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...