Microhydrodynamics

Principles and Selected Applications

Sangtae Kim, Seppo J. Karrila, Seppo J. Karrila

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486442198
• eBook ISBN: 9780486317670
• Topic: Technology & Engineering
• Subtopic: Applied Sciences
• Index: Technology & Engineering

"This book is well organized and comprehensive . . . an eloquent and enduring statement of significant hydrodynamic principles." — AIChE Journal
Microhydrodynamics concerns the flow and related phenomena pertinent to the motion of small particles suspended in viscous fluids. This text focuses on determining the motion of a particle or particles through a viscous fluid in bounded and unbounded flow. Its central theme is the mobility relation between particle motion and forces.
Microhydrodynamics: Principles and Selected Applications functions as a manual that explains methods for solving particulate flows at low-Reynolds number, from analytical to computational methods. The ever-increasing growth in computational power has resulted in a similar growth in the range of solvable problems in microhydrodynamics. Suitable for graduate students in engineering and applied mathematics, this text treats the mathematical foundations and highlights the interplay of both mathematical and physical insights, guiding readers through single particle theory and problems related to multiparticle analyses.

Part III

Hydrodynamic Interactions

Chapter 7

General Formulation of Resistance and Mobility Relations

7.1 Introduction

In our effort to understand the behavior of suspensions, we must take into account the effect of particle-particle interactions, for such interactions exist in all but very dilute suspensions. In this part of the book, we consider several methods for calculating hydrodynamic interactions between particles. Our task will be guided in part by the knowledge gained in Part II of the flow that induces or is induced by the motion of a particle.

What do we mean by hydrodynamic interaction? When two particles suspended in a viscous fluid approach each other, the motion of each particle is influenced by the other, even in the absence of interparticle interactions, such as van der Waals and electrostatic forces. The velocity field generated by the motion of one particle is transmitted through the fluid medium and influences the motion of, as well as the hydrodynamic force, torque, and stresslet on the other particle. Thus when two particles move towards each other, for example, as a result of an attractive colloidal force, the hydrodynamic interaction between the two retards the motion; using just the single particle mobility in the attractive force law overestimates the rate of aggregation.

For the sake of organization, we divide the discussion on interactions according to particle-particle and particle-wall interactions (see Figure 7. 1). For each, the method of attack depends on the separation between the surfaces. For widely separated particles (the distance between closest points on the surfaces is much greater than particle size), a general asymptotic method known as the method of reflection is available. The solution can be expressed analytically as a series in terms of (the small parameter) particle size over separation.

Surfaces near contact present a far more challenging problem, both from an analytical and computational viewpoint. For rigid surfaces in relative motion, the flow in the gap region dominates and lubrication theory provides the leading terms in an asymptotic expansion. Here, the small parameter ∈ is the gap distance divided by the characteristic size of the particle (usually proportional to the local radius of curvature). For squeezing motions of two rigid surfaces, this leading term is O(∈^–1) and truly dominates over higher order corrections. On the other hand, for shearing motions (curved surfaces sliding past each other) the leading term is only O(ln ∈^–1), and thus dominates the solution only in the mathematical sense. Usually, the next term is O(1), and even if we take ∈ based on the ratio of molecular to macroscopic dimensions, ln ∈^–...

Table of contents

• Cover
• Title
• Copyright
• Preface to the Dover Edition
• Contents
• Preface
• Organization Scheme
• I. Governing Equations and Fundamental Theorems
• II. Dynamics of a Single Particle
• III. Hydrodynamic Interactions
• IV Foundations of Parallel Computational Micro hydrodynamics
• Notation
• Index