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Journal of Non-Newtonian Fluid Mechanics 266 (2019) 46–58

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Journal of Non-Newtonian Fluid Mechanics

journal homepage: www.elsevier.com/locate/jnnfm

Electro-osmotic oscillatory flow of viscoelastic fluids in a microchannel

Samir H. Sadek, Fernando T. Pinho ∗

CEFT, Departamento de Engenharia Mecânica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

a r t i c l e i n f o

Keywords:

Microchannels

Electro-osmotic flow (EOF)

Viscoelastic fluids

Small amplitude oscillatory shear flow (SAOS)

Small amplitude oscillatory shear by

electro-osmosis (SAOSEO)

Multi-mode upper-convected Maxwell (UCM)

model

a b s t r a c t

This work presents an analytical solution for electro-osmotic flow (EOF) in small amplitude oscillatory shear

(SAOS) as a measuring tool suitable to characterize the linear viscoelastic properties of non-Newtonian fluids

in microchannel flow. The flow in the straight microchannel is driven by applying oscillating sinusoidal electric

potentials. Fourier series are used to derive an expression for the velocity field, under an externally imposed

generic potential field aimed at the practical application of SAOS in characterizing the rheological properties of

viscoelastic fluids. This extensive investigation covers a wide range of parameters and considers the multi-mode

upper-convected Maxwell (UCM) model, which represents the rheology of viscoelastic fluids in the limit of small

and slow deformations. Particular focus is given to two cases of practical interest: equal wall zeta potentials at

both channel walls and negligible zeta potential at one of the walls. The results show that in flows with thin

electric double layers (EDL), Reynolds numbers below 0.001, and Deborah numbers below 100, corresponding

to a viscoelastic Mach number of 0.32, the velocity field outside the EDLs is linear and has a large enough

amplitude of oscillation, which may allow the quantification of the storage and loss moduli in the linear regime.

This technique requires the use of significantly smaller sample sizes than the traditional SAOS in a rotational

rheometer.

L

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ist of abbreviations

DL electric double layer

OF electro-osmotic flow

ZP equal zeta potential

ZP negligible zeta potential

IV particle image velocimetry

TV particle tracking velocimetry

AOS small amplitude oscillatory shear

AOSEO small amplitude oscillatory shear by electro-osmosis

. Introduction

Transport phenomena at the micro-scale is increasingly of interest for

pplications in a variety of systems given the inherent savings in materi-

ls and energy, fast reaction times and the ability of today’s technology

o fabricate micro-systems with multi-purpose functions. In particular,

icro-scale systems are increasingly being used to process bio-fluids and

hemicals, in species separation, or mixing, among others. On moving

rom macro to micro-scale systems, the ratio of volume to surface forces

ramatically decreases and surface-based forcing mechanisms become

dvantageous relative to volume-based methods. Indeed, monitoring

nd controlling liquid transport accurately by electro-osmosis becomes

ncreasingly easier and more effective at the micro- and nano-scales,

hereas the use of the traditional pressure gradient driven flow becomes

∗ Corresponding author.

E-mail address: [email protected] (F.T. Pinho).

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ttps://doi.org/10.1016/j.jnnfm.2019.01.007

eceived 20 July 2018; Received in revised form 15 January 2019; Accepted 20 Janu

vailable online 13 February 2019

377-0257/© 2019 Elsevier B.V. All rights reserved.

ncreasingly less efficient as the size of the channels are reduced due to

he significant increase of the pressure gradients [1–3] .

Electro-osmosis is an electro-kinetic phenomenon, identified first by

euss [4] in the 19th century. In electro-osmosis, chemical equilibrium

etween a polar fluid and a solid dielectric wall results on a spontaneous

harge being acquired by the wall and the corresponding counter-charge

ccurring in near-wall layers on the liquid side. On the liquid side, a very

hin wall layer of immobile counter-ions develops and is followed by a

hicker layer of diffuse counter-ions, thus creating the so-called elec-

ric double layer (EDL). Upon application of an external electric poten-

ial field between the inlet and outlet of the microchannel the ensuing

otion of the diffuse layer counter-ions drags the remaining core fluid

n the channel by viscous forces. An overview of electro-osmosis and

f other electro-kinetic flow techniques can be found in [5–7] . Electro-

smosis offers special unique features over other types of pumping meth-

ds (e.g. micro-pumps), and has the ability of easily and very quickly

within the viscous time scale) change flow direction and magnitude by

hanging the applied potential field. The generated flow is defined by

he pattern of the imposed electric potential field, so it can be easily

riven at constant flow rate, or following an oscillatory pattern [8] .

As previously mentioned, chemical and biomedical lab-on-a-chip

ystems are the most frequent applications of micro-scale liquid flows.

he fluids are frequently made from complex molecules which ex-

ibit non-linear rheological behavior. These so-called non-Newtonian

uids have such rheological characteristics as variable viscosity and

iscoelasticity. The rheological characterization of non-Newtonian flu-

ds is performed with various controllable and quasi-controllable flows

ary 2019

https://doi.org/10.1016/j.jnnfm.2019.01.007http://www.ScienceDirect.comhttp://www.elsevier.com/locate/jnnfmhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.jnnfm.2019.01.007&domain=pdfmailto:[email protected]://doi.org/10.1016/j.jnnfm.2019.01.007

S.H. Sadek and F.T. Pinho Journal of Non-Newtonian Fluid Mechanics 266 (2019) 46–58

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i.e., flows with known kinematics) which is independent or weakly

ependent on the fluid properties to be measured [9] . One such flow

s the small amplitude oscillatory shear (SAOS) flow used to character-

ze the linear viscoelastic behavior of complex fluids. This Couette-type

ow is usually implemented in rotational rheometers, but linear vis-

oelasticity can also be assessed via other oscillatory flows provided the

mplitude of the oscillations are small and slow enough to make the

uid response independent of the amount of deformation and only de-

endent on the oscillating frequency and fluid material functions.

Oscillatory channel flow can be driven by an imposed harmonic mo-

ion of the sidewalls or an oscillating pressure gradient in the streamwise

irection. Even though these are qualitatively similar, from a mathemat-

cal point of view they are slightly different: whereas the first approach

mposes strain and monitors stress, the second one imposes the stress

nd monitors the strain. In this work, we explore the use of the oscil-

atory electro-osmotic flow (EOF) as a method to characterize the lin-

ar viscoelasticity of non-Newtonian fluids. This is achieved through a

eneral solution by adopting the multi-mode upper-convected Maxwell

UCM) model. It is well known that in the limit of small and slow de-

ormations the response of complex viscoelastic constitutive models re-

uces to that of the UCM model [10] , which in turn behaves as a linear

iscoelastic model in such limit. In turn, multi-mode models allow a bet-

er description of real fluids, which possess spectra of relaxation times

ather than single characteristic time scales [11] . The concept has been

round for more than a century in the form of the generalized Maxwell

odel (also called the Maxwell-Wiechert model) for linear viscoelastic-

ty.

The multi-mode UCM model has been used in a variety of practi-

al computational and especially experimental applications. Thurston

12] used it to describe the complex rheology of blood, Renardy [13] re-

ied on the multi-mode UCM model in his proof of existence of steady

low flows of viscoelastic fluids and, subsequently, Renardy [14] and

avelev and Renardy [15] addressed the conditions of flow controlla-

ility of viscoelastic fluids using again a multi-mode UCM fluid model.

omputational applications were first addressed by Rajagopalan et al.

16] , with the main convergence problem being always associated with

he mode with the highest relaxation time. Of direct relevance to this

ork are the contributions addressing the computation of the storage

nd loss moduli for multi-mode models [11,17] in SAOS (i.e., for rota-

ional rheometry). The use of the multi-mode UCM model to describe

omplex viscoelastic fluids and flows is also exemplified through the

ollowing diverse examples: predictions of the extrusion film casting

rocess using a polymeric resin of two different macromolecular chains

18] and of polymer crystallization kinetics under non-isothermal flow

onditions [19] ; examining steady creeping flow behavior [16] , large

mplitude oscillatory shear (LAOS) flow [20] and characterization of

omplex viscoelastic properties of bovine liver tissue [21] , and as well

s brain tissue [22] .

The remaining of this introduction addresses the contributions in

OF for non-Newtonian fluids in steady and especially unsteady flow.

tarting with steady flows, Das and Chakraborty [23] , and Chakraborty

24] were among the first to study analytically the momentum, heat

nd mass transfer in microchannel flows of non-Newtonian fluids driven

y electrokinetic forces, but their work was limited to the power-law

odel. Other analytical investigations of steady electro-osmosis chan-

el flows of power-law fluids were done by Zhao et al. [1] , who invoked

he Debye-Hückel approximation, and Zhao and Yang [25] , who com-

ined EO forcing with pressure-driven (PD) flow. These investigations

re not necessarily a limitation, because the steady flow characteristics

f viscoelastic fluids only depend on its viscosity law.

Park and Lee [26,27] were among the first to investigate numeri-

ally and analytically EOF with viscoelastic fluids. Of particular concern

n [26] was the evaluation of the Helmholtz-Smoluchowski velocity in

ure EOF by a simple cubic algebraic equation to be used as a wall func-

ion in the numerical calculations in order to bridge the electric double

ayer and avoid the use of very refined meshes near the walls. Park

47

nd Lee [27] extended their previous numerical study to investigate the

OF of viscoelastic fluids, described by the Phan-Thien —Tanner (PTT)

odel, through a rectangular duct with and without a pressure gradi-

nt, including the prediction of its secondary flow. Afonso et al. [28] ob-

ained the analytical solution for mixed electro-osmotic/pressure driven

EO/PD) channel flow for viscoelastic fluids described by the simplified

TT (sPTT) and the Finitely Extensible Non-linear Elastic (FENE-P) mod-

ls. Still on steady EO channel flows of viscoelastic fluids, Afonso et al.

29] considered asymmetric wall zeta potentials for the sPTT and FENE-

models, whereas Dhinakaran et al. [30] analysed the critical shear rate

nd Deborah number for the onset of fluid instabilities of constitutive

ature with the full PTT model.

As described previously, EOF motion starts once an external poten-

ial difference is applied across the electrodes embedded in a microchan-

el. The imposed driving potential may either correspond to a direct

urrent EOF (DCEOF) or an alternating current EOF (ACEOF), with all

revious steady flows concerning DCEOF. In ACEOF the flow depends

n the amplitude and frequency of the applied electric field in addition

o geometric, wall and fluid properties. Applying a non-uniform electric

eld leads to a nonzero time-average flow [31] , whereas the use of a

niform field results in a zero time-average flow. This latter case is a

pecial case of ACEOF, called time-periodic EOF, and the use of DCEOF

s a limit case with zero frequency [5,32] .

Dutta and Beskok [33] investigated analytically the two-dimensional

ow of Newtonian fluids driven by time-periodic EOF in straight mi-

rochannels and presented the corresponding velocity field distribu-

ion for weak potential (no interaction between electric double layers

EDLs)). Subsequently, Chakraborty and Srivastava [34] studied analyt-

cally the flow through straight microchannels with overlapped EDLs.

Of more direct relevance to this work, Liu et al. [35] presented an

nalytical solution for one-dimensional electro-osmotic flow between

scillating micro-parallel plates of viscoelastic fluids represented by a

ingle mode generalized Maxwell model. They considered a low zeta

otential which allows the linearization of the Poisson-Boltzmann equa-

ion. Their equations for the dimensionless velocity profile and volumet-

ic flow rate as a function of the oscillating Reynolds number, electro-

ynamic width, and normalized relaxation time, are useful to under-

tand the flow characteristics of this flow configuration.

Oscillating flows of small amplitude are used in rotational rheology

o probe the elastic nature of fluids in the linear regime. These are flow

onditions that are also present in some electro-osmotic flows and the

ain motivation of this work is precisely the assessment of the feasi-

ility and effectiveness of using oscillatory electro-osmotic shear flow

n the rheological characterization of viscoelastic fluids. As previously

xplained, real fluids have a spectrum of time scales and here the fluid

s precisely described by the multi-mode UCM model rather than by a

ingle mode model as done in [35] , thus to our best knowledge this prob-

em has not yet been solved. In addition to providing the EOF solution

or a multi-mode Maxwell fluid under the action of an oscillatory flow

orcing, this work addresses the use of the flow as a novel measuring

ool for small amplitude oscillatory shear flow driven by electro-osmosis

SAOSEO).

Section 2 starts with the continuity and momentum governing equa-

ions for the problem at hand along with the viscoelastic constitutive

ulti-mode UCM model being presented in Section 2.1 . The distribu-

ions of the induced EDL potential field 𝜓 and of the net electric-charge

ensity 𝜌𝑒 , with emphasis on two special cases corresponding to two

ifferent sets of wall boundary conditions, are presented in Section 2.2 .

ection 2.3 solves analytically the UCM constitutive equation and in

ection 2.4 the flow problem at hand is solved using Fourier series for

he more complex generic periodic signals of the imposed electric field.

ection 3 discusses the analytical solution derived in the previous sec-

ion and identifies the conditions that need to be satisfied for the ob-

ained solution to be useful for rheometric purposes, which is detailed

n Section 4 . Finally, Section 5 concludes and summarizes the main find-

ngs of this work.


Fig. 1. Schematic diagram, illustrating the microchannel di-

mensions, coordinate system, and the induced potential bound-

ary conditions.

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. Governing equations and analytical solution

The unsteady oscillatory shear flow of the incompressible viscoelas-

ic fluid under investigation is sketched in Fig. 1 , which shows the mi-

rochannel and the two-dimensional coordinate system used, with its

rigin located at the geometric symmetry plane. The channel height is

H , the length is L , and the depth of the channel in the z direction is

arge, i.e., the flow field is assumed independent of the z coordinate.

ig. 1 also defines the boundary conditions at the upper and lower

alls, where for the velocities there are no-slip boundary conditions,

( -H ) = u ( H ) = 0, and for the induced electric potentials we considerhe possibility of different zeta potentials at both walls, i.e., 𝜓( -H ) = 𝜓 1 nd 𝜓( H ) = 𝜓 2 .

The basic equations describing the flow under investigation are the

ontinuity, and the Cauchy momentum equations:

⋅ 𝐮 = 0 , D 𝐮 D 𝑡

= −∇ 𝑝 + ∇ ⋅ 𝛕 + 𝜌𝑒 𝐄 (1)

here 𝜌 is the fluid density (incompressible fluid assumption), t is the

ime, u is the velocity vector, p is the pressure, 𝛕 is the extra-stress ten-or, 𝜌𝑒 is the net electric-charge density associated with the spontaneous

ormation of electric double layers, and E is the applied external elec-

ric field. Assuming one-dimensional flow, the x -momentum equation

implifies to:

𝜕𝑢

𝜕𝑡 = 𝜕 𝜏𝑥𝑦

𝜕𝑦 + 𝜌𝑒 𝐸 𝑥 (2)

We assume also that the flow is driven externally by an electric field,

ithout any pressure gradients imposed, ( ∇ p = 0) and the flow is spa-ially fully-developed.

.1. Constitutive equation

In this work we use the multi-mode UCM model, where the total

xtra-stress is expressed as the sum of m modes [36] :

= 𝑚 ∑𝑛 =1

𝛕𝑛 (3)

As explained in the introduction, the use of a multi-mode UCM model

llows a more realistic description of fluid memory by a spectrum of

elaxation times [11] , including the contribution from a Newtonian sol-

ent by considering a null relaxation time in one of the modes. For each

ode, the extra-stress is expressed by the UCM model:

+ 𝜆∇ 𝛕 = 2 𝜂𝐃 (4)

here D = ( ∇ u T + ∇ u ) / 2 is the deformation rate tensor, 𝜆 is the re-axation time of the fluid, 𝜂 is the polymer viscosity coefficient (each

ode has different 𝜆 and 𝜂), and ∇ 𝛕 is the upper-convected derivative of

, defined by:

𝛕 = D 𝛕D 𝑡

− ∇ 𝐮 T ⋅ 𝛕 − 𝛕 ⋅ ∇ 𝐮 (5)

Assuming that the flow instantaneously becomes fully-developed, a

ood approximation for low Reynolds number flows, Eq. (4) simplifies

o:

48

𝑥𝑥 + 𝜆𝜕 𝜏𝑥𝑥

𝜕𝑡 = 2 𝜆𝜏𝑥𝑦

𝜕𝑢

𝜕𝑦 (6)

𝑥𝑦 + 𝜆𝜕 𝜏𝑥𝑦

𝜕𝑡 = 𝜂 𝜕𝑢

𝜕𝑦 (7)

𝑦𝑦 = 0 (8)

Since Eq. (7) is a first-order, linear differential equation, it can be

ntegrated, to give the UCM model in its single mode integral form [37] :

𝑥𝑦 =

𝑡

∫−∞

𝜂

𝜆𝑒 − (𝑡 − 𝑡 ′𝜆𝑛

)𝜕𝑢

𝜕𝑦 𝑑𝑡 ′ (9)

here t , t ′ and 𝜕 𝑢 ∕ 𝜕 𝑦 are respectively, the current time, the past timend the velocity gradient, in which u is a function of 𝑦 and t ′ . The shear

tress given by Eq. (9) is limited to linear viscoelastic fluids, which com-

rises motion with infinitesimal deformation gradients. Note also that

he differential Eq. (7) for the shear stress component of the UCM fluid,

r for each mode, is identical to the corresponding equation for the lin-

ar viscoelastic Maxwell model.

.2. Poisson-Boltzmann equation

When an electrolyte fluid is in contact with a dielectric wall, a spon-

aneous attraction of counter-ions to the wall surface and correspond-

ng repelling of co-ions takes place near the wall and an ionic charge

istribution arises between the wall and the fluid, leading to the forma-

ion of an electric double layer (EDL). By assuming that the EDL is thin

ear each wall, it is possible to consider the two EDL as independent

rom each other. This assumption is valid provided we are in the di-

ute regime, where the concentration of solute ions is sufficiently low to

revent ion interactions, therefore the ions follow Boltzmann distribu-

ion according to the classical electrostatics theory. This corresponds to

small ionic Péclet number ( 𝑃 𝑒 ), defined as 𝑃 𝑒 = 𝑅𝑒𝑆𝑐, where 𝑅𝑒 and𝑐 are the Reynolds and Schmidt numbers, respectively. The Schmidt

umber 𝑆𝑐 = 𝜂∕ ( 𝜌𝐷) is typically of the order of O( 10 3 ) or lower, sincehe ionic diffusivity ( D ) of many small ions in water is of the order of

( 10 −9 ) m 2 /s and higher. Therefore, in order for 𝑃 𝑒 ≤ 1 the Reynoldsumber must be below O( 10 −3 ) [38] . Note that these are values forteady flow, whereas we are dealing with unsteady flow and defining

n oscillatory Reynolds number ( 𝑅𝑒 = 𝜌𝜔 𝐻 2 ∕ 𝜂) . Such instantaneous “os-illatory Reynolds number ” is equal to the limiting numerical value of

𝑒 only twice in the cycle, i.e., by imposing 𝜌𝜔 𝐻 2 ∕ 𝜂 = 10 −3 most of theime the flow is in the dilute regime even if 𝑅𝑒 slightly exceeds this limit

alue. The induced EDL potential field 𝜓 can be expressed by means of

Poisson equation [39] :

2 𝜓 = − 𝜌e ∈

(10)

here ∈ denotes the dielectric constant of the solution. The net electric-harge density can be described by a Boltzmann equation [39] :

𝑒 = −2 𝑛 0 𝑒𝑧 sinh ( 𝑒𝑧

𝑘 B 𝑇 𝜓

) (11)


w

t

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e

2

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E

w

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𝜓

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𝜓

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𝜌

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i

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𝜏

w

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𝜌

o

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v

𝑢

w

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l

𝜌

𝑖

𝑖

∫

b

w

w

𝐴

i

𝛽

c

m

here 𝑛 0 is the ionic density, e is the elementary electric charge, z is

he valence of the ions, 𝑘 B is the Boltzmann constant, and T is the ab-

olute temperature. The induced EDL potential field depends only on 𝑦 ,

herefore, Eq. (10) simplifies to:

𝑑 2 𝜓

𝑑 𝑦 2 = −

𝜌𝑒

∈(12)

Substituting Eq. (11) into Eq. (12) leads to:

d 2 𝜓 d 𝑦 2

= 2 𝑛 0 𝑒𝑧 ∈

sinh ( 𝑒𝑧

𝑘 B 𝑇 𝜓

) (13)

For small values of ( 𝑒𝑧𝜓/ k B T ) Eq. (13) can be linearized since for

mall 𝑥 , sinh( 𝑥 ) ≅ 𝑥 ; this is termed the Debye–Hückel approximation,hich is valid when the electric energy is smaller than the thermal en-

rgy and for fluids such as water this limits the zeta potential to about

6 mV at room temperatures [40,41] . This also ensures that ionic finite

ize effects are negligible, since for dilute solutions, steric effects start to

e relevant at zeta potentials of the order of 330 mV for solutions based

n water [42] . With this linearization of the hyperbolic sine function,

q. (13) simplifies to:

d 2 𝜓 d 𝑦 2

= 𝜅2 𝜓 (14)

here 𝜅2 = (2 𝑛 0 𝑒 2 𝑧 2 ∕ ∈ 𝑘 B 𝑇 ) is the Debye-Hückel parameter, and 𝜅 rep-esents the inverse of the Debye layer thickness, 𝜅 = 1∕ 𝜉. Eq. (14) cane integrated for the given boundary conditions, 𝜓( -H ) = 𝜓 1 , 𝜓( H ) = 𝜓 2 ,eading to:

= (𝜓 2 e 𝜅𝐻 − 𝜓 1 e − 𝜅𝐻

)e 𝜅𝑦 −

(𝜓 2 e − 𝜅𝐻 − 𝜓 1 e 𝜅𝐻

)e − 𝜅𝑦

e 2 𝜅𝐻 − e −2 𝜅𝐻 (15)

hich can be rewritten in compact form as:

= 𝜓 2 (Ω1 e 𝜅𝑦 − Ω2 e − 𝜅𝑦

)(16)

here Ω1 = [ 𝑒 𝜅𝐻 − Πe − 𝜅𝐻 ] ∕ [ 2 sinh (2 𝜅𝐻) ] , Ω2 = [ 𝑒 − 𝜅𝐻 − Πe 𝜅𝐻 ] ∕ 2 sinh (2 𝜅𝐻) ] , and Π = 𝜓 1 ∕ 𝜓 2 . As a consequence, the electric chargeensity 𝜌𝑒 becomes:

𝑒 = − ∈ 𝜅2 𝜓 2 (Ω1 e 𝜅𝑦 − Ω2 e − 𝜅𝑦

)(17)

Eqs. (16) and (17) are the general formulae for evaluating the poten-

ial field, and the net electric charge density, across a 2D channel for any

iven pair of values of wall zeta potentials, 𝜓 1 and 𝜓 2 . Two particular

ases are of interest here, as follows:

- Equal wall zeta potentials : This is the typical situation, when both

walls are made from the same material and the boundary conditions

are 𝜓( -H ) = 𝜓( H ) = 𝜓 2 . Hence Π= 1, Ω1 = sinh ( 𝜅𝐻) ∕ [ sinh (2 𝜅𝐻) ] ,Ω2 = − Ω1 and Eqs. (16) and (17) simplify to:

𝜓 = 𝜓 2 cosh ( 𝜅𝑦 ) cosh ( 𝜅𝐻)

, 𝜌𝑒 = − ∈ 𝜅2 𝜓 2 cosh ( 𝜅𝑦 ) cosh ( 𝜅𝐻)

(18)

- Negligible zeta potential at one wall : This situation arises experimen-

tally when different materials are used in the upper and lower walls

with a special deposition treatment in one of the walls to pro-

vide a negligible zeta potential there. Here, this is represented by

a negligible zeta potential at the lower wall, 𝜓( -H ) = 0, and a fi-nite potential at the upper wall, 𝜓( H ) = 𝜓 2 . Therefore Π = 0, Ω1 =𝑒 𝜅𝐻 ∕ [ 2 sinh (2 𝜅𝐻) ] , Ω2 = 𝑒 − 𝜅𝐻 ∕ [ 2 sinh (2 𝜅𝐻) ] and Eqs. (16) and(17) simplify to:

𝜓 = 𝜓 2 sinh [ 𝜅( 𝐻 + 𝑦 ) ] sinh (2 𝜅𝐻)

, 𝜌𝑒 = − ∈ 𝜅2 𝜓 2 sinh [ 𝜅( 𝐻 + 𝑦 ) ] sinh (2 𝜅𝐻)

(19)

.3. Analytical solution for the multi-mode UCM model

For the multi-mode UCM model the shear stress can be written in its

ntegral form as the sum of m individual mode contributions, each one

iven by Eq. (9) [37] :

49

𝑥𝑦 =

𝑡

∫−∞

𝑚 ∑𝑛 =1

𝜂𝑛

𝜆𝑛 𝑒 − (𝑡 − 𝑡 ′𝜆𝑛

)𝜕𝑢

𝜕𝑦 𝑑𝑡 ′, (20)

here each mode has its specific polymer viscosity coefficient, 𝜂𝑛 , and

elaxation time, 𝜆𝑛 . Substituting Eq. (20) into Eq. (2) leads to:

𝜕𝑢

𝜕𝑡 =

𝑡

∫−∞

𝑚 ∑𝑛 =1

𝜂𝑛


)𝜕 2 𝑢

𝜕 𝑦 2 d 𝑡 ′ + 𝜌𝑒 𝐸 𝑥 (21)

We shall now impose an AC electric field in the form, 𝐸 𝑥 = 𝐸 0 cos ( 𝜔𝑡 ) ,r 𝐸 𝑥 = ℜ ( 𝐸 0 𝑒 𝑖𝜔𝑡 ) using complex variables, where 𝐸 0 is the maximummplitude of applied potential and 𝜔 is the frequency of oscillation. The

elocity field of the resulting periodic EOF can be written as:

= ℜ (𝑢 0 𝑒

𝑖𝜔𝑡 )

(22)

here 𝑢 0 is a complex velocity function in 𝑦 to be determined. By sub-

tituting both expressions for the electric and velocity fields in Eq. (21) ,

eads to:

𝜕

𝜕𝑡

(𝑢 0 𝑒

𝑖𝜔𝑡 )=

𝑡 ∫−∞

𝑚 ∑𝑛 =1

𝜂𝑛


)𝜕 2

𝜕 𝑦 2

(𝑢 0 𝑒

𝑖𝜔𝑡 ′)d 𝑡 ′ + 𝜌𝑒

(𝐸 0 𝑒

𝑖𝜔𝑡 ),

𝜌𝜔 𝑢 0 𝑒 𝑖𝜔𝑡 =

𝑡 ∫−∞

𝑚 ∑𝑛 =1

𝜂𝑛


)[𝜕 2 𝑢 0 𝜕 𝑦 2

𝑒 𝑖𝜔𝑡 ′]d 𝑡 ′ + 𝜌𝑒 𝐸 0 𝑒 𝑖𝜔𝑡

(23)

Replacing, s = t – t ′ , d s / d t ′ = –1, and t ′ = t – s leads to [35] :

𝜌𝜔 𝑢 0 𝑒 𝑖𝜔𝑡 =

𝜕 2 𝑢 0

𝜕 𝑦 2 𝑒 𝑖𝜔𝑡

∞

∫0

( 𝑚 ∑𝑛 =1

𝜂𝑛

𝜆𝑛 e − 𝑠 ∕ 𝜆𝑛 𝑒 − 𝑖𝜔𝑠

) d 𝑠 + 𝜌𝑒 𝐸 0 𝑒 𝑖𝜔𝑡 (24)

Integrating the middle term in Eq. (24) with respect to s , leads to:

∞

0

( 𝑚 ∑𝑛 =1

𝜂𝑛

𝜆𝑛 e − 𝑠 ∕ 𝜆𝑛 𝑒 − 𝑖𝜔𝑠

) d 𝑠 =

𝑚 ∑𝑛 =1

𝜂𝑛

1 + 𝑖 𝜆𝑛 𝜔 (25)

Substituting Eqs. (25) and (17) into Eq. (24) , and dividing both sides

y 𝑒 𝑖𝜔𝑡 𝑚 ∑𝑛 =1

𝜂𝑛

1+ 𝑖 𝜆𝑛 𝜔 , and rearranging, leads to:

𝜕 2 𝑢 0

𝜕 𝑦 2 −

𝑖𝜌𝜔 𝑢 0 𝑚 ∑𝑛 =1

𝜂𝑛

1+ 𝑖 𝜆𝑛 𝜔

= ∈ 𝜅2 𝜓 2 𝐸 0 𝑚 ∑𝑛 =1

𝜂𝑛


(Ω1 e 𝜅𝑦 − Ω2 e − 𝜅𝑦

)(26)

By simple algebra, the summation term 1∕ 𝑚 ∑𝑛 =1

[ 𝜂𝑛 ∕( 1 + 𝑖 𝜆𝑛 𝜔 ) ] can be

ritten as:

1 𝑚 ∑𝑛 =1

𝜂𝑛


=

𝑚 ∑𝑛 =1

𝜂𝑛

1+ 𝜆𝑛 2 𝜔 2 + 𝑖

𝑚 ∑𝑛 =1

𝜂𝑛 𝜆𝑛 𝜔

1+ 𝜆𝑛 2 𝜔 2 ( 𝑚 ∑𝑛 =1

𝜂𝑛

1+ 𝜆𝑛 2 𝜔 2

) 2 + (

𝑚 ∑𝑛 =1


1+ 𝜆𝑛 2 𝜔 2

) 2 = 𝐴 + 𝑖𝐵 𝐴 2 + 𝐵 2 (27) here

= 𝑚 ∑𝑛 =1

𝜂𝑛

1 + 𝜆𝑛 2 𝜔 2 , 𝐵 =

𝑚 ∑𝑛 =1


1 + 𝜆𝑛 2 𝜔 2 (28)

Substitution of the last term of Eq. (27) into Eq. (26) leads to:

𝜕 2 𝑢 0

𝜕 𝑦 2 − 𝑖𝜌𝜔 𝑢 0

( 𝐴 + 𝑖𝐵 𝐴 2 + 𝐵 2

) =∈ 𝜅2 𝜓 2 𝐸 0

( 𝐴 + 𝑖𝐵 𝐴 2 + 𝐵 2

) (Ω1 e 𝜅𝑦 − Ω2 e − 𝜅𝑦

)(29)

Eq. (29) can be written in dimensionless form, using the follow-

ng normalizations: �̄� = 𝑦 ∕ 𝐻 , �̄� = 𝜅𝐻 , �̄� 0 = 𝑢 0 ∕ 𝑢 sh , 𝑅𝑒 = 𝜌𝜔 𝐻 2 ∕ 𝜂0 , and

𝑛 = 𝜂𝑛 ∕ 𝜂0 (note that 𝑚 ∑𝑛 =1 𝛽𝑛 = 1 ), where 𝑢 sh = − ∈ 𝜓 2 𝐸 0 ∕ 𝜂0 is the Smolu-

howski velocity based on the upper channel wall zeta potential for the

aximum value of the applied potential field, 𝐸 . Eq. (29) becomes:

0


w

o

𝑖

e

𝛼

w

t

m

f

𝑢

w

𝐶

𝐶

𝐴

𝑢

w

ℜ+

ℑ++

A

o

𝑢

𝐸

t

𝑢

t

i

t

c

a

ℜ , ℑ

t

𝑢

a

t

𝑢

2

o

𝐸

w

t

𝐸

c

p

fi

t

𝑢

+ 𝑏 𝑗 ℜ �̄� 0 sin ( 𝜔 𝑗 𝑡 ) + ℑ �̄� 0 cos ( 𝜔 𝑗 𝑡 )

𝜕 2 �̄� 0

𝜕 ̄𝑦 2 − 𝑖𝑅𝑒

( �̄� + 𝑖 ̄𝐵 �̄� 2 + �̄� 2

) �̄� 0 = − ̄𝜅2

( �̄� + 𝑖 ̄𝐵 �̄� 2 + �̄� 2

) (Ω̄1 e �̄��̄� − Ω̄2 e − ̄𝜅�̄�

)(30)

here

�̄� = 𝑚 ∑𝑛 =1

𝛽𝑛

1 + 𝜆𝑛 2 𝜔 2 , �̄� =

𝑚 ∑𝑛 =1

𝛽𝑛 𝜆𝑛 𝜔

1 + 𝜆𝑛 2 𝜔 2 ,

Ω̄1 = 𝑒 �̄� − Πe − ̄𝜅2 sinh (2 ̄𝜅)

, Ω̄2 = 𝑒 − ̄𝜅 − Πe �̄�2 sinh (2 ̄𝜅)

(31)

Eq. (30) represents a complex second-order inhomogeneous

rdinary differential equation. For convenience, we replace

𝑅𝑒 [ ( �̄� + 𝑖 ̄𝐵 )∕( �̄� 2 + �̄� 2 ) ] by ( 𝛼 + 𝑖𝜎) 2 , which results in the followingxpressions for 𝛼 and 𝜎:

=

√ √ √ √ 𝑅𝑒 2

[ − ̄𝐵 ±

(�̄� 2 + �̄� 2

)1∕2 �̄� 2 + �̄� 2

] , 𝜎 =

√ √ √ √ 𝑅𝑒 2

[ �̄� ±

(�̄� 2 + �̄� 2

)1∕2 �̄� 2 + �̄� 2

] (32)

here the ± signs in 𝛼 and 𝜎 must be simultaneously positive, or nega-ive. Now Eq. (30) can be written as (the last term was also written in a

ore convenient form):

𝜕 2 �̄� 0

𝜕 ̄𝑦 2 − ( 𝛼 + 𝑖𝜎) 2 �̄� 0 = − ̄𝜅2

( �̄� + 𝑖 ̄𝐵 �̄� 2 + �̄� 2

) [(Ω̄1 − Ω̄2

)cosh ( ̄𝜅�̄� )

+( ̄Ω1 + Ω̄2 ) sinh ( ̄𝜅�̄� ) ]

(33)

The general solution of this 2 nd order inhomogeneous ordinary dif-

erential equation (ODE) takes the form:

̄ 0 = 𝐶 1 𝑒 ( 𝛼+ 𝑖𝜎) ̄𝑦 + 𝐶 2 𝑒 − ( 𝛼+ 𝑖𝜎) ̄𝑦 + 𝐴 1 sinh ( ̄𝜅�̄� ) + 𝐵 1 cosh ( ̄𝜅�̄� ) + 𝐶 3 (34)

ith the following coefficients:

1 = − ̄𝜅2 ( ̄Ω1 + ̄Ω2 ) cosh ( 𝛼+ 𝑖𝜎) sinh ( ̄𝜅)+( ̄Ω1 − ̄Ω2 ) sinh ( 𝛼+ 𝑖𝜎) cosh ( ̄𝜅)

( 𝛼2 − ̄𝜅2 − 𝜎2 + 𝑖 2 𝛼𝜎) sinh (2 𝛼+ 𝑖 2 𝜎) (�̄� + 𝑖 ̄𝐵 �̄� 2 + ̄𝐵 2

),

2 = �̄�2 ( ̄Ω1 + ̄Ω2 ) cosh ( 𝛼+ 𝑖𝜎) sinh ( ̄𝜅)−( ̄Ω1 − ̄Ω2 ) sinh ( 𝛼+ 𝑖𝜎) cosh ( ̄𝜅)

( 𝛼2 − ̄𝜅2 − 𝜎2 + 𝑖 2 𝛼𝜎) sinh (2 𝛼+ 𝑖 2 𝜎) (�̄� + 𝑖 ̄𝐵 �̄� 2 + ̄𝐵 2

),

1 = − ̄𝜅2 ( ̄Ω1 + ̄Ω2 ) �̄�2 − ( 𝛼+ 𝑖𝜎) 2

(�̄� + 𝑖 ̄𝐵 �̄� 2 + ̄𝐵 2

), 𝐵 1 =

− ̄𝜅2 ( ̄Ω1 − ̄Ω2 ) �̄�2 − ( 𝛼+ 𝑖𝜎) 2

(�̄� + 𝑖 ̄𝐵 �̄� 2 + ̄𝐵 2

), 𝐶 3 = 0

(35)

Eq. (34) can be rewritten in a compact form:

̄ 0 = ℜ (�̄� 0 )+ 𝑖 ℑ

(�̄� 0 )

(36)

ith the real and imaginary coefficients given as:

(�̄� 0 )= Φ1 { Φ2 sinh ( 𝛼�̄� ) sin ( σ�̄� ) + Φ3 cosh ( 𝛼�̄� ) cos ( σ�̄� )

Φ4 cosh ( 𝛼�̄� ) sin ( σ�̄� ) + Φ5 sinh ( 𝛼�̄� ) cos ( σ�̄� ) + Φ6 [(Ω̄1 + Ω̄2

)sinh ( ̄𝜅�̄� )

+ (Ω̄1 − Ω̄2

)cosh ( ̄𝜅�̄� )

]} ,

(�̄� 0 )= Φ1 { Φ7 sinh ( 𝛼�̄� ) sin ( σ�̄� )

Φ8 cosh ( 𝛼�̄� ) cos ( σ�̄� ) + Φ9 cosh ( 𝛼�̄� ) sin ( σ�̄� ) + Φ10 sinh ( 𝛼�̄� ) cos ( σ�̄� ) Φ11

[(Ω̄1 + Ω̄2

)sinh ( ̄𝜅�̄� ) +

(Ω̄1 − Ω̄2

)cosh ( ̄𝜅�̄� )

]}

(37)

The independent constant coefficients Φ𝑘 are presented inppendix A .

From Eq. (22) , written in dimensionless form ( ̄𝑢 = 𝑢 ∕ 𝑢 sh ) , we can nowbtain the expression for the EOF velocity field:

̄ = ℜ (�̄� 0 𝑒

𝑖𝜔𝑡 )= ℜ

(�̄� 0 )cos ( 𝜔𝑡 ) − ℑ

(�̄� 0 )sin ( 𝜔𝑡 ) (38)

Similarly, when imposing an external potential field of the form 𝐸 𝑥 = 0 sin ( 𝜔𝑡 ) = ℑ ( 𝐸 0 𝑒 𝑖𝜔𝑡 ) , the dimensionless velocity for this sine wave ex-

ernal forcing would be:

̄ = ℑ (�̄� 0 𝑒

𝑖𝜔𝑡 )= ℜ

(�̄� 0 )sin ( 𝜔𝑡 ) + ℑ

(�̄� 0 )cos ( 𝜔𝑡 ) (39)

The previous expressions represent the velocity field when an ex-

ernal applied potential field of the form of a cosine or a sine wave is

mposed in a microchannel with different wall zeta potentials, respec-

ively. Next, we analyse the two above-mentioned particular solutions

orresponding to the cases when both walls have the same zeta potential

nd when one of the walls has a negligible zeta potential, as follows:

50

- Equal wall zeta potentials : in this case, the real and imaginary terms

of �̄� 0 are respectively expressed as:

(�̄� 0 EZP

)= Θ1

[Θ2 cosh ( 𝛼�̄� ) cos ( σ�̄� ) + Θ3 sinh ( 𝛼�̄� ) sin ( σ�̄� ) + Θ4 cosh ( ̄𝜅�̄� )

]

(�̄� 0 EZP

)= Θ1

[Θ5 cosh ( 𝛼�̄� ) cos ( σ�̄� ) + Θ6 sinh ( 𝛼�̄� ) sin ( σ�̄� ) + Θ7 cosh ( ̄𝜅�̄� )

](40)

where the subscript EZP denotes "equal zeta potential" and the con-

stant coefficients Θ𝑘 are defined in Appendix B .

The resulting velocity field generated by an externally imposed po-

tential field of the form 𝐸 𝑥 = 𝐸 0 cos ( 𝜔𝑡 ) is given by:

�̄� = ℜ (�̄� 0 EZP

)cos ( 𝜔𝑡 ) − ℑ

(�̄� 0 EZP

)sin ( 𝜔𝑡 ) (41)

and when the applied external potential field is given by 𝐸 𝑥 =𝐸 0 sin ( 𝜔𝑡 ) , the resulting velocity field is:

�̄� = ℜ (�̄� 0 EZP

)sin ( 𝜔𝑡 ) + ℑ

(�̄� 0 EZP

)cos ( 𝜔𝑡 ) (42)

- Negligible zeta potentials at the lower wall : in this special case, the real

and imaginary terms of �̄� 0 are given by:

ℜ (�̄� 0 NZP

)= Ξ1 { Ξ2 cosh ( 𝛼�̄� ) sin ( σ�̄� ) + Ξ3 sinh ( 𝛼�̄� ) cos ( σ�̄� )

+ Ξ4 sinh ( 𝛼�̄� ) sin ( σ�̄� ) + Ξ5 cosh ( 𝛼�̄� ) cos ( σ�̄� ) + Ξ6 { cosh ( ̄𝜅) sinh ( ̄𝜅�̄� ) + sinh ( ̄𝜅) cosh ( ̄𝜅�̄� ) } } , ℑ (�̄� 0 NZP

)= Ξ1 { Ξ7 cosh ( 𝛼�̄� ) sin ( σ�̄� )

+ Ξ8 sinh ( 𝛼�̄� ) cos ( σ�̄� ) + Ξ9 sinh ( 𝛼�̄� ) sin ( σ�̄� ) + Ξ10 cosh ( 𝛼�̄� ) cos ( σ�̄� ) + Ξ11 { cosh ( ̄𝜅) sinh ( ̄𝜅�̄� ) + sinh ( ̄𝜅) cosh ( ̄𝜅�̄� ) } }

(43)

where the subscript NZP stands for "negligible zeta potential" and

the constant coefficients Ξ𝑘 are defined in Appendix C .

The resulting velocity field generated by an externally imposed po-

ential field of the form 𝐸 𝑥 = 𝐸 0 cos ( 𝜔𝑡 ) is given by:

̄ = ℜ (�̄� 0 NZP

)cos ( 𝜔𝑡 ) − ℑ

(�̄� 0 NZP

)sin ( 𝜔𝑡 ) (44)

nd when the applied external potential field is given by 𝐸 𝑥 = 𝐸 0 sin ( 𝜔𝑡 ) ,he resulting velocity field is:

̄ = ℜ (�̄� 0 NZP

)sin ( 𝜔𝑡 ) + ℑ

(�̄� 0 NZP

)cos ( 𝜔𝑡 ) (45)

.4. Analytical solution for generic periodic forcings

In general, assuming the applied potential field has a cyclic nature

f period T , it can be written as:

𝑥 ( 𝑡 ) = 𝐸 0 𝑓 (cos ( 𝜔 𝑗 𝑡 ) + 𝑖 sin ( 𝜔 𝑗 𝑡 )

)(46)

ith 𝜔 𝑗 = 2 𝜋𝑗∕ 𝑇 . Let the function 𝐸 𝑥 ( 𝑡 ) be defined in the range − 𝑇 ∕2 ≤ 𝑡 ≤ 𝑇 ∕2 , so

hat its Fourier series can be represented by:

𝑥 ( 𝑡 ) = 𝐸 0

[ 𝑎 0 +

∞∑𝑗=1 𝑎 𝑗 cos ( 𝜔 𝑗 𝑡 ) + 𝑏 𝑗 sin ( 𝜔 𝑗 𝑡 )

] (47)

The coefficients 𝑎 0 , 𝑎 𝑗 , and 𝑏 𝑗 can be computed as:

𝑎 0 = 1 𝑇

𝑇 ∕2

∫− 𝑇 ∕2

𝐸 𝑥 ( 𝑡 ) 𝑑 𝑡, 𝑎 𝑗 = 2 𝑇

𝑇 ∕2

∫− 𝑇 ∕2

𝐸 𝑥 ( 𝑡 ) cos ( 𝜔 𝑗 𝑡 ) 𝑑 𝑡,

𝑏 𝑗 = 2 𝑇

𝑇 ∕2

∫− 𝑇 ∕2

𝐸 𝑥 ( 𝑡 ) sin ( 𝜔 𝑗 𝑡 ) 𝑑𝑡 (48)

Due to the linearity of the governing equations and of the boundary

onditions, the resulting velocity field induced by this generic electric

otential can thus be written as a linear combination of the velocity

elds generated by each of the terms of the Fourier series, so that it has

he general form:

̄ = − 𝜂0 ∈𝜓 2 𝐸 0 𝑢 = 𝑎 0 ℜ (�̄� 0 )+

∞∑𝑗=1

{𝑎 𝑗 [ℜ (�̄� 0 )cos ( 𝜔 𝑗 𝑡 ) − ℑ

(�̄� 0 )sin ( 𝜔 𝑗 𝑡 )

][ ( ) ( ) ]} (49)


Fig. 2. Profiles of the normalized streamwise velocity component for a Newto-

nian fluid with Π = 𝜓 1 ∕ 𝜓 2 = −1 , 0 , and 1 at 𝑅𝑒 = 0.001, �̄�= 100, 𝜔 t = 0 and m = 1.

3

o

M

d

r

m

h

s

S

i

c

u

p

m

i

g

a

m

fl

q

𝑢

t

t

e

i

(

f

r

b

≠ i

t

a

t

o

p

𝜆

fi

T

s

a

a

(

a

a

c

F

p

n

r

t

n

t

a

t

a

l

i

d

fi

s

D

t

F

t

𝑅

t

fl

i

𝑀

r

i

t

t

w

i

o

c

t

I

t

c

[

w

t

l

l

i

M

s

h

e

s

f

o

a

o

w

f

e

F

h

t

. Results and discussion

In the previous section, the analytical solutions were obtained for

scillatory flow of a viscoelastic fluid, described by the multi-mode

axwell model under the sole influence of an oscillating electro-osmosis

riving force, when the channel walls are characterized by asymmet-

ic or symmetric zeta potentials. In this section the case of single mode

odel ( m = 1) is considered, unless otherwise stated. The main objectiveere is to discuss the influences of selected parameters on the oscillatory

hear flow to provide a better understanding about the practical use of

AOSEO, in particular to investigate the flow regime where the veloc-

ty profile is linear and has a magnified amplitude of oscillation. Such

haracteristics are necessary and useful, respectively, if one wishes to

se this flow as a measuring tool to characterize the linear rheological

roperties of viscoelastic fluids, such as the storage ( G ) and loss ( G ")

oduli. Liu et al. [35] has also used the single mode UCM model to

nvestigate periodically forced electro-osmotic flow, but they aimed at

enerally characterizing the velocity profiles and volumetric flow rate as

function of the oscillating Reynolds number, EDL thickness and nor-

alized relaxation time (Deborah number). Even though our general

ow features are similar for a single mode, the focus of our discussion is

uite different and in addition we will also look at a two-mode model.

A general formulation for the dimensionless velocity profile, �̄� = ∕ 𝑢 sh , is given by Eq. (38) as a function of dimensionless time, 𝜔 t , andhe dimensionless channel transverse coordinate, �̄� . Eq. (38) can be fur-

her simplified for simple cases, and we consider the limiting cases of

qual wall zeta potentials (EZP) and of negligible zeta potential (NZP)

n the lower wall, given by Eqs. (41) and (44) , respectively. Eqs. (38) ,

41) and (44) , result from an externally applied potential field of the

orm 𝐸 𝑥 = 𝐸 0 cos ( 𝜔𝑡 ) , which depends on the following dimensionless pa-ameters: �̄�, 𝜔 t , 𝜆𝑛 𝜔 , 𝑅𝑒 , Π and 𝛽𝑛 , where n represents the mode num-

er. Using the weighted-averaged relaxation time 𝜆 = 𝑚 ∑𝑛 =1 𝜆𝑛 𝛽𝑛 , where n

solvent [43] , we define the Deborah number, 𝐷𝑒 = 𝜆𝜔 [44] , whichs a fundamental independent dimensionless number in this flow. Addi-

ional dimensionless quantities that help to understand the flow physics

re the oscillatory Reynolds number ( 𝑅𝑒 = 𝜌𝜔 𝐻 2 ∕ 𝜂0 ) , and in particularhe viscoelastic Mach number ( 𝑀 =

√𝑅𝑒 𝐷𝑒 ), which gives an indication

f the propagation of shear waves (in elastic fluids at rest shear waves

ropagate with constant velocity, 𝑐 = √𝜂∕ 𝜌𝜆).

For purely driven electro-osmotic flow of a Newtonian fluid ( m = 1,= 0), Fig. 2 shows the influence of Π on the normalized velocity pro-les, plotted as a function of the dimensionless transverse coordinate.

he results are calculated from Eq. (38) for Π= –1, 0, and 1, under theame flow conditions of 𝑅𝑒 = 0.001, �̄�= 100, and 𝜔 t = 0. The cases Π= 1nd Π= –1 refer to equal and opposite wall zeta potentials at both uppernd lower walls, respectively, while Π= 0 refers to the case with a neutral

51

no charge) lower wall. As expected, Fig. 2 shows that Π= 1 results in symmetric velocity profile, while for Π= –1 and Π= 0 anti-symmetricnd asymmetric profiles are found, respectively.

The results shown in Figs. 3–6 are calculated using Eq. (44) , and

orrespond to the case of Π= 0 (NZP) and m = 1 (one mode), whereasig. 7 pertains to m = 2 (more specifically, a Newtonian solvent modelus one viscoelastic mode).

Fig. 3 illustrates the oscillatory flow behavior at 𝜔 t = 0, for a Newto-ian fluid on the left-hand side and for a viscoelastic fluid ( 𝜆𝜔 = 5) on theight-hand side, with 𝑅𝑒 varying from 0.001 to 10 and �̄� varying from 5

o 200. Since the fluid on the right plot is elastic, the viscoelastic Mach

umber can be computed and varies from 𝑀 = 0.07 to 𝑀 = 7.1, here dueo the variation of 𝑅𝑒 . Note that when 𝑅𝑒 exceeds the range of validity

ssociated with the existence of a thin EDL, discussed in Section 2.2 ,

his solution, assuming the existence of a thin EDL and the validity of

Boltzmann distribution of ions, no longer shows linearly behaved ve-

ocity profiles.

To further illustrate the influence of Reynolds number, we show

n Fig. 3 that for small 𝑅𝑒 (e.g. 𝑅𝑒 = 0.001) there are no detectableifferences between the Newtonian ( Fig. 3 -(A-i)) and viscoelastic pro-

les ( Fig. 3 -(A-ii)), regardless of the value of �̄�, but provided �̄� is the

ame in both cases. In addition, as the value of �̄� increases and the

ebye layer becomes progressively thinner the velocity profiles across

he channel become more linear. Furthermore, for the Newtonian fluid,

ig. 3 -(B-i) to 3-(D-i) show that at low 𝑅𝑒 oscillations are weak due

o viscous dampening, but oscillatory behavior is slowly enhanced as

𝑒 increases and this is accompanied by a gradual decay in the ampli-

ude of the normalized velocity profile. In contrast, for the viscoelastic

uid ( 𝜆𝜔 = 5) significant changes take place as 𝑅𝑒 and 𝑀 progressivelyncrease from 𝑅𝑒 = 0.01 and 𝑀 = 0.22 in Fig. 3 -(B-ii) to 𝑅𝑒 = 10 and= 7.1 in Fig. 3 -(D-ii). A wave behavior is clearly perceived to form al-

eady at ( 𝑅𝑒 , 𝑀) = (1, 2.2) and as the viscoelastic Mach number furtherncreases the flow becomes largely dominated by elastic waves rather

han by viscous effects with the amplitude of the waves and their spa-

ial frequency increasing with 𝑅𝑒 (this is shown as a compression of the

aves towards the plate with higher zeta potential). Furthermore, as 𝑅𝑒

ncreases to 𝑅𝑒 = 10 the amplitude of the propagating waves decay fastern moving away from the high zeta potential plate ( ̄𝑦 = 1 ), because vis-ous diffusion is not sufficiently fast to transport information towards

he other wall (note that 𝑅𝑒 is proportional to the oscillation frequency).

n contrast, at lower 𝑅𝑒 the elastic waves have a more uniform ampli-

ude, because viscous diffusion can act more effectively across the whole

hannel. This has similarities to what was observed by Cruz and Pinho

44] in their investigation of Stokes‘ second problem with UCM fluids,

ho showed that the penetration depth ( 𝑦 𝑝 ) varies in inverse proportion

o the Reynolds number very much as the boundary layer thickness in

aminar boundary layers for Newtonian fluid flows.

As mentioned above, at low Reynolds number the normalized ve-

ocity profiles for Newtonian and viscoelastic ( 𝜆𝜔 = 5) fluids are linearn �̄� , but for the latter case this is only true at low De (low viscoelastic

ach number). As shown in Fig. 4 , as 𝜆𝜔 progressively increases very

light deviations from linearity gradually appear for 𝑅𝑒 = 0.001 (cf. left-and side), and become more intense as the Reynolds number increases,

ven though those deviations are still small at 𝑅𝑒 = 0.01 (cf. right-handide (RHS)). The deviation from linearity shown on the RHS progresses

rom the high zeta potential wall and is stronger the higher the value

f 𝜆𝜔 , regardless of the value of �̄�. In order to recover linear behavior,

s the fluid elasticity is increased (fluid relaxation time increased) the

scillation frequency must be concomitantly reduced.

As discussed in Section 2.2 , the critical Reynolds number associated

ith the validity of a ionic Boltzmann distribution is 𝑅 𝑒 cr ≈ 0 . 001 , butor a Newtonian fluid the velocity profile would cease to behave lin-

arly for 𝑅 𝑒 cr ≈ 0 . 01 if the ionic Péclet number did not impose a limit.or viscoelastic fluids, the flow is controlled also by the fluid elasticity,

ere quantified by 𝜆𝜔 or 𝑀 . Viscoelasticity also affects the linearity of

he flow response so we need to consider for this fluid two additional


Fig. 3. Profiles of the normalized velocity for a Newtonian

fluid (left-hand side) and a viscoelastic fluid, De = 𝜆𝜔 = 5 (right-hand side) for 𝜔 t = 0, Π= 0 and m = 1, as a function of �̄�, Reynolds ( 𝑅𝑒 ) and viscoelastic Mach ( 𝑀) numbers: (A-i)

𝑅𝑒 = 0.001, 𝑀 = 0; (B-i) 𝑅𝑒 = 0.01, 𝑀 = 0; (C-i) 𝑅𝑒 = 1, 𝑀 = 0; (D-i) 𝑅𝑒 = 10, 𝑀 = 0 and (A-ii) 𝑅𝑒 = 0.001, 𝑀 = 0.07; (B-ii) 𝑅𝑒 = 0.01, 𝑀 = 0.22; (C-ii) 𝑅𝑒 = 1, 𝑀 = 2.2 and (D-ii) 𝑅𝑒 = 10, 𝑀 = 7.1

c

c

l

i

𝑀

t

p

c

i

T

𝑀

v

c

H

t

fl

ritical limits: the critical Deborah number ( 𝐷 𝑒 cr ) and the critical vis-

oelastic Mach number ( 𝑀 cr ) above which the velocity ceases to be

inear. Since all effects are coupled, for 𝑅 𝑒 cr ≈ 0 . 01 the correspond-ng critical Deborah and viscoelastic Mach numbers are 𝐷 𝑒 cr ≈ 10 and cr ≈ 0 . 32 , whereas for the critical 𝑅 𝑒 cr ≈ 0 . 001 we have 𝐷 𝑒 cr ≈ 100 and

he same 𝑀 cr ≈ 0 . 32 . The critical values of De were determined by im-osing a maximum deviation of 10% to the normalized velocity at the

hannel centerline ( ̄𝑦 = 0 ) in relation to the Newtonian linear veloc-

52

ty profile at the instant 𝜔 t = 0, the profile in Fig. 4 -(A-i) as �̄� → ∞.he critical values of 𝑀 are obtained from the following definition:

cr = √𝑅 𝑒 cr 𝐷 𝑒 cr . Incidentally, for those two Reynolds numbers, the

ariation of 𝐷 𝑒 cr with the value of the maximum deviation ( Δ in [%])riterion is approximately linear, i.e., 𝐷 𝑒 cr ≈ 0 . 01Δ∕ 𝑅𝑒 or 𝑀 cr ≈ 0 . 1

√Δ.

ence, to design a micro-rheometer for SAOSEO, we should not exceed

hese critical limits when using either Newtonian ( 𝑅 𝑒 cr ), or viscoelastic

uids ( 𝑅 𝑒 , 𝑀 and 𝐷 𝑒 ) in order to obtain the desired flow field.

cr cr cr


Fig. 4. Profiles of the normalized velocity components for dif-

ferent Deborah numbers ( 𝜆𝜔 ), viscoelastic Mach ( 𝑀) numbers

and �̄� values, at Reynolds number 𝑅𝑒 = 0.001 (left-hand side) and 𝑅𝑒 = 0.01 (right-hand side), at instant 𝜔 t = 0 with Π= 0 and m = 1: (A-i) 𝜆𝜔 = 0, 𝑀 = 0; (B-i) 𝜆𝜔 = 20, 𝑀 = 0.14; (C-i) 𝜆𝜔 = 40, 𝑀 = 0.2 and (D-i) 𝜆𝜔 = 60, 𝑀 = 0.24; (A-ii) 𝜆𝜔 = 0, 𝑀 = 0; (B-ii) 𝜆𝜔 = 20, 𝑀 = 0.45; (C-ii) 𝜆𝜔 = 40, 𝑀 = 0.63 and (D-ii) 𝜆𝜔 = 60, 𝑀 = 0.77.

a

a

t

b

m

w

N

d

a

n

fl

l

v

n

w

m

s

Fig. 5 presents the variation of the normalized velocity profile along

full cycle of oscillation ( 𝜔 t varies from 0 to 2 𝜋), for �̄� = 100, Π= 0nd 𝑅𝑒 = { 0.001, 10 } . Newtonian and viscoelastic fluids have plots onhe left and right-hand-side, respectively. At low 𝑅𝑒 , here represented

y 𝑅𝑒 = 0.001, Fig. 5 -(A) shows that both fluids fluctuate in a linearanner, but the viscoelastic fluid oscillates with a higher amplitude

hich changes with 𝜔 t especially when 𝜔𝑡 = 𝜋∕2 and 3 𝜋/2, while theewtonian fluid oscillates with a lower amplitude and less detectable

ifferences with the variation of 𝜔 t . Fig. 5 -(B), pertaining to 𝑅𝑒 = 10,

53

lready shows that even though the maximum amplitude of oscillation

ear the wall is the same as for the lower 𝑅𝑒 = 0.001 case, both fluidsuctuate now in a non-linear manner, with the viscoelastic fluid oscil-

ating more intensively due to the fluid elasticity (i.e. the waves decay

ery slowly with distance to the wall), while the waves of the Newto-

ian fluid are quickly dampened on going from the EDL wall to the wall

ith a negligible zeta potential. When critical values of the relevant di-

ensionless parameters are exceeded, this figure clearly shows that this

et-up should not be used for the purpose of SAOSEO.


Fig. 5. Profiles of the normalized velocity for a New-

tonian fluid (left-hand side) and a viscoelastic fluid,

De = 𝜆𝜔 = 5 (right-hand side) for �̄� = 100, Π = 0, m = 1, and as a function of 𝜔 t and Reynolds number: (A)

𝑅𝑒 = 0.001, (B) 𝑅𝑒 = 10.

Fig. 6. Variation of the normalized velocity at �̄� = 0 . 95 with 𝜔 t/ 2 𝜋 for 𝑅𝑒 = 0.001, �̄� = 100, Π= 0, m = 1 and as a function of De = 𝜆𝜔 .

n

i

Π v

p

N

t

l

t

m

i

v

t

i

t

a

r

o

h

p

1

4

i

(

v

m

c

n

o

c

p

c

b

0

i

Fig. 6 shows the influence of varying 𝜆𝜔 from 0.01 to 10 on the

ormalized velocity profile for one full cycle at the position �̄� = 0 . 95 ,.e. close to the high zeta potential surface for 𝑅𝑒 = 0.001, �̄� = 100, and= 0. The amplitude of the velocity profile increases with 𝜆𝜔 , and the

elocity becomes progressively out-of phase with the imposed electric

otential, due to the conspicuous increase in the fluid elasticity.

The addition of even a small amount of polymer to an otherwise

ewtonian fluid alters significantly the fluid rheology which is a func-

ion of fluid concentration. In Fig. 7 we consider the simplest polymer so-

ution, a two-mode fluid, where the first mode corresponds to the New-

onian solvent ( 𝜆1 = 0 and viscosity 𝜂1 ) and the second mode to the poly-er additive with relaxation time 𝜆 and viscosity 𝜂 . The total viscos-
2 2
54

ty, 𝜂0 = 𝜂1 + 𝜂2 , is kept constant, while varying the solvent and polymeriscosities to obtain solutions at different concentrations. The concen-

ration is proportional to 1 − 𝛽1 = 𝜂2 ∕ ( 𝜂1 + 𝜂2 ) and in the plots we assessts influence with values of 1 − 𝛽1 = 0.1, 0.2, 0.3, 0.4 and 0.5, respec-ively. The influence of concentration is investigated for low ( 𝜆𝜔 = 0.1)nd high ( 𝜆𝜔 = 5) Deborah numbers, as shown in Fig. 7 -(A) and (B),espectively. At low De identical Newtonian-like velocity profiles are

btained, while significant variation in the amplitude was realized for

igher De on account of the elastic amplification of elastic waves. The

lotted profiles are all at the instant of 𝜔 t = 0, 𝑅𝑒 = 0.001, Π= 0, and �̄�=00.

. On the use of electro-osmosis for SAOS rheology

The previous section discussed the analytical results obtained by

nspection of the velocity profiles as a function of relevant quantities

̄𝜅, 𝜔 t , 𝜆𝑛 𝜔 , 𝑅𝑒 , 𝑀 , Π) and defined critical numbers beyond which theelocity field is no longer linear in �̄� . This section discusses the develop-

ent of a microchannel rheometer for SAOS, but working on the prin-

iples of EOF as a measuring tool.

On characterizing the linear viscoelastic rheological properties of

on-Newtonian fluids by means of small amplitude oscillatory electro-

smotic shear flow (henceforth denoted SAOSEO), we quantify spe-

ific SAOSEO storage ( G ′ ) and loss ( G ″ ) moduli from which it will be

ossible to determine the spectra of relaxation times ( 𝜆𝑛 ) and of vis-osity coefficients ( 𝜂𝑛 ) as in standard SAOS. This requires SAOSEO toe imposed under the operational conditions of very small 𝑅𝑒 ( 𝑅𝑒<

.001), large �̄� (e.g. ( ̄𝜅≥ 100)) and low De ( 𝜆𝜔 ≤ 100), thus correspond-ng to low 𝑀 ( 𝑀 ≤ 0.32) to ensure a homogenous shear flow, with a


Fig. 7. Effect of 1 − 𝛽1 = 𝜂2 ∕ ( 𝜂1 + 𝜂2 ) on the profiles of the normalized velocity for a two-mode polymer solution

(mode 1: Newtonian solvent, mode 2: polymer additive)

at 𝑅𝑒 = 0.001, Π= 0, �̄�= 100 at instant 𝜔 t = 0 for Deborah ( 𝜆𝜔 ) and Mach ( 𝑀) numbers of: (A) 𝜆𝜔 = 0.1, 𝑀 = 0.01 and (B) 𝜆𝜔 = 5, 𝑀 = 0.07.

Fig. 8. Schematic diagram illustrating small ampli-

tude oscillatory electro-osmotic shear flow (SAOSEO)

under operating conditions of very small 𝑅𝑒 and large

�̄�, leading to a flow with similar characteristics to that

of SAOS in rotational shear.

t

s

o

T

a

t

i

a

e

F

g

s

I

E

i

𝑢

a

ℜ

ℑ

a

u

𝐸

t

𝑢

w

t

r

𝑢

w

a

𝜙

𝜙

s

t

𝑥

t

Δ

m

v

ime-dependent linear velocity profile. The fluid contained inside a

traight microchannel is forced by an externally imposed potential field

f the form 𝐸 𝑥 = 𝐸 0 sin ( 𝜔𝑡 ) between the microchannel inlet and outlet.he fluid inside the microchannel oscillates in a sinusoidal mode at an

ngular frequency 𝜔 , with an amplitude varying with time t , as illus-

rated in Fig. 8 , in a fashion that is similar to what was already described

n Fig. 5 -(A) at small 𝑅𝑒 = 0.001 and high �̄�= 100 for both Newtoniannd viscoelastic fluids. The velocity profile across the channel is linear,

xcept in the vicinity of the upper wall.

The profile of the normalized velocity, schematically shown in

ig. 8 for negligible zeta potential at the lower wall, Π= 0, isiven in Eq. (45) , but by imposing now the conditions of a very

mall 𝑅𝑒 and a large �̄�, that equation can be further simplified.

ndeed, the second term on the left hand-side of the momentum

q. (30) , 𝑖𝑅𝑒 [ ( �̄� + 𝑖 ̄𝐵 )∕( �̄� 2 + �̄� 2 ) ] , can be neglected so that by integrat-ng Eq. (30) leads to (we skip the details for conciseness):

̄ 0 = ℜ (�̄� 0 )+ 𝑖 ℑ

(�̄� 0 )

(50)

where the real and the imaginary terms of the complex velocity function

re given as:

(�̄� 0 )= 1 2

(�̄�

�̄� 2 + ̄𝐵 2

){ ( 1 + Π)

[1 − cosh ( ̄𝜅�̄� ) cosh ( ̄𝜅)

]+ ( 1 − Π)

[�̄� − sinh ( ̄𝜅�̄� ) sinh ( ̄𝜅)

]} ,

(�̄� 0 )= 1 2

(�̄�

�̄� 2 + ̄𝐵 2

){ ( 1 + Π)

[1 − cosh ( ̄𝜅�̄� ) cosh ( ̄𝜅)

]+ ( 1 − Π)

[�̄� − sinh ( ̄𝜅�̄� ) sinh ( ̄𝜅)

]} (51) For large �̄�, Eq. (51) can be further simplified by dropping the sinh

nd cosh, which are responsible for the sharp velocity gradient near the

pper wall due to the EDL thickness effect, leading to:

ℜ (�̄� 0 )= (

�̄�

�̄� 2 + �̄� 2

) { ( 1 + Π) + �̄� ( 1 − Π)

2

} ,

ℑ (�̄� 0 )= (

�̄�

�̄� 2 + �̄� 2

) { ( 1 + Π) + �̄� ( 1 − Π)

2

} (52)

55

So, as a result of imposing an external potential field of the form

𝑥 = 𝐸 0 sin ( 𝜔𝑡 ) , the ensuing velocity field is a function of time and ofhe dependent variable �̄� in the following way:

= 𝑢 sh ( 1 + Π) + �̄� ( 1 − Π)

2

{ �̄�

�̄� 2 + �̄� 2 sin ( 𝜔𝑡 ) + �̄�

�̄� 2 + �̄� 2 cos ( 𝜔𝑡 )

} (53)

here the first term of Eq. (53) is in-phase with the imposed poten-

ial 𝐸 𝑥 (viscous response) and the second term is out-of-phase (elastic

esponse). Alternatively, the previous equation can be written as:

= 𝑢 sh ( 1 + Π) + �̄� ( 1 − Π)

2 1 √

�̄� 2 + �̄� 2 sin ( 𝜔𝑡 + 𝜙) (54)

here 𝜙 represents the phase difference between the imposed potential

nd the resulting velocity profile, and it is defined as:

= cos −1 (

�̄� √�̄� 2 + �̄� 2

) = sin −1

( �̄� √

�̄� 2 + �̄� 2

) (55)

For a purely viscous fluid 𝜙 = 0 , while for a purely elastic material= 𝜋∕2 (in SAOS 𝛿 represents the phase difference between the imposed

train and the resulting shear stress, note that 𝛿 = 𝜋∕2 − 𝜙). The position of tracer particles can also be determined from the in-

egration of the velocity profile, 𝑥 − 𝑥 0 = 𝑡 ∫𝑡 0

𝑢 𝑑𝑡 , resulting in:

= 𝑥 0 − 𝑢 sh ( 1 + Π) + �̄� ( 1 − Π)

2 𝜔 √�̄� 2 + �̄� 2

{cos ( 𝜔 𝑡 + 𝜙) − cos

(𝜔 𝑡 0 + 𝜙

)}(56)

The maximum displacement of a particle over a full cycle of oscilla-

ion is given by:

𝑥 max = 𝑢 sh ( 1 + Π) + �̄� ( 1 − Π)

𝜔 √�̄� 2 + �̄� 2

(57)

In practice, the SAOSEO test can be easily implemented in straight

icrofluidic channels, by measuring the velocity using a particle image

elocimetry (PIV) system or by tracing the displacement of individual


t

𝑢

t

A

u

s

t

I

𝑢

𝑢

l

t

e

t √

i

e

l

i

m

Δ

a

r

w

a

d

K

E

s

𝐺

w

r

o

m

r

n

b

t

e

b

t

t

c

f

p

e

a

h

e

a

o

e

q

I

w

i

c

t

b

c

F

d

s

s

t

e

5

i

s

o

c

t

n

o

fl

e

h

m

1

b

fl

s

r

l

t

m

t

t

p

i

A

f

s

r

E

p

f

C

S

t

racer particles using particle tracking velocimetry (PTV). To evaluate

sh and Π for an applied 𝐸 0 potential, we need to know the zeta po-entials of the walls (besides ∈ and 𝜂0 ), which are not easily available.lternatively, we propose a simple technique that simultaneously eval-

ates 𝑢 sh and Π, by measuring the fully-developed velocity profile underteady flow for an applied 𝐸 0 potential (which should be kept low, as in

he SAOSEO, to guarantee that we are in the linear regime, i.e. 𝑢 sh ∝ 𝐸 0 ).n such condition, the steady-state velocity profile is given as:

ss = 𝑢 sh ( 1 + Π) + �̄� ( 1 − Π)

2 (58)

By fitting the linear velocity profile along �̄� , we can easily determine

sh and Π. Experimentally, if no PIV or PTV system is available, a simpleong exposure photography technique can be used, and by recording

he length of the particle path lines at each �̄� position, and knowing the

xposure time allows the calculation of the velocity profile.

Subsequently, for the same value of 𝐸 0 , applying a sinusoidal po-

ential difference, 𝐸 𝑥 = 𝐸 0 sin ( 𝜔𝑡 ) allows to determine easily the factor�̄� 2 + �̄� 2 (which has similarities with the complex modulus amplitude

n SAOS, defined as |𝐺 ∗ | = √𝐺 ′2 + 𝐺 ′′2 ) by using the same long timexposure technique. For each 𝜔 , the exposure time should be equal or

arger than a full period of oscillation ( 𝛿𝑡 ≥ 𝑇 = 2 𝜋∕ 𝜔 ), and by measur-ng the length of the path lines at each position Δ𝑥 max allows to deter-ine

√�̄� 2 + �̄� 2 since:

𝑥 max = [ 𝑢 sh

( 1 + Π) + �̄� ( 1 − Π) 2

] 2

𝜔 √�̄� 2 + �̄� 2

(59)

nd the function in square brackets was previously determined.

To obtain further information using the SAOSEO technique proposed

equires the use of a PIV or PTV system synchronized with the voltage

ave generator. By measuring the time evolution of the velocity profile

long the wave cycle, allows a fit to Eq. (54) to determine the phase

ifference angle 𝜙 and the parameter √�̄� 2 + �̄� 2 (if not yet determined).

nowing these quantities, �̄� and �̄� can be easily computed since (e.g.

q. (55) ):

�̄� √�̄� 2 + �̄� 2

= cos ( 𝜙) , �̄� √�̄� 2 + �̄� 2

= sin ( 𝜙) (60)

Now, recalling the definitions of �̄� and �̄� , we can easily compute the

torage and loss moduli:

′ = 𝑚 ∑𝑛 =1

𝜂𝑛 𝜆𝑛 𝜔 2

1 + (𝜆𝑛 𝜔

)2 = 𝜔𝐵 = 𝜂0 𝜔 ̄𝐵 , 𝐺 ′′ = 𝑚 ∑𝑛 =1

𝜂𝑛 𝜔

1 + (𝜆𝑛 𝜔

)2 = 𝜔𝐴 = 𝜂0 𝜔 �̄�(61)

here the total shear viscosity can be easily measured at very low shear

ates using a capillary viscometer.

With the proposed technique, we can easily determine the variation

f G ′ and G ″ with 𝜔 , as is usually done in SAOS, and by fitting a multi-

ode model determine 𝜂𝑛 and 𝜆𝑛 parameters. However, this procedure

equires the fluid to remain homogeneous throughout the microchan-

el, i.e., such effects as polymer wall depletion or adsorption must not

e present. In the presence of wall depletion, the fluid response will

end to approach that of a purely-viscous fluid, whereas wall adsorption

nhances elastic effects. The experimental assessment of these effects is

eyond the scope of this work, but the practical implementation of the

echnique calls for their minimization and such endeavor requires fu-

ure research, testing different types of viscoelastic fluids and different

hannel wall treatments in order to assess their influence on the possible

ormation of wall-depletion and wall-adsorption layers. Note that it is

ossible to account for some of these wall effects as was done in the very

laborate general analytical solution of Sousa et al. [45] for steady flow,

nd by Jian et al. [46] for transient EOF of generalized Maxwell fluids,

ere taking into account the depletion layer effects and the Maxwell

lectrical stress at the interface. For other investigations on wall effects

ssociated with polymers, the reader is referred to [47–50] .

56

Note also that in the use of an experimental technique that relies

n following/tracking a suspended particle, its motion is both due to

lectro-osmosis and electrophoresis and due account of the latter is re-

uired for accurate measurements as is well shown in Tatsumi et al. [51] .

n particular, if the zeta potentials for the particles and micro-channel

alls have the same sign, the electro-osmosis and electrophoretic veloc-

ties will have opposite signs for the same applied electric fields, which

an reduce, and even cancel, the particle motion. However, if both ma-

erials have zeta potentials of opposite sign the particle velocities will

e amplified. The time response of the particles will not be affected, be-

ause the characteristic response time of electrophoresis is very small.

or a typical channel of 100 μm width and suspended particles of 1 μm

iameter, time-scales of electro-osmosis (accounted for in the analytical

olution) and of electrophoresis are of the order of 10 ms and 1 μs, re-

pectively. The experimental investigation of Sadek et al. [52] discusses

hese issues and existing techniques to measure the electrophoretic and

lectro-osmotic mobilities are reviewed.

. Conclusions

Analytical solutions for the oscillatory shear flow of viscoelastic flu-

ds driven by electro-osmotic forcing were obtained for the case of a

traight microchannel with asymmetric wall zeta potentials. The rhe-

logical behavior of the fluid is described by the multi-mode upper-

onvected Maxwell model and the work investigates the influence of

he relevant dimensionless parameters ( ̄𝜅, 𝜔 t, 𝜆𝑛 𝜔 , Re, M and Π) on theormalized velocity profiles when imposing an externally potential field

f the form 𝐸 𝑥 = 𝐸 0 cos ( 𝜔𝑡 ) or 𝐸 𝑥 = 𝐸 0 sin ( 𝜔𝑡 ) . Results for viscoelasticuids showed that under certain operating conditions and outside the

lectric double layers the velocity field of the microchannel is linear and

as a large amplitude of oscillation. These conditions are found at si-

ultaneously low Reynolds number ( 𝑅 𝑒 cr < 0.001), thin EDL (e.g. ( ̄𝜅cr ≥00)), low Deborah number De ( 𝐷 𝑒 cr ≤ 100) and low elastic Mach num-er ( 𝑀 cr ≤ 0.32). The flow linearity and magnified amplitude of theseow conditions may allow the use of this small amplitude oscillatory

hear flow induced by electro-osmosis (denoted by SAOSEO) to perform

heological measurements aimed at identifying and measure the rheo-

ogical characteristics of viscoelastic fluids, such as the storage ( G ′ ) and

he loss ( G ″ ) moduli. Experimentally, in a straight microfluidic channel,

easurements can be performed using a micro particle image velocime-

ry (μ-PIV) system or a micro particle tracking velocimetry (μ-PTV) sys-

em, but if neither of these systems is available, a simple long exposure

hotography technique can be used, but reminding that this technique

s limited in terms of obtaining full measurements.

cknowledgments

Samir H. Sadek and Fernando Pinho acknowledge financial support

rom FCT ( Fundação para a Ciência e a Tecnologia ) through scholar-

hip SFRH/BD/85971/2012 and project PTDC/EMS-ENE/2390/2014 ,

espectively. Both authors acknowledge the support of CEFT ( Centro de

studos de Fenómenos de Transporte) and of funding by FCT through

roject UID/EMS/00532/2013 and are greatful to Prof. Manuel A. Alves

or helpful discussions.

onflict of interest statement

The authors declare no conflict of interest.

upplementary material

Supplementary material associated with this article can be found, in

he online version, at doi: 10.1016/j.jnnfm.2019.01.007 .

http://dx.doi.org/10.13039/501100005856http://dx.doi.org/10.13039/501100001871http://dx.doi.org/10.13039/501100001871https://doi.org/10.1016/j.jnnfm.2019.01.007


A

E

𝜎2 )2 62)

Φ

Φ

Φ

Φ

Φ

Φ

Φ

Φ

Φ

Φ

Λ

Λ

A

E

Θ 𝜎2

)

Θ

Θ

Θ

Θ

Θ

Θ

A

E

Ξ2 )2 +

Ξ

Ξ

Ξ

Ξ

Ξ

ppendix A

The ℜ ( ̄𝑢 0 ) and the ℑ ( ̄𝑢 0 ) constant coefficients appearing inq. (37) are:

Φ1 = �̄�2 (

�̄� 2 + �̄� 2 ){

[ cosh ( 2 𝛼) sin ( 2 𝜎) ] 2 + [ sinh ( 2 𝛼) cos ( 2 𝜎) ] 2 }[(

𝛼2 − �̄�2 −

2 = (Ω̄1 − ̄Ω2

){cosh ( 𝛼) sin ( σ)

[Λ1 sinh ( 2 𝛼) cos ( 2σ) + Λ2 cosh ( 2 𝛼) sin ( 2σ)

]+ sinh ( 𝛼) cos ( σ)

[− Λ1 cosh ( 2 𝛼) sin ( 2σ)

+ Λ2 sinh ( 2 𝛼) cos ( 2σ) ]}

cosh ( ̄𝜅) (63)

3 = (Ω̄1 − ̄Ω2

){sinh ( 𝛼) cos ( σ)

[− Λ1 sinh ( 2 𝛼) cos ( 2σ) − Λ2 cosh ( 2 𝛼) sin ( 2σ)

]+ cosh ( 𝛼) sin ( σ)

[− Λ1 cosh ( 2 𝛼) sin ( 2σ)

+ Λ2 sinh ( 2 𝛼) cos ( 2σ) ]}

cosh ( ̄𝜅) (64)

4 = (Ω̄1 + Ω̄2

){sinh ( 𝛼) sin ( σ)

[Λ1 sinh ( 2 𝛼) cos ( 2σ) + Λ2 cosh ( 2 𝛼) sin ( 2σ)

]+ cosh ( 𝛼) cos ( σ)

[− Λ1 cosh ( 2 𝛼) sin ( 2σ) + Λ2 sinh ( 2 𝛼) cos ( 2σ)

]}sinh ( ̄𝜅)

(65)

5 = (Ω̄1 + Ω̄2

) {cosh ( 𝛼) cos ( σ)

[− Λ1 sinh ( 2 𝛼) cos ( 2σ)

− Λ2 cosh ( 2 𝛼) sin ( 2σ) ]+ sinh ( 𝛼) sin ( σ)

[− Λ1 cosh ( 2 𝛼) sin ( 2σ)

+ Λ2 sinh ( 2 𝛼) cos ( 2σ) ]}

sinh ( ̄𝜅) (66)

6 = {[ cosh ( 2 𝛼) sin ( 2 𝜎) ] 2 + [ sinh ( 2 𝛼) cos ( 2 𝜎) ] 2

}[�̄� (𝛼2 − �̄�2 − σ2

)+ 2 �̄� 𝛼𝜎

](67)

7 = (Ω̄1 − Ω̄2

){sinh ( 𝛼) cos ( σ)

[− Λ1 sinh ( 2 𝛼) cos ( 2σ)

− Λ2 cosh ( 2 𝛼) sin ( 2σ) ]+ cosh ( 𝛼) sin ( σ)

[− Λ1 cosh ( 2 𝛼) sin ( 2σ)

+ Λ2 sinh ( 2 𝛼) cos ( 2σ) ]}

cosh ( ̄𝜅) (68)

8 = (Ω̄1 − Ω̄2

){cosh ( 𝛼) sin ( σ)

[− Λ1 sinh ( 2 𝛼) cos ( 2σ)

− Λ2 cosh ( 2 𝛼) sin ( 2σ) ]+ sinh ( 𝛼) cos ( σ)

[Λ1 cosh ( 2 𝛼) sin ( 2σ)

− Λ2 sinh ( 2 𝛼) cos ( 2σ) ]}

cosh ( ̄𝜅) (69)

9 = (Ω̄1 + Ω̄2

){cosh ( 𝛼) cos ( σ)

[− Λ1 sinh ( 2 𝛼) cos ( 2σ)

− Λ2 cosh ( 2 𝛼) sin ( 2σ) ]+ sinh ( 𝛼) sin ( σ)

[− Λ1 cosh ( 2 𝛼) sin ( 2σ)

+ Λ2 sinh ( 2 𝛼) cos ( 2σ) ]}

sinh ( ̄𝜅) (70)

1 = �̄�2

2 cosh ( ̄𝜅) (�̄� 2 + �̄� 2

){[ cosh ( 𝛼) cos ( 𝜎) ] 2 + [ sinh ( 𝛼) sin ( 𝜎) ] 2

}[(𝛼2 − �̄�2 −

1 = �̄�2

2 (�̄� 2 + �̄� 2

){[ sinh ( 2 𝛼) cos ( 2 𝜎) ] 2 + [ cosh ( 2 𝛼) sin ( 2 𝜎) ] 2

}[(𝛼2 − �̄�2 − 𝜎

57

+ 4 𝛼2 𝜎2 ] (

10 = (Ω̄1 + Ω̄2

){cosh ( 𝛼) cos ( σ)

[Λ1 cosh ( 2 𝛼) sin ( 2σ)

− Λ2 sinh ( 2 𝛼) cos ( 2σ) ]+ sinh ( 𝛼) sin ( σ)

[− Λ1 sinh ( 2 𝛼) cos ( 2σ)

− Λ2 cosh ( 2 𝛼) sin ( 2σ) ]}

sinh ( ̄𝜅) (71)

11 = {[ cosh ( 2 𝛼) sin ( 2 𝜎) ] 2 + [ sinh ( 2 𝛼) cos ( 2 𝜎) ] 2

}[�̄� (𝛼2 − �̄�2 − σ2

)− 2 �̄� 𝛼𝜎

](72)

1 = 2 �̄� (𝛼2 − �̄�2 − 𝜎2

)+ 4 �̄� 𝛼𝜎 (73)

2 = 2 ̄𝐵 (𝛼2 − �̄�2 − 𝜎2

)− 4 �̄� 𝛼𝜎 (74)

ppendix B

The ℜ ( ̄𝑢 0 EZP ) and the ℑ ( ̄𝑢 0 EZP ) constant coefficients appearing inq. (40) are:

2 + 4 𝛼2 𝜎2 ] (75)

2 = [− Λ1 cosh ( 𝛼) cos ( σ) − Λ2 sinh ( 𝛼) sin ( σ)

]cosh ( ̄𝜅) (76)

3 = [− Λ1 sinh ( 𝛼) sin ( σ) + Λ2 cosh ( 𝛼) cos ( σ)

]cosh ( ̄𝜅) (77)

4 = Λ1 {[ cosh ( 𝛼) cos ( σ) ] 2 + [ sinh ( 𝛼) sin ( σ) ] 2

}(78)

5 = [Λ1 sinh ( 𝛼) sin ( σ) − Λ2 cosh ( 𝛼) cos ( σ)

]cosh ( ̄𝜅) (79)

6 = [− Λ1 cosh ( 𝛼) cos ( σ) − Λ2 sinh ( 𝛼) sin ( σ)

]cosh ( ̄𝜅) (80)

7 = Λ2 {[ cosh ( 𝛼) cos ( σ) ] 2 + [ sinh ( 𝛼) sin ( σ) ] 2

}(81)

ppendix C

The ℜ ( ̄𝑢 0 NZP ) and the ℑ ( ̄𝑢 0 NZP ) constant coefficients appearing inq. (43) are:

4 𝛼2 𝜎2 ] (82)

2 = sinh ( 𝛼) sin ( σ) [Λ1 sinh ( 2 𝛼) cos ( 2σ) + Λ2 cosh ( 2 𝛼) sin ( 2σ)

]+ cosh ( 𝛼) cos ( σ)


] (83) 3 = cosh ( 𝛼) cos ( σ)


]+ sinh ( 𝛼) sin ( σ)


] (84) 4 = cosh ( 2 𝛼) sin ( 2σ)

[− Λ1 sinh ( 𝛼) cos ( σ) + Λ2 cosh ( 𝛼) sin ( σ)

]+ sinh ( 2 𝛼) cos ( 2σ)

[Λ1 cosh ( 𝛼) sin ( σ) + Λ2 sinh ( 𝛼) cos ( σ)

] (85) 5 = sinh ( 2 𝛼) cos ( 2σ)


]+ cosh ( 2 𝛼) sin ( 2σ)

[− Λ1 cosh ( 𝛼) sin ( σ) − Λ2 sinh ( 𝛼) cos ( σ)

] (86) 6 = Λ1

{[ sinh ( 2 𝛼) cos ( 2σ) ] 2 + [ cosh ( 2 𝛼) sin ( 2σ) ] 2

}∕ sinh ( 2 ̄𝜅) (87)


Ξ

Ξ

Ξ

Ξ

Ξ

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

7 = cosh ( 𝛼) cos ( σ) [− Λ1 sinh ( 2 𝛼) cos ( 2σ) − Λ2 cosh ( 2 𝛼) sin ( 2σ)



] (88)8 = cosh ( 𝛼) cos ( σ)

[Λ1 cosh ( 2 𝛼) sin ( 2σ) − Λ2 sinh ( 2 𝛼) cos ( 2σ)



] (89)9 = cosh ( 2 𝛼) sin ( 2σ) cosh ( 𝛼) sin ( σ)

[− Λ1 − Λ2 sinh ( 𝛼) cos ( σ)

]+ sinh ( 2 𝛼) cos ( 2σ)


] (90)10 = cosh ( 2 𝛼) sin ( 2σ)

[Λ1 sinh ( 𝛼) cos ( σ) − Λ2 cosh ( 𝛼) sin ( σ)

]+ sinh ( 2 𝛼) cos ( 2σ)

[− Λ1 cosh ( 𝛼) sin ( σ) − Λ2 sinh ( 𝛼) cos ( σ)

] (91)11 = Λ2

{[ sinh ( 2 𝛼) cos ( 2σ) ] 2 + [ cosh ( 2 𝛼) sin ( 2σ) ] 2

}∕ sinh ( 2 ̄𝜅) (92)

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