FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS

7/30/2019 FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS

1/482


2/482


3/482

FLUID MECHANICS AND PIPE FLOW:

TURBULENCE, SIMULATION

AND DYNAMICS

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form orby any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no

expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No

liability is assumed for incidental or consequential damages in connection with or arising out of information

contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in

rendering legal, medical or any other professional services.


4/482


5/482

FLUID MECHANICS AND PIPE FLOW:

TURBULENCE, SIMULATION

AND DYNAMICS

DONALD MATOS

AND

CRISTIAN VALERIO

EDITORS

Nova Science Publishers, Inc.

New York


6/482

Copyright 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or

transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical

photocopying, recording or otherwise without the written permission of the Publisher.

For permission to use material from this book please contact us:

Telephone 631-231-7269; Fax 631-231-8175

Web Site: http://www.novapublishers.com

NOTICE TO THE READER

The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or

implied warranty of any kind and assumes no responsibility for any errors or omissions. No

liability is assumed for incidental or consequential damages in connection with or arising out of

information contained in this book. The Publisher shall not be liable for any special,

consequential, or exemplary damages resulting, in whole or in part, from the readers use of, or

reliance upon, this material. Any parts of this book based on government reports are so indicatedand copyright is claimed for those parts to the extent applicable to compilations of such works.

Independent verification should be sought for any data, advice or recommendations contained in

this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage

to persons or property arising from any methods, products, instructions, ideas or otherwise

contained in this publication.

This publication is designed to provide accurate and authoritative information with regard to the

subject matter covered herein. It is sold with the clear understanding that the Publisher is not

engaged in rendering legal or any other professional services. If legal or any other expert

assistance is required, the services of a competent person should be sought. FROM A

DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THEAMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

Fluid mechanics and pipe flow : turbulence, simulation, and dynamics / editors, Donald Matos and Cristian

Valerio.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-61668-990-2 (E-Book)

1. Fluid mechanics. 2. Pipe--Fluid dynamics. I. Matos, Donald. II. Valerio, Cristian.

TA357.F5787 2009

620.1'06--dc22

2009017666

Published by Nova Science Publishers, Inc. New York


7/482

CONTENTS

Preface vii

Chapter 1 Solute Transport, Dispersion, and Separation in NanofluidicChannels

1

Xiangchun XuanChapter 2 H2O in the Mantle: From Fluid to High-Pressure Hydrous

Silicates27

N.R. Khisina, R. Wirth and S. Matsyuk

Chapter 3 On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

41

K. Mohanarangam and J.Y. Tu

Chapter 4 A Review of Population Balance Modelling for Multiphase Flows:Approaches, Applications and Future Aspects

117

Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu

Chapter 5 Numerical Analysis of Heat Transfer and Fluid Flow for Three-Dimensional Horizontal Annuli with Open Ends

171

Chun-Lang Yeh

Chapter 6 Convective Heat Transfer in the Thermal Entrance Regionof Parallel Flow Double-Pipe Heat Exchangersfor Non-Newtonian Fluids

205

Ryoichi Chiba, Masaaki Izumi and Yoshihiro Sugano

Chapter 7 Numerical Simulation of Turbulent Pipe Flow 231M. Ould-Rouis and A.A. Feiz

Chapter 8 Pipe Flow Analysis of Uranium Nuclear Heating with ConjugateHeat Transfer

269

G.H. Yeoh and M.K.M. Ho

Chapter 9 First and Second Law Thermodynamics Analysis of Pipe Flow 317Ahmet Z. Sahin


8/482

Contentsvi

Chapter 10 Single-Phase Incompressible Fluid Flowin Mini- and Micro-channels

343

Lixin Cheng

Chapter 11 Experimental Study of Pulsating Turbulent Flow

through a Divergent Tube

365

Masaru Sumida

Chapter 12 Solution of an Airfoil Design Inverse Problem for a Viscous FlowUsing a Contractive Operator

379

Jan imk and Jaroslav Pelant

Chapter 13 Some Free Boundary Problems in Potential Flow Regime Usingthe Level Set Method

399

M. Garzon, N. Bobillo-Ares and J.A. Sethian

Chapter 14 A New Approach for Polydispersed Turbulent Two-Phase Flows:The Case of Deposition in Pipe-Flows

441

S. ChibbaroIndex 455


9/482

PREFACE

Fluid mechanics is the study of how fluids move and the forces that develop as a result.Fluids include liquids and gases and fluid flow can be either laminar or turbulent. This book

presents a level set based methodology that will avoid problems in potential flow models withmoving boundaries. A review of the state-of-the-art population balance modelling techniques

that have been adopted to describe the nature of dispersed phase in multiphase problems ispresented as well. Recent works that are aimed at putting forward the main ideas behind anew theoretical approach to turbulent wall-bounded flows are examined, including a state-of-the-art review on single-phase incompressible fluid flow.

Recent breakthrough in nanofabrication has stimulated the interest of solute separation innanofluidic channels. Since the hydraulic radius of nanochannels is comparable to thethickness of electric double layers, the enormous electric fields inherent to the latter generatetransverse electromigrations causing charge-dependent solute distributions over the channelcross-section. As a consequence, the non-uniform fluid flow through nanochannels yieldscharge-dependent solute speeds enabling the separation of solutes by charge alone.In Chapter 1 we develop a theoretical model of solute transport, dispersion and separation in

electroosmotic and pressure-driven flows through nanofluidic channels. This model providesa basis for the optimization of solute separation in nanochannels in terms of selectivity andresolution as traditionally defined.

As presented in Chapter 2, infrared spectroscopic data show that nominally anhydrousolivine (Mg,Fe)2SiO4 contains traces of H2O, up to several hundred wt. ppm of H2O (Miller etal., 1987; Bell et al., 2004; Koch-Muller et al., 2006; Matsyuk & Langer, 2004) and thereforeolivine is suggested to be a water carrier in the mantle (Thompson, 1992). Protonation ofolivine during its crystallization from a hydrous melt resulted in the appearance of intrinsicOH-defects (Libowitsky & Beran, 1995). Mantle olivine nodules from kimberlites wereinvestigated with FTIR and TEM methods (Khisina et al., 2001, 2002, 2008). The results arethe following: (1) Water content in xenoliths is lower than water content in xenocrysts. Fromthese data we concluded that kimberlite magma had been saturated by H2O, whereas adjacent

mantle rocks had been crystallized from water-depleted melts. (2) Extrinsic water in olivine isrepresented by high-pressure phases, 10-Phase Mg3Si4O10(OH)2.nH2O and hydrous olivinen(Mg,Fe)2SiO4.(H2MgSiO4), both of which belong to the group of Dense HydrousMagnesium Silicates (DHMS), which were synthesized in laboratory high-pressureexperiments (Prewitt & Downs, 1999). The DHMS were regarded as possible mineral carriersfor H2O in the mantle; however, they were not found in natural material until quite recently.


10/482

Donald Matos and Cristian Valerioviii

Our observations demonstrate the first finding of the 10-Phase and hydrous olivine as amantle substance. (3) 10-Phase, which occurred as either nanoinclusions or narrow veins inolivine, is a ubiquitous nano-mineral of kimberlite and closely related to olivine. (4) There aretwo different mechanisms of the 10-Phase formation: (a) purification of olivine from OH-

bearing defects resulting in transformation of olivine to the 10 -Phase with the liberation of

water fluid; and (b) replacement of olivine for the 10-Phase due to hydrous metasomatismin the mantle in the presence of H2O fluid.

With the increase of computational power, computational modelling of two-phase flowproblems using computational fluid dynamics (CFD) techniques is gradually becomingattractive in the engineering field. The major aim of Chapter 3 is to investigate the TurbulenceModulation (TM) of dilute two phase flows. Various density regimes of the two-phase flowshave been investigated in this paper, namely the dilute Gas-Particle (GP) flow, Liquid-Particle (LP) flow and also the Liquid-Air (LA) flows. While the density is quite high for thedispersed phase flow for the gas-particle flow, the density ratio is almost the same for theliquid particle flow, while for the liquid-air flow the density is quite high for the carrier phaseflow. The study of all these density regimes gives a clear picture of how the carrier phase

behaves in the presence of the dispersed phases, which ultimately leads to better design andsafety of many two-phase flow equipments and processes. In order to carry out this approach,an Eulerian-Eulerian Two-Fluid model, with additional source terms to account for the

presence of the dispersed phase in the turbulence equations has been employed for particulateflows, whereas Population Balance (PB) have been employed to study the bubbly flows. Forthe dilute gas-particle flows, particle-turbulence interaction over a backward-facing stepgeometry was numerically investigated. Two different particle classes with same Stokesnumber and varied particle Reynolds number are considered in this study. A detailed studyinto the turbulent behaviour of dilute particulate flow under the influence of two carrier

phases namely gas and liquid was also been carried out behind a sudden expansion geometry.The major endeavour of the study is to ascertain the response of the particles within thecarrier (gas or liquid) phase. The main aim prompting the current study is the density

difference between the carrier and the dispersed phase. While the ratio is quite high in termsof the dispersed phase for the gas-particle flows, the ratio is far more less in terms of theliquid-particle flows. Numerical simulations were carried out for both these classes of flowsand their results were validated against their respective sets of experimental data. For theLiquid-Air flows the phenomenon of drag reduction by the injection of micro-bubbles intoturbulent boundary layer has been investigated using an Eulerian-Eulerian two-fluid model.Two variants namely the Inhomogeneous and MUSIG (MUltiple-SIze-Group) based onPopulation balance models are investigated. The simulated results were benchmarked againstthe experimental findings and also against other numerical studies explaining the variousaspects of drag reduction. For the two Reynolds number cases considered, the buoyancy withthe plate on the bottom configuration is investigated, as from the experiments it is seen that

buoyancy seem to play a role in the drag reduction. The under predictions of the MUSIGmodel at low flow rates was investigated and reported, their predictions seem to fair betterwith the decrease of the break-up tendency among the micro-bubbles.

Population balance modelling is of significant importance in many scientific andindustrial instances such as: fluidizations, precipitation, particles formation in aerosols,

bubbly and droplet flows and so on. In population balance modelling, the solution of thepopulation balance equation (PBE) records the number of entities in dispersed phase that


11/482

Preface ix

always governs the overall behaviour of the practical system under consideration. For themajority of cases, the solution evolves dynamically according to the birth and death

processes of which it is tightly coupled with the system operation condition. Theimplementation of PBE in conjunction with the Computational Fluid Dynamics (CFD) isthereby becoming ever a crucial consideration in multiphase flow simulations. Nevertheless,

the inherent integrodifferential form of the PBE poses tremendous difficulties on its solutionprocedures where analytical solutions are rare and impossible to be achieved. In Chapter 4,we present a review of the state-of-the-art population balance modelling techniques that have

been adopted to describe the phenomenological nature of dispersed phase in multiphaseproblems. The main focus of the review can be broadly classified into three categories: (i)Numerical approaches or solution algorithms of the PBE; (ii) Applications of the PBE inpractical gas-liquid multiphase problems and (iii) Possible aspects of the future developmentin population balance modelling. For the first category, details of solution algorithms basedon both method of moment (MOM) and discrete class method (CM) that have been proposedin the literature are provided. Advantages and drawbacks of both approaches are alsodiscussed from the theoretical and practical viewpoints. For the second category, applicationsof existing population balance models in practical multiphase problems that have been

proposed in the literature are summarized. Selected existing mathematical closures formodelling the birth and death rate of bubbles in gas-liquid flows are introduced.Particular attention is devoted to assess the capability of some selected models in predicting

bubbly flow conditions through detail validation studies against experimental data. Thesestudies demonstrate that good agreement can be achieved by the present model by comparingthe predicted results against measured data with regards to the radial distribution of voidfraction, Sauter mean bubble diameter, interfacial area concentration and liquid axial velocity.Finally, weaknesses and limitations of the existing models are revealed are suggestions forfurther development are discussed. Emerging topics for future population balance studies are

provided as to complete the aspect of population balance modelling.Study of the heat transfer and fluid flow inside concentric or eccentric annuli can be

applied in many engineering fields, e.g. solar energy collection, fire protection, undergroundconduit, heat dissipation for electrical equipment, etc. In the past few decades, these studieswere concentrated in two-dimensional research and were mostly devoted to the investigationof the effects of convective heat transfer. However, in practical situation, this problem should

be three-dimensional, except for the vertical concentric annuli which could be modeled astwo-dimensional (axisymmetric). In addition, the effects of heat conduction and radiationshould not be neglected unless the outer cylinder is adiabatic and the temperature of the flowfield is sufficiently low. As the author knows, none of the open literature is devoted to theinvestigation of the conjugated heat transfer of convection, conduction and radiation for this

problem. The author has worked in industrial piping design area and is experienced in thisfield. The author has also employed three-dimensional body-fitted coordinate systemassociated with zonal grid method to analyze the natural convective heat transfer and fluidflow inside three-dimensional horizontal concentric or eccentric annuli with open ends.Owing to its broad application in practical engineering problems, Chapter 5 is devoted to adetailed discussion of the simulation method for the heat transfer and fluid flow inside three-dimensional horizontal concentric or eccentric annuli with open ends. Two illustrative

problems are exhibited to demonstrate its practical applications.


12/482

Donald Matos and Cristian Valeriox

In Chapter 6, the conjugated Graetz problem in parallel flow double-pipe heat exchangersis analytically solved by an integral transform methodVodickas methodand an analyticalsolution to the fluid temperatures varying along the radial and axial directions is obtained in acompletely explicit form. Since the present study focuses on the range of a sufficiently largePclet number, heat conduction along the axial direction is considered to be negligible. An

important feature of the analytical method presented is that it permits arbitrary velocitydistributions of the fluids as long as they are hydrodynamically fully developed. Numericalcalculations are performed for the case in which a Newtonian fluid flows in the annulus of thedouble pipe, whereas a non-Newtonian fluid obeying a simple power law flows through theinner pipe. The numerical results demonstrate the effects of the thermal conductivity ratio ofthe fluids, Pclet number ratio and power-law index on the temperature distributions in thefluids and the amount of exchanged heat between the two fluids.

Many experimental and numerical studies have been devoted to turbulent pipe flows dueto the number of applications in which theses flows govern heat or mass transfer processes:heat exchangers, agricultural spraying machines, gasoline engines, and gas turbines forexamples. The simplest case of non-rotating pipe has been extensively studied experimentallyand numerically. Most of pipe flow numerical simulations have studied stability andtransition. Some Direct Numerical Simulations (DNS) have been performed, with a 3-Dspectral code, or using mixed finite difference and spectral methods. There is few DNS of theturbulent rotating pipe flow in the literature. Investigations devoted to Large EddySimulations (LES) of turbulence pipe flow are very limited. With DNS and LES, one canderive more turbulence statistics and determine a well-resolved flow field which is a

prerequisite for correct predictions of heat transfer. However, the turbulent pipe flows havenot been so deeply studied through DNS and LES as the plane-channel flows, due to the

peculiar numerical difficulties associated with the cylindrical coordinate system used for thenumerical simulation of the pipe flows.

Chapter 7 presents Direct Numerical Simulations and Large Eddy Simulations of fullydeveloped turbulent pipe flow in non-rotating and rotating cases. The governing equations are

discretized on a staggered mesh in cylindrical coordinates. The numerical integration isperformed by a finite difference scheme, second-order accurate in space and time. The timeadvancement employs a fractional step method. The aim of this study is to investigate theeffects of the Reynolds number and of the rotation number on the turbulent flowcharacteristics. The mean velocity profiles and many turbulence statistics are compared tonumerical and experimental data available in the literature, and reasonably good agreement isobtained. In particular, the results show that the axial velocity profile gradually approaches alaminar shape when increasing the rotation rate, due to the stability effect caused by thecentrifugal force. Consequently, the friction factor decreases. The rotation of the wall haslarge effects on the root mean square (rms), these effects being more pronounced for thestreamwise rms velocity. The influence of rotation is to reduce the Reynolds stress component

Vr'Vz' and to increase the two other stresses Vr'V' and V'Vz'. The effect of the Reynoldsnumber on the rms of the axial velocity (Vz'21/2) and the distributions ofVr'Vz' is evident,

and it increases with an increase in the Reynolds number. On the other hand, the Vr'V'-profiles appear to be nearly independent of the Reynolds number. The present DNS and LESpredictions will be helpful for developing more accurate turbulence models for heat transferand fluid flow in pipe flows.


13/482

Preface xi

The field of computational fluid dynamics (CFD) has evolved from an academic curiosityto a tool of practical importance. Applications of CFD have become increasingly important innuclear engineering and science, where exacting standards of safety and reliability are

paramount. The newly-commissioned Open Pool Australian Light-water (OPAL) researchreactor at the Australian Nuclear Science and Technology Organisation (ANSTO) has been

designed to irradiate uranium targets to produce molybdenum medical isotopes for diagnosisand radiotherapy. During the irradiation process, a vast amount of power is generated whichrequires efficient heat removal. The preferred method is by light-water forced convectioncoolingessentially a study of complex pipe flows with coupled conjugate heat transfer.Feasibility investigation on the use of computational fluid dynamics methodologies intovarious pipe flow configurations for a variety of molybdenum targets and pipe geometries aredetailed in Chapter 8. Such an undertaking has been met with a number of significantmodeling challenges: firstly, the complexity of the geometry that needed to be modeled.Herein, challenges in grid generation are addressed by the creation of purpose-built body-fitted and/or unstructured meshes to map the intricacies within the geometry in order toensure numerical accuracy as well as computational efficiency in the solution of the predictedresult. Secondly, various parts of the irradiation rig that are required to be specified ascomposite solid materials are defined to attain the correct heat transfer characteristics.Thirdly, the use of an appropriate turbulence model is deemed to be necessary for the correctdescription of the fluid and heat flow through the irradiation targets, since the heat removal isforced convection and the flow regime is fully turbulent, which further adds to the complexityof the solution. As complicated as the computational fluid dynamics modeling is, numericalmodeling has significantly reduced the cost and lead time in the molybdenum-target design

process, and such an approach would not have been possible without the continualimprovement of computational power and hardware. This chapter also addresses theimportance of experimental modeling to evaluate the design and numerical results of thevelocity and flow paths generated by the numerical models. Predicted results have been foundto agree well with experimental observations of pipe flows through transparent models and

experimental measurements via the Laser Doppler Velocimetry instrument.In Chapter 9, the entropy generation for during fluid flow in a pipe is investigated. The

temperature dependence of the viscosity is taken into consideration in the analysis. Laminarand turbulent flow cases are treated separately. Two types of thermal boundary conditions areconsidered; uniform heat flux and constant wall temperature. In addition, various cross-sectional pipe geometries were compared from the point of view of entropy generation and

pumping power requirement in order to determine the possible optimum pipe geometry whichminimizes the exergy losses.

Chapter 10 aims to present a state-of-the-art review on single-phase incompressible fluidflow in mini- and micro-channels. First, classification of mini- and micro-channels isdiscussed. Then, conventional theories on laminar, laminar to turbulent transition andturbulent fluid flow in macro-channels (conventional channels) are summarized. Next, a briefreview of the available studies on single-phase incompressible fluid flow in mini- and micro-channels is presented. Some experimental results on single phase laminar, laminar toturbulent transition and turbulent flows are presented. The deviations from the conventionalfriction factor correlations for single-phase incompressible fluid flow in mini and micro-channels are discussed. The effect factors on mini- and micro-channel single-phase fluid floware analyzed. Especially, the surface roughness effect is focused on. According to this review,


14/482

Donald Matos and Cristian Valerioxii

the future research needs have been identified. So far, no systematic agreed knowledge ofsingle-phase fluid flow in mini- and micro-channels has yet been achieved. Therefore, effortsshould be made to contribute to systematic theories for microscale fluid flow through verycareful experiments.

In Chapter 11, an experimental investigation was conducted of pulsating turbulent flow in

a conically divergent tube with a total divergence angle of 12. The experiments were carriedout under the conditions of Womersley numbers of=1040, mean Reynolds number ofReta=20000 and oscillatory Reynolds number ofReos =10000 (the flow rate ratio of = 0.5).Time-dependent wall static pressure and axial velocity were measured at several longitudinalstations and the distributions were illustrated for representative phases within one cycle. Therise between the pressures at the inlet and the exit of the divergent tube does not become toolarge when the flow rate increases, it being moderately high in the decelerative phase. The

profiles of the phase-averaged velocity and the turbulence intensity in the cross section arevery different from those for steady flow. Also, they show complex changes along the tubeaxis in both the accelerative and decelerative phases.

Chapter 12 deals with a numerical method for a solution of an airfoil design inverse

problem. The presented method is intended for a design of an airfoil based on a prescribedpressure distribution along a mean camber line, especially for modifying existing airfoils. Themain idea of this method is a coupling of a direct and approximate inverse operator. The goalis to find a pseudo-distribution corresponding to the desired airfoil with respect to theapproximate inversion. This is done in an iterative way. The direct operator represents asolution of a flow around an airfoil, described by a system of the Navier-Stokes equations inthe case of a laminar flow and by the k model in the case of a turbulent flow. There is arelative freedom of choosing the model describing the flow. The system of PDEs is solved byan implicit finite volume method. The approximate inverse operator is based on a thin airfoiltheory for a potential flow, equipped with some corrections according to the model used. Theairfoil is constructed using a mean camber line and a thickness function. The so far developedmethod has several restrictions. It is applicable to a subsonic pressure distribution satisfying a

certain condition for the position of a stagnation point. Numerical results are presented.Recent advances in the field of fluid mechanics with moving fronts are linked to the use

of Level SetMethods, a versatile mathematical technique to follow free boundaries whichundergo topological changes. A challenging class of problems in this context are those relatedto the solution of a partial differential equation posed on a moving domain, in which the

boundary condition for the PDE solver has to be obtained from a partial differential equationdefined on the front. This is the case of potential flow models with moving boundaries.Moreover, the fluid front may carry some material substance which diffuses in the front and isadvected by the front velocity, as for example the use of surfactants to lower surface tension.We present a Level Set based methodology to embed this partial differential equationsdefined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid

particles and its known limitations. To show the advantages of this approach in the field ofFluid Mechanics we present in Chapter 13 one particular application: the numericalapproximation of a potential flow model to simulate the evolution and breaking of a solitarywave propagating over a slopping bottom and compare the level set based algorithm with

previous front tracking models.


15/482

Preface xiii

Chapter 14 is basically a review of recent works that is aimed at putting forward the mainideas behind a new theoretical approach to turbulent wall-bounded flows, notably pipe-flows,in which complex physics is involved, such as combustion or particle transport. Pipe flowsare ubiquitous in industrial applications and have been studied intensively in the last century,

both from a theoretical and experimental point of view. The result of such a strong effort is a

good comprehension of the physics underlying the dynamics of these flows and theproposition of reliable models for simple turbulent pipe-flows at large Reynolds numberNevertheless, the advancing of engineering frontiers casts a growing demand for modelssuitable for the study of more complex flows. For instance, the motion and the interactionwith walls of aerosol particles, the presence of roughness on walls and the possibility of dragreduction through the introduction of few complex molecules in the flow constitute someinteresting examples of pipe-flows with some new complex physics involved. A goodmodeling approach to these flows is yet to come and, in the commentary, we support the ideathat a new angle of attack is needed with respect to present methods. In this article, weanalyze which are the fundamental features of complex two-phase flows and we point out thatthere are two key elements to be taken into account by a suitable theoretical model: 1) Theseflows exhibit chaotic patterns; 2) The presence of instantaneous coherent structures radicallychange the flow properties. From a methodological point of view, two main theoreticalapproaches have been considered so far: the solution of equations based on first principles(for example, the Navier-Stokes equations for a single phase fluid) or Eulerian models basedon constitutive relations. In analogy with the language of statistical physics, we consider theformer as a microscopic approach and the later as a macroscopic one. We discuss why weconsider both approaches unsatisfying with regard to the description of general complexturbulent flows, like two-phase flows. Hence, we argue that a significant breakthrough can beobtained by choosing a new approach based upon two main ideas: 1) The approach has to bemesoscopic (in the middle between the microscopic and the macroscopic) and statistical; 2)Some geometrical features of turbulence have to be introduced in the statistical model. We

present the main characteristics of a stochastic model which respects the conditions expressed

by the point 1) and a method to fulfill the point 2). These arguments are backed up with somerecent numerical results of deposition onto walls in turbulent pipe-flows. Finally, some

perspectives are also given.


16/482


17/482

In: Fluid Mechanics and Pipe Flow ISBN: 978-1-60741-037-9Editors: D. Matos and C. Valerio, pp. 1-26 2009 Nova Science Publishers, Inc.

Chapter 1

SOLUTE TRANSPORT, DISPERSION, AND SEPARATION

IN NANOFLUIDIC CHANNELS

Xiangchun Xuan*

Department of Mechanical Engineering, Clemson University,Clemson, SC 29634-0921, USA

Abstract

Recent breakthrough in nanofabrication has stimulated the interest of solute separation innanofluidic channels. Since the hydraulic radius of nanochannels is comparable to thethickness of electric double layers, the enormous electric fields inherent to the latter generatetransverse electromigrations causing charge-dependent solute distributions over the channelcross-section. As a consequence, the non-uniform fluid flow through nanochannels yieldscharge-dependent solute speeds enabling the separation of solutes by charge alone. In this

chapter we develop a theoretical model of solute transport, dispersion and separation inelectroosmotic and pressure-driven flows through nanofluidic channels. This model provides abasis for the optimization of solute separation in nanochannels in terms of selectivity andresolution as traditionally defined.

1. Introduction

Solute transport and separation in micro-columns (e.g., micro capillaries and chip-basedmicrochannels) have been a focus of research and development in electrophoresis andchromatography communities for many years. Recent breakthrough in nanofabrication hasinitiated the study of these topics among others in nanofluidic channels [1-3]. Since thehydraulic radius of nanochannels is comparable to the thickness of electric double layers

(EDL), the enormous electric fields inherent to the latter generate transverse electromigrationscausing charge-dependent solute distributions over the channel cross-section [4-7]. As aconsequence, the non-uniform fluid flow in nanochannels yields charge-dependent solutespeeds enabling the separation of solutes by charge alone [8,9]. Such charge-based solute

* E-mail address: [email protected]. Tel: (864) 656-5630. Fax: (864) 656-7299


18/482

Xiangchun Xuan2

separation was first proposed and implemented by Pennathur and Santiago [10] and Garcia etal. [11] in electroosmotic flow through nanoscale channels, termed nanochannelelectrophoresis. As a matter of fact, this separation may also happen in pressure-driven flowalong nanoscale channels, termed here as nanochannel chromatography for comparison,which was first demonstrated theoretically by Griffiths and Nilson [12] and Xuan and Li [13],

and later experimentally verified by Lius group [14].So far, a number of theoretical studies have been conducted on the transport [4-6,9-

13,15,19], dispersion [7,9,12,15-19] and separation [4,9-13,15,19] of solutes in free solutionsthrough nanofluidic channels. This chapter combines and unites the works from ourselves inthis area [6,13,15,17-19], and is aimed to develop a general analytical model of solutetransport, dispersion and separation in nanochannels. It is important to note that this modelapplies only to point-like solutes. For those with a finite size, one must consider thehydrodynamic and electrostatic interactions among solutes, electric field, and flow fluid, andas well the Steric interactions between solutes and channel walls etc [20].

2. Nomenclature

a channel half-height

Bi defined function, = exp(zi)cb bulk concentration of the background electrolyteci concentration of solute species iCi bulk concentration of solute species ici,0 concentration of solute species i at the channel centerlineCi,0 initial concentration of solute species i at the channel centerlineDi solute diffusion coefficient

iD effective solute diffusion coefficientE axial electric fieldEst streaming potential fieldF Faradays constanthi reduced theoretical plate heightj electric current densityKi hydrodynamic dispersionL channel lengthP hydrodynamic pressure drop per unit channel lengthPe Peclet numberrji solute selectivityR Universal gas constantRji resolution

t time coordinateT absolute temperatureui axial solute speed

iu mean solute speed

vi solute mobilityWi half width of the initially injected solute zone


19/482

Solute Transport, Dispersion, and Separation in Nanofluidic Channels 3

x streamwise or longitudinal coordinateXi the central location of the injected solute zoney transverse coordinatezi valence of ionsZ electrokinetic figure of merit

Greek Symbols

non-dimensional product of fluid properties, = b/RT

permittivity

apparent viscosity ratio

reciprocal of Debye length

i dispersion coefficient

b molar conductivity of the background electrolyte

fluid viscosity

electrical double layer potential non-dimensional EDL potential0 EDL potential at the channel centere net charge density

b bulk electric conductivity of background electrolyte

i standard deviation of solute peak distribution

t standard deviation of solute peak distribution in the time domain

zeta potential

* non-dimensional zeta potential

Subscripts

e electroosmosis relatedp pressure-driven relatedi solute species i

3. Fluid Flow in Nanochannels

Given the fact that the width (in micrometers) of state-of-the-art nanofluidic channels isusually much larger than their depth (in nanometers) [1-3], we consider the solute transport influid flow through a long straight nanoslit, see Figure 1 for the schematic. The flow may be

electric field-driven, i.e., electroosmotic, or pressure-driven. For simplicity, the electrolytesolution is assumed symmetric with unit-charge, e.g., KCl. As the time scale for fluid flow (inthe order of nanoseconds) is far less than that of solute transport (typically of tens ofseconds), we assume a steady-state, fully-developed incompressible fluid motion, which in aslit channel is governed by


20/482

Xiangchun Xuan4

x

y

u + viziFE au

Solute zone

Figure 1. Schematic of solute transport in a slit nanochannel (only the top half is illustrated due to thesymmetry).

2

20e

d uP E

dy + + = (3-1)

where is the fluid viscosity, u the axial fluid velocity,y the transverse coordinate originatingfrom the channel axis, P the pressure drop per unit channel length, and E the axial electricfield either externally applied in electroosmotic flow or internally induced in pressure-driven

flow (i.e., the so-called streaming potential field) [21-24]. The net charge density, e, issolved from the Poisson equation [25]

2

2e

d

dy

= (3-2)

where is the fluid permittivity and is the EDL potential. Invoking the no-slip condition forEq. (3-1) and the zeta potential condition for Eq. (3-2) on the channel wall (i.e., y = a), onecan easily obtain

p eu u u= + (3-3)

2 2

21

2p

a yu P

a

=

(3-4)

1eu E

=

(3-5)

where up is the pressure-driven fluid velocity, ue the electroosmotic fluid velocity, a the half-

height of the channel, and = F/RTand * = F/RT the dimensionless forms of the EDLand wall zeta potentials with Fthe Faradays constant,R the universal gas constant and Ttheabsolute fluid temperature. It is noted that the contribution of charged solutes to the net

charge density e has been neglected. This is reasonable as long as the solute concentration ismuch lower than the ionic concentration of the background electrolyte, which is fulfilled intypical solute separations. Under such a condition, it is also safe to assume a uniform zeta

potential on the channel wall.


21/482


The non-dimensional EDL potential in Eq. (3-5), , may be solved from the Poisson-Boltzmann equation [25]

( )2

2

2sinh

d

dy

= (3-6)

where 22b

F c RT = is the inverse of the so-called Debye screening length with cb the

bulk concentration of the background electrolyte. We recognize that the assumed Boltzmanndistribution of electrolyte ions in Poisson-Boltzmann equation might be questionable innanoscale channels, especially in those with strong EDL overlapping [26,27]. However, thisequation has been successfully used to explain the experimentally measured electricconductance and streaming current in variable nanofluidic channels [28-32], and is thus stillemployed here.

For the case of a small magnitude of (e.g., || < 25 mV or |*| < 1) which is actuallydesirable for sensitive solute separations in nanochannels as demonstrated by Griffiths and

Nilson [12], one may use the Debye-Huckel approximation to simplify Eq. (3-6) as [21,25]

22

2

d

dy

= (3-7)

It is then straightforward to obtain

( )( )

* cosh

cosh

y

a

= (3-8)

where a may be viewed as the normalized channel half-height. It is important to note that for

a given fluid and channel combination, the wall zeta potential will in general vary with a[28,31,32]. One option to address this is to use a surface-charge based potential parameter forscaling instead of zeta potential [16]. In this work and other studies [6-14], the zeta potentialis used directly, because it may be readily determined through experiment and provides adirect measure of the electroosmotic mobility.

The area-averaged fluid velocity u may be written in terms of the Poiseuille and

electroosmotic components

ep uuu += (3-9)

Pa

up3

2

= (3-10)


22/482

Xiangchun Xuan6

( )tanh1e

au E

a

=

(3-11)

where ( ) ( )0a

d y a= " "

signifies an area-averaged quantity.

3.1. Electroosmotic Flow

For electroosmotic flow, no pressure gradient is present, and so the fluid motion in Eq.(3-3) is described by

0pu = and( )( )

cosh1

coshe

yu E

a

=

(3-12)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Ele

ctroosmoticvelocityprofile

y/a

a = 1

2

5

1020

Figure 2. Radial profile of the normalized electroosmotic fluid velocity, ue/UHS= 1 cosh(y)/cosh(a),at different a values. All symbols are referred to the nomenclature.

Figure 2 shows the profile of electroosmotic velocity normalized by the so-called Helmholtz-

Smoluchowski velocity UHS = E/ [25], i.e., ue/UHS = 1 cosh(y)/cosh(a), in a slit

channel with different a values. When a > 10, the curves are almost plug-like except nearthe channel wall and the bulk velocity is equal to UHS. These are the typical features of

electroosmotic flow when there is little or zero EDL overlapping. The profiles at a < 5become essentially parabolic resembling the traditional pressure-driven flow. Moreover, themaximum velocity along the channel centerline is significantly lower than UHSand decreases

with a, indicating a vanishingly small electroosmotic mobility when a approaches 0.


23/482


3.2. Pressure-Driven Flow

For pressure-driven flow, the downstream accumulation of counter-ions results in thedevelopment of a streaming potential field [21-24]. This induced electric field, Est, can bedetermined from the condition of zero electric current though the channel. If an equal

mobility for the positive and negative ions of the electrolyte is assumed, the electric currentdensity,j, in pressure-driven flow is given as [21-24]

( )coshe b st j u E = + (3-13)

where b = cbb is the bulk conductivity of the electrolyte with b being the molarconductivity. Referring to Eqs. (3-2) and (3-3), one may rewrite the last equation as

( ) ( )2

2coshp e b b st

d RTj u u c E

dy F

= + + (3-14)

Note that the surface conductance of the outer diffusion layer in the EDL has been considered

through the cosine hyperbolic function in Eq. (3-14) (which reduces to 1 at = 0). Thecontribution of the inner Stern layer conductance [33] to the electric current is, however,ignored. Readers may be referred to Davidson and Xuan [34] for a discussion of this issue inelectrokinetic streaming effects.

Integrating j in Eq. (3-14) over the channel cross-section and using the zero electriccurrent condition in a steady-state pressure-driven flow yield

( )

*1

*22 3

st

b

gE P

c F g g

=

+(3-15)

( )a

ag

tanh11 = ,

( )( )aa

ag

22 cosh

1tanh= and ( )

=a

a

ydg

0

3 cosh (3-16)

Therefore, the fluid motion in pressure-driven flow is characterized as

2 2

21

2p

a yu P

a

=

and

( )( )

cosh1

coshe st

yu E

a

=

(3-17)

Area-averaging the two velocity components in the last equation and combining themwith Eq. (3-15) lead to

Zu

u

p

e = (3-18)


24/482

Xiangchun Xuan8

where Z is previously termed electrokinetic figure of merit as it gauges the efficiency ofelectrokinetic energy conversion [35,36], and defined as

( ) ( )

21

2 2

2 3

3gZ

a g g

=

+

(3-19)

RT

b

= (3-20)

where 2 = 2F2cb/RThas been invoked during the derivation, and is a non-dimensionalproduct of fluid properties whose reciprocal was termed Levine number by Griffiths and

Nilson [37]. Apparently, Z depends on three non-dimensional parameters, , a and *,

among which spans in the range of 2 10 and * spans in the rage of8 * 0 [33]for typical aqueous solutions. Moreover,Zis unconditionally positive and less than unity due

to the entropy generation in non-equilibrium electrokinetic flow [38]. The curves ofZat * =

1 (or25 mV) and = 2 and 10, respectively, are displayed in Figure 3 as a function ofa. One can see thatZachieves the maximum at around a = 2, indicating that nanochannelswith a strong EDL overlapping are the necessary conditions for efficient electrokinetic energyconversion. For more information about Z and its function in electrokinetic energyconversion, the reader is referred to Xuan and Li [36].

0.00

0.03

0.06

0.09

0.12

0.15

0.1 1 10 100

Z

a

= 2

= 10

Figure 3. The electrokinetic figure of merit as a function of a at * = 1 and = 2 and 10,respectively.


25/482


Based on Eq. (3-18), the effects of streaming potential on the average fluid speed, i.e.,

u in Eq. (3-9), in an otherwise pure pressure-driven flow, are characterized by

1p

uZ

u= (3-21)

This equation also provides a measure of the so-called electro-viscous effects inmicro/nanochannels [39]. If the concept of apparent viscosity is employed to characterize

such retardation effects, the apparent viscosity ratio is readily derived as

( )1 1 Z = (3-22)

For more information on this topic, the reader is referred to Li [39] and Xuan [40].

4. Solute Transport in Nanochannels

Solute transport in nanochannels is governed by the Nernst-Planck equation in theabsence of chemical reactions, which under the assumption of fully-developed fluid flow iswritten as

2 2

2 2i i i i

i i i i i i

c c c cu D D v z F c

t x x y y y

+ = + +

(4-1)

i e p i iu u u v z FE = + + (4-2)

where ci is the concentration of solute species i, tthe time coordinate, ui the local solute speed(a combination of electroosmosis, pressure-driven motion, and electrophoresis), x the axialcoordinate originating from the channel inlet (see Figure 1), Di the molecular diffusioncoefficient, vi the solute mobility, andzi the solute charge number. Note that the product viziFrepresents the solute electrophoretic mobility. Integrating Eq. (4.1) over the channel cross-section eliminates the last two terms on the right hand side due to the impermeable wallconditions

2

2

i i i i

i

c u c cD

t x x

+ =

(4-3)

where again ... indicates the area-average over the channel cross-section as defined earlier.

Since the time scale for transverse solute diffusion in nanochannels (characterized by

a2/Di, which is about 100 s when a = 100 nm and Di = 110

10 m2/s) is much shorter than


26/482

Xiangchun Xuan10

that for longitudinal solute transport (typically of tens of seconds), it is reasonable to assumethat solute species are at a quasi-steady equilibrium in they direction, i.e.,

2

20 ii i i i

cD v z F c

y y y

= +

(4-4)

Integrating Eq. (4.4) twice and using the Nernst-Einstein relation [33], vi =Di/RT, one obtains

( ) ( ) ( ),0 0, , , expi i ic x y t c x t z= (4-5)

where ci,0 is the solute concentration at the channel centerline where the local EDL potential

is defined as 0 (non-zero in the presence of EDL overlapping). Substituting Eq. (4.5) intoEq. (4.3) and considering the hydrodynamic dispersion due to the velocity non-uniformityover the channel cross-section [41,42], one may obtain

2,0 ,0 ,0

2

i i i

i i

c c cu D

t x x

+ =

(4-6)

i ip ie i iu u u v z FE = + + (4-7)

p i

ip

i

u Bu

B= and e iie

i

u Bu

B= (4-8)

wherei

u is the mean solute speed (i.e., zone velocity) withip

u andie

u being its components

due to pressure-driven and electroosmotic flows, iD the effective diffusion coefficient whichis a combination of molecule diffusion and hydrodynamic dispersion and will be addressed in

the next section, and Bi = exp(zi) the like-Boltzmann distribution of solutes in the cross-

stream direction. It is the dependence ofi

u on the charge numberzi that enables the charged-

based solute separation in nanochannels.For an initially uniform concentration Ci,0 of solute species i along the channel axis, a

closed-form solution to Eq. (4-6) is given by [5]

,0,0 erf erf

2 2 2

i i i i i ii

i i

C W x X u t x X u t c

D t D t

+ + = +

(4-9)

where erf denotes the error function, and Wi is the half width of the initial solute zone with itscenter being located at Xi. As such, the electrokinetic transport of solute species innanochannels is described by


27/482


( ) ( ), , erf erf exp2 2 2

i i i i i ii i

i i

C W x X u t x X u t c x y t z

D t D t

+ + = +

(4-10)

where ( ),0 0expi i iC C z= is the bulk solute concentration at zero potential outside the slitnanochannel, refer to Eq. (4-5).

zv= +2

zv= +1

zv= 0

zv=1

zv=2

Cmax0.8 Cmax0.6 Cmax0.4 Cmax0.2 Cmax

Figure 4. Transport of solutes with zi = [+2, 2] through a 100 nm deep channel in nanochannelchromatography. Other parameters are referred to the text. Reprinted with permission from [13].

Figure 4 illustrates the transport of an initially Wi = 1 m wide plug of solutes with zi =[+2, 2] (from top to bottom) through a 100 nm deep (i.e., a = 50 nm) channel 5 s after a

pressure gradient P = 1108 Pam-1 was imposed. Note that only the top half of the channel isshown due to symmetry. The ionic concentration of the background electrolyte is cb = 1 mM,

corresponding to a 5. The other two non-dimensional parameters are assumed to be = 4and * = 2 (or = 50 mV), both of which are typical to aqueous solutions as indicated

above. As to the validity of the Debye-Huckel approximation at * = 2, we have recentlydemonstrated using numerical simulation the fairly good accuracy of Eq. (3-7) in predictingthe solute migration velocity [5]. As shown, positive solutes are concentrated to near thenegatively charged wall due to the solute-wall electrostatic interactions [9,11], or in essencethe transverse electromigration in response to the induced EDL field [5,10,12]. Moreover, thehigher the charge number is, the closer the solutes are to the walls. As the fluid velocity nearno-slip walls is slower than its average, positive solutes move slower than neutral solutes thatare still uniformly distributed over the channel cross-section. Conversely, negative solutes arerepelled by the negatively changed walls and concentrated to the region close to the channelcenter. Hence, they move faster than neutral solutes as seen in Figure 4.

As the fluid velocity profile is available in Eq. (3-12) for electroosmotic flow and in Eq.

(3-17) for pressure-driven flow, the mean speed of solutes, iu , is readily obtained from Eq.(4-7). Figure 5 compares the mean speed of solutes withzi = [2, +2] in electroosmotic flow

with an electric field of 4 kV/m. One can see thati

u of all five solutes decreases when a

gets smaller. This reduction may be explained by the overall lower electroosmotic velocity at

a smallera, as demonstrated in Figure 2. When a > 100, negatively charged solutes moveslower than positive ones due to their opposite electrophoresis to fluid electroosmosis


28/482

Xiangchun Xuan12

(identical to the curve with zi = 0). When a gets smaller than 100, however, negativelysolutes start moving faster than positively solutes as the latter ones are concentrated to theEDL region within which the fluid has a slower speed than the bulk as explained above. The

relative magnitude ofi

u between the solutes of like charges is, however, a complex function

of both the charge numberzi, which determines the velocity component due to fluid flow, i.e.,ip ie

u u+ in Eq. (4-7), and the solute mobility vi, which determines the velocity component

due to solute electrophoresis, i.e., the most right term in Eq. (4-7). When a further decreasesto less than 1, the EDL potential becomes nearly flat due to the strong EDL overlapping (see

Figure 2), and so the order of iu for the three solutes at large a (i.e., microchannel

electrophoresis) is recovered.

Figure 5. Comparison of the mean speeds of solutes with zi = [2, +2] as a function of a innanochannel electrophoresis. The solute diffusivity is assumed to be constant, Di = 510

11 m2/s. Otherparameters are referred to the text.

Combining Eqs. (4-7) and (4-8) provides a measure of the streaming potential effects onthe solute mean speed in pressure-driven flow,

1 e i i i i st i

ip p i

u B v z F B E uu u B

+= + (4-11)

It is obvious that the last equation is reduced to Eq. (3-21) for neutral solutes, i.e., zi = 0 and

thus Bi = 1. Figure 6 shows the ratio i ipu u as a function of a. As expected, streaming


29/482


potential effects reduce the solute speed due to the induced electroosmotic backflow. This

reduction varies with the solute chargezi and attains the extreme at about a = 3, where thefigure of meritZ(refer to Figure 3) approaches its maximum indicating the largest streaming

potential effects. In both the high and low limits of a, streaming potential effectsbecome

negligible, i.e.,Z 0, see Figure 3. Accordingly,i ip

u u reduces to 1 for all solutes at large

a while varying withzi at small a because of the finite solute mobility [15].

0.7

0.8

0.9

1

1.1

0.1 1 10 100

ui/uip

a

zi = +2

+1

0

2

1

Figure 6. Effects of streaming potential on the solute mean speed in nanochannel chromatography.

5. Solute Dispersion in Nanochannels

As up and ue vary over the channel cross-section (refer to Eqs. (3-4) and (3-5), and Figure2), they both contribute to the spreading of solutes along the flow direction, which is termedhydrodynamic dispersion or Taylor dispersion [41,42]. The general formula for calculatingthis dispersion is given by [43,44]

( )2

1

2 0

y

i i i i

i

i i

B B u u dya

KD B

=

(5-1)

Referring to Eqs. (4-2) and (4-7), one may then rewrite the last equation as

( )2

2 2

i ip p ipe p e ie e

i

aK F u F u u F u

D= + + (5-2)


30/482

Xiangchun Xuan14

( )2

11

0

y

ip i i p ip iF B B u u dy B

= (5-3a)

( )

211

0

y

ie i i e ie iF B B u u dy B

= (5-3b)

( ) ( )11

0 02

y y

ipe i p ip i e ie i iF B u u B dy u u B dy B = (5-3c)

where m m mu u u = and im im mu u u

= (m =p and e). Note that the three terms, Fip, Fipeand Fie in Eq. (5-2) represent the contributions to dispersion due to the pressure-driven flow,the coupling between pressure-driven and electroosmotic flows, and the electroosmotic flow,respectively.

Hydrodynamic dispersion is often expressed in terms of a non-dimensional dispersion

coefficienti [45],

iiii DPeK2= (5-4)

where the Peclet number Pei in this case may be defined with respect to the mean solute

speed, i.e., iii DauPe = [12,19], or to the area-averaged fluid velocity, i.e.,

iepi DauuPe += [15-18,45]. Using the solute speed-based Peclet number, i

becomes dependent on the solute diffusivity Di which complicates the analysis. This is

because the solute mobility vi in iu [see Eq. (4-7)] is coupled to Di via the Nernst-Einstein

relation, vi = Di/RT. Such dependence doesnt occur ifi is defined using the fluid velocity-based Peclet number. Here, we employ the latter definition in keeping with the dispersion

studies of neutral solutes in the literature [45]. As such, the dispersion coefficient i may beeasily obtained from Eq. (5-2) as

( )

2 2

2

ip p ipe p e ie e

i

p e

F u F u u F u

u u

+ +

=+

(5-5)

It is important to note that i is independent of the solute speed or the driving force of the

flow while Ki

(in the unit ofDi) not. Instead,

iis primarily determined by the flow type

(pressure- or electric field-driven), channel structure (including shape and size) and solutecharge number.


31/482


5.1. Electroosmotic Flow

In electroosmotic flow, the hydrodynamic dispersion in Eq. (5-2) is reduced to

22

ei ie

i

a uK F

D= (5-6)

Accordingly, the dispersion coefficient in Eq. (5.5) is simplified as

i ieF = (5-7)

0.0001

0.001

0.01

0.1

0.1 1 10 100

i

fornanochannelelectro

phoresis

a

zi = 0

+1

+2

1

2

Figure 7. Illustration of dispersion coefficient i of solutes with zi = [2, +2] in nanochannelelectrophoresis as a function ofa.

Figure 7 shows i of solutes with zi = [2, +2] in nanochannel electrophoresis as afunction ofa. We see that in the entire range of a, i of positive solutes is larger than that

of neutral ones while i of negative solutes is smaller than the latter. This is because positivesolutes are concentrated to near the channel walls where the velocity gradients are large whilenegative ones are concentrated to the channel centerline where the velocity gradients are

small (refer to Figure 4). Moreover, the higher the charge numberzi, the larger is i for

positive solutes and the smaller for negative ones. In the low limit of a (i.e., the narrowestchannel), the EDL potential is essentially uniform over the channel cross-section (refer toFigure 2), and so is the solute distribution regardless of the charge number. As a consequence,

i of all charged solutes approach that of neutral solutes, i.e., 2/105. Note that this value isequal to the dispersion coefficient of neutral solutes in a pure pressure-driven flow indicatingthe resemblance between pressure-driven and electroosmotic flows in very small


32/482

Xiangchun Xuan16

nanochannels. This aspect will be revisited shortly. In the high limit of a (i.e., the widestchannel), the EDL thickness is so thin compared to the channel height that the solutedistribution becomes once again uniform across the channel (the EDL potential is, here,

uniformly zero while equal to the wall zeta potential in the low limit of a). Therefore, thehydrodynamic dispersion in electroosmotic flow, or the so-called electrokinetic dispersion

[46], decreases with the square ofa and ultimately converges to zero [47,48].

5.2. Pressure-Driven Flow

In pressure-driven flow with consideration of streaming potential, the hydrodynamicdispersion is obtained from Eq. (5-2) as

( )22

22 31

p

i ip i i

i

a uK F Z Z

D = + (5-8)

2i ipe ipF F = and 3i ie ipF F = (5-9)

during which Eqs. (3-17) and (3-18) have been invoked andZis the electrokinetic figure ofmerit as defined in Eq. (3-19). It is apparent that the streaming potential inducedelectroosmotic backflow produces two additional dispersions in pressure-driven flow: one is

the electrokinetic dispersion due to electroosmotic flow itself, the term with i3 in Eq. (5-8),which tends to increase the total dispersion, and the other is due to the coupling of pressure-

driven and electroosmotic flows, the term with i2 in Eq. (5-8), which tends to decrease thetotal dispersion. The latter phenomenon has been employed previously to reduce thehydrodynamic dispersion in capillary electrophoresis where a pressure-driven backflow is

intentionally introduced to partially compensate the non-uniformity in electroosmotic velocityprofile [49,50]. If streaming potential effects are ignored, i.e., for a pure pressure-driven flowwithZ= 0, Eq. (5-8) reduces to

22p

ip ip

i

a uK F

D= (5-10)

such that

22 31

ii i

ip

KZ Z

K = + (5-11)

Similarly, the dispersion coefficient in Eq. (5-5) for real pressure-driven flow is obtainedas

( )

22 3

2

1

1i i

i ip

Z ZF

Z

+=

(5-12)


33/482


where Fip = ip is the dispersion coefficient for a pure pressure-driven flow by analogy to Eq.(5-7) in a pure electroosmotic flow. We thus have

( )

222 3

2

1

1

i i i i

ip ip

Z Z K

KZ

+= =

(5-13)

Therefore, i/ip differs from Ki/Kip by only the square of the apparent viscosity ratio , see

the definition in Eq. (3-22). As is independent of the solute charge numberzi, it is expected

that the variation ofi/ip with respect tozi will be identical to that ofKi/Kip.

0.001

0.01

0.1

1

0.1 1 10 100

i

fornanochannelchromatography

a

zi = 0

+1

+2

1

2

Figure 8. Dispersion coefficient i of solutes with zi = [2, +2] in nanochannel chromatography as afunction ofa.

Figure 8 shows i of solutes with zi = [2, +2] as a function of a in nanochannelchromatography. Due to the same reason as stated above for nanochannel electrophoresis, iof positive solutes is larger than that of neutral ones while i of negative solutes is the

smallest. In both the high and low limits of a, the flow-induced streaming potential isnegligible, see Eq. (3-18) and Figure 3. Hence, the electroosmotic back flow and the induced

solute electrophoresis vanish, yielding i = 2/105 regardless of the solute charge [51]. It isimportant to note that i of neutral solutes in pressure-driven flow is not uniformly 2/105 as

accepted in the literature. Due to the effects of flow-induced streaming potential, i is

increased by the electroosmotic backflow [i.e., i3Z2 term in Eq. (5-12)] even though the

coupled dispersion term [i.e., i2Zterm in Eq. (5-12)] drops for neutral solutes [17].


34/482

Xiangchun Xuan18

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0.1 1 10 100

Ki/

Kipori/ip

a

zi

zi

Ki/Kip

i/ip

.

Figure 9. Effects of streaming potential on the ratio of solute dispersion, Ki/Kip, and the ratio of

dispersion coefficient, i/ip, in nanochannel chromatography as a function of a. Adapted withpermission from [18].

Figure 9 displays the effects of streaming potential on the ratio of solute dispersion,

Ki/Kip, and the ratio of dispersion coefficient, i/ip, in nanochannel chromatography as a

function of a. In all cases, Ki/Kip is less than 1 indicating that streaming potential effectsresult in a decrease in hydrodynamic dispersion. This reduction, as a consequence of theinduced electroosmotic backflow, gets larger (i.e., Ki/Kip deviates further away from 1) when

the solute chargeziincreases. The optimum a at which Ki/Kip achieves its extreme increasesslightly with zi. In contrast to the decrease in solute dispersion, the dispersion coefficient is

increased by the effects of streaming potential, i.e., i/ip > 1. These dissimilar trends stem

from the dependence of on a, see Eqs. (3-19), (3-22) and (5-13). As streaming potentialeffects increase (or in other words, the electrokinetic figure of merit Z increases), theelectroosmotic backflow increases causing a decrease in Ki/Kip (and Ki/Kip < 1) while an

increase in (and > 1). The net result is the observed variation ofi/ip with respect to a.

The increase in i/ip is more sensitive to zi than the decrease in Ki/Kip. However, the trend

that i/ip varies with respect toziis consistent with Ki/Kip as pointed out earlier. In addition,

i/ip attains a maximum at a larger value ofa than that at which Ki/Kip is minimized.

5.3. Neutral Solutes

For neutral solutes (zi = 0), closed-form formulae are available for the functions Fm (m =

ip, ipe, ie) defined in Eq. (5-3) which in turn determine i2 and i3 through Eq. (5-9).Specifically, we find [17]


35/482


2 105ipF = (5-14)

( )( ) ( )

2 4

2 2 6 1

1 15ipeF

a a

= +

(5-15)

( ) ( ) ( ) ( ) ( )

2

2 2 2 22 2

1 2 3 1

31 2 2 coshieF

a a a a

= +

(5-16)

( )tanh a

a

= (5-17)

Figure 10 compares the magnitude of i2 and i3 [see their definitions in Eq. (5-9)] for

neutral solutes as a function ofa. As shown, i2 is always larger than i3. In the low limit of

a, i2 approaches 2 while i3 approaches 1, and the square bracketed terms in Eq. (5-2) thusreduces to ( + )

2 reflecting the similarity of pressure-driven and electroosmotic flow

profiles in very narrow nanochannels. In the high limit ofa, both i2 and i3 approach zerobecause the streaming potential is negligible and the electroosmotic velocity profile becomesessentially plug-like. Note that Eq. (5-14) gives the well-known hydrodynamic dispersioncoefficient of neutral solutes in a pure pressure-driven flow between two parallel plates [51].Moreover, Eq. (5-16) is identical to that derived by Griffiths and Nilson [47,48] which givesthe electrokinetic dispersion coefficient of neutral solutes in a pure electroosmotic flow

between two parallel plates.

0

0.5

1

1.5

2

0.1 1 10 100 1000

a

i2

i3

Figure 10. Plot ofi2 and i3 for neutral solutes as a function ofa. Adapted with permission from [17].


36/482

Xiangchun Xuan20

6. Solute Separation in Nanochannels

Solute separation is typically characterized by retention, selectivity, plate height (or platenumber), and resolution [43], of which retention and plate height are related to only one typeof solutes. In contrast, selectivity and resolution are both dependent on two types of solute

species, and thus provide a direct measure of the separation performance of solutes. As plateheight is involved in the definition of resolution, see Eq. (6-8), it will still be considered

below along with the selectivity and resolution for a comprehensive understanding of soluteseparation in nanofluidic channels.

In order to emphasize the advantage of electrophoresis and chromatography innanochannels over those taking place in micro-columns, we focus on the solutes with asimilar electrophoretic mobility, or specifically, viziF = constant. This is equivalent toassuming a constant charge-to-size ratio or a constant product, Dz = Dizi, of solute chargeand diffusivity because solute size is inversely proportional to its diffusivity via the Nernst-Einstein relation [33]. Such solutes are unable to be separated in free solutions through

pressure-driven or electroosmotic microchannel flows. A typical value of the solute charge-

diffusivity product, Dz = 11010

m2

/s, was selected in the following demonstrations whilethe solute charge numberzi may be varied from 4 to +4. The ratio of channel length to half-height was fixed at L/a = 104 for convenience even though we recognize that fixing thechannel length might be a wiser option when the channel height is varied.

6.1. Selectivity

Selectivity, rji, is defined as the ratio of the mean speeds of solutes i andj

jiji uur = (6-1)

and should be larger than 1 as traditionally defined [43]. A larger rji indicates a betterseparation. Figure 11 compares the selectivity, rji, of (a) positive and (b) negative solutes innanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). Itis important to note that the indices ofrji, which indicate the charge values of the two solutesto be separated, are switched between positive and negative solutes in order that rji > 1 astraditional defined [43]. Specifically, we use r21, r32, and r43 for positive solutes (or more

generally, solutes with zi* < 0) as those with higher charges are concentrated in a region of

smaller fluid speed (i.e., closer to the channel wall) and thus move slower. Note that thesolute electrophoretic mobility has been assumed to remain unvaried. In contrast, negative

solutes (or solutes with zi* > 0) with higher charges appear predominantly in the region of

larger fluid speed (closer to the channel center) and thus move faster. Therefore, we need to

use r12, r23, and r34 for negative solutes. This index switch also applies to the resolution, Rji,which will be illustrated in Figure 12.


37/482


1

2

3

4

0.1 1 10 100

Selectivity,rji

a

(a)

r21

r32

r43

r21

r32

r43

1.0

1.1

1.2

0.1 1 10 100

S

electivity,rji

a

(b)

r12

r23

r34

r12

r23

r34

Figure 11: Selectivity, rji, of (a) positive and (b) negative solutes in nanochannel chromatography (solidlines) and nanochannel electrophoresis (dashed lines). Reprinted with permission from [19].

One can see in Figure 11a that the selectivity, rji, of positive solutes in nanochannelchromatography is always greater than that of the same pair of solutes in nanochannelelectrophoresis. This discrepancy gets larger when the solute charge numberzi increases.

Meanwhile, the optimal a value at which rji is maximized increases for both chromatographyand electrophoresis though it is always smaller in the former case. The discrepancy between


38/482

Xiangchun Xuan22

these two optimal a also increases with the rise ofzi. For negative solutes, Figure 11b showsa significantly lowerrji than positive solutes in nanochannel chromatography. Moreover, rjidecreases when the solute charge number increases. The optimal a at which rji is maximizedis also smaller than that for positive solutes, and decreases (but only slightly) with zi. Allthese results apply equally to rji of negative solutes in nanochannel electrophoresis except at

around a = 0.6 where rji varies rapidly with a. Within this region ofa, the electrophoretic

velocity of negative solutes is close to the fluid electroosmotic velocity [more accurately, ieu

in Eq. (4-7)] while in the opposite direction. Therefore, the real solute speed is essentially so

small that even a trivial difference in the solute speed (essentially the difference in ieu as

the solute electrophoretic velocity is constant due to the fixed charge-to-size ratio) could yielda large rji.

It is, however, important to note that the speed of negative solutes could be reversed in

nanochannel electrophoresis when a is less than a threshold value (e.g., a = 0.6 in Figure11b). In other words, solutes migrate to the anode side instead of the cathode side along withthe electrolyte solution. In such circumstances, it is very likely that only one of the two solute

species migrates toward the detector no matter the detector is placed in the cathode or theanode side of the channel. Another consequence is that the maximum rji in nanochannelelectrophoresis might be achieved with a fairly long analysis time, which makes theseparation practically meaningless. We therefore expect that solutes with a constantelectrophoretic mobility can be better separated in nanochannel chromatography than in

nanochannel electrophoresis. Moreover, solutes with zi* < 0 can be separated more easily

than can those withzi* > 0.

6.2. Plate Height

Plate height,Hi, is the spatial variance of the solute peak distribution,2i , divided by the

migration distance, L, within a time period of ti. It is often expressed in the followingdimensionless form of a reduced plate height, hi [42,45]

i

iiiiii

ua

D

aL

tD

aLa

Hh

=

===

222(6-2)

( )21i i i i i iD D K D Pe = + = + (6-3)

where iD is the effective diffusion coefficient due to a combination of hydrodynamic

dispersion Ki [see Eq. (5-4)] and molecular diffusionDi. Note that other sources of dispersionsuch as injection and detection (refer to [46,52,53] for detail) have been neglected forsimplicity.

Following Griffiths and Nilsons analysis [12], we may combine Eqs. (6-2) and (6-3) torewrite the reduced plate height as


39/482


( )iiii PePeh += 12 (6-4)

Therefore, hi attains its minimum

iih 4min, = at ioptiPe 1, = (6-5)

In other words, there exists an optimal value for the mean solute speed, iiopti aDu =, ,

and thus an optimal electric field in nanochannel electrophoresis or an optimal pressuregradient in nanochannel chromatography, at which the separation efficiency is maximized. Ashi is a function of solely the dispersion coefficient i that has been demonstrated in Figures 7and 8 for nanochannel electrophoresis and chromatography, respectively, its variations with

respect tozi and a are not repeated here for brevity.

6.3. Resolution

Resolution, Rji, can be defined in two different ways: the one introduced by Giddings[54], i.e., Eq. (6-6), and the one adopted by Huber [55] and Kenndler et al. [56-58], i.e., Eq.(6-7),

( )jtit

ij

ji

ttR

,,2 +

= (6-6)

it

ij

ji

ttR

,

= (6-7)

where tis the migration time as defined in Eq. (6-2) and t is the standard deviation of solutepeak distribution in the time domain. Consistent with the solute selectivity rji, a larger value

ofRji indicates a better separation. Substituting ii uLt = , jj uLt = and iiit u =, into

the last equation leads to

( ) ( )1 1ji ji jii i

L aLR r r

h= = (6-8)

Referring back to Eq. (6-5), it is straightforward to obtain

( ),max,min

1ji ji

i

L aR r

h= at ioptiPe 1, = (6-9)


40/482

Xiangchun Xuan24

because the selectivity rji is independent of the solute Peclet number. Therefore, when theplate height of one type of solute is minimized, the separation resolution of this solute fromanother type of solute may reach the maximum value.

Figure 12 compares the maximum resolution, Rji,max, of positive and negative solutes (aslabeled) in nanochannel chromatography (solid lines) and nanochannel electrophoresis

(dashed lines). The indices ofRji,max are assigned following those of the selectivity, rji, inFigure 11, to ensureRji,max > 0. One can see thatRji,max of positive solutes in chromatography

is larger than that of negative ones throughout the range of a. In electrophoresis, the former

also yield a better resolution ifa > 1. When a < 1,Rji,max of negative solutes increases and

reaches the extremes at a = 0.6 due to the sudden rise in the selectivity (refer to Figure 11b)

as explained above. Within the same range of a, Rji,max of positive solutes continues

decreasing when a decreases and thus becomes smaller than that of negative solutes.Interestingly, chromatography and electrophoresis offer a comparable resolution for both

types of solutes in nanoscale channels ifa > 1.

1

10

100

0.1 1 10 100

Maximumresolution,Rji,max

a

R32R43

R21

R21

R32

R43

R34

R12

R23Negative solutes

Positivesolutes

Figure 12. Maximum resolution, Rji,max, of positive and negative solutes in nanochannelchromatography (solid lines) and nanochannel electrophoresis (dashed lines). Reprinted with

permission from [19].

It is also noted in Figure 12 that the optimum channel size for both separation methods

appears to be 1 < a < 10. In other words, the best channel half-height for solute separation innanochannels will be 10 nm < a < 100 nm if 1 mM electrolyte solutions are used. In this

context, the optimum Peclet number to achieve the maximum resolution in a channel ofa =

5 (ora 50 nm) will be Pei,opt = O(4) because hi,min = O(1). Although this Peclet number(corresponding to the mean solute speed of the order of 8 mm/s) seems a little too high incurrent nanofluidics, it indicates that large fluid flows are preferred in both nanochannel


41/482


chromatography and nanochannel electrophoresis for high throughputs and separationefficiencies.

7. Conclusion

We have developed an analytical model to study the transport, dispersion and separationof solutes (both charged and non-charged) in electroosmotic and pressure-driven flowsthrough nanoscale slit channels. This model explains why solutes can be separated by chargein nanochannels, and provides compact formulas for calculating the migration speed andhydrodynamic dispersion of solutes. It also presents a simple approach to optimizing theseparation performance in nanochannels, which has been applied particularly to solutes with asimilar electrophoretic mobility. In addition, we would like to point out that the model or theapproach developed in this work can be readily extended to one-dimensional round nanotubes[12,15-17,19] and to even two-dimensional rectangular nanochannels [7,59,60].

References

[1] Prakash, S.; Piruska, A.; Gatimu, E. N.; Bohn, P. W. et al. IEEE Sensor. J. 2008, 8, 441-450.

[2] Abgrall, P.; and Nguyen, N. T.Anal. Chem. 2008, 80, 2326-2341.[3] Schoh, R. B.; and Han, J. Y.; Renaud, P.Rev. Modern Phys. 2008, 80, 839-883.[4] Pennathur, S.; and Santiago, J. G.Anal. Chem. 2005, 77, 6772-6781.[5] Petsev, D. N.J. Chem. Phys. 2005, 123, 244907.[6] Xuan, X.; and Li, D.Electrophoresis 2006, 27, 5020-5031.[7] Dutta, D.J. Colloid. Interf. Sci. 2007, 315, 740-746.[8] Pennathur, S.; Baldessari, F.; Santiago, J. G.; Kattah, M. G. et al.Anal. Chem. 2007, 79,

8316-8322.[9] Das, S.; and Chakraborty, S.Electrophoresis, 2008, 29, 1115-1124.[10] Pennathur, S.; and Santiago, J. G.Anal. Chem. 2005, 77, 6782-6789.[11] Garcia, A. L.; Ista, L. K.; Petsev, D. N. et al. Lab Chip 2005, 5, 1271-1276.[12] Griffiths, S. K.; and Nilson, R. N.Anal. Chem. 2006, 78, 8134-8141.[13] Xuan, X.; and Li, D.Electrophoresis 2007, 28, 627-634.[14] Wang, X.; Kang , J.; Wang, S.; Lu, J.; and Liu, S. J Chromatography A 2008, 1200,

108-113.[15] Xuan, X.J. Chromatography A 2008, 1187, 289-292.[16] De Leebeeck, A.; and Sinton, D.Electrophoresis 2006, 27, 4999-5008.[17] Xuan, X.; and Sinton, D.Microfluid. Nanofluid. 2007, 3, 723-728.[18] Xuan, X.Anal. Chem. 2007, 79, 7928-7932.[19] Xuan, X.Electrophoresis 2008, 29, 3737-3743.[20] Israelachvili, J. Intermolecular & Surface Forces, Academic Press, 2nd edition, San

Diego, CA, 1991.[21] Burgreen, D., and Nakache, F. R.,J. Phys. Chem. 1964, 68, 1084-1091.[22]Rice, C. L. and Whitehead, R.,J. Phys. Chem. 1965, 69, 4017-4024.[23] Hildreth, D.,J. Phys. Chem. 1970, 74, 2006-2015.


42/482

Xiangchun Xuan26

[24] Li, D. Colloid. Surf. A 2001, 191, 35-57.[25] Hunter, R. J. Zeta potential in colloid science, principles and applications, Academic

Press, New York, 1981.[26] Qu, W.; and Li, D.J. Colloid and Interface Sci. 2000, 224, 397-407.[27] Taylor, J.; and Ren, C. L.Microfluid. Nanofluid. 2005, 1, 356-363.[28] Stein, D.; Kruithof, M.; and Dekker, C. Phys. Rev. Lett. 2004, 93, 035901.[29] Karnik, R.; Fan, R.; Yue, M. et al.Nano Lett. 2005, 5, 943-948.[30] Fan, R.; Yue, M.; Karnik, R. et al. Phys. Rev. Lett. 2005, 95, 086607.[31] Van der Heyden, F. H.J.; Stein, D.; and Dekker, C. Phys. Rev. Lett. 2005, 95, 116104.[32] Van Der Hayden, F. H. J.; Bonthius, D. J.; Stein, D.; Meyer, C.; and Dekker, C. Nano

Lett. 2007, 7, 1022-1025.[33] Probstein, R. F. Physicochemical hydrodynamics, John Willey & Sons, New York,

1995.[34] Davidson, C., and Xuan, X.,Electrophoresis 2008, 29, 1125-1130.[35] Morrison, F. A.; and Osterle, J. F.J. Chem. Phys. 1965, 43, 2111-2115.[36] X. Xuan, and D. Li,J. Power Source, 156 (2006) 677-684.[37]

Griffiths, S. K.; and Nilson, R. H.Electrophoresis 2005, 26, 351-361.[38] Xuan, X.; and Li, DJ Micromech. Microeng. 2004, 14, 290-298.

[39] Li, D.,Electrokinetics in Microfluidics, Elsevier Academic Press,Burlington, MA 2004.[40] Xuan, X.Microfluid. Nanofluid. 2008, 4, 457-462.[41] Taylor, G. I. Proc. Roy. Soc. London A 1953, 219, 186-203.[42] Aris, R. Proc. Roy. Soc. London A 1956, 235, 67-77.[43] Giddings, J. C. Unified separation science, John Wiley & Sons, Inc., New York, 1991.[44] Martin, M.; Giddings, J. C.J. Phys. Chem. 1981, 85, 727-733.[45] Dutta, D.; Ramachandran, A.; and Leighton, D. T. Microfluid. Nanofluid. 2006, 2,

275-290.[46] Ghosal, S.,Annu. Rev. Fluid Mech. 2006, 38, 309-338.[47] Griffiths, S. K., Nilson, R. H.Anal. Chem. 1999, 71, 5522-5529.[48] Griffiths, S. K., and Nilson, R. N.,Anal. Chem. 2000, 72, 4767-4777.[49] Datta, R.Biotechnol. Prog. 1990, 6, 485-493.[50] Datta, R.; and Kotamarthi, V. RAICHE J. 1990, 36. 916-926.[51] Wooding, R. A.,J. Fluid. Mech. 1960, 7, 501-515.[52] Gas, B.; Stedry, M.; and Kenndler, E.Electrophoresis 1997, 18, 2123-2133.[53] Gas, B.; Kenndler, E.Electrophoresis 2000, 21, 3888-3897.[54] Giddings, J. C. Sep. Sci. 1969, 4, 181-189.[55] Huber, J. F. K. Fresenius' Z. Anal. Chem. 1975, 277, 341-347.[56] Kenndler, E.J Cap. Elec. 1996, 3, 191-198.[57] Schwer, C.; and Kenndler, E. Chromatographia 1992, 33, 331-335.[58] Kenndler, E.; and Fridel, W.J Chromatography 1992, 608, 161-170.[59]

Dutta, D.

Electrophoresis2007, 28, 4552-4560.[60] Dutta, D.Anal. Chem. 2008, 80, 4723-4730.


43/482

In: Fluid Mechanics and Pipe Flow ISBN: 978-1-60741-037-9Editors: D. Matos and C. Valerio, pp. 27-39 2009 Nova Science Publishers, Inc.

Chapter 2

H2O IN THE MANTLE: FROM FLUID

TO HIGH-PRESSURE HYDROUS SILICATES

N.R. Khisina1,*

, R. Wirth2

and S. Matsyuk3

1Institute of Geochemistry and Analytical Chemistry of Russian Academy of Sciences,Kosygin st. 19, 119991 Moscow, Russia

2GeoForschungZentrum Potsdam, Germany3Institute of Geochemistry, Mineralogy and Ore Formation, National Academy of

Sciences of Ukraine, Paladin Ave., 34, 03680 Kiev-142, Ukraine

Abstract

Infrared spectroscopic data show that nominally anhydrous olivine (Mg,Fe)2SiO4 containstraces of H2O, up to several hundred wt. ppm of H2O (Miller et al., 1987; Bell et al., 2004;Koch-Muller et al., 2006; Matsyuk & Langer, 2004) and therefore olivine is suggested to be awater carrier in the mantle (Thompson, 1992). Protonation of olivine during its crystallizationfrom a hydrous melt resulted in the appearance ofintrinsic OH-defects (Libowitsky & Beran,1995). Mantle olivine nodules from kimberlites were investigated with FTIR and TEMmethods (Khisina et al., 2001, 2002, 2008). The results

FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS

Documents

Transcript of FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS