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ISBN-13: 9781849736718
Publisher: Royal Society of Chemistry, The
Publication date: 12/16/2014
Pages: 574
Product dimensions: 6.14(w) x 9.21(h) x (d)

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Microscale Acoustofluidics


By Thomas Laurell, Andreas Lenshof

The Royal Society of Chemistry

Copyright © 2015 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-84973-706-7



CHAPTER 1

Governing Equations in Microfluidics


HENRIK BRUUS

Department of Physics, Technical University of Denmark, Lyngby, Denmark E-mail: bruus@fysik.dtu.dk


1.1 Introduction

Microfluidics deals with the flow of fluids and suspensions in channels of sub-millimetre-sized cross-sections under the influence of external forces. Here, viscosity dominates over inertia, ensuring the absence of turbulence and the appearance of regular and predictable laminar flow streams, which implies an exceptional spatial and temporal control of solutes. The combination of laminar flow streams and precise control of external forces acting on particles in solution, has resulted in particle handling methods useful for analytical chemistry and bioanalysis, based on different physical mechanisms including inertia, electrokinetics, dielectrophoretics, magneto-phoretics, as well as mechanical contact forces.

Acoustofluidics, i.e. ultrasound-based external forcing of microparticles in microfluidics, has attracted particular attention because it allows gentle, label-free separation based on purely mechanical properties: size, shape, density, and compressibility. The early acoustophoretic microparticle filters were soon developed into the first successful on-chip acoustophoretic separation devices. Many different biotechnical applications of acoustophoresis have subsequently emerged including cell trapping, plasmapheresis, forensic analysis, food analysis, cell sorting using surface acoustic waves, cell synchronization, and cell differentiation.

Furthermore, substantial advancements in understanding the fundamental physics of microsystems acoustophoresis have been achieved through full-chip imaging of acoustic resonances, surface acoustic wave generation of standing waves, multi-resonance chips, advanced frequency control, on-chip integration with magnetic separators, acoustics-assisted micro-grippers, in situ force calibration, automated systems, and full 3D characterization of acoustophoresis.

In this chapter, adapted from Bruus, we study the governing equations in microfluidics formulated in terms of the classical continuum field description of velocity v, pressure p, and density ρ. We also present some of the basic flow solutions, equivalent circuit modeling useful for predicting the flow rates in networks of microfluidic channels, and scaling laws for various microfluidic phenomena.


1.2 The Basic Continuum Fields

In the following we use the so-called Eulerian picture of the continuum fields, where we observe how the fields evolve in time at each fixed spatial position r. Consequently, the position r and the time t are independent variables. The Eulerian picture is illustrated by the velocity field in Figure 1.1. In general, the value of any field variable F(r, t) is defined as the average value <(Fmol(r', t')> of the corresponding molecular quantity for all the molecules contained in some liquid particle of volume ΔV(r) around r at time t

[MATHEMATICAL EXPRESSION OMITTED] (1.1)

If we, for brevity, let mi and vi be the mass and the velocity of molecule i, respectively, and furthermore let i]member of]ΔV stand for all molecules i present inside the volume ΔV(r) at time t, then the definition of the density ρ(r, t) and the velocity field v(r,t) can be written as

[MATHEMATICAL EXPRESSION OMITTED] (1.2a)

[MATHEMATICAL EXPRESSION OMITTED] (1.2b)

Here, we have introduced the symbol "[equivalent to]" to mean "equal-to-by-definition". Notice how the velocity is defined through the more fundamental concept of momentum.

The dependent field variables in microfluidics can be scalars (such as density ρ, viscosity η, pressure p, temperature T, and free energy Φ), vectors (such as velocity v, current density J, pressure gradient [nabla]p, force density f, and electric fields E), and tensors (such as stress tensor σ and velocity gradient [nabla]v).


1.3 Mathematical Notation

The mathematical treatment of microfluidic problems is complicated due to the presence of several scalar, vector and tensor fields and the non-linear partial differential equations that govern them. To facilitate the treatment, some simplifying notation is called for, and here we follow Bruus. First, a suitable co-ordinate system must be chosen. We shall mainly work with Cartesian co-ordinates (x, y, z) with corresponding basis vectors ex, ey, and ez of unity length and that are mutually orthogonal. The position vector r = (rx, ry, rz) = (x, y, z) written as

[MATHEMATICAL EXPRESSION OMITTED] (1.3)

Any vector v can be written in terms of its components vi (where for Cartesian co-ordinates i = x, y, z) as

[MATHEMATICAL EXPRESSION OMITTED] (1.4)

In the last equality we have introduced the Einstein summation convention: a repeated index implies a summation over that index (unless noted otherwise). Other examples of this compact, so-called index notation, are the vector scalar product u·v, the length v of a vector v, and the ith component of a vector u given as a matrix M multiplied by a vector v,

[MATHEMATICAL EXPRESSION OMITTED] (1.5a)

[MATHEMATICAL EXPRESSION OMITTED] (1.5b)

[MATHEMATICAL EXPRESSION OMITTED] (1.5c)

For the partial derivatives of a given function F(r, t), we use the symbols [partial derivative]i, with i = x, y, z, and [partial derivative]t

[MATHEMATICAL EXPRESSION OMITTED] (1.6)

The vector differential operator nabla [nabla] contains the three partial space derivatives [partial derivative]i. It plays an important role in differential calculus, and it is defined by

[MATHEMATICAL EXPRESSION OMITTED] (1.7)

The Laplace operator, which appears in numerous partial differential equations in theoretical physics, is just the scalar product of the nabla operator with itself,

[MATHEMATICAL EXPRESSION OMITTED] (1.8)

In terms of [nabla], the total time derivative of a quantity F(r(t), t) flowing along with the fluid can be written as

[MATHEMATICAL EXPRESSION OMITTED] (1.9)

Since [nabla] is a differential operator, the order of the factors does matter in a scalar product containing it. So, whereas v · [nabla] in the previous equation is a scalar differential operator, the product [nabla] · v with the reversed order of the factors is a scalar function. The latter appears so often in mathematical physics that it has acquired its own name, namely the divergence of the vector field,

[MATHEMATICAL EXPRESSION OMITTED] (1.10)

Concerning integrals, we denote the 3D integral measure by dr, so that in Cartesian coordinates we have dr = dx dy dz. We also consider definite integrals as operators acting on integrands, thus we keep the integral sign and the associated integral measure together to the left of the integrand. As an example, the integral in spherical coordinates (r, θ, φ) over a spherical body with radius a of the scalar function S(r) is written as

[MATHEMATICAL EXPRESSION OMITTED] (1.11)

When working with vectors and tensors it is advantageous to use the following two special symbols: the Kronecker delta δij and the Levi-Civita symbol εijk,

[MATHEMATICAL EXPRESSION OMITTED] (1.12a)

[MATHEMATICAL EXPRESSION OMITTED] (1.12b)

In the index notation, the Levi-Cevita symbol appears directly in the definition of the ith component of the cross-product u x v of two vectors u and v, and of the rotation [nabla] x v,

[MATHEMATICAL EXPRESSION OMITTED] (1.13)

When evaluating the rotation of a rotation, the "pairing-minus-antipairing" theorem is useful,

[MATHEMATICAL EXPRESSION OMITTED] (1.14)

As a last mathematical subject, we mention Gauss's theorem, which we shall employ repeatedly below. For a given vector field V(r), it relates the volume integral in a given region Ω of the divergence [nabla] · V to the integral over the surface [partial derivative] Ω of the flux V · n da through an area element da with the surface normal n,

[MATHEMATICAL EXPRESSION OMITTED] (1.15)

By definition, the surface normal n of a closed surface is an outward-pointing unit vector perpendicular to the surface, hence the name divergence. For a tensor Tjk of rank 2 the theorem states [MATHEMATICAL EXPRESSION OMITTED] with a straightforward generalization to higher ranks.


1.4 Governing Equations

1.4.1 The Continuity Equation

The first governing equation to be derived is the continuity equation, which expresses the conservation of mass. We consider a compressible fluid, i.e. the density ρ may vary as a function of space and time, and a fixed, arbitrarily shaped region Ω in the fluid as sketched in Figure 1.2. The total mass M(Ω, t) inside Ω can be expressed as a volume integral over the density ρ.

[MATHEMATICAL EXPRESSION OMITTED] (1.16)

Since mass can neither appear nor disappear spontaneously in non-relativistic mechanics, M(Ω, t) can only vary due to a mass flux through the surface [partial derivative]Ω of the region Ω. The mass current density J is defined as the mass density ρ times the advection velocity v, i.e. the mass flow per oriented area per time (kg m-2 s-1),

J(r, t) = ρ(r, t)v(r, t). (1.17)

As the region Ω is fixed, [partial derivative]tM(Ω, t) can be calculated either by differentiating the volume integral eqn (1.16),

[MATHEMATICAL EXPRESSION OMITTED] (1.18)

or as a surface integral over [partial derivative]Ω of the mass current density using eqn (1.17) and Figure 1.2,

[MATHEMATICAL EXPRESSION OMITTED] (1.19)

The last expression is obtained by applying Gauss's theorem eqn (1.15) to the vector field V [equivalent to] ρv. We have used -n because this is the direction of entering the region. It follows immediately from eqn (1.18) and (1.19) that

[MATHEMATICAL EXPRESSION OMITTED] (1.20)

This result is true for any choice of region Ω. However, this is only possible if the integrands are identical. Thus we have derived the continuity equation,

[MATHEMATICAL EXPRESSION OMITTED] (1.21)

In many cases, especially in microfluidics, where the flow velocity is much smaller than the velocity of speed of sound (pressure waves), the fluid appears incompressible. This means that any volume change [MATHEMATICAL EXPRESSION OMITTED] in the time interval Δt for an arbitrary region Ω must be zero, or by Gauss's theorem [MATHEMATICAL EXPRESSION OMITTED], and the continuity equation simplifies to.

[MATHEMATICAL EXPRESSION OMITTED] (1.22)

In Chapter 2, compressibility ρ = ρ(p) will be discussed further in connection with acoustics.


1.4.2 The Navier–Stokes Equation

The second governing equation, the Navier–Stokes equation, is the equation of motion for the Eulerian velocity field directly related to momentum conservation and the momentum density ρv. It is derived using an approach similar to that which led us to the continuity equation. We consider the ith component Pi(Ω, t) of the total momentum of the fluid inside an arbitrarily shaped, but fixed, region Ω. In analogy with the mass eqn (1.18), the rate of change of the momentum is given by

[MATHEMATICAL EXPRESSION OMITTED] (1.23)

In contrast to the mass inside Ω, which according to eqn (1.19) can only change by advection through the surface [partial derivative]Ω, the momentum Pi(Ω, t) can change both by advection and by the action of forces given by Newton's second law. These forces can be divided into body forces that act on the interior of Ω, e.g. gravitational and electrical forces, and contact forces that act on the surface [partial derivative]Ω of Ω, e.g. pressure and viscosity forces. Thus, the rate of change of the ith component of the momentum can be written as

[MATHEMATICAL EXPRESSION OMITTED] (1.24)

A body force fbody is an external force that acts throughout the entire body of the fluid. The change in the momentum of Ω due to fbody, e.g. gravity in terms of the density ρ and the acceleration of gravity g, is.

[MATHEMATICAL EXPRESSION OMITTED] (1.25)

For the advection of momentum ρv into Ω, we note that it is described in terms of the tensor (ρv)v, just as advection of density ρ is described by the vector (ρ)v. Considering the ith momentum component, we see that the flux of momentum into Ω through the infinitesimal area da is given by (ρvi)v· (-n)da. Thus the total change [partial derivative]t Padvi (Ω, t) of momentum in Ω due to advection is

[MATHEMATICAL EXPRESSION OMITTED] (1.26)

The rate of momentum change due to pressure is the sum of the pressure force p(-n)da from the surroundings on each infinitesimal area da of the surface [partial derivative]Ω. Thus, for the ith component of the momentum along the unit vector ei we obtain

[MATHEMATICAL EXPRESSION OMITTED] (1.27)

In the last equation we use [MATHEMATICAL EXPRESSION OMITTED], whereby n can be ascribed the same free index j differing from the momentum component index i as in eqn (1.26).

The momentum in Ω is also changed by viscous friction at the surface [partial derivative]Ω from the surrounding fluid. The frictional force dF on a surface element da with the normal vector n must be characterized by a tensor rank of two as two vectors are involved. This tensor is denoted as the viscous stress tensor σ'ij and it expresses the ith component of the friction force per area acting on a surface element oriented with its surface normal parallel to the jth unit vector ej. So we have dFi = σ'ijnj da, which leads directly to the change in the momentum of Ω due to the viscous forces at the surface [partial derivative]Ω,

[MATHEMATICAL EXPRESSION OMITTED] (1.28)

The internal friction is only non-zero when fluid particles move relative to each other, hence the viscous stress tensor σ'ij depends only on the spatial derivatives of the velocity. For the small velocity gradients encountered in microfluidics we can safely assume that only first-order derivatives are relevant. Thus, σ'ij must thus depend linearly on the velocity gradients [partial derivative]ivj. Further analysis shows that it must be symmetric, and one way of writing the tensor of rank two satisfying these conditions is

[MATHEMATICAL EXPRESSION OMITTED] (1.29)

where the first term relates to the dynamic shear viscosity η of an incompressible fluid, and the second term appears when compressibility-induced dilatational viscosity (proportional to [nabla] · v) cannot be neglected. The value of η is determined experimentally, and for water we have

[MATHEMATICAL EXPRESSION OMITTED] (1.30)


(Continues...)

Excerpted from Microscale Acoustofluidics by Thomas Laurell, Andreas Lenshof. Copyright © 2015 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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Table of Contents

Governing equations in microfluidics; Regular perturbation theory for acoustic fields; Linear continuum mechanics; Linear piezoelectric transducers; Building microfluidic acoustic resonators; Experimental characterization of devices; Acoustic radiation force; Applications in acoustophoresis; Modelling and applications of planar resonant devices; Scaling-laws and time scales in acoustofluidics; Affinity acoustophoresis; Biocompatibility and cell viability in microfluidic acoustic resonators; Analysis of acoustic streaming by singular perturbation; Applications in acoustic streaming; Streaming with ultrasound waves interacting with solid particles; Acoustics streaming near liquid-gas interfaces: drops and bubbles; SAW devices; Microscopy for lab-on-a-chip devices; Particle manipulation in acoustic cavities; Applications in acoustic trapping; Enhanced immunoassays and particle sensors; Multiwavelength devices and scale dependent properties; Acoustic manipulation combined with other techniques

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