Microfluidics and Nanofluidics E-Book

Clement Kleinstreuer

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 392

Veröffentlichungsjahr: 2013

Beschreibung

Fluidics originated as the description of pneumatic and hydraulic control systems, where fluids were employed (instead of electric currents) for signal transfer and processing. Microfluidics and Nanofluidics: Theory and Selected Applications offers an accessible, broad-based coverage of the basics through advanced applications of microfluidics and nanofluidics. It is essential reading for upper-level undergraduates and graduate students in engineering and professionals in industry.

Leseprobe

Table of Contents

Title Page

Copyright

Dedication

Preface

Part A: A Review of Essentials in Macrofluidics

Chapter 1: Theory

1.1 Introduction and Overview

1.2 Definitions and Concepts

1.3 Conservation Laws

1.4 Homework Assignments

Chapter 2: Applications

2.1 Internal Fluid Flow

2.2 Porous Medium Flow

Chapter 2: Applications

2.3 Mixture Flows

2.4 Heat Transfer

Chapter 2: Applications

2.5 Convection-Diffusion Mass Transfer

2.6 Homework Assignments

References (Part A)

Part B: Microfluidics

Chapter 3: Microchannel Flow Theory

3.1 Introduction

3.2 Basic Concepts and Limitations

3.3 Homework Assignments

Chapter 4: Applications in Microfluidics

4.1 Introduction

4.2 Micropumps and Microchannel Flow

4.3 Micromixing

4.4 Laboratory-on-a-Chip Devices

4.5 Homework Assignments and Course Projects

References (Part B)

Part C: Nanofluidics

Chapter 5: Fluid Flow and Nanofluid Flow in Nanoconduits

5.1 Introduction

5.2 Liquid Flow in Nanoconduits

5.3 Rarefied Gas Flow in Nanochannels

5.4 Homework Assignments and Course Projects

Chapter 6: Applications in Nanofluidics

6.1 Introduction

6.2 Nanoparticle Fabrication

6.3 Forced Convection Cooling with Nanofluids

6.4 Nanodrug Delivery

6.5 Homework Assignments and Course Projects

References (Part C)

Part D: Computer Simulations of Fluid-Particle Mixture Flows

Chapter 7: Modeling and Simulation Aspects

7.1 Introduction

7.2 Mathematical Modeling

7.3 Computer Simulation

Chapter 8: Computational Case Studies

8.1 Introduction

8.2 Model Validation and Physical Insight

8.3 Solid Tumor Targeting with Microspheres

8.4 Homework Assignments and Course Projects

References (Part D)

Appendices

Appendix A

A.1 TENSOR CALCULUS

A.2 DIFFERENTIATION

A.3 INTEGRAL TRANSFORMATIONS

A.4 ORDINARY DIFFERENTIAL EQUATIONS

A.5 TRANSPORT EQUATIONS (CONTINUITY, MOMENTUM, AND HEAT TRANSFER)

Appendix B

B.1 CONVERSION FACTORS

B.2 PROPERTIES

B.3 DRAG COEFFICIENT: (A) Smooth Sphere and (B) An Infinite Cylinder as a Function of Reynolds Number

B.4 MOODY CHART

References (Appendices)

Index

Cover image: Courtesy of the Folch Lab, University of Washington

Cover design: Anne-Michele Abbott

This book is printed on acid-free paper.

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To my family,

Preface

“Fluidics” originated as the description of pneumatic and hydraulic control systems, where fluids were employed (instead of electric currents) for signal transfer and processing. Fluidics then broadened and now comprises the technique of handling fluid flows from the macroscale down to the nanoscale. In turn, micro-/nanofluidics is a relatively small but very important part of nanoscience and technology, as indicated by the growing number of subject-oriented engineering and physics journals.

This textbook is written primarily for mature undergraduates in engineering and physics. However, it should be of interest to first-year graduate students and professionals in industry as well. Part A reviews key elements of classical fluid mechanics topics, with the main focus on laminar internal flows as needed for the remaining Chapters 3 to 8. The goal is to assure the same background for all students and hence the time spent on the material of Chapter 1, “Theory,” and Chapter 2, “Applications,” may vary somewhat from audience to audience. Part B, “Microfluidics,” is the heart of the book, in terms of depth and extent, because of the accessibility of the topic and its wide range of engineering applications (see Chapters 3 and 4). Dealing with the more complex transport phenomena in “Nanofluidics” (see Part C) is much more challenging because advanced numerical solution tools are still not readily available to undergraduate/graduate students for course assignments. Thus, Chapters 5 and 6 are more descriptive and discuss only solutions to rather simple nanoscale problems. Nevertheless, for those interested in pursuing solutions to real-world problems in micro-/nanofluidics, Part D provides some introductory math modeling aspects with computer applications (see Chapters 7 and 8).

When compared to current books, e.g., Tabeling (2005), Nguyen & Wereley (2006), Zhang (2007), Bruus (2007), or Kirby (2010), the present material is in content and form more transport phenomena oriented and accessible to advanced undergraduates and first-year graduate students. Most other books on microfluidics are topic-specific reviews of the exponentially growing literature. Examples include Microfluidics edited by S. Colin (2010) and a handbook edited by S. K. Mitra & S. Chakraborty (2011). While some of the cited books also describe elements of nanofluidics, only the recent texts by Das et al. (2007), Rogers et al. (2008), and Hornyak et al. (2008), focus exclusively on nanotechnology with chapters on nanofluids and nanofluidics. Cited references in the preface appear in the list at the end of Part A.

The main learning objectives are to gain a solid knowledge base of the fundamentals and to acquire modern application skills. Furthermore, this eight-chapter exposure should provide students with a sufficient background for advanced studies in these fascinating and very future-oriented engineering areas, as well as for expanded job opportunities. Pedagogical elements include a 50/50 physics-mathematics approach when introducing new material, illustrating concepts, showing graphical/tabulated results as well as links to flow visualizations, and, very important, providing professional problem solution steps. Specifically, the problem solution format follows strictly: System Sketch, Assumptions/Postulates, and Concept/Approach—before starting the solution phase which consists of symbolic math model development (see Sect. A.1 and A.2), analytic (and occasionally) numerical solution, graphs, and comments on “physical insight.” After some illustrative examples, most solved text examples have the same level of difficulty as assigned homework sets listed in Section 2.6. In general, homework assignments are grouped into “concept questions” to gain physical insight, engineering problems to hone independent problem solution skills, and/or course projects. Concerning course projects, the setup, suggestions, expectations, and rewards appear at the end of Chapters 4 to 6 and 8. They are probably the most important learning experience when done right. A Solutions Manual is available for instructors adopting the textbook.

The ultimate goals after course completion are that the more serious student can solve traditional and modern fluidics problems independently, can provide physical insight, and can suggest (say, via a course project) system design improvements.

As all books, this text relied on numerous open-source material as well as contributions provided by research associates, graduate students, and former participants of the author's course “Microfluidics and Nanofluidics” at North Carolina State University (NCSU). Special thanks go to Dr. Jie Li for typing, generating the graphs and figures, checking the example solutions, formatting the text, and obtaining the cited references. The Index was generated by Zelin Xu, who also reformatted the text; the proofreading of the text was performed by Tejas Umbarkar; while Chapter 8 project results were supplied by Emily Childress and Arun Varghese Kolanjiyil, all presently PhD students in the Computational Multi-Physics Lab <http://www.mae.ncsu.edu/cmpli> at NCSU. Some of the book manuscript was written when the author worked as a Visiting Scholar at Stanford University during summers. The engaging discussions with Prof. John Eaton and his students (Mechanical Engineering Department) and the hospitality of the Dewes, Krauskopf and Tidmarsh host families are gratefully acknowledged as well.

For critical comments, constructive suggestions, and tutorial material, please contact the author via [email protected].

Part A

A Review of Essentials in Macrofluidics

The review of macrofluidics repeats mostly undergraduate-level theory and provides solved examples of transport phenomena, i.e., traditional (meaning conventional macroworld) fluid mechanics, heat and mass transfer, with a couple of more advanced topics plus applications added. Internal flow problems dominate and for their solutions the differential modeling approach is preferred. Specifically, for any given problem the basic conservation laws (see Sect. A.5) are reduced based on physical understanding (i.e., system sketch plus assumptions), sound postulates concerning the dependent variables, and then solved via direct integration or approximation methods. Clearly, Part A sets the stage for most of the problems solved in Part B and Part C.

Chapter 1

Theory

Clearly, the general equations describing conservation of mass, momentum, and energy hold for transport phenomena occurring in all systems/devices from the macroscale to the nanoscale, outside quantum mechanics. However, for most real-world applications such equations are very difficult to solve and hence we restrict our analyses to special cases in order to understand the fundamentals and develop skills to solve simplified problems.

This chapter first reviews the necessary definitions and concepts in fluid dynamics, i.e., fluid flow, heat and mass transfer. Then the conservation laws are derived, employing different approaches to provide insight of the meaning of equation terms and their limitations.

It should be noted that Chapters 1 and 2 are reduced and updated versions of Part A chapters of the author's text Biofluid Dynamics (2006). The material (used with permission from Taylor & Francis Publishers) is now geared towards engineering students who already have had introductory courses in thermodynamics, fluid mechanics and heat transfer, or a couple of comprehensive courses in transport phenomena.

1.1 Introduction and Overview

Traditionally, “fluidics” referred to a technology where fluids were used as key components of control and sensing systems. Nowadays the research and application areas of “fluidics” have been greatly expanded. Specifically, fluidics deals with transport phenomena, i.e., mass, momentum and heat transfer, in devices ranging in size from the macroscale down to the nanoscale. As it will become evident, this modern description implies two things:

i. Conventional fluid dynamics (i.e., macrofluidics) forms a necessary knowledge base when solving most microfluidics and some nanofluidics problems.

ii. Length scaling from the macroworld (in meters and millimeters) down to the micrometer or nanometer range (i.e., while ) requires new considerations concerning possible changes in fluid properties, validity of the continuum hypothesis, modified boundary conditions, and the importance of new (surface) forces or phenomena.

So, to freshen up on macrofluidics, this chapter reviews undergraduate-level essentials in fluid mechanics and heat transfer and provides an introduction to porous media and mixture flows. Implications of geometric scaling, known as the “size reduction effect,” are briefly discussed next.

The most important scaling impact becomes apparent when considering the area-to-volume ratio of a simple fluid conduit or an entire device:

1.1

Evidently, in the micro/nanosize limit the ratio becomes very large, i.e., , where such as the hydraulic diameter, channel height, or width. This implies that in micro/nanofluidics the system's surface-area-related quantities, e.g., pressure and shear forces, become dominant. Other potentially important micro/nanoscale forces, rightly neglected in macrofluidics, are surface tension as well as electrostatic and magnetohydrodynamic forces. To provide a quick awareness of other size-related aspects, the following tabulated summary characterizes flow considerations in macrochannels versus microchannels. Specifically, it contrasts important flow conditions and phenomena in conduits of the order of meters and millimeters vs. those in microchannels being of the order of micrometers (see Table 1.1).

Brief Comments Regarding Table 1.1.

Fortunately, the continuum mechanics assumption holds (i.e., a fluid is homogeneous and infinitely divisible) for most microchannel flows. Hence, reduced forms of the conservation laws (see Sect. 1.3) can be employed to solve fluid flow and heat/mass transfer problems in most device geometries (see Sect. 2.1 and Chapters 3 and 4). The boundary condition of “no velocity slip at solid walls” is standard in macrofluidics. However, microchannels fabricated with hydrophobic material and/or having rough surfaces may exhibit liquid velocity slip at the walls. Considering laminar flow, the entrance length of a conduit can be estimated as:

1.2

where the hydraulic diameter is defined as , with A being the cross-sectional area and P the perimeter, the Reynolds number , and for macroconduits and 0.5 for microchannels. For fully turbulent flow, . Considering that typically , entrance effects can be important. For example, if the favorite simplification “fully developed flow” cannot be assumed anymore (see Sect. 1.4). The Reynolds number is the most important dimensionless group in fluid mechanics. However, for microsystems employed in biochemistry as well as in biomedical and chemical engineering, the Reynolds number is usually very low, i.e., . In contrast, microscale cooling devices, i.e., heat exchangers, require high Reynolds numbers to achieve sufficient heat rejection. Onset to turbulence, mainly characterized by random fluctuations of all dependent variables, may occur earlier in microsystems than in geometrically equivalent macrosystems. In some cases, surface roughness over, say, 3% of the channel height may cause interesting flow phenomena near the wall, such as velocity slip and/or transition to turbulence. For microsystems with heavy liquids and high velocity gradients, energy dissipation due to viscous heating should be considered. The temperature jump condition at the wall may be applicable when dealing with convection heat transfer of rarefied gases (see Chapters 2 and 3). The last three entries in Table 1.1, i.e., diffusion, surface tension, and electrokinetics, are of interest almost exclusively in microfluidics and nanofluidics (see Part B and Part C).

Table 1.1 Comparison of Flows in Macrochannels vs. Microchannels

Condition/Phenomenon

Flow in Macroconduits

Flow in Microconduits

Continuum Mechanics Hypothesis

Any fluid is a continuum

Continuum assumption holds for most liquid flows when and for gases when

Type of Fluid(i.e., liquid versus gas)

Special considerations for compressible and/or rarefied gases

Differentiate between Newtonian and non-Newtonian liquids

May have to treat gas flow and liquid flow differently because of the impact of a given fluid's molecular structure and behavioral characteristics

No-Slip Condition

Can generally be assumed

Liquid-solid slip may occur on hydrophobic surfaces. Velocity slip and temperature jump may occur with rarefied gases

Entrance Effects

Entrance length is usually negligible when compared to the length of the conduit

Entrance length may be on the order of a microchannel length

Reynolds Number

Important to evaluate laminar vs. transitional vs. turbulent flows

Typically justifies Stokes flow and allows for nonmechanical pumps driving fluid flow

Turbulence

Transition varies with geometry of domain, but often requires larger Re numbers than in microchannel flow. Example

Transition to turbulence may occur earlier, e.g., at

Surface Roughness

Is often negligible or included in the friction factor (see Moody chart in App. B)

May need to be considered due to manufacturing limitations at this small scale; roughness may be comparable to dimensions of the system and hence causes complex flow fields near walls

Viscous Heating

Is often small/negligible

May become a major player due to high velocity gradients in tiny channels with viscous fluids

Wall Temperature Condition

Usually thermodynamic equilibrium is assumed

For rarefied gas flows, there may be a temperature jump between the solid wall and the gas

Diffusion

Present, but often very slow; therefore, often negligible

Due to the small size of channels, diffusion is important and can be used for mixing

Surface Tension

Is often negligible

May become a major contributing force, and hence is being used for small fluid volume transfer

Electrohydrodynamic effects, such as electroosmosis (EO)

Negligible

In a liquid electrolyte an electric double layer (EDL) can be formed, which is set into motion via an applied electric field

Fluidics, as treated in this book, is part of Newtonian mechanics, i.e., dealing with deterministic, or statistically averaged, processes (see Branch A in Figure 1.1).

Figure 1.1 Branches of physics waiting for unification

For fluid flow in nanoscale systems the continuum mechanics assumption is typically invalid because the length scales of fluid molecules are on the order of nanochannel widths or heights. For example, the intermolecular distance for water molecules is 0.3-0.4 nm while for air molecules it is 3.3 nm, with a mean-free path of about 60 nm. Hence, for rarefied gases, not being in thermodynamic equilibrium, the motion and collision of packages of molecules have to be statistically simulated or measured. For liquids in nanochannels, molecular dynamics simulation, i.e., the solution of Newton's second law of motion for representative molecules, is necessary.

1.2 Definitions and Concepts

As indicated in Sect. 1.1, a solid knowledge base and good problem-solving skills in macroscale fluid dynamics, i.e., fluid flow plus heat and mass transfer, are important to model most transport phenomena in microfluidics and some in nanofluidics. So, we start out with a review of essential definitions and then revisit basic engineering concepts in macrofluidics. The overriding goals are to understand the fundamentals and to be able to solve problems independently.

1.2.1 Definitions

Elemental to transport phenomena is the description of fluid flow, i.e., the equation of motion, which is also called the momentum transfer equation. It is an application of Newton's second law, which Newton postulated for the motion of a particle. For most realistic engineering applications the equation of motion is three-dimensional (3-D) and nonlinear, the latter because of fluid inertia terms such as , etc. (see App. A.5). However, it is typically independent of the scalar heat transfer and species mass equations, i.e., fluid properties are not measurably affected by changes in fluid temperature and species concentration, the latter in case of mixture flows. In summary, the major emphasis in Chapters 1 and 2 are on the description, solution, and understanding of the physics of fluid flow in conduits.

Here is a compilation of a few definitions:

A

fluid

is an assemblage of gas or liquid molecules which deforms continuously, i.e., it

flows

under the application of a shear stress.

Note:

Solids do not behave like that; but what about borderline cases, i.e., the behavior of materials such as jelly, grain, sand, etc.?

Key

fluid properties

are density ρ, dynamic viscosity μ, thermal conductivity

k

, species diffusivity , as well as heat capacities

c

p

and

c

v

. In general, all six are usually temperature dependent. Very important is the viscosity (see also kinematic viscosity ) representing frictional (or drag) effects. Certain fluids, such as polymeric liquids, blood, food stuff, etc., are also shear rate dependent and hence called

non-Newtonian fluids

(see Sect. 2.3.4).

Flows, i.e., fluid in motion powered by a force or gradient

, can be categorized into:

Internal Flows

and

External Flows

- oil, air, water or steam in pipes and inside devices

- blood in arteries/veins or air in lungs

- water in rivers or canals

- gas in pipelines

- air past vehicles, buildings, and planes

- water past pillars, submarines, etc.

- polymer coating on solid surfaces

Driving forces for fluid flow

include gravity, pressure differentials or gradients, temperature gradients, surface tension, electroosmotic or electromagnetic forces, etc.

Forces

appear either as body forces (e.g., gravity) or as surface forces (e.g., pressure). When acting on a fluid element they can be split into

normal

and

tangential forces

leading to pressure and normal/shear stresses. For example, on any surface element:

1.3

while

1.4

Recall: As Stokes postulated, the total stress depends on the spatial derivative of the velocity vector, i.e., (see App. A.2). For example, shear stress occurs due to relative frictional motion of fluid elements (or viscous layers). In contrast, the total pressure sums up three pressure forms, where the mechanical (or thermodynamic) pressure is experienced when moving with the fluid (and therefore labeled “static” pressure and measured with a piezometer). The dynamic pressure is due to the kinetic energy of fluid motion (i.e., ), and the hydrostatic pressure is due to gravity (i.e., ρgz):

1.5a,b

where

1.6a,b

From the fluid statics equation for a stagnant fluid body (or reservoir), where h is the depth coordinate, we obtain:

1.7

Clearly, the hydrostatic pressure due to the fluid weight appears in the momentum equation as a body force per unit volume, i.e., . On the microscopic level, fluid molecules are randomly moving in all directions. In the presence of a wall, collisions, i.e., impulse per time, cause a fluctuating force on the wall. This resulting push statistically averaged over time and divided by the impact area is the pressure.

In general:

Any fluid flow is described by its

velocity

and

pressure

fields. The velocity vector of a fluid element can be written in terms of its three scalar components:

1.8a

or

1.8b

or

1.8c

Its total time derivative is the fluid element acceleration (see Example 1.1 or Sect. A.1):

1.9

where Eq. (1.9) is also known as the Stokes, material, or substantial time derivative.

Streamlines

for the visualization of flow fields are lines to which the local velocity vectors are tangential. In steady laminar flow streamlines and fluid-particle pathlines are identical. For example, for steady 2-D flow (see Sect. 1.4):

1.10

where the 2-D velocity components have to be given to obtain, after integration, the streamline equation y(x).

Dimensionless groups

, i.e., ratios of forces, fluxes, processes, or system parameters, indicate the importance of specific transport phenomena. For example, the Reynolds number is defined as (see Example 1.2):

1.11

where v is an average system velocity, L is a representative system “length” scale (e.g., the tube diameter D), and is the kinematic viscosity of the fluid.

Other dimensionless groups with applications in engineering include the Womersley number and Strouhal number (both dealing with oscillatory/transient flows), Euler number (pressure difference), Weber number (surface tension), Stokes number (particle dynamics), Schmidt number (diffusive mass transfer), Sherwood number (convective mass transfer ) and Nusselt number, the ratio of heat conduction to heat convection (see Sect. A.3). The most common source (i.e., derivation) of these numbers is the nondimensionalization of partial differential equations describing the transport phenomena at hand, or alternatively via scale analysis (see Example 1.2).

Example 1.1:

Derive the material (or Stokes) derivative, operating on the velocity vector, describing the “total time rate of change” of a fluid flow field.

Hint:

For illustration purposes, use an arbitrary velocity field, , and form its total differential.

Recall

The total differential of any continuous and differentiable function, such as , can be expressed in terms of its infinitesimal contributions in terms of changes of the independent variables:

Solution

Dividing through by dt and recognizing that , and are the local velocity components, we have:

Substituting the “particle dynamics” differential with the “fluid element” differential yields:

Example 1.2:

Generation of Dimensionless Groups

a. Scale Analysis

As outlined in Sect. 1.3, the Navier-Stokes equation (see Eq. (1.63)) describes fluid element acceleration due to several forces per unit mass, i.e.,

Now, by definition:

Employing the scales and where v may be an average velocity and L a system characteristic dimension, we obtain:

Similarly, taking

we can write with system time scale T (e.g., cardiac cycle:)

which is the Strouhal number. For example, when , and hence the process, or transport phenomenon, is quasi-steady.

b. Nondimensionalization of Governing Equations

Taking the transient boundary-layer equations (see Sect. 1.3, Eq. (1.63)) as an example,

we nondimensionalize each variable with suitable, constant reference quantities. Specifically, approach velocity , plate length , system time T, and atmospheric pressure are such quantities. Then,

Note:

Commonly, is defined as , where is the varying boundary-layer thickness.

Inserting all variables, i.e., etc., into the governing equation yields

Dividing the entire equation by, say, generates:

Comments:

In a way three goals have been achieved:

the governing equation is now dimensionless;the variables vary only between 0 and 1; andthe overall fluid flow behavior can be assessed by the magnitude of three groups, i.e., Str, Eu, and Re numbers.

1.2.2 Flow Field Description

Any flow field can be described at either the microscopic or the macroscopic level. The microscopic or molecular models consider the position, velocity, and state of every molecule, or representative packages of molecules, of a fluid at all times. In contrast, averaging discrete-particle information (i.e., position, velocity, and state) over a local fluid volume yields macroscopic quantities, e.g., the velocity field in time and space of the flow field. The advantages of the molecular approach include general applicability, i.e., no need for submodels (e.g., for the stress tensor, heat flux, turbulence, wall conditions, etc.) and an absence of numerical instabilities (e.g., due to steep flow field gradients). However, considering myriads of molecules, atoms, or nanoparticles requires enormous computer resources, and hence only simple channel or stratified flows with a finite number of interacting molecules (assumed to be solid spheres) can be presently analyzed.

For example, in a 1-mm cube there are about 34 billion water molecules (about a million air molecules at STP); these high numbers make molecular dynamics simulation prohibitive but, on the other hand, intuitively validate the continuum assumption if the flow domain is sufficiently large.

Continuum Mechanics Assumption.

As alluded to in Sect. 1.2.1 (see Table 1.1), fundamental to the description of all transport phenomena are the conservation laws, concerning mass, momentum, and energy, as applied to continua. In general, solid structures and fluid flow fields are continua as long as the local material properties can be defined as averages computed over material elements/volumes sufficiently large when compared to microscopic length scales of the solid or fluid but small relative to the (macroscopic) structure. Variations in solid-structure or fluid flow quantities can be obtained via solutions of differential equations describing the interactions between forces (or gradients) and motion. Specifically, the continuum mechanics method is an effective tool to physically explain and mathematically describe various transport phenomena without detailed knowledge of their internal molecular structures. In summary, continuum mechanics deals with three aspects:

Kinematics

, i.e., fluid element motion regardless of the cause

Dynamics

, i.e., the origin and impact of forces and fluxes generating fluid motion and waste heat, e.g., the stress tensor or heat flux vector, leading to entropy increase

Balance principles

, i.e., the mass, momentum, and energy conservation laws

Also, usually all flow properties are in local thermodynamic equilibrium, implying that the macroscopic quantities of the flow field can adjust swiftly to their surroundings. This local adjustment to varying conditions is rapidly achieved if the fluid has very small characteristic length and time scales of molecular collisions, when compared to the macroscopic flow variations. However, as the channel (or tube) size, typically indicated by the hydraulic diameter Dh, is reduced to the microscale, the surface-area-to-volume ratio becomes larger because and wall surface effects may become important, as mentioned in Sect. 1.1.

Flow Dynamics and Fluid Kinematics.

Here, the overall goal is to find and analyze the interactions between fluid forces, e.g., pressure, gravity/buoyancy, drag/friction, inertia, etc., and fluid motion, i.e., the velocity vector field and pressure distribution from which everything else can be directly obtained or derived (see Figure 1.2). In turn, scalar transport equations, i.e., convection mass and heat transfer, can be solved based on the velocity field to obtain critical magnitudes and gradients (or fluxes) of species concentrations and temperatures, respectively.

Figure 1.2 Dynamics and kinematics of fluid flow: (a) force-motion interactions; and (b) 2-D fluid kinematics

In summary, unbalanced surface/body forces and gradients cause motion in the form of fluid translation, rotation, and/or deformation, while temperature or concentration gradients cause mainly heat or species mass transfer. Note that flow visualization CDs plus web-based university sources provide fascinating videos of complex fluid flow, temperature, and species concentration fields. Please check out the following links: http://en.wikipedia.org/wiki/Flow_visualizationhttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.100.6782&rep=rep1&type=pdf

Balance Principles.

The fundamental laws of mass, energy, and momentum conservation are typically derived in the form of mass, energy, and force balances for a differential fluid volume (i.e., a representative elementary volume) generating partial differential equations (PDEs). When a much larger open system is considered for mass, momentum, and/or energy balances, integral expressions result. Such balances in integral form, known as the Reynolds Transport Theorem (RTT), can be readily transformed into PDEs by employing the Divergence Theorem (see Sect. 1.3 and App. A).

Within the continuum mechanics framework, two basic flow field descriptions are of interest, i.e., the Lagrangian viewpoint and the Eulerian (or control volume) approach (see Figure 1.3, where and ).

Figure 1.3 Closed versus open systems

For the Lagrangian description (see Figure 1.3a) one could consider first just a particle moving on a pathline with respect to a fixed Cartesian coordinate system. Initially, the particle position is at and a moment later at where based on vector algebra . Following the particle's motion for , the position vector is in general:

1.12

In the limit the time rate of change in location is the particle (or fluid element) velocity, i.e.,

1.13

and after a second time derivation:

1.14

Now, the “material point concept” is extended to a material volume with constant identifiable mass, forming a “closed system” that moves and deforms with the flow but no mass crosses the material volume surface, because it is closed (see Figure 1.3a, second example). Again, the system is tracked through space, and as time expires it is of interest to know what the changes in system mass, momentum, and energy are. This can be expressed in terms of the system's extensive property Bsystem which could be mass m, momentum , or total energy E. Thus, the key question is: “How can we express the fate of the Bsystem” or, in mathematical shorthand, what is “”? Clearly, the material time (or Stokes) derivative (see Example 1.1) follows the closed system and records the total time rate of change of whatever is being tracked.

In order to elaborate on the material derivative (see Example 1.1) as employed in the Lagrangian description, a brief illustration of the various time derivatives is in order, i.e., (local), (total of a material point or solid particle), and (total of a fluid element). Their differences can be illustrated using acceleration (see Example 1.1 and App. A):

, where

u

is the fluid element velocity in the

x

-direction,

is employed in solid particle dynamics,

whereas

is the total fluid element acceleration.

In the Eulerian frame, an “open system” is considered where mass, momentum, and energy may readily cross boundaries, i.e., being convected across the control volume surface and local fluid flow changes may occur within the control volume over time (see Figure 1.3b). The fixed or moving control volume may be a large system/device with inlet and outlet ports, it may be small finite volumes generated by a computational mesh, or it may be in the limit a “point” in the flow field. In general, the Eulerian observer fixed to an inertial reference frame records temporal and spatial changes of the flow field at all “points” or, in case of a control volume, transient mass, momentum, and/or energy changes inside and fluxes across its control surfaces.

In contrast, the Lagrangian observer stays with each fluid element or material volume and records its basic changes while moving through space. Section 1.3 employs both viewpoints to describe mass, momentum, and heat transfer in integral form, known as the Reynolds Transport Theorem (RTT). Thus, the RTT simply links the conservation laws from the Lagrangian to the Eulerian frame. In turn, a surface-to-volume integral transformation then yields the conservation laws in differential form (i.e., PDEs) in the Eulerian framework, also known as the control volume approach.

1.2.3 Flow Field Categorization

Exact flow problem identification, especially in industrial settings, is one of the more important and sometimes the most difficult first task. After obtaining some basic information and reliable data, it helps to think and speculate about the physics of the fluid flow, asking:

i. What category does the given flow system fall into, and how does it respond to normal as well as extreme changes in operating conditions? Figure 1.4 may be useful for categorization of real fluids and types of flows.

ii. What variables and system parameters play an important role in the observed transport phenomena, i.e., linear or angular momentum transfer, fluid mass or species mass transfer, and heat transfer?

iii. What are the key dimensionless groups and what are their expected ranges?

Figure 1.4 Special cases of viscous fluid flows

Answers to these questions assist in grouping the flow problem at hand. For example, with the exception of “superfluids,” all others are viscous, some more (e.g., syrup) and some less (e.g., rarefied gases). However, with the advent of Prandtl's boundary-layer concept the flow field, say, around an airfoil has been traditionally divided into a very thin (growing) viscous layer and beyond that an unperturbed inviscid region (see Schlichting & Gersten, 2000). This paradigm helped to better understand actual fluid mechanics phenomena and to simplify velocity and pressure as well as drag and lift calculations. Specifically, at sufficiently high approach velocities a fluid layer adjacent to a submerged body experiences steep gradients due to the “no-slip” condition and hence constitutes a viscous flow region, while outside the boundary layer frictional effects are negligible (see Prandtl equations versus Euler equation in Sect. 1.3.3.3). Clearly, with the prevalence of powerful CFD software and engineering workstations, such a fluid flow classification is becoming more and more superfluous for practical applications.

While, in addition to air, water and almost all oils are Newtonian, some synthetic motor oils are shear rate dependent and that holds as well for a variety of new (fluidic) products. This implies that modern engineers have to cope with the analysis and computer modeling of non-Newtonian fluids (see Sect. 2.3.4). For example, Latex paint is shear thinning, i.e., when painting a vertical door rapid brush strokes induce high shear rates () and the paint viscosity/resistance is very low. When brushing stops, locally thicker paint layers (due to gravity) try to descend slowly; however, at low shear rates the paint viscosity is very high and hence “tear-drop” formation is avoided and a near-perfect coating can dry on the vertical door.

All natural phenomena change with time and hence are unsteady (i.e., transient) while in industry it is mostly desirable that processes are steady, except during production line start-up, failure, or shut-down. For example, turbines, compressors, and heat exchangers operate continuously for long periods of time and hence are labeled “steady-flow devices”; in contrast, pacemakers, control systems, and drink dispensers work in a time-dependent fashion. In some cases, like a heart valve, devices change their orientation periodically and the associated flows oscillate about a mean value. In contrast, it should be noted that the term uniform implies “no change with system location,” as in uniform (i.e., constant over a cross section) velocity or uniform particle distribution, which all could still vary with time.

Mathematical flow field descriptions become complicated when laminar flow turns unstable due to high speed and/or geometric irregularities ranging from surface roughness to complex conduits. The deterministic laminar flow turns transitional on its way to become fully turbulent, i.e., chaotic, transient 3-D with random velocity fluctuations, which help in mixing but also induce high apparent stresses. As an example of “flow transition,” picture a group (on bikes or skis) going faster and faster down a mountain while the terrain gets rougher. The initially quite ordered group of riders/skiers may change swiftly into an unbalanced, chaotic group. So far no universal model for turbulence, let alone for the transitional regime from laminar to turbulent flow, has been found. Thus, major efforts focus on direct numerical simulation (DNS) of turbulent flows which are characterized by relatively high Reynolds numbers and chaotic, transient 3-D flow pattern.

Basic Flow Assumptions and Their Mathematical Statements.

Once a given fluid dynamics problem has been categorized (Figure 1.4), some justifiable assumptions have to be considered in order to simplify the equations describing the flow system's transport phenomena. The three most important ones are time dependence, dimensionality, and flow (or Reynolds number) regime. Especially, if justifiable, steady laminar 1-D (parallel or unidirectional or fully developed) flow simplifies a given problem analysis (see Sect. 1.3.3.3 with Table 1.2 as well as Table 2.1).

Table 1.2 Solutions of Special Cases of the N-S Equations

1.2.4 Thermodynamic Properties and Constitutive Equations

Thermodynamic Properties.

Examples of thermodynamic properties are mass and volume (extensive properties) as well as velocity, pressure, and temperature (intensive properties), all essential to characterize a general system, process, or device. In addition, there are transport properties, such as viscosity, diffusivity, and thermal conductivity, which are all temperature dependent and may greatly influence, or even largely determine, a fluid flow field. Any extensive, i.e., mass-dependent, property divided by a unit mass is called a specific property, such as the specific volume (where its inverse is the fluid density) or the specific energy (see Sect. 1.3). An equation of state is a correlation of independent intensive properties, where for a simple compressible system just two describe the state of such a system. A famous example is the ideal-gas relation, , where and R is the gas constant.

At the microscopic level (based on kinetic theory), the fluid temperature is directly proportional to the kinetic energy of the fluid's molecular motion (Probstein, 1994). Specifically, , where k is the Boltzmann constant, m is the molecular mass, and is the fluctuating velocity vector. The pressure, as indicated, is the result of molecular bombardment. The density depends macroscopically on both pressure and temperature and microscopically on the number of molecules per unit volume; for example, there are air molecules in 1 cm3. Comparing the compressibility of liquids vs. gases, it takes to achieve the same fractional change in density.

Constitutive Equations.

Looking ahead (see Sect. 1.3), when considering the conservation laws for fluid flow and heat transfer, it is apparent that additional relationships must be found in order to solve for the variables: velocity vector , fluid pressure p, fluid temperature T, and species concentration c as well as stress tensor , heat flux vector , and species flux . Thus, this is necessary for reasons of (i) mathematical closure, i.e., a number of unknowns require the same number of equations, and (ii) physical evidence, i.e., additional material properties other than the density are important in the description of system/material/fluid behavior. These additional relations, or constitutive equations, are fluxes which relate via “material properties” to gradients of the principal unknowns. Specifically, for basic linear proportionalities we Recall:

Stokes' postulate, i.e., the fluid stress tensor

1.15

where is the dynamic viscosity;

Fourier's law, i.e., the heat conduction flux

1.16

where k is the thermal conductivity; and

Fick's law for the species mass flux

1.17

where is the binary diffusion coefficient.

Of these three constitutive equations, the total stress tensor is in macrofluidics the most important and complex one (see Sect. 1.3.3.2 for more details). For, say, one-dimensional (1-D) cases to move fluid elements relative to each other, a shear force is necessary. Thus, for simple shear flows (see Figure 1.5) . In general, the shear stress is proportional to and the dynamic viscosity is just temperature dependent for Newtonian fluids (e.g., air, water, and oil) or shear rate dependent for polymeric liquids, paints, blood (at low shear rates), food stuff, etc.

Figure 1.5 Illustration of the shear stress derivation for simple shear flow

1.3 Conservation Laws

After proper problem recognition and classification (see Sect. 1.2.3), central to engineering analysis are the tasks of realistic, accurate, and manageable modeling followed by analytical (or numerical) solution. While in most cases a given system's conservation laws are known, to solve the equations, often subject to complex boundary conditions and closure models, is quite a different story. So, after highlighting different approaches to derive conservation laws for mass, linear momentum, and energy, several examples in this section as well as in Chapter 2 illustrate basic phenomena and solution techniques.

1.3.1 Derivation Approaches

Derivation of the conservation laws describing all essential transport phenomena is very important because they provide a deeper understanding of the underlying physics and implied assumptions, i.e., the power and limitations of a particular mathematical model. Of course, derivations are regarded by most as boring and mathematically quite taxing; however, for those, it's time to become a convert for the two beneficial reasons stated. Furthermore, one should not forget the power of dimensional analysis (DA) which requires only simple algebra when nondimensionalizing governing equations and hence generating dimensionless groups. Alternatively, scale analysis (SA) is a nifty way of deriving dimensionless groups as demonstrated in this chapter (see Example 1.2). Both DA and SA are standard laboratory/computational tools for estimating dominant transport phenomena, graphing results, to evaluate engineering systems, and to test kinematic/dynamic similarities between a physical model and the actual prototype.

Outside the cutting-edge research environment, fluid mechanics problems are solved as special cases, i.e., the conservation equations are greatly reduced based on justifiable assumptions on a case-by-case basis (see Sect. 1.3.3.3). Clearly, the simplest case is fluid statics where the fluid mass forms a “whole body,” either stationary or moving without any relative velocities (see Eq. (1.7)). The popular (because very simple) Bernoulli equation, for frictionless fluid flow along a representative streamline, balances kinetic energy , flow work , and potential energy and hence in some cases provides useful pressure-velocity-elevation correlations. The most frequently used equations in macrofluidics and microfluidics are the Navier-Stokes (N-S) equations, describing momentum and heat transfer for constant-property fluids, assuming that the flow is a continuum. After some basic fluid flow applications, relatively new material is introduced to broaden the student's knowledge base and provide a higher skill level to cope with today's engineering problems encountered in industry or graduate school.

There are basically four ways of obtaining specific equations expressing the conservation laws:

i.Molecular Approach: Fluid properties and transport equations can be obtained from kinetic theory and the Boltzmann equation, respectively, employing statistical means. Alternatively, is solved for each molecule using direct numerical integration.

ii.Integral Approach: Starting with the RTT for a fixed open control volume (Euler), specific transport equations in integral form can be obtained (see Sect. 1.3.2).

iii.Differential Approach: Starting with 1-D balances over an REV (representative elementary volume) and then expanding them to 3-D, the mass, momentum, and energy transfer equations in differential form can be formulated. Alternatively, the RTT is transformed via the divergence theorem, where in the limit the field equations in differential form are obtained (see Sect. 1.3.3).

iv.Phenomenological Approach: Starting with balance equations for an open system, transport phenomena in complex transitional, turbulent, or multiphase flows are derived largely based on empirical correlations and dimensional analysis considerations. A very practical example is the description of transport phenomena with fluid compartment models. These “compartments” are either well-mixed, i.e., transient lumped-parameter models without any spatial resolution, or transient with a one-dimensional resolution in the axial direction.

Especially for the (here preferred) differential approach (iii), the system-specific fluid flow assumptions have to be carefully stated and justified.

1.3.2 Reynolds Transport Theorem

Consider to be an arbitrary extensive quantity of a closed system, say, a moving material volume. In general, such a system could be an ideal piston-cylinder device with enclosed (constant) gas mass, a rigid tank without any fluid leaks, or an identifiable pollutant cloud—all subject to forces and energy transfer (see Figure 1.3a). In any case, represents the system's mass, momentum, or energy.

Task 1 is to express in the Lagrangian frame the fate of in terms of the material derivative, , i.e., the total time rate of change of (see Sect. 1.2.2 reviewing the two system approaches and Example 1.1 discussing the operator ). Specifically, based on:

Conservation of mass

1.18a

Conservation of momentum (or Newton's second law)

1.18b

Conservation of energy or first law of thermodynamics

1.18c

In Task 2 the conservation laws, in terms of , are related to an open system, i.e., in the Eulerian frame. Here, for a fixed control volume () with material streams flowing across the control surface (C.S.), and possibly accumulating inside , we observe with specific quantity , or :

or in mathematical shorthand:

1.19

Equation (1.19), which is formally derived in any undergraduate fluids text, is the RTT for a fixed control volume. Clearly, the specific quantity β can be expressed as:

1.20a-c

Extended Cases.

For a moving control volume the fluid velocity is replaced by (see Example 1.6). The operator , acting on the first term on the right-hand-side (RHS) of Eq. (1.19), has to be replaced by when the control volume is deformable, i.e., the C.S. moves with time (see Example 1.5). For a noninertial coordinate system, for example, when tracking an accelerating system such as a rocket, of Eq. (1.18b) is expressed as:

1.21a

where accounts for noninertial effects (e.g., arbitrary acceleration):

1.21b

In case of . rotation,

1.21c

Specifically, for a rotating material volume, the fluid angular momentum per unit volume has to be considered. The law of conservation of angular momentum states that the rate of change of angular momentum of a material volume is equal to the resultant moment on the volume (see any undergraduate fluids text for more details).

Setting Up the Reynolds Transport Theorem.

There are a few sequential steps necessary for tailoring the general RTT toward a specific flow system description and solving the resulting integral equations:

1. Identify the extensive quantity , e.g., the mass of the identifiable material, linear (or angular) momentum, or total energy. As a result, the specific property of the closed system is known (see Eqs. (1.18 and 1.20)).

2. Determine for each conservation case, i.e., mass, momentum, or energy (see Eq. (1.18)).

3. Select a “smart” control volume and determine if: the control volume is fixed, or moving at , or accelerating at , or accelerating and rotating (see “Extended Cases” above);the flow problem is steady or transient, i.e., is or (note, when the rate of change of inside the is negligible);a control surface moves, i.e., we have a deformable where ;the fluid properties are constant or variable;the inflow/outflow velocity fields are constant, i.e., uniform, or a function of inlet/exit space variables; andthe resulting integral balance equations for mass and momentum (or energy) are decoupled or not.

1. Set up the momentum, i.e., force balance, equation for each coordinate direction.

2. Solve the volume and/or surface integrals (use integration tables, if necessary).

3. Follow the inflow/outflow sign convention (see Figure 1.6), i.e., and

4. Check the results for correctness, i.e., apply common sense.

Figure 1.6 Sign convention for the “net efflux” RTT term (Recall: ; and )

1.3.2.1 Fluid Mass Conservation in Integral Form

Conservation of mass is very intuitive and standard in daily-life observations. A given mass of a fluid may change its thermodynamic state, i.e., liquid or gaseous, but it can neither be destroyed nor created. This, as the other two conservation laws, can be expressed in integral form for a control volume of any size and shape or derived in differential form.

In order to track within the Lagrangian frame an identifiable constant mass of fluid, we set (see Figure 1.3a and Eq. (1.20a))

The conservation principle requires that, with , and hence Eq. (1.19) reads:

1.22

Thus, we just completed Steps (i) and (ii) of the “setting-up-the-RTT” procedure. Aspects of Step (iii) are best illustrated with a couple of examples.

Example 1.3:

Volumetric Flow Rate

Consider a liquid-filled tank (depth H) with a horizontal slot outlet (height 2h and width w) where the short-tube locally varying outlet velocity can be expressed as:

A constant fluid mass flow rate, , is added to maintain the liquid depth H. The z