Modeling Microscale.fluids

Chapter 1Some Fundamental Aspects of Fluid Mechanicsover Microscopic Length Scales

Jeevanjyoti Chakraborty and Suman Chakraborty

Abstract In this chapter we recapitulate some of the fundamental theories of fluidflow at the micro-scale and then introduce the theoretical models of various phenom-ena which enrich the mechanics of this fluid flow. Specifically, we concentrate on therudiments of electrokinetics, surface tension, non-Newtonian fluid, and acoustoflu-idics and their applications in fluid flow at the micro-scale. We resort to analyticallyaddressable mathematical treatments for convenience of an introductory reading.

1 Introduction

The uninitiated reader, with a healthy background in engineering science, may won-der that “flow physics at the micro-scale” is nothing but fluid mechanics appliedat the scale of micrometers. In reality, though, this topic is much broader than isadmitted by such a viewpoint. Indeed, the traditional way of manipulating fluids inconduits/channels has been through the application of mechanically applied pres-sure gradients or through exploiting the natural effects of gravity. These drivingagents are on the total volume of the fluid. However, as dimensions scale down,the effects of the interfaces between the fluid and the confining surfaces becomeincreasingly prominent, which is in keeping with the intuitive expectation that sur-face effects (as compared to volume effects) should become more important asphysical dimensions decrease. This gives us a cue that with increasing levels ofminiaturization, driving agents which exploit surface effects (in contrast to pres-sure gradients and/or gravity) might become increasingly important. In fact, that isso. Electrokinetic and surface tension effects are two extremely important surfaceeffects. These are vast areas in themselves. This, together with the fact that thereexist myriad ways to exploit these effects for flow manipulation, gives us an ideaabout the magnificent breadth of micro-scale flow physics.

It follows naturally from this preliminary discussion that flow physics at themicro-scale is truly an interdisciplinary subject. But, what, after all, is the motivationbehind these studies and of bringing them all together? The answer lies in the mod-ern technology of microfluidics, which is a broad term subsuming within it all the

S. Chakraborty (B)Department of Mechanical Engineering, Indian Institute of Technology (IIT), Kharagpur, Indiae-mail: [email protected]

1S. Chakraborty (ed.), Mechanics Over Micro and Nano Scales,DOI 10.1007/978-1-4419-9601-5_1, C© Springer Science+Business Media, LLC 2011

2 J. Chakraborty and S. Chakraborty

science and technology that is concerned with fluid flows over micron or sub-micronlengths. Specifically, it pertains to the precise manipulation and control of minutevolumes of fluids through miniaturized conduits. The prime mover behind the emer-gence of microfluidics has been the advancements in micro- and nano-fabricationtechnologies. The application of microfluidic technology itself is diverse – rangingfrom the most popular biotechnological/biomedical engineering realm to inkjetprinting and thermal management of electrokinetic devices. Many of these microflu-idic applications are at the bleeding edge of research. The basics of these advancedapplications are pertinently the fundamental principles of micro-scale flow physics.

We begin with a revision of the fundamentals of fluid mechanics with specialemphasis on the applicability of the equations of fluid motion at the micro-scale.

2 Recapitulation of Fundamentals

Newton formulated his laws of mechanics in his book Principia (1687). However, itwas not until 1752 that a mathematically clear description of fluid mechanics (albeitinviscid) was put forward by Euler. It was finally Cauchy who in 1822 introducedthe concept of stress tensor and incorporated it into Euler’s laws, thus presentinga very general theory for the motion of any continuous body [1]. These equationsof Cauchy were not just limited to a fluid medium. It is important to note that theframe of reference used in these formulations is an Eulerian one. This is a frameof reference that focuses attention on a particular region in space as opposed to aLagrangian frame of reference which focuses attention on a particular set of materialpoints. Due to the inherent flowing nature of fluids it is the Eulerian frame that isthe preferred one in fluid medium.

Instead of a presentation following the chronology of historical development, wewill take up a more pedagogical approach with a uniform mathematical framework.The backbone of such a mathematical framework is the Reynolds transport theorem.This theorem relates the system approach with the control volume approach:

DA

Dt

∣∣∣∣sys

= ∂

∂t

∫

CVρadV +

∫

CSρavr · ndS (1.1)

where A is any scalar or vector function denoting an extensive property, a is A perunit mass, vr is the velocity relative to the control surface, dV is a differentially smallelement of the control volume, dS is an elemental arc on the control surface with aunit normal n, and ρ is the density of the fluid.

2.1 Conservation of Mass

To use the Reynolds transport theorem, the relevant extensive property is the totalmass. Thus, A = M (the total mass) so that a = 1. Now, mass conservation, bydefinition, means

1 Some Fundamental Aspects of Fluid Mechanics 3

DM

Dt= 0

or,

∫

CV

D (ρ�u)Dt

dV = 0 (1.2)

Therefore,

∂

∂t

∫

CVρdV +

∫

CSρ�u · ndS = 0 (1.3)

Using Gauss’ divergence theorem, we get

∂

∂t

∫

CVρdV +

∫

CV∇ · (ρ�u) dV = 0

or, for a non-deformable control volume

∫

CV

{∂ρ

∂t+ ∇ · (ρ�u)

}

dV = 0 (1.4)

Now, this is true for any arbitrary non-deformable control volume of elementaryvolume dV implying the necessary condition

∂ρ

∂t+ ∇ · (ρ�u) = 0 (1.5)

For a stationary control volume, �u is the fluid flow velocity.

2.2 Conservation of Linear Momentum

The relevant extensive property, this time, is the linear momentum so that A = M�uand a = �u. Now, using Reynolds’ transport theorem for the motion of any continu-ous media, and assuming, as before, that the control volume is non-deformable, wehave

D (M�u)Dt

∣∣∣∣sys

=∫

CV

∂ (ρ�u)∂t

dV +∫

CSρ�u�u · ndS (1.6)

Noting that

D (M�u)Dt

=∫

CV

D (ρ�u)Dt

dV


we have from Cauchy’s general theory of motion for any continuous medium,

D (ρu)

Dt= ∇ · [σ ] + �fB (1.7)

Therefore, from (1.6) and (1.7), we get

∫

CV

∂ (ρ�u)∂t

dV +∫

CSρ�u�u · ndS =

∫

CV

{

∇ · [σ ] + �fB}

dV (1.8)

Now, using Gauss’ divergence theorem as before, we obtain

∫

CV

∂ (ρ�u)∂t

dV +∫

CV∇ · (ρ�u�u) dV =

∫

CV

{

∇ · [σ ] + �fB}

dV (1.9)

This must be true for any arbitrary control volume, implying the necessarycondition

∂ (ρ�u)∂t

+ ∇ · (ρ�u�u) = ∇ · [σ ] + �fB (1.10)

We reiterate, here, that this is the general equation of motion of a fluid inan inertial frame of reference. It is valid irrespective of the nature of the fluidmaterial.

Now, let us simplify the left-hand side of (1.10), i.e., ∂ (ρ�u)/∂t + ∇ · (ρ�u�u). Letus first focus our attention on the ∇ · (ρ�u�u) term. This is a vector, and thus, ρ�u�umust be a tensor. It is a special product called the dyadic product and is explicitlydenoted by the symbol ⊗. Thus, ρ�u�u = ρ�u ⊗ �u

We expand this as follows:

ρ�u ⊗ �u =⎡

⎣

ρuρvρw

⎤

⎦ [u v w]

=⎡

⎣

ρuu ρuv ρuwρvu ρvv ρvwρwu ρwv ρww

⎤

⎦


Now,

∇ · (ρ�u ⊗ �u) =⎛

⎝

[∂

∂x

∂

∂y

∂

∂z

]⎡

⎣

ρuu ρuv ρuwρvu ρvv ρvwρwu ρwv ρww

⎤

⎦

⎞

⎠

T

=

⎡

⎢⎢⎢⎢⎢⎣

∂ (ρuu)

∂x+ ∂ (ρvu)

∂y+ ∂ (ρwu)

∂z∂ (ρuv)

∂x+ ∂ (ρvv)

∂y+ ∂ (ρwv)

∂z∂ (ρuw)

∂x+ ∂ (ρvw)

∂y+ ∂ (ρww)

∂z

⎤

⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎣

ρu∂ (u)

∂x+ u

∂ (ρu)

∂x+ ρv

∂ (u)

∂y+ u

∂ (ρv)

∂y+ ρw

∂ (u)

∂z+ u

∂ (ρw)

∂z

ρu∂ (v)

∂x+ v

∂ (ρu)

∂x+ ρv

∂ (v)

∂y+ v

∂ (ρv)

∂y+ ρw

∂ (v)

∂z+ v

∂ (ρw)

∂z

ρu∂ (w)

∂x+ w

∂ (ρu)

∂x+ ρv

∂ (w)

∂y+ w

∂ (ρv)

∂y+ ρw

∂ (w)

∂z+ w

∂ (ρw)

∂z

⎤

⎥⎥⎥⎥⎥⎥⎦

= [ρ�u · ∇u + u∇ · (ρ�u) ρ�u · ∇v + v∇ · (ρ�u) ρ�u · ∇w + w∇ · (ρ�u)]T

= ρ�u · ∇�u + �u∇ · (ρ�u)

Therefore,

∂ (ρ�u)∂t

+ ∇ · (ρ�u�u) = ρ∂�u∂t

+ �u∂ρ∂t

+ ρ�u · ∇�u + �u∇ · (ρ�u)

= ρ∂�u∂t

+ ρ�u · ∇�u + �u(∂ρ

∂t+ ∇ · (ρ�u)

)

= ρ∂�u∂t

+ ρ�u · ∇�u (using the continuity equation (1.5))

The general equation of motion of a fluid (of any material property) in an inertialEulerian frame of reference is, thus,

ρ∂�u∂t

+ ρ�u · ∇�u = ∇ · [σ ] + �fB (1.11)


2.3 The Navier–Stokes Equation

We now focus our attention on a special class of fluid, called Newtonian fluid, wherethe shear stress is proportional to the rate of shearing strain. The general constitutivelaw for a homogeneous, isotropic Newtonian fluid is, in indicial notation,

σij = −pdδij + λεkkδij + 2μεij (1.12)

or, in tensorial notation,

[σ ] = −pd [I] + λ tr ([ε]) [I] + 2μ [ε] (1.13)

where pd is the thermodynamic pressure, δij is the Kronecker delta symbol, [ε] isthe strain tensor and tr ([ε]) is the trace of the strain tensor, λ is the volume dilationcoefficient, and μ is the viscosity coefficient.

From kinematics it follows that

[ε] = 1

2∇ ⊗ �u + 1

2(∇ ⊗ �u)T

= 1

2

⎡

⎢⎢⎢⎢⎢⎢⎣

2∂u

∂x

∂v

∂x+ ∂u

∂y

∂w

∂x+ ∂u

∂z∂u

∂y+ ∂v

∂x2∂v

∂y

∂w

∂y+ ∂v

∂z∂u

∂z+ ∂w

∂x

∂v

∂z+ ∂w

∂y2∂w

∂z

⎤

⎥⎥⎥⎥⎥⎥⎦

Therefore,

tr ([ε]) = εkk = ∂u

∂x+ ∂v

∂y+ ∂w

∂z= ∇ · �u

Using this expression in (1.13) we get

[σ ] = −pd [I] + λ∇ · �u [I] + 2μ [ε] (1.14)

Now we note from the general equation of motion (1.10) that we need the term∇ · [σ ]. For a Newtonian fluid,

∇ · [σ ] = ∇ · (−pd [I] + λ∇ · �u [I] + 2μ [ε])

= −∇pd + λ∇ (∇ · �u)+ μ∇ (∇ · �u)+ μ∇2�u= −∇pd + μ∇2�u + (λ+ μ)∇ (∇ · �u)

The combination λ + 23μ is referred to as the bulk viscosity coefficient and

denoted by μv. Therefore,


∇ · [σ ] = −∇pd + μ∇2�u +(

μv + 1

3μ

)

∇ (∇ · �u)

The general equation of motion, then, becomes

ρ

[∂�u∂t

+ �u · (∇�u)]

= −∇pd + μ∇2�u +(

μv + 1

3μ

)

∇ (∇ · �u)+ �fB (1.15)

2.3.1 Further Simplifications

Let us denote σxx + σyy + σzz = −3p, where p is called the mechanical pressure.Using (1.14) we write

σxx + σyy + σzz = −3pd + 3λ∇ · �u + 2μ

(∂u

∂x+ ∂v

∂y+ ∂w

∂z

)

or,

− 3p = −3pd + 3λ∇ · �u + 2μ

(∂u

∂x+ ∂v

∂y+ ∂w

∂z

)

or,

− 3p = −3pd + 3λ∇ · �u + 2μ∇ · �u

or,

3pd − 3p = (3λ+ 2μ)∇ · �u

Now, there are two ways in which the thermodynamic pressure can become equalto the mechanical pressure:

(a) Incompressible flow: For an incompressible flow ∇ · �u = 0, so that 3 (pd − p)= 0 implying pd = p. For an incompressible flow the bulk viscosity remainsunspecified and the equation of motion becomes

ρ

[∂�u∂t

+ �u · (∇�u)]

= −∇p + μ∇2�u + �fB (1.16)

(b) Stokes’ hypothesis: This hypothesis states that for a wide class of fluid flowproblems 3λ + 2μ = 0. It holds true when the characteristic timescales in thesystem are large compared to the molecular relaxation time. It is also true formonoatomic gases. Using λ + 2

3μ = 0, i.e., μv = 0, we obtain the generalequation of motion as

ρ

[∂�u∂t

+ �u · (∇�u)]

= −∇p + μ∇2�u + 1

3μ∇ (∇ · �u)+ �fB (1.17)


2.4 Poiseuille Flow

The steady-state flow actuated solely by a pressure difference between the two endsof a channel is called Poiseuille flow. The mathematical modeling is simplified witha couple of considerations:

(a) The gradient of the flow velocity along the flow direction is zero.(b) The only non-zero component of the flow velocity is the one along the flow

direction, i.e., all components normal to the flow direction are zero.

With these considerations, for an incompressible flow, we have

0 = −∂p

∂x+ μ∇2u (1.18)

where x and u are, respectively, the coordinate component and the velocity com-ponent along the flow direction. Here p is piezometric pressure which containsthe effect of gravity as well. The zero velocity components perpendicular to thisflow direction imply zero gradients of the pressure along those directions. Hence,p ≡ p (x) and ∂p

/

∂x − ≡ dp/

dx. Now, dp/

dx = −�p/

L where�p is the pressuredifference between the two ends of the channel of length L. From (1.18) we have

μ

(∂2u

∂y2+ ∂2u

∂z2

)

= −�p

L

(∂u

∂x= 0

)

This simple equation is surprisingly successful in correctly modeling fluid flowin a wide variety of microfluidic channel shapes.

The simplest geometrical shape that can be considered within this class ofPoiseuille flow is that of the slit channel which is a two-dimensional idealizationof a three-dimensional rectangular channel where the width is “infinite,” meaningthat the width is so large that it does not significantly affect the flow profile. In thiscase,

μ∂2u

∂y2= −�p

L(1.19)

where y is the coordinate along the channel height. This governing equation togetherwith the no-slip boundary condition at the walls (u=0 at y=0 and H) gives us thevelocity profile:

u = − �p

2μLy (y − H) (1.20)

It is informative and useful (from the point of view of the experiment) to find arelation between the total volumetric flow rate and the applied pressure difference.


Q = ∫ H0 udy

= −∫ H

0

�p

2μLy (y − H) dy

= �pH3

12μL

Thus, �p = (12μL/

H3)

Q. The factor 12μL/

H3 is referred to as the hydraulicresistance for this particular channel. Similar expressions for the hydraulic resis-tance may be derived for other channel shapes.

It may be noted from (1.18) that the governing differential equation has beenreduced from the general nonlinear form of the Navier–Stokes equation to a linearone. Such a reduction based on consideration (a) (mentioned just before (1.18)) iswhat is referred to as the fully developed flow condition. Interestingly, even whenthe flow is not fully developed but the Reynolds number is small, the inertia termsin the Navier–Stokes equation become negligible (though not identically zero), ascompared to the viscous terms. Under such circumstances, the nonlinear terms in theNavier–Stokes equation become virtually inconsequential. The resultant simplifiedequation without the nonlinear terms is known as the Stokes equation, which is oftenthe starting point of analyzing low Reynolds number flows.

2.5 Physical Justification of Linearization

Let us consider the Navier–Stokes equation for a steady, incompressible flow,without any body force term:

ρ�u · ∇�u = −∇p + μ∇2�u (1.21)

A systematic way of investigating the order of the different terms present in anequation is to non-dimensionalize it using characteristic scaling parameters. Let thepertinent parameters in this equation be L and V for scaling length and velocity,respectively. Using these, we can write

ρ�u · ∇�u = ρ �u∼ ·∇∼ �u∼(

V2

L

)

(1.22)

μ∇2�u = μ∇2∼ �u∼(

V

L2

)

(1.23)

where symbols with tilde (~) under them represent dimensionless terms.


Substituting these expressions, in (1.21) we get

ρ �u∼ ·∇∼ �u∼(

V2

L

)

= −∇p + μ∇2∼ �u∼(

V

L2

)

or,

(ρVL

μ

)

�u∼ ·∇∼ �u∼ = − L

μV∇∼ p + ∇2

∼ �u∼ (1.24)

The combination ρVL/

μ is non-dimensional and is known by the name Reynoldsnumber Re. If Re � 1 then, clearly, the nonlinear term can be neglected. This fact isof utmost importance in microfluidics because in many situations the length scalesand velocity regimes are such that the Reynolds number is indeed much less than 1.It is in appreciation of this fact that the study of microfluidics is often termed as lowReynolds number hydrodynamics.

The physical justification of the Navier–Stokes equation linearization (this lin-earized version is called the Stokes equation) for most microfluidic modelingpurposes must be appreciated within certain caveats:

(a) Unsteady case:A valid temporal scale when there is no external temporal perturbation imposedon the system is T = L

/

V . Then

ρ∂�u∂t

= ρ

∂ �u∼∂ t∼

(V2

L

)

A comparison of this expression with (1.22) clearly shows that the unsteadyterm will also be present in the non-dimensional equation with Re as a coef-ficient. Since Re is small, the term with this temporal dependence can beneglected.

It is extremely important to understand the fact that such a scaling would notbe valid if there is an externally imposed temporal variation in the system. If,for example, one of the boundaries is moved periodically with frequency ω, thecorrect temporal scale would be 1

/

ω and not L/

V which is inherent to the flow.In such a case, the velocity scale could be expressed in terms of the length scaleand the timescale so that V=Lω. Then,

ρ∂�u∂t

= ρ

∂ �u∼∂ t∼

(ω2L)

ρ�u · ∇�u = ρ �u∼ ·∇∼ �u∼(ω2L)


and

μ∇2�u = μ∇2∼ �u∼(ω

L

)

The unsteady non-dimensional equation becomes

R e

⎛

⎝

∂ �u∼∂ t∼

+ �u∼ ·∇∼ �u∼

⎞

⎠ = − 1

μω∇∼ p + ∇2

∼ �u∼ (1.25)

where R e = ρωL2/

μ. Just as before, if R e = ρωL2/

μ � 1 then all theinertial terms (i.e., the left-hand side of the equation) can be neglected.

(b) Different length scales:We have hitherto defined the Reynolds numbed based on a characteristic lengthscale of the system. However, if the system under consideration is characterizedby more than one length scale (a common occurrence in many real systems) thenhow do we define the Reynolds number? The convention in such a scenariois to choose the smallest length scale for the Reynolds number definition. Toillustrate this point, let us consider the example of steady flow of a thin film ofviscous fluid between two rigid boundaries at z=0 and z = h(x, y). The verticallength scale is h while the horizontal length scale is L. Here, h � L so theReynolds number (by convention) is R e = ρVh

/

μ.Now,

∂

∂z ∂

∂x,∂

∂y

(

because1

h 1

L

)

Therefore,

∇2�u ≈ ∂2u

∂z2

Furthermore, the incompressibility condition ∇ · �u = 0, i.e.,

∂u

∂x+ ∂v

∂y+ ∂w

∂z= 0 implies w ∼ h

LV

Therefore,

ρ�u · ∇�u ∼ ρ

(V2

L,

V2

L,

V2h

L2

)

= ρV2

L

(

1, 1,h

L

)

μ∇2�u ∼ μ∂2�u∂z2

= μ

(V

h2,

V

h2,

Vh

h2L

)

= μV

h2

(

1, 1,h

L

)


Then,

|ρ�u · ∇�u|∣∣μ∇2�u∣∣ =

ρV2/

LμV/

h2= ρVh2

μL= ρVh

μ

(h

L

)

= R e

(h

L

)

Thus, the nonlinear terms can be neglected if R e(

h/

L) � 1 even if Re is

not � 1. We could say that the linearization can be done provided the effectiveReynolds number R eeff = R e

(

h/

L)� 1.

3 Electrokinetics

We had mentioned the importance of surface effects as it particularly pertainsto micro-scales in Section 1. Of all the flow manipulation techniques whichexploit surface effects, electrokinetic effects are, arguably, the most popular ones inmicrofluidics. Fundamental to the use of electrokinetic effects is an understandingof the electrical double layer (EDL). Generally, most solid surfaces tend to acquire anet surface charge (positive or negative) when brought into contact with an aqueous(polar) medium. There are various mechanisms behind this surface charging phe-nomenon such as ionization of covalently bound surface groups or ion adsorption.We will not delve deeper into these mechanisms; instead we direct the interestedreader to comprehensive expositions of the topic in [4, 13].

Aqueous solutions invariably have dissolved ions (e.g., from dissolved salts ordissociated water groups) present in them (even distilled water is not perfectlydevoid of ions). The ions which are charged oppositely to the charged surface arecalled counterions, and the ions which have the same charge polarity as the sur-face are called co-ions. The charged surface, naturally, attracts the counterions andrepels the co-ions. If there were no thermal motion of the ions, the charged surfacewould be perfectly shielded by a layer of counterions stacked against the surface.However, ions have non-zero absolute temperature and the concomitant randomthermal motion of the ions precludes such a physical picture. What happens, then, isthat a balance is established between the electrostatic forces and the thermal inter-actions so that, at equilibrium, a certain charge distribution prevails adjacent to thesurface (of course, with the predominant presence of the counterions in the vicin-ity of the surface). This charge distribution with the predominant distribution ofcounterions is called the electrical double layer (EDL).

There are various theories which attempt to give models for the physical pic-ture of the EDL.1 We present here the Gouy–Chapman model with the Sternmodification (GCS model).

1For a historical development of the attempts to theoretically model the EDL, the interested readermay refer to [16].

arora115

Highlight

arora115

Highlight

arora115

Highlight

arora115

Highlight

arora115

Highlight


A schematic depicting the charge and the potential distribution within an EDLis shown in Fig. 1.1. A layer of immobile counterions is present just next to thecharged surface. This layer is known as the compact layer or the Stern layer or theHelmholtz layer. The thickness of this layer is about a few Angstroms and, hence,the potential distribution within it may be assumed to be linear. From this Sternlayer to the electrically neutral bulk liquid, the ions are mobile. This layer of mobileions beyond the Stern layer is called the Gouy–Chapman layer or the diffuse layerof the EDL. Besides this, there is a plane called the shear plane or surface which isconsidered to be the plane at which the mobile portion of the EDL can flow past thecharged surface. The potential at the shear plane is called the zeta potential (ζ ). Thecharacteristic thickness of the EDL is known as the Debye length (λ) which is the

Fig. 1.1 Distribution of counterions and co-ions in an EDL (top) and potential profile screeningthe surface charge (bottom)

arora115

Highlight

arora115

Highlight


length from the shear plane over which the EDL potential reduces to 1/e of ζ (wheree is the Euler number).

The four primary electrokinetic effects are as follows:

1. Electroosmosis: It refers to the relative movement of liquid over a stationarycharged surface, with an external electric field acting as the actuator.

2. Streaming potential: It refers to the electric potential that is induced when a liq-uid, containing ions, is driven by a pressure gradient to flow along a stationarycharged surface.

3. Electrophoresis: It refers to the movement of a charged surface (e.g., a chargedparticle) relative to a stationary liquid due to the application of an externalelectric field. The phenomena of electrophoresis and electroosmosis are closelyrelated. It is also important to note that, strictly following the definition, thesephenomena are not dependent on the establishment of the EDL. They just dependon the surface being charged irrespective of the mechanism bringing about thecharge.

4. Sedimentation potential: It refers to the potential that is induced when a chargedparticle moves relative to a stationary liquid (for example, under the effect ofgravity).

Besides these, closely related electrokinetic phenomena like dielectrophoresisand diffusioosmosis are also important. In what follows, we will concentrate onelectroosmosis, streaming potential, electrophoresis, and dielectrophoresis becausethese have received the greatest attention (due to the surfeit of practical applica-tions) in the microfluidics community. But, before we can delve into these topicsindividually, we need to develop a mathematical description of the EDL.

We begin the mathematical description of the EDL by considering a single plateor surface in an infinite liquid phase. For the system to be in equilibrium, the elec-trochemical potential of the ions needs to be constant everywhere to ensure that thesystem is in equilibrium. Thus, for a one-dimensional system of a solvent in contactwith a planar surface with the y-coordinate representing the direction normal to thesurface, we have

dμi

dy= 0 (1.26)

where μi is the electrochemical potential of ions of type i. The electrochemicalpotential is defined as

μi = μi + zieψ (1.27)

where μi and zi are, respectively, the chemical potential and valence of ions of typei and e is the protonic charge. Now, from thermodynamics, the chemical potentialcan be expressed as

μi = μ0i + kBT ln ni (1.28)


where μ0i is a constant for ions of type i, kB is the Boltzmann constant, T is the

absolute temperature of the solution, and ni is the ionic number concentration oftype i. Differentiating (1.28) with respect to y

dμi

dy= kBT

ni

dni

dy(1.29)

Therefore, from (1.27), we have

dμi

dy= dμi

dy+ zie

dψ

dy

or,

0 = kBT

ni

dni

dy+ zie

dψ

dy(using (1.29))

or,

dni

ni= − zie

kBTdψ (1.30)

Now, (1.30) can be solved in conjunction with appropriate boundary conditions.A very common scenario is one where the potential far away from the surface is0 and the ionic number concentration corresponds to a bulk value indicating theabsence of any surface effect. In mathematical terms, this physical scenario trans-lates to the following: at y → ∞, ni = n0

i and ψ = 0. The solution, with such aboundary condition, is

ni = n0i exp

(

− zieψ

kBT

)

(1.31)

This is the Boltzmann distribution of ions near a charged surface. The assump-tions underlying this derivation are as follows:

• The system is in equilibrium with no macroscopic advection/diffusion of ions• The solid surface is microscopically homogeneous• The far-stream boundary condition is applicable, meaning that the charged sur-

face is in contact with a large enough liquid medium such that at a distance far inbulk no surface effect is experienced

Surprisingly, however, the Boltzmann distribution is successfully employed inscenarios where fluid flow is taking place, i.e., a system which is certainly notin thermodynamic equilibrium. Although this blatant violation of an underlyingassumption might seem like a paradox, it can be shown that for low velocities withR e � 10, the Boltzmann distribution is, indeed, a good approximation. Let ussee how.


We use the ionic species conservation equation (for the steady case) whichdescribes the flux of the ith species as

∇ ·�ji = 0 (1.32)

Now, �ji can be described by

�ji = ni�u − Di∇ni −(

zieDini

kBT

)

∇ψ (1.33)

Using (1.33) in (1.32), we have the Nernst–Planck equation

∇ · (ni�u) = ∇ · (Di∇ni)+ ∇ ·[(

zieDini

kBT

)

∇ψ]

(1.34)

For low velocities, the advection term can be neglected, thus

Di∇2ni + Di∇ ·[(

zieni

kBT

)

∇ψ]

= 0 (for constant Di) (1.35)

In the case of a one-dimensional situation (along y)

d2ni

dy2+ d

dy

(zieni

kBT

dψ

dy

)

= 0 (1.36)

Integrating (1.36) and imposing the condition dψ/

dy = 0, dni/

dy = 0 asy → ∞ gives

dni

dy+(

zieni

kBT

)dψ

dy= 0 (1.37)

This is, basically, the same as (1.30) derived previously. This shows that, indeed,for low velocities the Boltzmann distribution is valid even though the fluid flowviolates the equilibrium condition.

Now, in order to get a full picture of the ionic distribution we need to know thepotential distribution. With this aim we start off from Gauss’ law:

∫

S

�E · nds = Qenclosed

ε0

or, using the Gauss’ divergence theorem to transform the surface integral to avolume integral:

∫

V

(

∇ · �E − ρe

ε0

)

dV = 0

where Qenclosed is the total charge enclosed within the volume V bound by thesurface S and ρe is the charge density.


This must be true for any arbitrary volume, so

∇ · �E = ρe

ε0(1.38)

It is important to note that since the fluid medium within which the electric field iscalculated is a dielectric one, we have to account for the induced (or, bound) chargeappearing as a result of polarization. This means, ρe = ρe,free +ρe,bound where ρe,freeis volumetric charge density of the free ions in the dielectric medium while ρe,boundis the volumetric density of the bound charges. It is known that ρe,bound = −∇ · �Pwhere �P is the polarization vector. Assuming that the dielectric medium is linearand isotropic �P = ε0χe �E where χe is the electrical susceptibility. Therefore,

ε0∇ · �E = ρe,free − ∇ · (ε0χe �E)

or,

∇ · {�E (ε0 + ε0χe)} = ρe,free

or,

∇ · (ε�E) = ρe,free (1.39)

where ε = ε0 (1 + χe) is the permittivity of the dielectric medium. If ε is assumedto be invariant spatially then from (1.39) we have

∇ · �E = ρe,free

ε(1.40)

Since the electric field �E may be expressed in terms of a potential field ψ as�E = −∇ψ , (1.40) may be expressed as

∇2ψ = −ρe,free

ε= −e

∑zini

ε(1.41)

where the summation is taken over all species of ions. For a symmetric electrolytezi = z = z+ = −z− = constant and (1.41) reduces to

∇2ψ = −e (n+z+ + n−z−)ε

= −ez (n+ − n−)ε

(1.42)

It is important to note that the special case of a symmetric electrolyte is a usefulconsideration because it is possible to treat most electrolytes as though they weresymmetric with valency z equal to the counterion valency.2

2For a full-fledged justification of this assertion refer to Section 2.3.3.3 of [9].


3.1 Electroosmosis

The presence of charges (usually in the form of ions) in the liquid can be use-fully exploited to actuate bulk motion of the liquid by the application of an electricfield. Basically, the electric field acts upon these charges. As they move due to theelectrical force they drag the liquid along with them. This phenomenon is calledelectroosmosis; the fluid flow is referred to as electroosmotic flow (EOF).

In order to model this fluid actuation we need to incorporate an extra body forceterm in the momentum equation for a fluid, i.e., the Navier–Stokes equation. Closeto the surface of a substrate, it is the double layer formation that results in the ionicdistribution on which the electric field acts. Let us see, in detail, how the body forcecomes about.

The Maxwell stress, neglecting magnetic effects, is given by [8]

[T] = ε

{

�E ⊗ �E − 1

2�E · �E [I]

}

(1.43)

The body force on the fluid due to the Maxwell stress is

�FE = ∇ · [T]

= ε∇ · {�E ⊗ �E}− ε1

2∇ · (�E · �E [I]

)

(assuming, as before, that ε is

spatially invariant)

= ε�E (∇ · �E)+ ε(�E · ∇) �E − ε 1

2∇ (�E · �E)

= ε�E (∇ · �E)+ ε(�E · ∇) �E − ε 1

2 × 2�E · ∇ �E (please see footnote3)

= ε�E (∇ · �E)

Here, �E is the total electrical field and may be expressed in terms of a potentialas �E = −∇ϕ. Using this expression of �E, we obtain

�FE = ε∇ϕ∇2ϕ (1.44)

In addition to this contribution from the Maxwell stress, there is another contri-bution from the osmotic pressure to the body force term. The osmotic pressure isdefined as the pressure required to be exerted on a solution to prevent the perco-lation of solvent across a semi-permeable membrane from another solution havinglower concentration. In the most general case, the osmotic pressure arises when-ever there is a gradient of concentration such as in the present case of non-uniformion distribution in the EDL. In mathematical terms, osmotic pressure is defined to be

3The equality ∇ (�E · �E) = 2(�E · ∇) �E is established in the context of dielectrophoresis (see

(1.124)–(1.126)).

arora115

Highlight

arora115

Highlight


∏ = nkBT . The body force on the fluid due to this osmotic pressure is �FO = −∇∏,and assuming isothermal conditions we obtain

�FO = −kBT∇ (n+ + n−)

= −kBT

[

− ez

kBTn+ + ez

kBTn−]

∇ψ

= ez[

n+ − n−]∇ψ

We use (1.41) to express ez (n+ − n−) as −ε (d2ψ/

dy2)

. Thus,

�FO = −ε∇2ψ∇ψ (1.45)

We note that the potential field ϕ is made up of two components: ϕ0 due to theexternally applied electric field and ψ due to the charged substrate, i.e., ϕ = ϕ0 +ψ .Moreover, the potential ϕ0 satisfies the Laplace equation ∇2ϕ0 = 0, so that ∇2ϕ =∇2ϕ0 + ∇2ψ = ∇2ψ . Therefore, the total body force on the fluid is

�F = �FE + �FO

= ε∇ϕ∇2ϕ − ε∇2ψ∇ψ(

∇2ϕ = 0 + ∇2ψ)

= ε∇ϕ0∇2ψ

= −ρe,free∇ϕ0

Finally,

�F = ρe,free �Eext (1.46)

Moreover, if the permittivity depends on the fluid density ρ, then an extra termgets added to the body force,4 thus

�F = ρe,free �E + ∇(

1

2�E · �Eρ ∂ε

∂ρ

)

(1.47)

Let us incorporate this body force in the equation of motion of the fluid flowingin a microchannel. The coordinate normal to the wall is y while the coordinate alongthe flow direction is x. Again for flow in a microchannel with low values of Re weneglect the inertial terms, thus

0 = μ∇2�u + ρe,free �Eext (1.48)

It can be noted here that in the absence of any imposed pressure gradient and/or agradient of zeta potential, the pressure gradient terms get dropped in the equation of

4The derivation of this extra term is beyond the scope of this book. However, it may be read in[12].

arora115

Highlight


motion depicted by (1.48). The case with non-zero pressure gradient can be analyzedby using the Stokes equation without any electrokinetic body force term. The flowoccurring under the combined effect of an imposed pressure gradient and an externalelectric field may be analyzed by considering the resultant effect to be dictated bya linear superposition of the following: (i) Stokes equation with a pressure gradientterm but without an electrokinetic body force term and (ii) Stokes equation withan electrokinetic body force term but without a pressure gradient term. In case theosmotic pressure gradient term is not considered as a separate body force term in themomentum equation, its contribution may be clubbed with the traditional pressuregradient term so that one may write −∇P = −∇(p+�) in which case the electricalbody force must be written as ρe, free �E, where �E is the total electric field due, andnot ρe, free �Eext. To delineate the flow characteristics due to pure electroosmotic flow(governed by (ii) mentioned as above), we solve (1.48).

For the special case of flow in a slit-type microchannel, the flow can be suitablymodeled as one-dimensional. Thus,

0 = μd2u

dy2+ ρe,freeEx

where Ex is the externally applied electrical field parallel to the flow direction. Using(1.41) for ρe, we get from (1.48)

0 = μd2u

dy2− ε

d2ψ

dy2Ex (1.49)

Now, using the boundary conditions du/

dy = 0 and dψ/

dy = 0 at y=H andu=0 and ψ = ζ at y=0, we obtain the velocity profile as

u = − 1

μεζEx

(

1 − ψ

ζ

)

(1.50)

The combination −εζEx/

μ has the dimension of velocity and is referred to asthe Helmholtz–Smoluchowski velocity, uHS. However, the velocity cannot be deter-mined just yet because the distribution of ψ is unknown. To find ψ , we use thePoisson equation (1.41) together with the Boltzmann distribution (1.31) to obtainthe celebrated Poisson–Boltzmann distribution for a z:z symmetric electrolyte:

∇2ψ = −n0ez

(

exp

(

−ezψ

kBT

)

− exp

(ezψ

kBT

))

(1.51)

Considering y as the coordinate normal to the substrate over which the EDL isestablished, (1.51) reduces to

d2ψ

dy2= 2n0ez sinh

(ezψ

kBT

)

(1.52)

arora115

Highlight

arora115

Highlight

arora115

Highlight

arora115

Highlight

arora115

Highlight


Equation (1.52) may be solved subject to various boundary conditions. A few ofthese are exemplified in the following cases, considering EDL development adjacentto the solid substrate of a parallel plate without having any EDL overlap:

Case I: The zeta potential ζ at both walls (y=0 and y=2H) is specified. ζ is smallsuch that

∣∣ezζ/

kBT∣∣� 1. In this condition,

sinh

(ezζ

kBT

)

≈ ezζ

kBT

This artifice is called the Debye–Hückel linearization. In practice, this linearizationis invoked up to |ζ | = 25 mV (for T=300 K). Accordingly, (1.52) gets simplified to

d2ψ

dy2= 2n0e2z2

εkBTψ (1.53)

Here,(

2n0e2z2/

εkBT)−1/2 = λ, called the Debye length, is a characteristic thick-

ness of the EDL. Equation (1.53) together with the boundary conditions ψ = ζ aty=0 and y=2H is used to obtain

ψ = ζcosh[

y−Hλ

]

cosh[Hλ

] (1.54)

Case II: Instead of the zeta potential, as in Case I, the surface charge (per unitwall area), i.e., surface charge density σ , is specified.

In the case of the Debye length being much smaller than half the channel height,the EDL distribution may be considered to be independent at each wall. In sucha case, to maintain the condition of electroneutrality, the surface charge on a wallmust be equal and opposite to the total (unbalanced) charge in the EDL near thatwall. This means

σ |y=0 = −∫ H

0ρedy

=∫ H

0ε

d2ψ

dy2dy (using the Poisson equation (1.41))

(from symmetry condition at the centre-line)

Finally,

σ = −εdψ

dy

∣∣∣∣y=0

(1.55)


Similarly,

σ |y=2H = εdψ

dy

∣∣∣∣y=2H

(1.56)

We again invoke the Debye–Hückel linearization artifice in order to be able touse the simplified (1.53). Using this equation, together with the conditions in (1.55)and (1.56), we obtain

ψ =(σλ

ε

) cosh[

y−Hλ

]

sinh[Hλ

] (1.57)

The zeta potential can be calculated to be ζ = ψ (y = 0, 2H) = σλ/

ε. So therestriction within which the Debye–Hückel linearization is valid is

∣∣∣∣

ζez

kBT

∣∣∣∣=∣∣∣∣

σλez

εkBT

∣∣∣∣� 1

Case III: Again, as in Case I, the zeta potential is specified but we no moreconsider it to be small so that the Debye–Hückel linearization cannot be used. Insuch a case, the governing equation to be used is the nonlinearized Poisson equation(one-dimensional):

d2ψ

dy2= 2n0ez

εsinh

(ezψ

kBT

)

(1.58)

Equation (1.58) is subject to the boundary conditions ψ = ζ at y = 0 anddψ/

dy = 0 at y = H. In order to solve (1.58) analytically, we first non-dimensionalize it using the scheme ψ = ezψ

/

4kBT and y = y/

H. Then,

1

H2

d2ψ

dy2= 1

4λ2sinh(

4ψ)

or,

d2ψ

dy2= K2

4sinh(

4ψ)

(1.59)

where K = H/

λ is a ratio denoting the relative half-height of the channelto the EDL characteristic thickness. Now, multiplying both sides by 2

(

dψ/

dy)

,we get

2dψ

dy

d2ψ

dy2= K2

42

dψ

dysinh(

4ψ)


or,

d

dy

((dψ

dy

)2)

= K2

8

d

dy

(

cosh(

4ψ))

(1.60)

On integrating (1.60) once, we get

(dψ

dy

)2

= K2

8cosh(

4ψ)+ C1

For most microfluidic applications, the electrical potential is zero at some dis-tance slightly beyond the EDL region. This is not true for the case when the EDLoriginating at the two walls overlaps in the mid-channel region. Therefore, for thecase when there is no overlap, the boundary conditions specified above can beextended to ψ = 0 at y = 1.

Using this extra condition gives C1 = −K2/

8. Thus, we have

(dψ

dy

)2

= K2

8

[

cosh(

4ψ)− 1

]

(1.61)

Now, using the identity cosh (2x) = 1 + 2sinh2 (x), we obtain, from (1.61)

(dψ

dy

)2

= K2

4sinh2 (2ψ

)

or,

(dψ

dy

)

= −√

K2

4sinh2 (2ψ

)

or,

dψ

dy= −K

2sinh(

2ψ)

(1.62)

The negative sign indicates that if the zeta potential is negative, the potentialwill increase from the zeta potential at the shear plane to zero in the bulk solution.Proceeding to solve the differential equation (1.62), we get

dψ

sinh(

2ψ) = −K

2dy

or,


cosh2 (ψ)− sinh2 (ψ

)

2 sinh(

ψ)

cosh(

ψ) dψ = −K

2dy

(using the identity cosh2 (x)− sinh2 (x) = 1)

or,

1 − tanh2 (ψ)

tanh(

ψ) dψ = −Kdy

or,

sech2 (ψ)

tanh(

ψ) dψ = −Kdy

(using the identity sech2 (x)+ tanh2 (x) = 1)

or,

d(

tanh(

ψ))

tanh(

ψ) = −Kdy

or,

tanh(

ψ) = C2 exp (−Ky)

Using the boundary condition ψ = ζ at y = 0, we get

tanh(

ψ) = tanh

(

ζ)

exp (−Ky)

Finally,

ψ = tanh−1 [tanh(

ζ)

exp (−Ky)]

(1.63)

In the case when the EDLs are thick, the two electrostatic potentials generatedby the two plates may be taken as additive:

ψ (y) = tanh−1 [tanh(

ζ)

exp (−Ky)]+ tanh−1 [tanh

(

ζ)

exp (−2 K + Ky)]

(1.64)

The cases discussed here, though not exhaustive, are certainly very significantones from the point of modeling real situations. Another realm not touched uponhere is that under the case of EDL overlap. When the EDL charge distribution issuch that the ratio K = H

/

λ < 4, the center-line condition ψ (y = H) = 0 is nolonger valid. In that case, no analytical closed form solution of (1.58) can be found.

It is important to remember that in the description of the EDL through Poisson–Boltzmann equation, the ions are assumed to be point charges and non-interacting.


This means that the real finite size of ions is neglected or, in other words,steric effects are not considered. Indeed, the Boltzmann distribution of (1.31) pre-dicts an unbounded and physically impossible growth in counterion concentration.This aphysical picture is particularly manifested at high values of zeta potential.To overcome these limitations of the Poisson–Boltzmann equation, modificationsincorporating the finite size of the ions have been invoked in the literature, leadingto modified Poisson–Boltzmann equations [10]. The concerned derivations are notreproduced in this introductory reading.

Having an idea about the potential profile in the channel, we can proceed tofind the velocity profile using (1.50). We consider the simple case correspondingto Case I discussed previously. The non-dimensional velocity profile in that caseis shown in Fig. 1.2. Note that the velocity u has been non-dimensionalized bythe Helmholtz–Smoluchowski velocity −εζE

/

μ, while the coordinate y along thechannel height has been non-dimensionalized by the half-channel height H. Thenon-dimensional parameter K = H

/

λ represents the relative height of the channel(half-height) compared to the characteristic thickness of the EDL.

It can be seen from the figure that the velocity profile is a plug-like one (exceptfor a thin region near the walls) with magnitude equal to uHS. Thus, in the caseof thin EDL, the electroosmotic flow can be thought of as having a constant pro-file with a slip length equal to uHS. In sharp contrast to this, the velocity profile intraditional pressure gradient-driven flow is a parabolic one. The immediate advan-tage of this plug-like EOF profile over the parabolic case is that shear-induced axialdispersion is significantly reduced. Furthermore, the EOF velocity is not dependenton the geometrical dimensions unlike the pressure gradient-driven velocity whichdecreases with the second power of channel size. This, along with the eliminationof moving components as well as easy integrability with the electrical/electronic

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

u

K = 10

K = 50

Fig. 1.2 Non-dimensionalvelocity in electroosmoticflow

arora115

Highlight

arora115

Highlight

arora115

Highlight

arora115

Highlight


circuitry in lab-on-a-chip devices, is the primary motivation of using EOF in minia-turized systems. However, EOF is not without its share of disadvantages: It has astrong dependence on the surface chemistry and the physico-chemical properties ofthe solution. The plug-like velocity profile of ideal EOF may be disturbed whenstray pressure gradients are induced due to non-uniformities on the surface lead-ing, in turn, to non-negligible axial dispersions. Not only that, EOF can give riseto significant Joule heating problems that may destabilize thermally labile biologi-cal samples. Be that as it may, EOF continues to be the most popular alternative totraditional pressure gradient-driven flow in the microfluidic realm.

3.2 Induced Charge Electroosmosis (ICEO)

It was discussed at the beginning of Section 3.1 that the genesis of EOF lies in acloud of charge getting formed and the application of an electric field in the properdirection to induce motion of the charges (in this cloud) and, concomitantly, thesuspending fluid. How the charge cloud comes about does not dictate the existenceof EOF. With this fresh insight, let us look at a “different” kind of EOF.

When a conductor is placed in an electric field �E as shown in Fig. 1.3, the con-duction electrons reorient themselves in such a way that the net electric field withinthe conductor is zero, and all the induced charges appear on the surface of the con-ductor. The net effect is that a conductor placed in an electric field develops a surfacecharge (satisfying, of course, the initial electroneutrality condition within the wholeconductor body). Now, if an aqueous solution of ions is present around this con-ductor, these surface charges will attract the oppositely charged ions, leading to thedevelopment of a cloud of ions that screens the surface charge on the conductor. It isextremely important to understand that if there were no ions present in the fluid (ide-ally), the electric lines of force would have met the surface of the conductor at rightangles. But the presence of the oppositely charged screening cloud of ions “deflects”these lines of force such that very near to the surface these lines are almost tangen-tial. The gross effect is that the distribution of the electric lines of force around theconductor together with its screening cloud of ions appears to be identical to that inthe case of an ideal non-conductor.

If we now focus on the region in the close vicinity of the conductor, the situationis not much unlike what was present in the traditional EOF. We have a screening

Fig. 1.3 Schematic ofinduced chargeelectroosmosis

arora115

Highlight

arora115

Highlight


cloud of ions (predominantly charged opposite to the surface charge) together withan electric field that is tangential to the surface. In this situation, it is intuitive toexpect that the ions in the screening cloud will be actuated to move parallel to thesurface of the conductor, inducing, in turn, the suspending liquid to move with themdue to drag force. And that is exactly what happens, indeed.

The foregoing qualitative understanding of the “different” EOF has been devel-oped by drawing analogies with the “traditional” EOF. The fluid actuation mech-anism is, indeed, the same physically. Be that as it may, it would be naïve not toappreciate the differences between the two situations. In the case of the “traditional”EOF, the surface charging mechanism and the distribution of the screening cloud ofcharges (the EDL) was independent of the application of the external electric field.In fact, the same EDL distribution is established (given the same surface and solu-tion conditions) irrespective of the fact that the external electric field is applied ornot. However, in the present situation, the surface charging, and thus the develop-ment of the screening cloud, is totally dependent on the applied electric field. Insimple terms, it might be said that the electric field actuates what it itself induces.And that is a clue to why this kind of flow is called induced charge electroosmo-sis (ICEO). Even from this qualitative description, it may be appreciated that sincethe surface charge and the screening cloud distribution is induced by the externalelectric field, the quantitative description would result in this distribution being afunction of the external field. Again, since the flow actuation is brought about bythe influence of the very same electric field, the resultant flow is to be expected tohave a nonlinear dependence on the external field. This is indeed so, and thus, ICEOcomes under the purview of nonlinear electrokinetic phenomena.

3.3 Streaming Potential

It is not necessary to apply an external electric field in order for electrokinetic effectsto play a significant role in micro-scale flows. Even in purely pressure-driven flowsin the presence of a charge distribution in the form of a double layer, electrokineticeffects are manifested. As the ions are advected along with the flow, they build upin the reservoirs and create a back potential. The current that is generated as a resultof the advection of the ions is called streaming current. The back potential that isdeveloped as a result of this streaming current is called streaming potential. Thisis depicted schematically in Fig. 1.4. Again, a current is generated in response tothis streaming potential and it is called the conduction current. The direction ofthe conduction current is opposite to that of the streaming current. At equilibrium,since no external electric field is applied, the net ionic current in the system must bezero. Indeed, it is this condition of zero ionic current that is utilized to calculate theelectric field associated with the streaming potential. Let us see how.5

We start with the general Boltzmann distribution from (1.31) with the additionalconsideration of a symmetric electrolyte:

5We follow the mathematical treatment presented in [6].

arora115

Highlight

arora115

Highlight

arora115

Highlight


Fig. 1.4 Generation of streaming potential

n± = n0 exp

(

∓ezψ

kBT

)

This, in turn, can be employed to estimate the total ionic current in a micro/nano-channel of height 2H (with y varying from 0 to 2H) as

Iionic = e∫ 2H

0(u+z+n+ + u−z−n−) dy (1.65)

For a z:z symmetric electrolyte such that z+ = −z− = z,

Iionic = ez∫ 2H

0(u+n+ − u−n−) dy (1.66)

Here, u+ (u−) refer to the axial velocities of the cations (anions), expressed as

u± = u ± zeE

f±(1.67)

Under the assumption of identical values of cationic and/or anionic friction coef-ficient of charge f = f+ = f−, the expression for Iionic from (1.66) simplifies to

Iionic = ez∫ 2H

0(n+ − n−) udy + e2z2E

f

∫ 2H

0(n+ + n−) dy (1.68)

As discussed previously, for pure pressure-driven transport, Iionic becomesidentically zero; the corresponding value of E is called the streaming field Es.

The velocity field is the result not only of the pressure gradient drive but also ofthe streaming potential. If we disregard the origin of this back potential, the situationis just like a flow taking place under the combined effect of a pressure gradient andan electric field (albeit, an induced one). In such a situation, we can take a cue from(1.49) and write the governing equation for the velocity as


0 = −dp

dx+ μ

∂2u

∂y2+ ρeEs (1.69)

We make use of the Poisson equation, to obtain from (1.69)

μ∂2u

∂y2= dp

dx+ ε

∂2ψ

∂y2Es (1.70)

The boundary conditions to be applied are, as before, u=0 and ψ = ζ at y=0and u=0 and ψ = ζ at y=2H. Using these, we get

u = − 1

2μ

dp

dx

(

2Hy − y2)

︸︷︷︸

up

−εζEs

μ

(

1 − ψ

ζ

)

︸︷︷︸

ue

= up + ue

(1.71)

When we substitute this expression of u in (1.68) and set Iionic to zero, we cansolve for the expression for the induced streaming potential field Es as

Es =n0ez∫ 2H

0 up sinh(

ezψkBT

)

dy

n0e2z2

f

∫ 2H0 cosh

(ezψkBT

)

dy + n0ezεζμ

∫ 2H0

(

1 − ψζ

)

sinh(

ezψkBT

)

dy(1.72)

Using the expression of up from (1.71) in (1.72), we get

Es =−f

2ezμdpdx

∫ 2H0

(

2Hy − y2)

sinh(

ezψkBT

)

dy∫ 2H

0 cosh(

ezψkBT

)

dy + f εζμez

∫ 2H0

(

1 − ψζ

)

sinh(

ezψkBT

)

dy(1.73)

We use the non-dimensional scheme y = y/

H, ψ = ezψ/

4kBT to obtain thefollowing form of the streaming potential field:

Es =−f H2

2ezμdpdx

∫ 20

(

2y − y2)

sinh(

4ψ)

dy∫ 2

0 cosh(

4ψ)

dy + 4f εkBTμe2z2 ζ

∫ 20

(

1 − ψ

ζ

)

sinh(

4ψ)

dy

or,

Es = E0I1

I2 + RI3(1.74)


where

I1 = ∫ 20

(

2y − y2)

sinh(

4ψ)

dy, I2 = ∫ 20 cosh

(

4ψ)

dy

I3 =∫ 2

0ζ

(

1 − ψ

ζ

)

sinh(

4ψ)

dy, R = 4f εkBT

μe2z2, and E0 = − f H2

2ezμ

dp

dx

The ionic friction coefficient of charge f may be expressed as f = e2NA/

F2�

where NA is the Avogadro number, F is the Faraday constant, and � is the ionicmobility. f may also be expressed as f = kBT

/

D where D is the same diffusivityused in the Nernst–Planck equation (1.34).

It is important to note that the streaming potential is responsible for inhibitingthe flow. The decreased volumetric flow rate may be attributed, on a gross scale,to an increased viscosity of the fluid, due to the electrokinetic effects. This is oftenreferred to as the electroviscous effect. Since the electric field due to streamingpotential, Es, is a function of the applied pressure gradient, the velocity, explicitlyconsidering the streaming potential effects, can be expressed as

u1 = − 1

2μ

dp

dxfe (y)

If the streaming potential effects are incorporated within an effective increasedviscosity (the electroviscous effect), then

u2 = − 1

2μev

dp

dx

(

2Hy − y2)

The volumetric rate should be equal in both these for them to be equivalent. Thus,

∫ 2H

0u1dy =

∫ 2H

0u2dy

μev

μ= 2H3

/

3∫ 2H

0 fedy(1.75)

Just as a back electric potential develops in a purely pressure-driven flow, aback pressure gradient may develop when a purely electroosmotic flow takes placebetween two reservoirs due to the changing liquid levels in the two reservoirs. Atequilibrium, the flow due to the back pressure is equal to the flow due to the applica-tion of the external electric field and there is no net flow. This is known as the finitereservoir size effect.


3.4 Electrophoresis

It is not difficult to understand electrophoresis if one has grasped the idea of elec-troosmosis. Both these phenomena refer to the relative movement of a chargedsurface and a liquid. In electroosmosis, the charged surface is held stationary withthe liquid moving past it, while in electrophoresis it is the charged surface that movesrelative to the liquid. In both the cases, an external electric field is the actuator of themotion. The similarity between electroosmosis and electrophoresis becomes strik-ing if we fix our frame of reference on the moving electrophoretic particle. Fromsuch a viewpoint the relative motion manifests itself just like electroosmosis of theliquid past the particle. Indeed, this is what is done in the mathematical treatment ofelectrophoresis as will be clear in the analysis that follows.

The electrophoretic particle can be any charged body – a colloid, a macro-molecule, or even a microorganism. This charge may be intrinsic (like in a DNAmolecule) or it may be an induced one (like EDL formation). Consequently, in themost general case, the particle can be of any irregular shape with charge that can beintrinsic or induced. However, as a body representative of many real physical situ-ations and for the ease of mathematical analysis,6 we will restrict ourselves to theelectrophoresis of a spherical particle. We will also restrict ourselves to a situationwhere the charge on this spherical particle is brought about due to EDL formation.

The electrical potential around a spherical particle in a polar medium can befound from the Poisson’s equation (reduced form due to spherical symmetry):

1

r2

d

dr

(

r2 dψ

dr

)

= −ρe

ε(1.76)

or,

1

r2

d

dr

(

r2 dψ

dr

)

= 2n0ez

εsinh

(ezψ

kBT

)

(1.77)

Invoking the Debye–Hückel linearization for small potential values, we get

1

r2

d

dr

(

r2 dψ

dr

)

= 2n0e2z2

εkBTψ

1

r2

d

dr

(

r2 dψ

dr

)

= 1

λ2ψ

(1.78)

where 1/

λ2 = 2n0e2z2/

εkBT . In order to solve (1.78), we make the substitutionξ = rψ . Using this, we obtain from (1.78)

6We follow the structure of the development presented in [15].


1

r

d2ξ

dr2= 1

λ2

ξ

r

or,

d2ξ

dr2= ξ

λ2(1.79)

The general solution of (1.79) is of the form ξ = A exp(

r/

λ) + B exp

(−r/

λ)

.The boundary condition ψ → 0 as r → ∞ implies ξ is finite when r → ∞. Fromthis, we get A = 0. Again, the boundary condition ψ = ζ when r = R impliesξ = ζR when r=R. From this, we get B = Rζ exp

(

R/

λ)

. Therefore,

ψ = ζR

rexp

(

− r − R

λ

)

(1.80)

The total surface charge on the spherical particle is (by the condition of elec-troneutrality) negative of the total charge distribution in the double layer. Thiscondition is used to find an expression for the surface charge:

qsurface = −qEDL = −∫ ∞

R4πr2ρedr (1.81)

From (1.76),

ρe = −ε 1

r2

d

dr

(

r2 dψ

dr

)

So, from (1.81), we get

(1.82)From (1.80), we can see that

dψ

dr

∣∣∣∣R

= −ζ(

1

R+ 1

λ

)

So,

qsurface = 4πεR2ζ

(1

R+ 1

λ

)

(1.83)

Case I: λ R

In the case where the Debye length λ is large compared to R, the particle may betreated as a point charge. The electrical force may, then, be equated with the Stokesdrag on the particle to find its velocity (for steady motion). Thus,

qsurfaceEx = 6πμRU (1.84)


where U is the velocity of steady motion of the particle and Ex is the unperturbedelectric field.

Using (1.84) in (1.83) and simplifying, we obtain

U = 2

3

ζε(

1 + R/

λ)

Ex

μ(1.85)

Since we are analyzing the situation where R � λ, we have

U ≈ 2

3

ζεEx

μ(1.86)

This is known as the Hückel equation.

Case II: λ � R

When the Debye length is very small compared to the radius of the particle, inregions very close to the surface of the particle, the curvature of the particle canbe neglected so that the EDL may be viewed just as it was in the case of the planarsubstrate of an EOF. Indeed, drawing an analogy with EOF taking place in a parallelplate geometry where the plug-like flow profile (for very thin EDL) magnitude isgiven by the Helmholtz–Smoluchowski velocity, we may write the fluid velocityparallel to the particle surface as

U|| = −εζE||μ

(1.87)

where E|| represents the electric field just close to the particle surface and U|| is thetangential velocity of the fluid. Of course, it is important to note that U|| and E|| varyalong the surface of the particle. Moreover, this velocity of the fluid is written froma reference frame fixed on the particle, so that from this frame, the fluid appears toflow past the surface. In order to find the full solution of the velocity we will followa subtle logic. The rigorous mathematical treatment of this logical deduction canbe read in [14]. We first consider the electric field distribution as it stands with theparticle embedded in the field. The electric field may be represented by the potentialas �E = −∇φ. Since there are no free charges (ρe = 0), the application of Poisson’sequation ∇ · �E = −ρe/ε gives

∇2φ = 0 (1.88)

The boundary conditions which this standard Laplace equation is subjected toare the no penetration n · ∇φ = 0 (i.e., ∂φ

/

∂n = 0) and the “far-stream” φ∞(corresponding to the unperturbed electric field in the region far away from the par-ticle). In order to solve this equation, we start with the expanded form of the Laplaceequation (note the reduction to two variables from symmetry considerations):

∂

∂r

(

r2 ∂φ

∂r

)

+ 1

sin θ

∂

∂θ

(

sin θ∂φ

∂θ

)

= 0 (1.89)


This equation may be solved using separation of variables. Let us assume asolution of the form φ = G (r)H (θ). Then,

∂

∂r

(

r2 dG

drH

)

+ 1

sin θ

∂

∂θ

(

G sin θdH

dθ

)

= 0 (1.90)

or,

∂

∂r

(

r2 dG

drH

)

= − 1

sin θ

∂

∂θ

(

G sin θdH

dθ

)

= K (say) (1.91)

From (1.91) we get the following two equations:

1

sin θ

d

dθ

(

sin θdH

dθ

)

+ KH = 0 (1.92)

and

1

G

d

dr

(

r2 dG

dr

)

= K (1.93)

Let us assume that K can be denoted as n(n+1). Then, from (1.93), we get

(

r2G′)′= n (n + 1)G

⇒ r2G′′ + 2rG′ − n (n + 1)G = 0 (1.94)

This equation is in the Euler–Cauchy form. Thus, we can assume a solution ofthe form G = rα . Substituting this form in (1.94), we get

α (α − 1)+ 2α − n (n + 1) = 0 (1.95)

The solutions of (1.95) are α = n, −1−n. Therefore, the two solutions of (1.94)are G1 (r) = rn and G2 (r) = 1

/

rn+1.Now, let us turn our attention to the solution of (1.92). Let cos θ = w, then

sin2θ = 1 − w2 and

d

dθ= d

dw

dw

dθ= − sin θ

d

dw

Therefore,

1

sin θ

d

dθ

(

sin θdH

dθ

)

+ KH = 0

or,


− d

dw

[

sin θ

(

− sin θdH

dw

)]

+ KH = 0

or,

− d

dw

[(

1 − w2) dH

dw

]

+ n (n + 1)H = 0

or,

(

1 − w2) d2H

dw2− 2w

dH

dw+ n (n + 1)H = 0 (1.96)

This is the form of Legendre’s equation. In order that the solution (togetherwith its derivatives) of the Laplace equation (1.89) be continuous, it is necessaryto restrict n to integer values (following Kreyszig [11]).

For n=0, 1, . . . the Legendre polynomials H = Pn (w) = Pn (cos θ) are solutionsof Legendre’s equation. Thus, there are two series of solution of (1.96):

φn1 (r, θ) = AnrnPn (cos θ) and φn2 (r, θ) = Bn1

rn+1Pn (cos θ)

Thus, the general solution can be expressed as

φ (r, θ) =∞∑

n=0

AnrnPn (cos θ)+∞∑

n=0

Bn1

rn+1Pn (cos θ) (1.97)

The boundary conditions that must be satisfied are

limr→∞φ (r, θ) = −Exr cos θ and

∂φ (r, θ)

∂r

∣∣∣∣r=R

= 0

The first boundary condition gives us a clue that φ should be of the form

φ = −Exr cos θ +∞∑

n=0

Bn1

rn+1Pn (cos θ)

meaning A1 = −Ex and An=0 for n=0 and ∀n ≥ 2. And, from the second boundarycondition,

− Ex cos θ − B0P0 (cos θ)

R2− 2B1P1 (cos θ)

R3− · · · = 0 (1.98)

We note that P0 (x) = 1, P1 (x) = x, and so on. Using this in (1.98) and equatingthe like powers of cos θ , we get


B0 = 0, B1 = 1

2A1R3 = −1

2ExR3, and Bn = 0 ∀n ≥ 2

Therefore, from (1.97), we obtain

φ = −Exr cos θ − 1

2ExR3 cos θ

r2= −Ex cos θ

(

r + R3

2r2

)

(1.99)

From this solution, we can see that

E|| = 1

r

∂φ

∂θ

∣∣∣∣r=R

= −3

2Ex sin θ

together with E⊥ = 0 (true to the boundary condition utilized).The velocity profile should be such that it satisfies the Navier–Stokes equations

along with the slip boundary condition (1.87) and ensures that the flow does notexert any force or moment on the particle. It has been shown by Morrison [14]that irrotational flow satisfies these conditions. An irrotational flow velocity field isderivable from a potential �u = −∇�. Furthermore, the velocity should satisfy the nopenetration boundary condition n · �u = 0 or n ·∇� = 0. It may, now, be immediatelyobserved that the differential equations and the boundary conditions are identical forthe electric potential and the velocity potential. With the appreciation of this fact,we can take a clue from (1.87) (relating the slip velocity to the tangential electricalfield) and write the relation between the potential � corresponding to the velocityfield and the potential φ corresponding to the electrical field as

� = −εζμφ (1.100)

Writing this in terms of the velocity and electric field, we get

U = −εζEx

μ(1.101)

So, the velocity of the particle is, in a reference fixed in space,

Uparticle = εζEx

μ(1.102)

This is the Helmholtz–Smoluchowski equation.We have, until now, considered two special cases – first, when the Debye length

is much larger than the radius of the particle and, second, when the Debye lengthis much smaller than the radius. These are the extreme cases of a general situationwhere the Debye length and the radius of the particle are of comparable dimensions.We will not provide the mathematical treatment of this general case, instead directthe interested reader to [15].


3.5 Dielectrophoresis

The force acting on a charged particle placed in an electric field, resulting in elec-trophoresis, could be intuitively understood. However, a force may also act on anuncharged but polarizable particle if it is placed in a non-uniform electric field,and this force actuates the translational motion of the particle. This phenomenonis called dielectrophoresis. It is important to note that the nature of the dielec-trophoretic force is strongly dependent on the non-uniformity of the electric fieldand the dielectric properties of both the particle and the surrounding medium. Beforeproceeding to a full-fledged mathematical treatment of dielectrophoresis, let us firstunderstand qualitatively the physical picture. Under the action of an electric field apolarizable particle will polarize so that a dipole is induced in it. If the electric fieldwere uniform, no net force would have acted on the particle. But, since the appliedelectric field is spatially non-uniform, different forces act on the two ends of the“dipole” so that a net electric force does act on the particle.

Following arguments similar to the case of electrophoresis, we consider a spher-ical particle as shown in Fig. 1.5 and present a mathematical analysis of thedielectrophoretic force acting on it. Let the permittivity of the particle be ε2 andthat of the medium be ε1. The electric potential in the region interior to the particleis φi while that in the surrounding medium is φe. The radius of the spherical particleis R. The governing differential equation, just like in the case of electrophoresis, isthe Laplace equation ∇2φ = 0 (φ (r ≤ R) ≡ φi and φ (r ≥ R) ≡ φe). The boundaryconditions that must be satisfied are7

Fig. 1.5 Dielectrophoresis ofa spherical particle

7We start with the presentation structure found in the chapter on dielectrophoresis in [2]; however,for pedagogical reasons, we give, here, a more detailed mathematical derivation.


φi (0, θ) is finite (1.103)

φi (R, θ) = φe (R, θ) (1.104)

ε2∂φi (R, θ)

∂r= ε1

∂φe (R, θ)

∂r(1.105)

and

limr→∞φe (r, θ) = −E0r cos θ (1.106)

From inspection, it is clear that in order to satisfy the fourth boundary condition(1.106), the solution has to be of the form

φe = −E0r cos θ +∞∑

n=0

Bne1

rn+1Pn (cos θ)

meaning A1e = −E0 and Ane = 0 for n = 0 and ∀n ≥ 2. And, from the first boundarycondition (1.103) that φi should be finite at r = 0 gives us φi =∑∞

n AnirnPn (cos θ)meaning Bni = 0 ∀n = 0, 1, .... Now, using the second boundary condition (1.104),we get

−E0r cos θ + B0e1

RP0 (cos θ)+ B1e

1

R2P1 (cos θ)+ · · · = A0iP0 (cos θ)

+ A1iRP1 (cos θ)+ · · ·(1.107)

We know that P0 (x) = 1, P1 (x) = x, and so on. Utilizing this in (1.107), we get

− E0r cos θ + B0e1

R+ B1e

1

R2cos θ + · · · = A0i + A1eR cos θ + · · ·

Comparing the like powers of cos θ , we get

A0i = 0, A1i = −E0 + B1e

R3, B0e = 0, Bne = 0 ∀n ≥ 2, and Ani = 0 ∀n ≥ 2 (1.108)

Using the third boundary condition (1.105)

ε1∂φe

∂r

∣∣∣∣r=R

= ε2∂φi

∂r

∣∣∣∣r=R

or,

ε1

[

A1eP1 (cos θ)− 2B1e

R3P1 (cos θ)

]

= ε2A1iP1 (cos θ)


or,

ε1

ε2

[

A1e − 2B1e

R3

]

= A1i (1.109)

Comparing (1.109) and the second equality in (1.108), we get

−E0 + B1e

R3= ε1

ε2

[

−E0 − 2

R3B1e

]

or,

B1e = E0R3 ε2 − ε1

ε2 + 2ε1

Therefore, using (1.109), we get

A1i = −E0 + E0ε2 − ε1

ε2 + 2ε1or A1i = −E0

3ε1

ε2 + 2ε1(1.110)

Thus, the potential φ is given by

φ =

⎧

⎪⎪⎨

⎪⎪⎩

−E0r cos θ + E0R3

r2

(ε2 − ε1

ε2 + 2ε1

)

cos θ , r > R

−E03ε1

ε2 + 2ε1, r < R

(1.111)

Let us, now, find the potential at a distance r (from the origin of a coordinatesystem) due to a charge distribution of density ρe within the domain bounded by �as shown in Fig. 1.6.

The permittivity of the medium is ε1. The charge distribution domain is verysmall such that max

[

r′] � r, where r′ is the distance of an elemental volume from

Fig. 1.6 Potential at a pointdue to a charge distribution inspace


the origin. The potential due to this elemental volume dτ of the charge density isgiven by

dV = 1

4πε1

ρedτ

l(1.112)

where l is the distance of the point of interest from the elemental volume. Therefore,the potential due to the total charge distribution is

V = 1

4πε1

∫ρe

ldτ (1.113)

From Fig. 1.6, we can write

cos θ = r2 + r′2 − l2

2rr′ (using cosine identity from trigonometry)

So,

l2 = r2 + r′2 − 2rr′ cos θ

or,

l2 = r2

[

1 + r′2

r2− 2

r′

rcos θ

]

or,

l = r√

1+ ∈

where

∈= r′

r

(r′

r− 2 cos θ

)

� 1

or,

1

l= 1

r(1+ ∈)−1/2 = 1

r

(

1 − 1

2∈ +3

8∈2 − · · ·

)

(using binomial expansion)

Thus, from (1.113), after expanding and simplifying, we obtain8

8This is, basically, the multipole expansion method. A detailed description of this approach isfound in the classical textbook of Griffiths [8].


V = 1

4πε1

[1

r

∫

ρdτ + 1

r2

∫

r′ cos θρdτ + 1

r3

∫

r′2(

3

2cos2θ − 1

2

)

ρdτ + · · ·]

(1.114)

For a charge distribution, where the total charge is zero (as it is in this case), thefirst term called the monopole term vanishes. The contribution of the dipole term is

Vdip = 1

4πε1

1

r2

∫

r′ cos θρdτ (1.115)

Here, r′ cos θ = r · �r′. Therefore, from (1.115), we can write

Vdip = 1

4πε1

1

r2r ·∫

�r′ρdτ (1.116)

The integral �p = ∫ �r′ρdτ is called the dipole moment of the distribution. Thus,

Vdip = 1

4πε1

r · �pr2

= 1

4πε1

p cos θ

r2(1.117)

Interestingly, if we take a physical dipole as shown in Fig. 1.7 (as a special case ofthe general charge distribution with total charge zero), we find the same magnitudeof the potential as just found in (1.117). Let us see how. We first note that

V = 1

4πε1

∣∣∣∣

q

l+− q

l−

∣∣∣∣

Now, l2± = r2 + r2± − 2rr± cos θ and l± ≈ [1 − 2(

r±/

r)

cos θ]1/2. Therefore,

Fig. 1.7 Potential due to anactual dipole


V = 1

4πε1

∣∣∣∣∣

q

r

(

1 − 2r+r

cos θ)−1/2 − q

r

(

1 − 2r−r

cos θ)−1/2

∣∣∣∣∣

= 1

4πε1

q

r

|r+ − r−|r

cos θ

(1.118)

Now, s2 = r2+ + r2− − 2r+r− cosα ≈ r2+ + r2− − 2r+r− = (r+ − r−)2. So,s = |r+ − r−|. Therefore, from (1.118), we get

V = qs cos θ

4πε1r2= q

4πε1

�s · r

r2= �p · r

4πε1r2= p cos θ

4πε1r2(1.119)

Having shown that the dipole contribution of a charge distribution is equivalent tothat of a physical dipole, we follow Bruus [2] to argue that if a given potential con-tains a component of the form B

(

cos θ/

r2)

, then it implies that a dipole of strengthp = 4πε1B is located at the center of the coordinate system. The motivation behindsuch a subtle argument is the form of the potential as found in (1.117) (or (1.119)).

Now, if we look at (1.111), we see that there, too, is a termE0R3

(

(ε2 − ε1)/

(ε2 + 2ε1)) (

cos θ/

r2)

in the expression of the potential outsidethe spherical particle. Thus, by the artifice, just mentioned, we can say that theredoes exist a dipole of dipole moment

p = 4πε1E0R3(ε2 − ε1

ε2 + 2ε1

)

(1.120)

The combination K = (ε2 − ε1/

(ε2 + 2ε1))

is called the Clausius–Mossottifactor.

If the electric field is non-uniform, the force on a dipole is given by

F = F+ − F−

= q(�E�l+d�l − E�l)

= q

[

∂ �E∂x

dx + ∂ �E∂y

dy + ∂ �E∂z

dz

]

= q(

dxi + dyj + dzk)

· ∇ �E = (�p · ∇) �E

(1.121)

We had previously shown that the dipole moment associated with the chargedistribution in the dielectric sphere is identical to that of a physical dipole. We had,of course, limited ourselves to dipole terms of the general multipole expansion.Thus, as long as the sphere is very small, it may be replaced by a physical dipole.Another significance of the very small particle assumption is that the non-uniformfield does not change the dipole moment expression of (1.120) which was derivedfor and is, strictly speaking, valid for a uniform field. Now, we may proceed to write


the expression for the dielectric force on the small dielectric spherical particle, using(1.121):

�FDEP = (�p · ∇) �E (1.122)

with the understanding that �p is the dipole moment of the charge distribution:

�FDEP = 4πε1ε2 − ε1

ε2 + 2ε1R3 (�E0 · ∇) �E0 (1.123)

It is important to note that we have written �E = �E0 by neglecting the higher orderterms in the Taylor expansion containing the gradient terms.

Now, we simplify the(�E0 · ∇) �E0 term in (1.123) using the vector identity

(�A · ∇) �B = ∇

(�A · �B)

− (�B · ∇) �A − �A × (∇ × �B)− �B ×(

∇ × �A)

(1.124)

In this case, �A = �E0 and �B = �E0. Therefore,

(�E0 · ∇) �E0 = ∇ (�E0 · �E0)−(�E0 · ∇) �E0−�E0×(∇ × �E0

)−�E0×(∇ × �E0)

(1.125)

We know that ∇ × �E0 = 0 because E0 is the gradient of a scalar, and curl of thegradient of a scalar identically vanishes. Therefore, from (1.125), we obtain

2(�E0 · ∇) �E0 = ∇ (�E0 · �E0

)

or,

(�E0 · ∇) �E0 = 1

2∇ (�E0 · �E0

)

(1.126)

Using (1.126) in (1.123), we can write

�FDEP = 2πε1ε2 − ε1

ε2 + 2ε1R3∇ (�E0 · �E0

)

(1.127)

Up to this point, we have been considering ideal dielectrics (both particle andthe suspending medium) having zero conductivities. But, for real dielectrics, wehave to take into consideration the conductivities. This is done by incorporating theconductivities within a complex permittivity. Thus,

εn = εn − iσn

ωwith n = 1, 2 (1.128)


where σ n is the conductivity and ω is the angular frequency. Then,

�FDEP = 2πRe

[

ε1(

ε2 − ε1)

ε2 + 2ε1

]

R3∇ (�E0 · �E0)

(1.129)

where Re denotes the real part.Using Stokes law, the hydrodynamic drag force on a particle of radius R far away

from any wall is given by

FHYD = 6πμRU (1.130)

where U is the speed of the particle through the medium having coefficient of viscos-ityμ. To find this speed U we equate the dielectrophoretic force to the hydrodynamicforce, thus

FDEP = FHYD

or,

2πε1ε2 − ε1

ε2 + 2ε1R3∇ (�E0 · �E0

) = 6πμRU (1.131)

considering ideal dielectric particle and suspending medium. Finally, we get

U = 1

3με1ε2 − ε1

ε2 + 2ε1R2∇ (�E0 · �E0

)

(1.132)

This is the velocity of a non-accelerating spherical particle undergoing dielec-trophoresis.

4 Surface Tension-Driven Flows

It was mentioned in Section 1 that surface effects become progressively dominantas dimensions scale down. We discussed the electrokinetic surface effects in theprevious section. In this section, we discuss fluid flow actuation and control throughthe manipulation of surface tension forces. There are a number of agents which maybe used to bring about this manipulation, namely hydrodynamic, thermal, chemical,electrical, or optical. Pertinently, the possibility of fluid actuation exploiting surfacetension is contingent on the existence of an interface – be it a free surface or aliquid–fluid interface.

Surface tension (γ ) is the force per unit length acting along the interface ofimmiscible phases. The microscopic origin of surface tension is most easily illus-trated in the case of a liquid–gas system. Molecules in the liquid bulk (sufficientlydistant from any surface so as not to “feel” its presence) experience equal forces


(from the neighboring molecules) in all directions. In contrast to this, at the liquid–gas interface, the molecules experience a higher force from the liquid side comparedto the vapor side. The net force on the interface molecules is thus directed inwardtoward the liquid side. In order that the interface is sustained, it must possess a cer-tain amount of energy to overcome this net force. This energy per unit interfacialarea is known as the surface energy.

4.1 Interfaces – Young and Laplace Equation

In many practical situations, there are liquid–gas interfaces present adjacent to asolid phase. To characterize such three-phase systems, reference is made to thecontact angle as shown in Fig. 1.8.

A simple force balance along the solid surface gives

γsl + γlv cos θ = γsv

or,

cos θ = γsv − γsl

γlv(1.133)

This is the famous Young’s equation. Considering the liquid to be water, the con-tact angle may be used to characterize the surface. Specifically, when 0 ≤ θ ≤ 90◦,the solid is termed hydrophilic. When θ > 90◦, the substrate is termed hydrophobic.The derivation of Young’s equation may give the impression that the vertical com-ponent of the force is left unbalanced but in reality this is balanced by the normalstress in the solid substrate.

Furthermore, from the perspective of fluid mechanics, the presence of an inter-face creates a jump in pressure across a curved interface. The pressure is higher onthe concave side. A quantitative estimate of this pressure difference follows:

Let us consider a portion ABCD of a curved surface as shown in Fig. 1.9. Thisportion is generated when two sets of mutually perpendicular planes cut the surface.The intersection of each of these panes with the surface is an arc. The radii of cur-vature of the arcs of length x and y as seen in the figure are, respectively, R1 and R2.Now, let us consider a movement of this curved surface outward by a small amount

Fig. 1.8 Evaluation ofcontact angle


Fig. 1.9 Stretching of asurface – evaluation of thepressure difference across acurved interface

dz such that in this new position the portion which was ABCD becomes A′B′C′D′.The arc lengths now become x+dx and y+dy. The increase in area on moving fromABCD to A′B′C′D′ is

dA = (x + dx) (y + dy)− xy

≈ xdy + ydx (1.134)

Now, the change in interfacial free energy in the process is

dG = γ dA = γ (xdy + ydx) (1.135)

Equating the change in energy with the work done due to the pressure differential�p, it follows that

dG = dw

or,

dG = �pdV

or,

γ (xdy + ydx) = �pxydz (1.136)

We notice from Fig. 1.9 that


x = R1dα

and

x + dx = (R1 + dz) α

Therefore,

x + dx

R1 + dz= x

R1

or,

dx

xdz= 1

R1(1.137)

Similarly,

y + dy

R2 + dz= y

R2

or,

dy

ydz= 1

R2(1.138)

Using (1.137) and (1.138) in (1.136), we get

γ

(

xydz

R2+ yx

dz

R1

)

= �pdxdz

or,

γ

(1

R1+ 1

R2

)

= �p (1.139)

This is the famous Laplace equation.The mathematical development we have presented until now is restricted to cases

where bulk internal forces due to gravity, electric field, and so on can be neglected.In order to analyze a situation which includes such forces, we take recourse tothe thermodynamic description of the droplet from a fundamental viewpoint forillustration. The general form of the free energy of a droplet is

E =∑

i �=j

Aijγij +∑

k

Uk − λV (1.140)


where V is the constant droplet volume, λ is a Lagrange multiplier to enforce theconstant volume constraint, Aij is the interfacial area that demarcates the phases i andj, with the corresponding surface energy being γ ij, Uk is the contribution to the freeenergy from the kth external force (such as gravity and electric field). If the coor-dinates used to represent the droplet are qm then the Young’s equation (1.133) andLaplace equation (1.139) can be recovered by minimizing the free energy (1.140).In most common situations (where complex surface morphologies are absent), thecondition of extremization are usually sufficient:

∂E

∂qm= 0 (1.141)

For a more complete description of this approach considering a spherical droplet,the interested reader is referred to [5].

We have, hitherto, been concerned with the equilibrium description of a droplet.There are multifarious applications of droplets in the microfluidic realm, andthese necessitate the use of more advanced mathematical treatments addressing thedynamics of droplets. However, beyond droplets, surface tension effects are alsoimportant for fluid flow through channels and tubes at the micro-scale. Recognizingthis importance, we present the theory of fluid transport in a microtube/capillaryunder the action of surface tension in the following section.

4.2 Surface Tension-Driven Flow in Microchannels/Capillary

The equilibrium height to which a liquid will rise in a vertical capillary can befound by equating the vertical component of the surface tension force on the liquidmeniscus to the body force on the liquid column due to gravity:

2πaγ cos θ = πa2Hρg

or,

H = 2γ cos θ

ρgα(1.142)

where H is the equilibrium height reached by the capillary.Now, it is also important to track how this equilibrium height is reached with time

(see Fig. 1.10). This means we need to record the transient conditions in the processof achieving the equilibrium height. It may first be assumed that at any instant, theliquid flow is a fully developed Poiseuille flow, in which case


Fig. 1.10 Rise of fluid due tocapillary action

u

u= 2

(

1 − r2

a2

)

(1.143)

where u is the average velocity. The wall shear stress is given by:

σw = −μdu

dr

∣∣∣∣r=a

= 4μ

au (1.144)

A force balance on the liquid column, which has a length L (t) at any instant of time,and neglecting inertial effects reads:

σw2πaL − 2πaγ cos θ + πa2ρLg = 0. (1.145)

Furthermore, noting that u = dL/dt, we have from (1.144) and (1.145)

dL (t)

dt= a2

8μL (t)

[2γ cos θ

a− ρgL (t)

]

or,

dL (t)

dt= γ

8μ

[2a cos θ

L (t)− ρga2

γ

]


or,

dL (t)

dt= ρgHa2

8μL− ρga2

8μ(using (1.142)) (1.146)

This is the Lucas–Washburn equation. When t is very small, L � H, i.e.,H/

L 1, and the Lucas–Washburn equation reduces to

dL (t)

dt= ρga2

8μ

(H

L

)

(1.147)

Equation (1.147) has the following solution:

L (t) =√

ρgH

4μ

√t (1.148)

i.e.,

L (t) ∝ √t

Although this solution gives a good estimate, it suffers from a fundamental dis-crepancy. For t → 0, dL (t)

/

dt → ∞, implying an infinite velocity, when the liquidjust starts rising in the capillary. When t is very large, L → H so that L = H − δL,where δL � H. Then,

dL (t)

dt= ρga2

8μ

[H

H − δL− 1

]

or,

− d (δL)

dt= ρga2

8μ

δL

H − δL

or,

− d (δL)

dt≈ ρga2

8μ

δL

H(1.149)

The solution of (1.149) is

δL (t) = C exp

(

−ρga2

8μHt

)

(1.150)

where C is an arbitrary constant of integration. Using (1.150) in the expression L =H − δL, we get


L (t) = H − C exp

(

−ρga2

8μHt

)

(1.151)

This means that, at large times, the capillary height L(t) approaches the equilib-rium height H asymptotically as an exponential saturation.

It is important to realize that the formulation of the Lucas–Washburn equationstems from a quasi-steady approximation – an assumption which precludes the useof any inertial effect. A model, including the inertial terms, was presented by Zhmudet al. [17]. It is a statement of Newton’s second law of motion as applied to the liquidmoving up under the combined forces of surface tension, viscous forces, and gravity.Thus,

d

dt(MVc) = FSurface tension + FViscous + FGravity (1.152)

where M = ρπa2L, Vc = dL/

dt, FSurface tension = 2πaγ cos θ , FViscous =−�pπa2, and FGravity = ρgLπa2. Now, as before, we assume a fully developedflow profile, such that Q = (πa4

/

8μ) (

�p/

L)

or(

πa2)

�p = 8μπL(

dL/

dt)

.Therefore, FViscous = −8μπL

(

dL/

dt)

. Using these in (1.152) we obtain

d

dt

(

ρπa2LdL

dt

)

= 2πaγ cos θ − 8μπLdL

dt+ ρgLπa2

ρ

[

Ld2L

dt2+(

dL

dt

)2]

= 2γ

acos θ − 8

a2μL

dL

dt− ρgL (1.153)

Now, although the natural initial condition to consider might seem to be L (0) = 0and dL (0)

/

dt = 0, such conditions give rise to an ill-posed problem with infiniteinitial acceleration. However, such a fundamental drawback can be circumvented byconsidering an added mass ρπa2λ inducted initially so that (1.153) becomes

ρ

[

(L + λ)d2L

dt2+(

dL

dt

)2]

= 2γ

acos θ − 8

a2μL

dL

dt− ρgL (1.154)

Here, λ is obtained from potential flow solution for a droplet of radius r readyto enter a capillary and its value depends on the capillary geometry. For the presentcase of a cylindrical tube λ ≈ 3ρπr3

/

8.The complete physical argument behind this artifice of considering an added

mass can be found in Zhmud et al. [17]. Furthermore, the steps required to solvethis equation numerically are also elaborated in the same reference.

We have only scratched the surface of this important phenomenon and the mod-els pertaining to various microfluidic applications – electrowetting, electrocapillary,thermocapillary, to name just a few. The detailed considerations of these are cer-tainly beyond the scope of this book. Yet, the fundamentals of surface tension


effects as described here remain fundamentally immutable. The extra considera-tions of electrokinetic, thermal, optical, and other physico-chemical effects exploitthese fundamentals of surface energy to manipulate, actuate, and control fluid flow.

5 Non-Newtonian Fluids

We have, hitherto, been discussing the mechanics of Newtonian fluids, i.e., fluidsin which the shear stress varies linearly with the velocity gradient. But, there arecertain fluids which do not follow this behavior. Even fluids which are otherwiseNewtonian in nature may show non-Newtonian flow characteristics because of thepresence of suspensions in them. Typical examples are biofluids like blood, whichare common in microfluidics applications for medical diagnostics.

Although there do not exist any fundamental constitutive equation to universallymodel non-Newtonian fluids, many empirical relations have been proposed. Formany engineering applications, these relations may be adequately represented bythe power law model. For a one-dimensional flow, the power law model is

τ = k

(du

dy

)n

(1.155)

where k is the consistency index and n is the flow behavior index.In order to draw a parallel with Newtonian fluid constitutive behavior, the shear

stress is written as

τ = k

∣∣∣∣

du

dy

∣∣∣∣

n−1 (du

dy

)

(1.156)

The absolute value is used to ensure that τ has the same sign as du/

dy.

Continuing the parallel with the Newtonian fluid, the term k∣∣du/

dy∣∣n−1 is given

a special name: apparent viscosity. There can be three cases:

(i) n<1: In this case, the apparent viscosity decreases as the rate of deformationincreases. These are called pseudoplastic fluids, of which, probably, the mostsignificant example is blood. The apparently simplistic power law model givesus an important insight into the in vivo blood flow biophysics. Blood flowsthrough extremely narrow “tubes” within the body. Within these tubes, thegradient of velocity is extremely high at the near-wall regions. This meansthat the apparent viscosity is substantially low in these regions – this factis tremendously significant from the point of view of expediting the bloodpumping.

(ii) n=1: This case reduces to the Newtonian fluid.(iii) n>1: In this case, the apparent viscosity increases as the rate of deformation

increases. These are called dilatant fluids.


There is a certain class of non-Newtonian fluid which deforms under appliedstress but returns, partially, to its original shape after the applied stress is released.Since it exhibits characteristics of both a viscous fluid and an elastic medium,it is termed viscoelastic. A number of biofluids come under the purview of thisclass; hence, viscoelasticity is particularly significant from the point of view ofmicrofluidics.

A widely accepted mathematical model for viscoelastic fluids is the Phan-Thien–Tanner model.

f (τkk) [τ ] + λ[

τ∇] = 2μ [D] (1.157)

where D = 12

(∇ ⊗ �uT + ∇ ⊗ �u), λ is the relaxation time of the fluid,[

τ∇] is upperconvected derivative of τ , defined as

[

τ∇] = (D [τ ]/

Dt)−∇ ⊗�uT · [τ ]− [τ ] ·∇ ⊗�u,

f (τkk) = 1 + (ελ/μ) τkk is the stress coefficient function, with τkk = tr ([τ ]). Whenε = 0 the upper convected model is recovered.

For a fully developed flow

�u = [u (y) 0]T

Therefore,

So,

D = 1

2

⎡

⎢⎢⎣

0∂u (y)

∂y

∂u (y)

∂y0

⎤

⎥⎥⎦

∇ ⊗ �uT · [τ ] =⎡

⎣

0 0∂u (y)

∂y0

⎤

⎦

[

τxx τxy

τxy τyy

]

=⎡

⎢⎣

τxy∂u (y)

∂yτyy∂u (y)

∂y

0 0

⎤

⎥⎦ (1.158)

Similarly,


[τ ] · ∇ ⊗ �u =

⎡

⎢⎢⎣

τxy∂u (y)

∂y0

τyy∂u (y)

∂y0

⎤

⎥⎥⎦

(1.159)

Using (1.158) and (1.159) we can write out the components from (1.157) as

f (τkk) τxx − 2λτxy∂u (y)

∂y= 2μ× 1

2× 0

or,

f (τkk) τxx = 2λτxy∂u (y)

∂y(1.160)

Similarly,

f (τkk) τyy = 0

or,

τyy = 0 (1.161)

and

(1.162)

Dividing (1.160) by (1.162), we obtain:

f (τkk) τxx

f (τkk) τxy= 2λτxy

μ

or,

τxx = 2λ

μτ 2

xy (1.163)

The argument of the stress coefficient function, now, becomes:

(1.164)

Now, Cauchy’s equation for a steady flow (with negligible inertia terms) is

−∇p + ∇ · [τ ] = 0


Since the flow is fully developed, the equation along the x (axial)-direction is

− ∂p

∂x+ ∂τxy

∂y= 0 (1.165)

Equation (1.165), in conjunction with the boundary condition τxy = 0 at thecenter-line (y=0), gives

τxy = ∂p

∂xy (1.166)

Using (1.166) in (1.163) we get

τxx = 2λ

μ

(∂p

∂x

)2

y2 (1.167)

Now, from (1.162), we have

f (τkk) τxy = μ∂u (y)

∂y

or,

(

1 + ελ

μτxx

)

τxy = μ∂u (y)

∂y(1.168)

Using (1.167) in (1.168), we get

∂u (y)

∂y=[

1 + 2ελ2

μ2

(∂p

∂x

)2

y2

]

∂p

∂x

(y

μ

)

(1.169)

This equation may be readily solved by using the boundary conditions relevantfor a particular problem.

The fact that surface effects become important with increasing levels ofminiaturization is not changed by the consideration of non-Newtonian fluids. Thus,in flows of non-Newtonian fluids, too, electrokinetic and capillary effects have asignificant role to play [3, 7].

6 Acoustofluidics

Acoustofluidics is the application of acoustics in microfluidics. When acousticwaves are propagating in a fluid there arise rapidly oscillating pressure and velocityfields in it. Simultaneously, there also arises a slow non-oscillating velocity compo-nent. It is true that under normal circumstances, in keeping with our macroscopicintuition, such effects are of minuscule significance. But, at the micro-scale even


such small effects can have non-negligible consequences. More importantly, sucheffects can be usefully exploited for fluidic actuation.

The slow non-oscillating velocity component that arises in periodic acousticallydriven flows is one example of the general class of streaming flows. The physicsthat lies, fundamentally, at the heart of such phenomena is something that we havebeen neglecting in our mathematical development – albeit on the basis of soundphysical justifications with contextual relevance. These are the nonlinearity and thecompressibility. In the particular case of acoustofluidics, however, such physics canno longer be viably neglected.

We start with the understanding that the changes in pressure, density, and veloc-ity brought about by the acoustic actuation are only small perturbations. Thus, weproceed, for the sake of analysis, with perturbation theory. Specifically, we considerperturbations up to second order. It will be seen that the time-invariant streamingvelocities are achieved at this order. In the mathematical treatment that follows, wefollow the structure of development presented in Bruus [2]. We first express the pres-sure, density, and velocity as asymptotic expansions in terms of the small parameterε such that ε � 1.

Equations of motion in acoustofluidics may be developed on the basis of second-order asymptotic expansion of the field variables appearing in the Navier–Stokesequation. The corresponding expanded variables have the following mathematicalstructures:

p = p0 + εp1 + ε2p2 (1.170)

ρ = ρ0 + ερ1 + ε2ρ2 (1.171)

and

�v = �0 + ε�v1 + ε2�v2 (1.172)

Here, p0 and ρ0 denote the values of the fluid pressure and density, respectively,in the undisturbed state with zero velocity.

Let us, first, consider the general form of the mass conservation equation (1.5):

∂ρ

∂t= −∇ · (ρ�v)

Using (1.171) and (1.172), and neglecting terms with coefficients having powersof ε higher than 2, we get

∂ρ0

∂t+ ε

∂ρ1

∂t+ ε2 ∂ρ2

∂t= −∇ ·

[

ερ0�v1 + ε2ρ0�v2 + ε2ρ1�v1

]

(1.173)

From (1.173) it is clear that

Zeroth-order equation:∂ρ0

∂t= 0 (1.174)


First-order equation:∂ρ1

∂t= −∇ · (ρ0�v1) (1.175)

Second-order equation:∂ρ2

∂t= −∇ · (ρ0�v2 + ρ1�v1) (1.176)

Let us, next, consider the general linear momentum conservation equationwithout invoking the condition of incompressibility as in (1.15):

ρ∂�v∂t

+ ρ�v · ∇�v = −∇p + μ∇2�v +(

μv + 1

3μ

)

∇ (∇ · �v) (1.177)

Using (1.170), (1.171), and (1.172) in (1.177) and neglecting terms with coeffi-cients having powers of ε higher than 2, we get for the various terms

ρ∂�v∂t

= ερ0∂�v1

∂t+ ε2ρ0

∂�v2

∂t+ ε2ρ1

∂�v1

∂t

ρ�v · ∇�v = ε2ρ0�v1 · ∇�v1

∇p = ∇p0 + ε∇p1 + ε2∇p2

μ∇2�v = �0 + ε∇2�v1 + ε2∇2�v2 and

∇ (∇ · �v) = ε∇ (∇ · �v1)+ ε2∇ (∇ · �v2)

Equating the terms having as coefficients the same powers of ε, we get

− ∇p0 = 0 (1.178)

ρ0∂�v1

∂t= −∇p1 + μ∇2�v1 +

(

μv + 1

3μ

)

∇ (∇ · �v1) (1.179)

and

ρ0∂�v2

∂t+ρ1

∂�v1

∂t+ρ0�v1 ·∇�v1 = −∇p2 +μ∇2�v2 +

(

μv + 1

3μ

)

∇ (∇ · �v2) (1.180)

Next, considering the pressure, and using Taylor’s expansion about the equilib-rium pressure p0, we obtain


p = p0 + ∂p

∂ρ

∣∣∣∣0

ρ − ρ0

1! + ∂2p

∂ρ2

∣∣∣∣0

(ρ − ρ0)2

2! + higher order terms

= p0 + ∂p

∂ρ

∣∣∣∣0

(

ερ1 + ε2ρ2

)

+ ∂

∂ρ

(∂p

∂ρ

)∣∣∣∣0

(

ερ1 + ε2ρ2)2

2+ higher order terms

= p0 + εc20ρ1 + ε2c2

0ρ2 + ε2 1

2

∂c2

∂ρ

∣∣∣∣0ρ2

1 + higher order terms (1.181)

On comparing (1.181) with (1.170), we get

p1 = c20ρ1 (1.182)

Finally, from (1.181), (1.176), and (1.180), the second-order equations are,respectively,

p2 = c20ρ2 + 1

2

∂c2

∂ρ

∣∣∣∣0ρ2

1 (1.183)

∂ρ2

∂t= −ρ0∇ · �v2 − ∇ · (ρ1�v1) (1.184)

and

ρ0∂�v2

∂t= −c2

0∇ρ2 − 1

2

∂c2

∂ρ

∣∣∣∣0∇ρ2

1 − ρ1∂�v1

∂t

−ρ0�v1 · ∇�v1 − μ∇2�v2 +(

μv + 1

3μ

)

∇ (∇ · �v2)

(1.185)

We assume that the time dependence of all first-order fields is harmonic,exp (−iωt). Then, we take the time average of each of (1.184) and (1.185). Thus,

⟨∂ρ2

∂t

⟩

= 〈−ρ0∇ · �v2 − ∇ · (ρ1�v1)〉


or,

∇ · 〈�v2〉 = − 1

ρ0∇ · 〈ρ1�v1〉 (1.186)

and

⟨

ρ0∂

∂t(∇ · �v2)

⟩

=⟨

−c20∇2ρ2 − ∇ · ρ1

∂�v1

∂t− ρ0∇ · [(�v1 · ∇) �v1

] · · ·

−1

2

∂c2

∂ρ

∣∣∣∣0∇2ρ2

1 + μ∇2 (∇ · �v2)+(

μv + 1

3μ

)

∇2 (∇ · �v2)

⟩

⇒ 1

c20

ρ0∂

∂t(∇ · 〈�v2〉) = −∇2 〈ρ2〉 − ∇ ·

⟨

ρ1

c20

(−iω) �v1

⟩

− ρ0

c20

∇ · 〈(�v1 · ∇) �v1〉

− 1

2c2

0∂c2

∂ρ

∣∣∣∣0∇2⟨

ρ21

⟩

+ 1

c20

(

μv + 4

3μ

)

∇2 (∇ · 〈�v2〉)

⇒ ∇2 〈ρ2〉 = iω

c20

∇ · 〈ρ1�v1〉 − ρ0

c20

∇ · 〈(�v1 · ∇) �v1〉

− 1

2c20

∂c2

∂ρ

∣∣∣∣0∇2⟨

ρ21

⟩

−(

μv + 4/3μ)

ρ0c20

∇2 (∇ · 〈ρ1�v1〉)(1.187)

Equations (1.186) and (1.187) denote, respectively, the streaming components ofthe velocity and density that arise due to the system oscillations. From the point ofview of microfluidics, these streaming components are of great significance becausethey can be exploited to bring about transport at such micro-scales.

7 Conclusions

We have, in each of the sections, described models of phenomena which are indi-vidually responsible for enriching the mechanics of fluid flow. However, beyondthese preliminary considerations, in real microfluidic applications, these variouscases might be significant in a combined manner. The corresponding mathematicalmodels of these intricate systems would necessitate the incorporation of the indi-vidual models in a cohesive manner. But, in the general case even this is perhapsan oversimplified statement. It must be appreciated that the co-existence of morethan one effect – for example, electrokinetics together with surface tension – mightimply not just a straightforward superimposition. Indeed, the simultaneous presenceof two or more effects might lead to results that do not follow from intuitive expec-tations. Genuine insights into such intertwined phenomena can be gained only ifone has truly grasped the fundamentals of flow physics as we have tried to delineatein a very rudimentary way here in this chapter. Furthermore, it must be appreciated


that the models of flow physics are strongly rooted in the developments of variousphysico-chemical hydrodynamics developed over the last couple of centuries or per-haps even more. Yet, as so often happens that the technology drives the developmentof science, the special considerations of microfluidic applications have led to newand fundamental insights especially in the last couple of decades. This, in turn, hasmotivated fruitful innovations, thus making the realm of micro-scale flow physics atruly potent and fertile field. As the continual feedback between theory and applica-tion goes on, the story of micro-scale flow physics has spiraled into an epic – and itis far from over!

References

1. Acheson, D. J., Elementary Fluid Dynamics, Oxford University Press, New York, NY (2003).2. Bruus, H., Theoretical Microfluidics, Oxford University Press, Oxford (2008).3. Chakraborty, S., Electroosmotically Driven Capillary Transport of Typical Non-Newtonian

Biofluids in Rectangular Microchannels, Analytica Chimica Acta, 605, 175–184 (2007).4. Chakraborty, S., Electric Double Layer, in Li, D. Encyclopedia of Microfluidics and

Nanofluidics (pp. 444–453), Springer, New York, NY (2008).5. Chakraborty, S., Surface Tension Driven Flow, in Li, D., Encyclopedia of Microfluidics and

Nanofluidics (pp. 1955–1968), Springer, New York, NY (2008).6. Chakraborty, S., Das, S., Streaming-Field-Induced Convective Transport and Its Influence on

the Electroviscous Effects in Narrow Fluidic Confinement Beyond the Debye-Hückel Limit,Physical Review E, 77, 037303 (2008).

7. Das, S., Chakraborty, S., Analytical Solutions for Velocity, Temperature and ConcentrationDistribution in Electroosmotic Microchannel Flows of a Non-Newtonian Bio-Fluid, AnalyticaChimica Acta, 559, 15–24 (2006).

8. Griffiths, D., Introduction to Electrodynamics, Prentice Hall, New Delhi (1998).9. Hunter, R. J., Zeta Potential in Colloid Science, Academic, New York, NY (1981).

10. Kilic, M. S., Bazant, M. Z., Ajdari, A., Steric Effects in the Dynamics of Electrolytes atLarge Applied Voltages. II. Modified Poisson-Nernst-Planck Equations, Physical Review E,75, 021503 (2007).

11. Kreyszig, E., Advanced Engineering Mathematics, 8th ed., Wiley, Singapore (2004).12. Landau, L. D., Lifshitz, E. M., Electrodynamics of Continuous Media, Pergamon, Oxford

(1984).13. Li, D., Electrokinetics in Microfluidics, Elsevier, London (2004).14. Morrison, F. A., Electrophoresis of a Particle of Arbitrary Shape, Journal of Colloid and

Interface Science, 34, 2 (1970).15. Probstein, R. F., Physicochemical Hydrodynamics, Wiley-Interscience, Hoboken, NJ (2003).16. Wall, S., The History of Electrokinetic Phenomena, Current Opinion in Colloid and Interface

Science, 15, 119–124 (2010).17. Zhmud, B. V., Tiberg, F., Hallstensson, K., Dynamics of Capillary Rise, Journal of Colloid

and Interface Science, 228, 263–269 (2000).

Modeling Microscale.fluids

Documents

surface effects

fluid flows

volume effects

fluid mechanics appliedat

surface tension effects

scale of micrometers

flow manipulation

nonnewtonian fluid