the Cap

117

(3)

Surface Velocity Distribution over the Clean Surface

120

(1)

Effect of Insoluble Surfactant When Both Surface Convection and Diffusion Are Important

121

(12)

Introduction

121

(2)

Expansion for Small Surface Peclet Number

123

(1)

Expansion for Small Elasticity Number

124

(1)

Series Improvement Using Pade Approximants

125

(2)

Numerical Solution

127

(6)

Effects of Newtonian Surface Rheology and Temperature Gradients Induced by Motion

133

(4)

Introduction

133

(1)

Motion of a Drop with a Newtonian Interface Subjected to an Arbitrary Interfacial Tension Gradient

134

(1)

Contribution Due to Stretching and Shrinkage of Interfacial Area Elements

135

(2)

Motion Due to a Source of Energy at the Drop Surface

137

(2)

Introduction

137

(1)

Analysis

137

(2)

Unsteady Motion

139

(6)

Introduction

139

(1)

Governing Equations and Initial and Boundary Conditions

140

(1)

Solution by Laplace Transforms

141

(3)

Discussion

144

(1)

Influence of Convective Transport of Energy - Asymptotic Analysis for Small Marangoni Number

145

(10)

Introduction

145

(2)

The Regular Perturbation Approach

147

(2)

Solution by the Method of Matched Asymptotic Expansions

149

(6)

Influence of Convective Transport of Energy - Asymptotic Analysis for Large Marangoni Number in the Gas Bubble Limit

155

(14)

Introduction

155

(1)

Governing Equations

156

(1)

Solution by the Method of Matched Asymptotic Expansions

157

(2)

The Case of Large Reynolds Number

159

(5)

Temperature Field in the Thermal Wake

164

(2)

The Case of Negligible Reynolds Number

166

(3)

Influence of Convective Transport of Energy - Asymptotic Analysis for Large Marangoni Number for a Drop

169

(13)

Introduction

169

(1)

Governing Equations and Asymptotic Scalings

170

(4)

Leading Order Outer Temperature Fields

174

(3)

Leading Order Inner Temperature Fields

177

(4)

The Gas Bubble Limit

181

(1)

Migration When Convective Transport Is Important - Results from Numerical Solution of the Governing Equations

182

(13)

Introduction

182

(2)

Results in the Gas Bubble Limit

184

(5)

Results in the Case of Drops

189

(2)

Influence of Shape Deformation

191

(4)

Experimental Results on Isolated Bubbles and Drops

195

(20)

Introduction

195

(2)

Experiments on Earth

197

(8)

Experiments in Reduced Gravity

205

(10)

Flow Structures

215

(14)

Introduction

215

(1)

Motion Driven by Thermocapillarity

215

(5)

Motion under the Combined Influence of Gravity and Thermocapillarity

220

(9)

PART THREE. INTERACTIONS OF BUBBLES AND DROPS

General Solutions

229

(8)

Solutions of Laplace's and Stokes's Equations in Bispherical Coordinates

229

(8)

Introduction

229

(1)

Bispherical Coordinates

229

(3)

Solution of Laplace's and Stokes's Equations in Bispherical Coordinates

232

(3)

Solution by the Method of Reflections

235

(2)

Interactions When Motion Is Driven by a Body Force

237

(15)

Motion Normal to a Plane Surface

237

(7)

Introduction

237

(1)

Theoretical Analysis

237

(7)

Comparison with Experimental Results

244

(1)

Interaction between Two Drops

244

(8)

Introduction

244

(1)

Motion along the Line of Centers

245

(2)

Motion Normal to the Line of Centers

247

(3)

Concluding Remarks

250

(2)

Interactions When Motion Is Driven by Thermocapillarity

252

(109)

Motion Normal to a Plane Surface

252

(16)

Introduction

252

(1)

Analysis

252

(4)

Some Results from the Analysis

256

(5)

Some Related Problems

261

(1)

The Case When the Drop Is Very Close to the Plane Surface

262

(3)

Comparison of Theoretical Predictions with Experimental Data

265

(3)

Motion of a Bubble in an Arbitrary Direction Near a Rigid Plane Surface

268

(6)

Introduction

268

(1)

Analysis

269

(2)

Results

271

(3)

Axisymmetric Motion of Two Interacting Drops

274

(11)

Introduction

274

(1)

Analysis in Bispherical Coordinates

275

(3)

Alternative Approaches

278

(3)

Results in the Axisymmetric Case

281

(4)

Related Problems

285

(1)

Motion of a Pair of Bubbles under the Combined Action of Gravity and Thermocapillarity

285

(13)

Introduction

285

(2)

Predictions for Bubbles of Equal Size

287

(2)

Predictions for Bubbles of Unequal Size

289

(3)

Flow Structures

292

(4)

Experimental Results

296

(2)

Interactions between Two Bubbles Oriented Arbitrarily with Respect to the Temperature Gradient

298

(9)

Introduction

298

(1)

Analysis

299

(4)

Results

303

(4)

Interactions among Three or More Drops

307

(10)

Introduction

307

(1)

Pairwise-Additive Approximation

307

(1)

Results for a Chain of Three Bubbles

308

(4)

Behavior of a Suspension of Drops Moving under the Influence of Gravity or Thermocapillarity

312

(5)

Axisymmetric Motion of a Droplet within a Stationary Drop - Concentric Case

317

(9)

Introduction

317

(1)

Analysis

318

(4)

Results

322

(4)

Axisymmetric Motion of a Droplet within a Stationary Drop - Eccentric Case

326

(11)

Introduction

326

(1)

Analysis

327

(4)

Results for the First Few Legendre Modes of the Surface Temperature

331

(6)

Axisymmetric Motion of a Compound Drop - Concentric Case

337

(6)

Introduction

337

(1)

Analysis

338

(3)

Results

341

(2)

Axisymmetric Motion of a Compound Drop - Eccentric Case

343

(6)

Introduction

343

(1)

Analysis

344

(1)

Results

345

(4)

Interactions When Convective Transport Effects Are Not negligible

349

(12)

Introduction

349

(1)

Analysis

350

(6)

Experimental Observations

356

(5)

PART FOUR. RELATED TOPICS

Mass Transfer between a Bubble or Drop and a Continuous Phase

361

(30)

Introduction

361

(2)

Governing Equations for the Dissolution or Growth of a Stationary Sphere in a Large Liquid Body

363

(4)

Introduction

363

(1)

The Case When Density Variations Are Permitted in the Continuous Phase

364

(2)

The Limiting Case of a Gas Bubble

366

(1)

Stationary Sphere in a Large Liquid Body - The Quasi-Stationary Solution

367

(3)

Introduction

367

(1)

Quasi-Stationary Concentration Field

368

(1)

Time Evolution of the Radius of the Sphere

369

(1)

Stationary Sphere in a Large Liquid Body - Similarity Solution for Growth from Zero Initial Size

370

(3)

Introduction

370

(1)

Analysis

370

(3)

Stationary Sphere in a Large Liquid Body - Asymptotic Expansions

373

(8)

Introduction

373

(1)

Governing Equations

373

(1)

Expansion in the Parameters Na and Nb

374

(2)

Expansion in the Square Root of Time

376

(2)

Comparison with a Numerical Solution

378

(2)

Extensions to Include Other Effects

380

(1)

Mass Transport from a Drop in Thermocapillary Motion at Small Compsitional Peclet Number

381

(6)

Introduction

381

(1)

Governing Equations for the Concentration Field

382

(1)

The Local and Average Mass Transfer Coefficients and the Governing Equation for the Evolution of the Radius of the Drop

383

(1)

Solution for Small Compositional Peclet Number

384

(3)

Mass Transport from a Drop in Thermocapillary Motion at Large Compositional Peclet Number

387

(4)

Introduction

387

(1)

Analysis

387

(4)

Motion Driven by the Interface in a Body of Fluid

391

(58)

Introduction

391

(2)

Flow in a Rectangular Liquid Layer

393

(9)

Introduction

393

(2)

Velocity Field

395

(3)

Shape of the Free Surface

398

(1)

Temperature Field

399

(3)

Transient Development of the Flow and Free Surface Shape in a Rectangular Liquid Layer

402

(14)

Introduction

402

(1)

Governing Equations and Time Scales

402

(3)

Evolution of the Velocity Field for Small Values of Time

405

(1)

Unsteady Deformation of the Free Surface for Large Bond Number

406

(2)

Simultaneous Evolution of the Flow and Free Surface Deformation for Large Bond Number

408

(3)

Unsteady Deformation of the Free Surface for Small Bond Number

411

(2)

Simultaneous Evolution of the Flow and Free Surface Deformation for Small Bond Number

413

(3)

Flow in Multiple Rectangular Layers

416

(3)

Introduction

416

(1)

Motion Driven by a Single Fluid-Fluid Interface

417

(1)

Motion Driven by Two Interfaces

418

(1)

Flow Near an End Wall in a Thin Rectangular Liquid Layer

419

(7)

Introduction

419

(1)

Analysis

420

(5)

Results

425

(1)

Effects of Inertia and Convective Energy Transport on the Flow in a Deep Liquid Layer

426

(9)

Introduction and Governing Equations

426

(2)

Scaling Analysis

428

(1)

Solution for the Scaled Streamfunction and Temperature Fields

429

(3)

Results

432

(3)

Flow in Cylindrical Layers and Float Zones

435

(6)

Introduction

435

(1)

Flow in Cylindrical Layers

436

(3)

Flow in Float Zones

439

(2)

Flow in the Spherical Geometry

441

(8)

Introduction

441

(1)

Flow in a Stationary Spherical Liquid Drop

441

(5)

Flow Induced by Bubble Attached to a Wall

446

(3)

References

449

(18)

Index

467

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The Motion of Bubbles and Drops in Reduced Gravity

Summary

Table of Contents

Supplemental Materials