Microfluidics and Nanofluidics

Microfluidics and NanofluidicsTheory and Selected Applications

Clement Kleinstreuer

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To my family,

Christin, Nicole, and Joshua

vii

Contents

Preface..........................................................................................................xv

Part A: A REVIEW OF ESSENTIALS IN MACROFLUIDICS .................. 1

CHAPTER 1 Theory ..................................................................................... 3

1.1 Introduction and Overview ........................................................................ 3

1.2 Definitions and Concepts ........................................................................... 81.2.1 Definitions ........................................................................................ 81.2.2 Flow Field Description ....................................................................141.2.3 Flow Field Categorization ................................................................181.2.4 Thermodynamic Properties and Constitutive Equations ..................20

1.3 Conservation Laws ...................................................................................231.3.1 Derivation Approaches ....................................................................231.3.2 Reynolds Transport Theorem ...........................................................24

1.3.2.1 Fluid Mass Conservation in Integral Form ................................271.3.2.2 Momentum Conservation in Integral Form ...............................331.3.2.3 Conservation Laws of Energy and Species Mass .......................37

1.3.3 Conservation Equations in Differential Form ..................................401.3.3.1 Fluid Mass Conservation...........................................................401.3.3.2 Linear Momentum Conservation ..............................................451.3.3.3 Reduced Forms of the Momentum Equation ............................551.3.3.4 Energy and Species Mass Conservation .....................................60

1.3.4 Entropy Generation Analysis ...........................................................65

1.4 Homework Assignments ...........................................................................741.4.1 Physical Insight ................................................................................741.4.2 Text Problems ..................................................................................76

CHAPTER 2 Applications ............................................................................79

2.1 Internal Fluid Flow ..................................................................................792.1.1 Problem-Solving Steps .....................................................................80

viii Contents

2.1.2 Sample Solutions of the Reduced Navier-Stokes Equations .............83

2.2 Porous Medium Flow .............................................................................108

2.3 Mixture Flows .......................................................................................1182.3.1 Introduction ...................................................................................1182.3.2 Modeling Approaches ....................................................................1192.3.3 Homogeneous Flow Equations ......................................................1272.3.4 Non-Newtonian Fluid Flow ...........................................................1292.3.5 Particle Transport ..........................................................................139

2.4 Heat Transfer ........................................................................................1512.4.1 Forced Convection Heat Transfer...................................................1512.4.2 Convection Heat Transfer Coefficient ............................................153

2.5 Convection-Diffusion Mass Transfer ......................................................1622.5.1 Modeling Approaches ....................................................................1622.5.2 Compartmental Modeling ..............................................................168

2.6 Homework Assignments .........................................................................1762.6.1 Definitions, Concepts, and Physical Insight ...................................1762.6.2 Text Problem ..................................................................................1782.6.3 Homework Sets ..............................................................................179

2.6.3.1 Homework Set Ia .....................................................................1792.6.3.2 Homework Set Ib ....................................................................1812.6.3.3 Homework Set IIa ...................................................................1822.6.3.4 Homework Set IIb ...................................................................184

References (Part A) .....................................................................................186

Part B: MICROFLUIDICS ........................................................................189

CHAPTER 3 Microchannel Flow Theory ....................................................191

3.1 Introduction ...........................................................................................1913.1.1 Microfluidic System Components ..................................................1913.1.2 Microfluidic System Integration .....................................................1923.1.3 Microfluidic System Challenges .....................................................194

3.2 Basic Concepts and Limitations .............................................................195

Contents ix

3.2.1 Scaling Laws ..................................................................................1953.2.2 Fluid Properties and Surface Tension Effects .................................2113.2.3 Wall Slip Velocity and Temperature Jump ......................................2163.2.4 Electrokinetic Phenomena ..............................................................222

3.2.4.1 Electroosmosis .........................................................................2223.2.4.2 Electrostatics ...........................................................................2263.2.4.3 Electrophoresis ........................................................................2343.2.4.4 Nernst-Planck Equation ..........................................................238

3.2.5 Magnetohydrodynamics .................................................................239

3.3 Homework Assignments .........................................................................2513.3.1 Physical Insight ..............................................................................2513.3.2 Text Problems ................................................................................252

CHAPTER 4 Applications in Microfluidics .................................................255

4.1 Introduction ...........................................................................................255

4.2 Micropumps and Microchannel Flow ......................................................2564.2.1 Micropumps ...................................................................................2564.2.2 Liquid Flow in Microchannels .......................................................2664.2.3 Gas Flow in Microchannels ...........................................................274

4.3 Micromixing ..........................................................................................280

4.4 Laboratory-on-a-Chip Devices ...............................................................2844.4.1 LoC Processing Steps .....................................................................2854.4.2 LoC Applications ...........................................................................286

4.5 Homework Assignments and Course Projects .........................................2884.5.1 Text-related Questions and Tasks ...................................................2884.5.2 Set-Up for Course Projects (CPs) ...................................................289

References (Part B) .....................................................................................290

Part C: NANOFLUIDICS ..........................................................................293

CHAPTER 5 Fluid Flow and Nanofluid Flow in Nanoconduits ....................295

x Contents

5.1 Introduction ...........................................................................................2955.1.1 Overview ........................................................................................2965.1.2 Nanostructures ..............................................................................2975.1.3 Nanothermodynamics ....................................................................300

5.2 Liquid Flow in Nanoconduits ..................................................................3035.2.1 Introduction and Overview ............................................................3035.2.2 Nontraditional Simulation Methods ..............................................3035.2.3 Summary of Nanoscale Phenomena and Descriptive Equations ....3105.2.4 Nanochannel Flow Examples ........................................................315

5.3 Rarefied Gas Flow in Nanochannels .......................................................3285.3.1 Overview ........................................................................................3285.3.2 Nanochannel Flow Examples ........................................................329

5.4 Homework Assignments and Course Projects .........................................3355.4.1 Text-Related Questions and Tasks ..................................................3355.4.2 Set-Up for Course Projects .............................................................336

CHAPTER 6 Applications in Nanofluidics ..................................................339

6.1 Introduction ...........................................................................................339

6.2 Nanoparticle Fabrication ........................................................................3406.2.1 Metals and Metal Oxides for Cooling ............................................3406.2.2 Drug Carriers in Nanomedicine .....................................................340

6.3 Forced Convection Cooling with Nanofluids ............................................3426.3.1 Nanofluid Properties ......................................................................3436.3.2 Thermal Nanofluid Flow ...............................................................3476.3.3 Friction Factor and Pressure Drop Results ....................................3486.3.4 Convective Heat Transfer ...............................................................349

6.4 Nanodrug Delivery .................................................................................3516.4.1 Types of Drug-Loaded Nanoparticles ............................................3516.4.2 Mechanisms of Nanodrug Targeting .............................................352

6.5 Homework Assignments and Course Projects .........................................3566.5.1 Text-Related Questions and Tasks ..................................................356

Contents xi

6.5.2 Set-Up for Course Projects .............................................................357

References (Part C) .....................................................................................358

Part D: COMPUTER SIMULATIONS OF FLUID-PARTICLE MIXTURE FLOWS ...................................................................................361

CHAPTER 7 Modeling and Simulation Aspects ..........................................363

7.1 Introduction ...........................................................................................363

7.2 Mathematical Modeling .........................................................................365

7.3 Computer Simulation .............................................................................3677.3.1 Result Interpretation ......................................................................3707.3.2 Computational Design Aspects ......................................................371

CHAPTER 8 Computational Case Studies ..................................................375

8.1 Introduction ...........................................................................................375

8.2 Model Validation and Physical Insight ...................................................3768.2.1 Transient Laminar Flow in a Single Bifurcation .............................3768.2.2 Fluid-Particle Dynamics in a Bifurcation .......................................381

8.3 Solid Tumor Targeting with Microspheres ..............................................3868.3.1 Direct Targeting Methodology .......................................................3868.3.2 Optimal Liver Tumor Targeting Study ...........................................386

8.4 Homework Assignments and Course Projects .........................................3908.4.1 Mathematical Modeling .................................................................3908.4.2 Set-Up for Course Projects .............................................................391

References (Part D) .....................................................................................393

APPENDICES ...........................................................................................395

xii Contents

APPENDIX A ............................................................................................397

A.1 Tensor Calculus .....................................................................................397A.1.1 Definitions ....................................................................................397A.1.2 Operations with ∇ .........................................................................398A.1.3 Some Tensor Identities ..................................................................402

A.2 Differentiation ......................................................................................403A.2.1 Differential Time Operators ..........................................................403A.2.2 The Total Differential....................................................................404A.2.3 Truncated Taylor Series Expansions and Binomial Theorem ........404A.2.4 Hyperbolic Functions ...................................................................405

A.3 Integral Transformations .......................................................................407A.3.1 The Divergence Theorem ..............................................................407A.3.2 Leibniz Rule ..................................................................................408A.3.3 Error Function ..............................................................................409A.3.4 Integral Methods ..........................................................................409

A.4 Ordinary Differential Equations ............................................................411

A.5 Transport Equations (Continuity, Momentum, and Heat Transfer).........415A.5.1 Continuity Equation .....................................................................415A.5.2 Equation of Motion (or Linear Momentum Equation).................415A.5.3 Momentum Equation for Constant-Property Fluids .....................416A.5.4 Heat Transfer Equation for Constant-Property Fluids ..................417A.5.5 Stresses: and Fluxes: ...................418A.5.6 Dissipation Function for Newtonian Fluids ..................................419

APPENDIX B ............................................................................................420

B.1 Conversion Factors ................................................................................420

B.2 Properties ..............................................................................................423

B.3 Drag Coefficient: (A) Smooth Sphere and (B) An Infinite Cylinder as a Function of Reynolds Number ...............................................................427

Contents xiii

B.4 Moody Chart .........................................................................................428

References (Appendices)...............................................................................429

INDEX .......................................................................................................431

xv

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.

xvi Preface

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/cmpl/> 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)

Preface xvii

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 ck@ncsu.edu.

1

Part A: A REVIEW OF ESSENTIALS IN MACROFLUIDICS

The review of macrofl uidics repeats mostly undergraduate-level theory and provides solved examples of transport phenomena, i.e., traditional (meaning conventional macroworld) fl uid mechanics, heat and mass transfer, with a couple of more advanced topics plus applications added. Internal fl ow problems dominate and for their solutions the differential modeling approach is preferred. Specifi cally, 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.

3

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 enginee ring 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 OverviewTraditionally, “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

3

4 Chapter 1 Theory

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 knowl-edge 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 qua ntities, 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 con sid-er ations 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 micro-channel 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 geome-tries (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

1.1 Introduction and Overview 5

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 fl uid is a continuumContinuum assumption holds for most liquid fl ows when and for gases when

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

• Special considerations for compressible and/or rarefi ed gases

• Differentiate between Newtonian and non-Newtonian liquids

May have to treat gas fl ow and liquid fl ow differently because of the impact of a given fl uid’s molecular structure and behavioral characteristics

6 Chapter 1 Theory

Table 1.1 Continued

Condition/Phenomenon Flow in Macroconduits Flow in Microconduits

No-Slip Condition Can generally be assumed

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

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

Entrance length may be on the order of a microchannel length

Reynolds NumberImportant to evaluate laminar vs. transitional vs. turbulent fl ows

Typically justifi es Stokes fl ow and allows for nonmechanical pumps driving fl uid fl ow

Turbulence

Transition varies with geometry of domain, but often requires larger Re numbers than in microchannel fl ow. 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 fl ow fi elds near walls

Viscous Heating Is often small/negligibleMay become a major player due to high velocity gradients in tiny channels with viscous fl uids

Wall Temperature Condition

Usually thermodynamic equilibrium is assumed

For rarefi ed gas fl ows, there may be a temperature jump between the solid wall and the gas

DiffusionPresent, 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 negligibleMay become a major contributing force, and hence is being used for small fl uid 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 fi eld

1.1 Introduction and Overview 7

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.

8 Chapter 1 Theory

1.2 Defi nitions and ConceptsAs 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 Defi nitions

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 cp and cv. 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- air past vehicles, buildings, and

planes- blood in arteries/veins or air in

lungs- water past pillars, submarines, etc.

- water in rivers or canals - polymer coating on solid surfaces- gas in pipelines

1.2 Definitions and Concepts 9

⦁ 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., �gz):

(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)

10 Chapter 1 Theory

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:

<rectangular coordinates> (1.8a)

or

<cyli ndrical coordinates> (1.8b)

or

<spherical coordinates> (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).