pressure sensor positioning in an electrokinetic microrheometer device: simulations of...

RESEARCH PAPER

Pressure sensor positioning in an electrokinetic microrheometerdevice: simulations of shear-thinning liquid flows

T. J. Craven • J. M. Rees • W. B. Zimmerman

Received: 5 November 2009 / Accepted: 21 January 2010 / Published online: 5 February 2010

� Springer-Verlag 2010

Abstract A novel design for a microrheometer is simu-

lated and tested using finite element modeling techniques.

Non-Newtonian fluid obeying the Carreau viscosity model

is driven through a microchannel T-junction using electro-

osmosis. A range of shear rates, and hence viscosities, is

produced as the fluid is forced to turn the corner of the

T-junction. Thus, the design has the potential to enable the

constitutive viscous parameters to be determined from a

single microfluidic experiment. Three-dimensional simula-

tions are performed for a broad range of Carreau constitutive

parameters. The pressure fields on the microchannel walls,

floor, and ceiling are shown to be sensitive to the Carreau

parameters that determine the fluid’s shear-thinning

behavior. The fluid dynamics theory and numerical results

described in this article pave the way for a detailed analysis

of the corresponding inverse problem, that is, to determine

the values of the Carreau constitutive parameters from the

pressure field measured by optimal positioning of cheap

piezo electric pressure transducers embedded into the inner

surface of the microchannel network.

Keywords Microfluidics � Rheometry � Non-Newtonian �Electrokinetic flow � Computational fluid dynamics

1 Introduction

The use of computational modeling for the design of

microfluidic devices is widespread. Owing to their laminar

nature and precise controllability, microchannel flows lend

themselves well to numerical simulation. Models for

electrokinetic and pressure-driven flows of Newtonian

liquids in microchannels are well known and have been

experimentally validated (Gao et al. 2005; Rawool and

Mitra 2006; Rawool et al. 2006). Results from accurate and

reliable computational models can be used to determine

quantities of interest that are not directly measurable, such

as the constitutive parameters of the viscosity model, by

solving an inverse problem (Gavrus et al. 1996; Szeliga

et al. 2006; Pujos et al. 2007).

A two-dimensional numerical study by Zimmerman

et al. (2006) demonstrated that such an inverse problem was

globally unique and solvable for a wide range of constitutive

parameters of the more complicated Carreau viscosity

model, using two-dimensional simulations of electro-

osmotically driven flows in a microchannel T-junction. The

mean and variance of the back wall pressure profile were

found to map uniquely to the time relaxation constant and

the exponential index. These concepts are expanded in this

article with a view to using cheap piezo electric pressure

transducers for measuring the pressure field on microchan-

nel surfaces. Bandulasena et al. (2008, 2009) used micron

resolution particle image velocimetry (l-PIV) to validate

numerical simulations of the pressure-driven flow of power

law fluids in a microchannel T-junction. This coupled

experimental–computational approach was used to suc-

cessfully infer the constitutive parameters of a power law

fluid from the simulated velocity and pressure fields using

inverse methods. However, the equipment is expensive, not

portable, and requires the microchannel to be optically

T. J. Craven (&) � W. B. Zimmerman

Department of Chemical and Process Engineering, University

of Sheffield, Newcastle Street, Sheffield S1 3JD, UK

e-mail: [email protected]

W. B. Zimmerman


J. M. Rees

Department of Applied Mathematics, University of Sheffield,

Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK


123

Microfluid Nanofluid (2010) 9:559–571

DOI 10.1007/s10404-010-0573-8

transparent. If a device could use pressure only to infer

constitutive parameters then it would be cheaper, more

portable and could be fabricated from opaque materials if

desired.

Since the viscosity of non-Newtonian fluids is a function

of the shear rate, either multiple experiments at different

shear rates need to be performed in order to obtain the

viscous parameters (Kang et al. 2005; Guillot et al. 2006)

or a range of shear rates needs to be set up within a single

experiment (Lee and Tripathi 2005; Srivastava and Burns

2006; Degre et al. 2006). A microchip-based method for

extracting viscous parameters of polymer solutions was

developed by Lee and Tripathi (2005). Fluorescent inten-

sities were used to obtain direct measurements of the

channel dilution ratio. Since the pressures of the chip’s

reservoirs were known they were able to use the momen-

tum balance equations to calculate the viscosities for each

channel of the chip. Srivastava and Burns (2006) designed

a self-calibrating microfluidic capillary viscometer for

obtaining power law parameters of non-Newtonian fluids.

As the fluid flowed into a long capillary tube, a limited

range of shear rates was produced. Degre et al. (2006) used

the visualization technique of particle image velocimetry to

extract flow stress and shear rates from measured velocity

profiles. The constitutive relations were obtained from

stress and strain relationships.

The novelty of our approach is to use a T-junction

geometry to set up a flow system exhibiting many shear rates

simultaneously, hence the viscous information of the fluid

viscosity function is present in a single flow (Bandulasena

et al. 2010). This information is then extracted indirectly by

measuring some of the flow characteristics, such as the

pressure at different positions in the channel (Zimmerman

et al. 2006), or the average velocity vector in a given region

of the flow (Bandulasena et al. 2008). In this article, we

consider a fully three-dimensional model of electro-

osmotically driven flow of a Carreau fluid. The micro-

channel has a curved depth profile because of the etching

process used in its manufacture, causing the flow field to

differ from the idealized two-dimensional case considered

in Zimmerman et al. (2006). The electric double-layer

movement at the channel walls is also present on the channel

floor and ceiling, acting to constrain the fluid more tightly

than in the two-dimensional case (Craven et al. 2008). The

resulting pressure profiles on the walls, floor, and ceiling

provide much information and their use for inferring the

Carreau parameters is demonstrated. We perform a para-

metric study solving the model for a wide range of values of

the Carreau parameters k and n and suggest optimal posi-

tioning for piezo electric pressure transducers.

The characteristics of electrokinetic flow are outlined in

Sect. 2. The computational fluid dynamics model used is

described in Sect. 3. Results are presented in Sect. 4 and

their implications are discussed in Sect. 5. Finally, the

conclusions are summarized in Sect. 6.

2 Electrokinetic flow in a microchannel T-junction

Figure 1 shows the channel cross-section and T-junction

layout considered in this work. Microchannels are etched

into the chip material (usually glass or PDMS) using

photolithography. When the channel has been produced, a

second chip is bonded to the first in an oven to produce a

ceiling for the microchannel. The channel is then com-

pletely sealed to the outside world except at the inlet and

outlet openings. The etching process produces channels

with curved sides, which for simplicity we have modeled as

quarter circles. In reality, the etching process may produce

profiles that exhibit a degree of non-uniformity along the

length of a channel. In the present study, we assume that

the channel profile is uniform along its entire length. The

microchannel considered in this work has a depth of 80 lm

and a width of 200 lm. Typical microchips have inlet and

outlet channel sections of around 10 mm, in which the flow

is uniform and can therefore be approximated by analytical

solutions if required.

2.1 Electro-osmotic flow

Flows in microchannels can be pressure-driven or may be

induced through electro-osmosis, where thin charged layers

Fig. 1 Cross-section of microchannel and T-junction configuration

560 Microfluid Nanofluid (2010) 9:559–571

123

of fluid close to the microchannel walls experience a

driving force in the presence of an applied electric field. In

the latter case, an electric double layer (EDL) forms when a

fluid containing free ions is in contact with a charged

surface. Trapped charges in the wall material are balanced

in the fluid region adjacent to the wall by the redistribution

of ions, resulting in the formation of an oppositely charged

layer of fluid next to the wall. Away from the charged wall,

the free ions balance one another out, so that the fluid has

zero net charge. If a potential difference is applied along

the channel, the charged fluid in the EDL experiences an

electric body force, and begins to move along the channel.

The fluid outside the EDL is brought into motion through

viscous forces, and is dragged along by the motion of the

fluid in the EDL resulting in an electro-osmotic flow. The

EDL is very thin, with a typical width of around 10 nm for

an ion concentration of 0.01 mol m-3 (Horiuchi and Dutta

2004; Craven et al. 2008).

Electro-osmotic flows can be controlled precisely

through the potential difference at the electrodes, while the

flat plug flow profile that results allows species to be

transported without the lateral mixing effects of pressure-

driven flows (Hunter 1992). In an electro-osmotic flow, the

shear rate outside the EDL is zero across the channel, and

therefore the fluid viscosity remains constant in straight

channel sections. In contrast, for pressure-driven flows,

velocity gradients across the width of the channel produce

a range of shear rates, and therefore the viscosity of such a

fluid changes across the width of the channel. At micro-

scales inertial effects are negligible, and fluid flow is

laminar and deterministic in nature. A typical electro-

osmotic flow velocity for an aqueous buffer is around

1 mm s-1 (MacInnes 2002), which corresponds to a Rey-

nolds number of around Re = 10-3 in a microchannel with

a characteristic length of 100 lm. In practice, the strength

of the applied electric field is set to achieve the desired

flow velocity. The strength of the field required to achieve

a certain velocity depends on the wall zeta potential, which

is linked to the concentration of free ions (measured by the

pH) of the solution. Such flows lend themselves particu-

larly well to numerical simulation owing to their laminar

nature.

2.2 Non-Newtonian viscosity model

Polymer solutions exhibit non-Newtonian flow behavior,

where the response to shear and strain forces is non-linear.

For example, long chain polymer molecule solutions, such

as xanthan gum, offer a smaller resistance to deformation

as the rate of shearing is increased. The apparent viscosity

of these so-called shear-thinning liquids varies with the

shear rate, leading to a generalized Newtonian model for

the stress tensor. In more concentrated polymer solutions,

viscoelastic behavior is also common, where fluid elements

react in an elastic fashion to the strain. Several rheological

models have been employed to model shear-thinning

polymeric liquids with considerable success. The Carreau

viscosity model is an empirical formula that relates the

apparent viscosity of a fluid element to the shear rate

experienced by that element. The model falls into the

generalized Newtonian class of models because it assumes

a symmetric, Newtonian stress tensor, with only the scalar

viscosity itself varying as a function of the shear rate. More

complicated models include strain tensors whose compo-

nents must satisfy additional conservation equations, and

can represent both shear rate dependent and visco-elastic

fluid behavior.

The Carreau viscosity (Richardson and Chhabra 2008)

of a fluid is governed by the shear rate and four model

parameters; the viscosity at zero shear rate l0, viscosity at

infinite shear rate l?, the time relaxation constant k, and

the exponential index n. It is given by the expression

lð _cÞ ¼ l1 þ l0 � l1ð Þ 1þ ðk _cÞ2h in�1

2

; ð1Þ

where _c is the shear rate. When the shear rate is zero, the

exponential term is equal to one and therefore lð _cÞ ¼ l0;

the Carreau viscosity is equal to the viscosity at zero shear

parameter l0. When the shear rate is large, the term

1þ ðk _cÞ2h in�1

2

tends to zero (with n\ 1), and the viscosity

approaches l?, the viscosity at infinite shear rate. In

practice, for shear-thinning fluids, the zero shear viscosities

of polymer solutions obeying the Carreau model are very

small, of the order of 10-4 l0. Its main purpose in the

model is to avoid numerical problems that can arise as a

result of a near-zero viscosity value in Cauchy’s moment

equation. The model parameters l0 and l? control the

minimum and maximum viscosities attained by the fluid.

In order to model a shear-thinning fluid, we set l0 [ l?.

The other two parameters k and n determine the shape of

the viscosity curve over a range of shear rates. In a steady

microchannel flow, a finite range of shear rates is present.

We focus on how the Carreau viscosity model behaves

across the shear-thinning regime (i.e., not on the upper/

lower limiting shear rates).

For a shear-thinning fluid, the viscosity curve falls away

from l0 at _c ¼ 0 and decays exponentially, tending to the

lower limit l? at high values of k_c: The steepness of the

viscosity curve is controlled by the exponential index n,

with smaller values of n giving steeper viscosity curves.

The value of k serves to scale the viscosity curve along the

shear rate axis. For small values of k the viscosity curve is

stretched out along the shear rate axis, resulting in a more

gentle drop in viscosity over the shear range. For large kthe viscosity curve is compressed along the shear rate axis,

leading to a sharp drop in the viscosity curve across the

Microfluid Nanofluid (2010) 9:559–571 561

123

shear range. When k ? 0 or n ? 1, the Carreau viscosity

becomes constant, with a Newtonian viscosity l0.

In combination, k and n determine the shape of the vis-

cosity curve while the range of shear rates present in the flow

determine the range of viscosities. We wish to use the vis-

cous information present in a flow to determine the Carreau

parameters through solving an inverse problem (this will

form the subject of a subsequent study). It is therefore

important that the shape of the viscosity curve displays

sensitivity to changes in k and n across the range of values in

which we are interested. It is shown in Sect. 3.3 that, given a

known shear rate range, there are upper and lower limits for

the values of k and n beyond which the Carreau viscosity

curve becomes insensitive to changes in the Carreau

parameters. We chose the parameter ranges over which we

performed our simulations accordingly to ensure that the

viscosity curve was sensitive to k and n across the whole

range. For a more detailed analysis of the behavior of the

Carreau viscosity function, see Zimmerman et al. (2006).

3 Computational model

In this section we describe a mathematical model for the

electoosmotic flow of non-Newtonian fluids in micro-

channels, its associated boundary conditions, and the finite

element methods used to solve the flow in the T-junction.

3.1 Model equations

The electro-osmotic flow of a fluid driven by an applied

electric field is governed by the conservation equations for

applied electric potential, fluid momentum, and mass. We

assume here that temperature variation in the flow has a

negligible effect on the viscosity, density, and electrical

conductivity of the fluid. We assume also that there is zero

net charge on the fluid, so that electrophoresis of charged

particles does not occur, and that the flow is steady state.

The equations are solved in non-dimensional form using

the reference scales l0, q, E, L, and U for viscosity, den-

sity, electric field, length, and velocity, respectively. Den-

sity is taken as constant, the Carreau zero shear viscosity is

used as the reference viscosity, and pressure is scaled

according to viscous forces owing to the laminar nature of

microchannel flows. The electric potential is scaled with

the applied electric field strength to produce non-dimen-

sional potential gradients of the order of unity in the

channel. The non-dimensional variables are

x� ¼ x

L; u� ¼ u

U; p� ¼ pL

l0U; /� ¼ /

EL; ð2Þ

where x*, u*, p*, and /* are the non-dimensional length,

velocity, pressure, and applied electric potential,

respectively. In practice, the slip velocity U is set through

varying the strength of the applied electric field E. The two

quantities are related by the Helmholtz-Smoluchowski

relation

U ¼ � efE

ls

; ð3Þ

where e and f are the electrical permittivity of the fluid and

the zeta potential at the edge of the EDL. The solvent

viscosity, ls, is used in this expression for the velocity at

the wall because the EDL is sufficiently thin that the

macromolecules that give rise to the higher zero shear

viscosity of the non-Newtonian fluid are not present in the

EDL owing to their radius of gyration being of comparable

size to the layer itself. This known as the depletion effect,

and is a valid assumption so long as the molecules do not

have a tendency to attach to the wall material (Tuinier and

Taniguchi 2005).

Dimensional analysis shows that the flow is character-

ized by the four dimensionless parameters

Re ¼ qUL

l0

; l�1 ¼l1l0

; k� ¼ kU

L; n� ¼ n; ð4Þ

where Re, l�1, k*, and n* are the non-dimensional

Reynolds number, Carreau viscosity ratio, time shear

relaxation parameter, and exponential index, respectively.

Together, these four parameters determine the flow profile.

Hereafter we refer only to non-dimensional variables

unless otherwise indicated. Dropping the * notation for

clarity, the set of non-dimensionalized governing equations

for /, u, and p are

r � r/ ¼ 0; ð5ÞRe u � ruð Þ � r � r ¼ 0; ð6Þr � u ¼ 0; ð7Þ

where r denotes the total stress tensor

rðp; uÞ ¼ �pIþ 2lð _cÞeðuÞ; ð8Þ

where I is the identity matrix, lð _cÞ is the Carreau shear

rate-dependent viscosity, and eðuÞ is the rate of strain

tensor, which is defined as

eðuÞ ¼ 1

2ruþ ðruÞTh i

: ð9Þ

The generalized shear rate (Owens and Phillips 2005) is the

square root of the contraction of the rate of strain tensor

with itself, and is an invariant of the rate of strain tensor. It

is given by

_c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieðuÞ : eðuÞ

p: ð10Þ

Finally, the Carreau viscosity is given in non-dimensional

form by the expression


123

lð _cÞ ¼ l1 þ 1� l1ð Þ 1þ ðk_cÞ½ �n�1

2 : ð11Þ

3.2 Boundary conditions

Electro-osmotic flow arises as a result of the movement of

a nanometer-scale charged layer of fluid at the micro-

channel walls, known as the EDL. The computational

effort required to resolve the layer directly is substantial

(Craven et al. 2008). Instead, it is common practice not to

resolve the layer, instead replacing it with a locally one-

dimensional approximation to the flow velocity at the edge

of the layer (Ermakov et al. 1998; MacInnes 2002; Zim-

merman et al. 2006). The fluid velocity at the edge of the

EDL is related to the electric field through the Helmholtz-

Smoluchowski condition Eq. 3, which is given in the non-

dimensional form by

u ¼ �r/: ð12Þ

Under the assumptions of constant electric permittivity and

zero charge on the fluid away from the walls, the applied

electric potential field is independent of the velocity and

pressure fields. This allows the two fields to be solved

sequentially, requiring less memory space and processing

time than if the equations were solved simultaneously. We

induce an electric field in the T-junction microchannel by

imposing potentials at the inlet and outlet boundaries.The

inlet and outlet potentials are

/ ¼ 4 at inlet; / ¼ 0 at outlet: ð13Þ

The inlet potential is set higher than the outlet potential by

the approximate non-dimensional length of the shortest

path between the inlet and the outlet; this results in the

potential gradient in the inlet channel being approximately

equal to -1, so that the electro-osmotic velocity is unity in

the inlet channel section. The fluid pressure is set equal to

zero at the inlet and outlet boundaries to achieve a zero

pressure difference along the channel, so that the flow is

driven solely by electro-osmosis, i.e., p = 0. It is assumed

that the microchannel wall material is electrically

insulating, so that at the walls there is a zero potential

gradient across the boundary,

r/ � n ¼ 0: ð14Þ

3.3 Numerical methods

The equations were discretized using the Galerkin finite

element method. The microchannel T-junction has a line of

symmetry down its center, and this was exploited to reduce

the number of grid elements needed in the finite element

mesh. Symmetry boundary conditions were used along the

channel center, i.e., r/ � n ¼ 0; u � n̂: The inlet and outlet

channel sections extend two channel heights from the

junction. The actual inlet and outlet channel sections are

much longer, but this shortening in the model is necessary

to keep the mesh to a manageable size. When inlet and

outlet channels are short, the boundary conditions at the

inlet and outlet can influence the solution within the

T-junction domain. In practice, a balance must be found

between the number of mesh elements used and the accu-

racy of the approximate inlet and outlet boundary condi-

tions. This was achieved through preliminary simulations

and a trial and error approach.

The finite element mesh used is shown in Fig. 2. The

mesh is made up of 40,010 tetrahedral elements, with the

element sizes in the region of the sharp corner specified as

1% of the channel depth. The largest flow gradients occur

at the right angled corner of the T-junction, with the higher

mesh concentration in this area needed to resolve the larger

gradients.

The electric field and the velocity and pressure fields

were solved sequentially, since the electric field is inde-

pendent of the velocity and pressure fields under the model

assumptions of constant electric permittivity and zero

charge on the fluid. The electric potential / was discretized

using Lagrange quadratic elements, leading to a system of

linear equations with 59,115 degrees of freedom (DOF).

The velocity and pressure fields were discretized using

Lagrange quadratic and Lagrange linear elements, respec-

tively, resulting in a linear system with 185,494 DOF.

The electro-osmotic flow of a Carreau non-Newtonian

fluid is a highly non-linear problem, which produces a

system of linear equations which becomes less well-posed

as k is increased and n is decreased. This increasing ill-

posedness causes convergence problems for memory effi-

cient iterative solvers such as GMRES. At higher values of

k (around k[ 3), the memory requirements of the pre-

conditioner algorithm become as large as the memory

required to solve the system using a direct solver. This

scenario compelled us to use a direct solver based on the

multi frontal method and LU factorisation (SPOOLES

Fig. 2 T-junction half-domain. Half of the full T-junction is modeled

to take advantage of the line of symmetry along the channel center

line, reducing the computational effort required to solve the model.

The inlet and outlet channels have a length of 1.5 L, where L is the

channel height


123

2.21). Computations were carried out using the Comsol

Multiphysics finite element PDE engine controlled using

custom scripts in MATLAB (Zimmerman 2006). The

hardware used was a 3 GHz machine running the Linux

operating system. Peak memory usage was around 2.5 GB;

memory usage increased as the condition number of the

linear system worsened at higher values of k and n.

3.4 Operating parameter ranges

The flow solution is dependent on four non-dimensional

quantities introduced in Eq. 4; the Reynolds number

Re, l�1, and the non-dimensional Carreau constitutive

parameters k and n. In this work, we vary the constitutive

parameters of the fluid while assuming constant values for

Re and l�1:While the zeta potential that governs the electro-osmotic

slip velocity of a fluid is not known a priori for different

mixtures of polymers and macromolecules, in practice the

desired slip velocity can be attained by varying the strength

of the applied electric field in the microchannel. Experi-

mental measurement of the slip velocity would require a

dedicated component in the microchannel network situated

before the T-junction. This, however, is a separate prob-

lem, which can be considered as part of future experi-

mental work. In order to keep the dimensionality of the

parameter space to two, we assume that Re and l�1 are

fixed and small, in accordance with electro-osmotic and

non-Newtonian flows in the literature:

Re ¼ 10�3; l�1 ¼ 10�4: ð15Þ

A characteristic of polymer solutions is their large zero

shear viscosity, which is usually several orders of magni-

tude larger than the solvent viscosity. For example, human

blood (Devarakonda et al. 2007), mayonnaise, and dilute

polyethelene glycol solutions (Guillot et al. 2006) and

dilute linear polystyrene solutions (Tanner 2002) all satisfy

the Carreau model and have viscosity ratios in the region

10-5 B l�1 B 10-2. The ranges of the Carreau parameters

were chosen to include a broad range of fluids. The max-

imum value for k was chosen at the limit of model con-

vergence. We were unable to achieve convergence for

k = 6 and n \ 0.5, so we took k = 5 as the upper limit for

the parametric study. The value of n for various fluids

always falls in the range 0 \ n \ 1, with lower values

corresponding to a sharper drop in viscosity under

shearing.

With the above considerations in mind, we varied the

Carreau parameters over the ranges

0:1\k\5; ð16Þ0:1\n\0:9: ð17Þ

It was decided to resolve the k range using two different

step lengths. A step size of dk = 0.05 was used between

0.1 B k B 1 to examine the behavior in this region in

detail, and a larger step size of dk = 1 was used for the

range 1 B k B 5 in order to cover a broader k-range while

keeping computational requirements to a manageable level.

We used a step size of dn = 0.1 for all n values. The model

was run for all combinations of the parameter values

expressed in the vectors

k¼½0:10;0:15;0:20; . . .;0:85;0:90;0:95;1:0;2:0;3:0;4:0;5:0�;ð18Þ

n ¼ ½0:9; 0:8; 0:7; 0:6; 0:5; 0:4; 0:3; 0:2; 0:1�: ð19Þ

In total, 208 separate simulations were performed, requir-

ing a computation time of approximately 400 h.

3.5 Solution strategy

The non-linear nature of the model equations combined

with the large mesh sizes used in this study mean that for

each distinct pair of Carreau parameter values, solving the

model takes several hours. As the shear-thinning effect

becomes more pronounced at high values of k, the number

of iterations required by the Newton solver increases, and

the closeness of the initial guess to the true solution starts

to have a significant effect on the computation time. In

extreme cases close to the limit of model convergence, the

solution can fail to converge at all if the starting guess is

too far from the actual solution. It is therefore important

when performing a large number of simulations to ensure

that all available information is used in order to speed up

computations. As an example, the time taken to solve the

model for the parameter values k = 3, n = 0.5 was around

180 min when an initial solution vector of zeros was used.

Using the solution at k = 3, n = 0.6 as an initial guess, the

solution time is cut in half to 90 min.

Since the aim is to collect solution data at equally

spaced points in parameter space, it makes sense to use the

closest available already computed solution as the initial

guess for he next, more non-Newtonian parameter point.

By choosing to find solutions for n-values decreasing from

n = 1.0, which corresponds to Newtonian flow (regardless

of the value of k), the previously computed solution can be

used as an initial condition for the current solution. Each

time the value of k is changed, the value of n is stepped

starting from n = 1.0 again, so that the stored Newtonian

solution can be used as a good initial guess to the actual

solution.

1 SPOOLES is a library for solving sparse real and complex linear

systems of equations, written in the C language, and is available from

http://www.netlib.org/linalg/spooles/.


123

Another important factor in the design of a solution

strategy for a large parametric problem is the amount of

available memory and data storage. The finite element

node solution data for each pair of Carreau parameters is

quite large at around 3 MB, and so it is impractical to store

many solutions in working memory while the model is

being solved. Instead, past solutions can be saved to the

hard disk for later retrieval and analysis. As we are prin-

cipally interested here in the pressure profile on the walls of

the T-junction microchannel, this information was extrac-

ted from the model solution at each step of the process.

3.6 Pressure data collection

Zimmerman et al. (2006) showed that the pressure profile

on the end wall of a two-dimensional T-junction is sensi-

tive to the values of the Carreau parameters k and n. In

order to test whether this hypothesis holds in three

dimensions, we wish to extract the pressure profile from all

of the inner surfaces of the T-junction microchannel. The

extraction is complicated slightly by the curved shape of

the channel walls, which necessitate the use of polar

coordinates in order to obtain the pressure profile as a ‘flat’

function parameterized by two spatial variables. We wish

to unravel the curved walls of the T-junction to create a

geometry ‘‘net’’ on which the wall pressure profiles can be

plotted. The curved wall surface is a quarter-circle with

radius L, as shown in Fig. 3. Taking h as the wall angle

increasing from the channel floor, points on the wall with

equal separation are traced out by varying h, so that a small

change in the wall angle corresponds to a small change

along the wall surface; dh = L dx. Using this method the

pressure can be sampled at many points on the wall and

plotted on a geometry ‘‘net’’, as shown in Fig. 6.

Owing to the irregular shape of the walls near the cor-

ners, where the inlet and outlet channel sections intersect in

a curved edge, we chose to take the corner at the channel

ceiling as the cut-off point for pressure sampling. The

pressure data on the lower parts of the curved walls in the

corner region (seen in Fig. 4 as the square area containing

the diagonal edge where the walls meet) were ignored to

simplify the task of pressure sampling. The wall pressure

points obtained from each flow profile are stored in

matrices to facilitate data analysis and plotting. For

example, the curved inlet wall has a pressure matrix Piw

with 23 rows and 30 columns, representing an area of

150 lm (the length of the inlet channel) by 115 lm (the

length of the curved wall). The individual matrices for the

curved walls, ceiling, and floor sections were then assem-

bled to produce the T-junction geometry net seen in Fig. 6.

With the pressure data stored in a matrix format, post

processing and analysis is made considerably easier.

4 Results

Given the parametric nature of this study, a large number

of flow profiles were computed. As we are interested here

primarily in how the pressure profile changes with different

k and n values, we have selected a typical flow profile at

k = 3, n = 0.5, which serves to illustrate the features of

the flow including the velocity, viscosity, and electric field

profiles. These values are within the typical range of those

for non-Newtonian flows, e.g., for blood, k = 3.313,

n = 0.3568 (Devarakonda et al. 2007).

Fig. 3 The pressure profile on the curved walls is extracted by

referring to points on the wall using cylindrical polar coordinates. The

wall pressure profile can be unravelled onto a flat ‘‘net’’ using the

transformation x = L h, where x is the distance along the wall surface

from the channel floor (from point A towards point B)

Fig. 4 Top–down (xy) view showing isosurfaces of electric potential.

Surfaces of equal potential appear aslines near the inlet and outlet

boundaries as they are near vertical. At the junction, the curved profile

of the channel walls result in the potential field having a small

z-component, so that the electric field points upwards at the corner


123

4.1 Typical flow profile

The applied electric potential distribution in the T-junction

is shown in Fig. 4. The potential falls from a value of 4 at

the inlet boundary to zero at the outlet boundary, resulting

in a potential gradient of r/ = -1 in the inlet channel

section. The potential gradient falls to around r/ ¼ � 12

in

the outlet channel section as the channel volume doubles

after the junction. While the isosurfaces are parallel to the

vertical near the inlet and outlet boundaries, in the vicinity

of the corner they are curved and have a small vertical

component, indicating that the electric field exerts a body

force on the EDL in the z-direction.

The z-component of the applied electric field is signifi-

cant near the corner of the T-junction, and in the center of

the channel (along the symmetry boundary) near the end

wall. The z-component is largest in magnitude at the walls

immediately upstream and downstream of the corner,

where the vertical force exerted by the field on the EDL is

of the same order of magnitude as the force in the

streamwise direction in the inlet channel. The EDL

z-velocity is positive upstream of the corner, and negative

downstream with a magnitude of 1.5 units. Fluid elements

rise and fall as they turn the corner, an effect caused by the

curved channel cross-section and the curved edge that

results when the T-junction is formed. This significant

vertical flow is a marked departure from two-dimensional

simulations (Zimmerman et al. 2006), and produces a

stronger pressure on the channel floor and ceiling than on

the side walls, as we see in Sect. 4.2.

The familiar plug flow velocity field that is character-

istic of electro-osmosis is observed in the inlet and outlet

channels. The volumetric flow rate in the outlet channel is

half of that in the inlet channel as the flow splits in two at

the junction, resulting in the velocity in the outlet channel

being approximately half of that in the inlet channel. There

is a stagnation point at the center of the channel on the end

wall, which coincides with a point of zero electric field

(zero potential gradient r/ = 0) so that there is zero body

force on the EDL at this point.

As witnessed in previous numerical studies (Ermakov

et al. 1998; MacInnes 2002; Zimmerman et al. 2006), a

numerical singularity occurs in the electric field at the

channel corners. The magnitude of the potential gradient in

the vicinity of the curved corner edge is limited only by

grid density at the corner. The fixed grid used here resulted

in maximum predicted velocities at the corner of about 10

units. We observed that numerical singularities in the

electric field exist not just at the point where the corner

edge joins the channel ceiling, but along the whole top part

of the edge. In other words, a line of electric field singu-

larities exist all along the corner edge, although the largest

magnitude occurs at the ceiling. It was shown by doubling

the mesh density at the corner from 1 to 0.5% of the

channel height that the corner singularity is localized, so

that the increase in mesh resolution did not increase the

magnitude of the electric field outside 0.2 channel heights

of the corner edge. This numerical phenomenon has been

studied in depth in Craven et al. (2008).

The viscosity profile of a Carreau fluid with constitutive

parameters k = 3, n = 0.5 is shown in Fig. 5. In the inlet

and outlet channels, there is a zero lateral velocity gradient

across the channel owing to the plug flow profile of electro-

osmotic flow. The viscosity is therefore constant, taking the

value l = l0 = 1.0. As fluid turns the corner, a range of

shear rates are experienced by the fluid elements as the

EDL drags them around the corner. This results in a

complex viscosity profile, which is illustrated using sur-

faces of constant viscosity in the figure. The maximum

shear rates occur at the corner of the junction, where the

velocity rises sharply. As the fluid is shear-thinning, its

viscosity falls to its minimum value at at corner where the

shear rate is greatest.

The presence of a range of viscosities in a single flow

implies that the viscosity function has been ‘‘sampled’’

over a range of shear rates, so that the information con-

tained within the flow profile can be used to infer the values

of the Carreau parameters k and n. We chose to extract this

Fig. 5 Isosurfaces of viscosity and velocity vectors for k = 3 and

n = 0.5. The viscosity falls from the zero shear viscosity l = 1 at the

inlet as shearing occurs in the corner region. Many shear rates are

present as the fluid is dragged around the corner by the electro-

osmotic flow at the channel walls, resulting in an information-rich

viscosity profile. The smallest viscosity, which corresponds to the

largest shear rate, occurs at the join between the corner edge and the

channel ceiling. Velocity vectors show the electro-osmotic plug flow

profile in the inlet and outlet channels and a stagnation point at the

center of the end wall of the T-junction


123

information by measuring the pressure profile on the T-

junction walls, as described in Sect. 4.2.

4.2 Pressure field

The pressure field on the walls, floor, and ceiling of the

T-junction is shown for k = 3, n = 0.5 in Fig. 6. The pres-

sure rises from the inlet pressure as fluid approaches the

junction, resulting in an area of higher pressure in the inlet

channel. As the fluid turns the corner and diverges, the

pressure drops resulting in an area of fluid with lower pres-

sure than the outlet. As the fluid approaches the outlet, the

pressure balances to the outlet value. The pressure rises and

falls sharply in the vicinity of the channel corner owing to the

electro-osmotic forces dragging the fluid around the corner.

We may convert non-dimensional pressure differences

into physical pressure differences by inverting the original

scaling from Eq. 2:

p ¼ l0U

Lp� � 10l0p�; ð20Þ

where p* is the non-dimensional pressure difference and l0 is

the zero-shear viscosity of the fluid. There is a direct relation-

ship between the zero-shear viscosity of the fluid and the

magnitude of the pressures on the microchannel walls. Typical

values of the zero-shear viscosity range from 10-1 Pa s for

blood, to 102 Pa s and higher for solutions of polymers such as

linear polystyrene. These correspond to pressure differences in

the microchannel in the range 1 � p*\ p\1000 � p* Pa.

Pressure variations along the channel walls are of the

order of 2 non-dimensional units, and change with the

values of the Carreau relaxation time k. As the value of kincreases, the magnitude of the pressure variation on the

channel walls decreases, with the pressure difference

around 1 pressure unit observed for k = 5, n = 0.5 com-

pared with pressure differences of 2 pressure units for

CHANNEL CEILING

y po

sitio

n µm

x position µm100 200 300

100

200

CHANNEL FLOOR

x position µm

y po

sitio

n µm

100 200 300 400

100

200

−1.5

−1

−0.5

0

0.5

1

1.5

−1.5

−1

−0.5

0

0.5

1

1.5

A

B

CD E

Fig. 6 Pressure displayed on

the geometry ‘‘net’’ including

the channel ceiling, walls, and

floor for k = 3 and n = 0.5.

Thetop image shows contours of

absolute pressure on the ceiling

(region A) and inlet side wall of

the microchannel (region B),

while the bottom image shows

pressure contours on the outlet

inside wall (region C), channel

floor (region D), and back wall

(region E). The T-junction was

unraveled by taking advantage

of the circular channel profile,

so that curved walls can be

described using polar

coordinates


123

k = 1, n = 0.5. These magnitudes imply that the physical

pressure variation on the microchannel walls falls in the

range 1–1000 Pa, depending on the zero-shear viscosity of

the particular fluid.

Very large pressures are observed along the corner edge

boundary, which appears to be a line of mesh-dependent

pressure singularities. The largest occurs at the point where

the edge joins the channel ceiling, where the magnitude is

800 units. There is a jump in the pressure from much larger

than the inlet pressure to much smaller than the inlet

pressure at the corner edge, although this behavior is

restricted to an area within a radius of 0.2 units of the

corner at the ceiling, and this radius reduces further moving

down the edge (as z decreases), so that the pressure jump

lower down the wall is much more localized. We exclude

these areas from our results, based on the fact that they are

artificial features of the pressure field resulting from the

approximate electro-osmotic velocity boundary condition,

Eq. 12.

4.3 Pressure sensitivity

Pressure profiles on the inner microchannel surfaces for a

range of constitutive parameters were simulated with the

aim of selecting the optimal positions at which to position

micropressure transducers. This data could then be used to

infer the values of the Carreau parameters by solving an

inverse problem. In order to be able to infer k and n

accurately, we require that the pressure profiles are sensi-

tive to the changing values of k and n across the range of

parameter values of interest (0.1 B k B 5 and 0.1 B n

B 0.9). In other words, we require that a small change in

the values of the Carreau parameters result in an appre-

ciable change in the wall pressure profile.

Herein we refer to the spatial position on the wall

pressure profile net using the coordinates (x, y). The pres-

sure data can be described as a function with four inputs p

= p(x, y, k, n), where x, y are surface coordinates and k, n

are the Carreau time relaxation constant and exponential

index respectively. The sensitivities of the pressure to

changes in the Carreau parameters are just the partial

derivatives of pressure with respect to k and n:

skðx; y; k; nÞ ¼op

ok; ð21Þ

snðx; y; k; nÞ ¼op

on: ð22Þ

We are interested in a measure of the total sensitivity of the

pressure at a particular point on the interior surface over the

whole range of Carreau parameters of interest. A natural

measure is therefore the sum over all k and n values of the

squared parameter sensitivities. The interior surface

regions with larger sensitivities summed over all

constitutive values will be more sensitive to changes in kand n over the entire parameter range. For a continuous

pressure profile p(x, y, k, n), we may express the total

sensitivity of the pressure to both of the Carreau parameters

at a given point on the surface as an integral of the sum of

the squared sensitivities over the entire Carreau parameter

range

Wðx; yÞ ¼Z

X

wks2k þ wns2

ndkdn; ð23Þ

where X = (k = 0.1, k = 5) 9 (n = 0.1, n = 0.9) is the

Carreau parameter set of interest, and wk, wn are weighting

constants that are set according to the relative magnitude or

importance attached to the respective parameters. The total

sensitivity W(x, y) is a measure of the sensitivity of the

pressure at each point on the wall to k and n over the entire

Carreau parameter range.

As described in Sect. 3.6, the pressure profile on the

inner microchannel surfaces is stored as a matrix P(k,n) of

discretely sampled pressure data with rows and columns

corresponding to the y and x positions on the wall. We

convert this data into a piece-wise continuous function

p(k,n)(x, y) by linearly interpolating between the pressure

values in the matrix.

The sensitivities sk(k,n) (x, y) and sn

(k,n) (x, y) are calculated

using the central difference approximation for derivatives,

and produce matrices of pressure sensitivities, with each

entry in the matrix corresponding to the sensitivity of a

single pressure point:

Sðk;nÞk ¼ Pðkþdk;nÞ � Pðk�dk;nÞ

2dk; ð24Þ

Sðk;nÞn ¼ Pðk;nþdnÞ � Pðk;n�dnÞ

2dn; ð25Þ

where dk and dn are the constant step sizes between k and n

data points, respectively. The central difference derivative

at a data point depends on the values at the previous and

following data points, and therefore cannot be used at the

first or last data points. Rather than using lower accuracy

forward and backward differencing at these points, we

chose to consider only the interior points and do not cal-

culate the sensitivities at the first and last data points for

both k and n. The central difference approximation is

accurate to an error term that is quadratic in the step size.

The error associated with the approximation is therefore

small since we used step sizes of dk = 0.05, dk = 1, and

dn = 0.1 for our pressure sampling.

The integral of the total sensitivity Eq. 23 is approxi-

mated using the trapezoidal rule of integration. Let S2ðk;nÞk

and S2ðk;nÞn be matrices whose elements are the squared

sensitivities, so that ðS2Þij ¼ s2ij: If we define a matrix of the

weighted sum of squared sensitivities on the walls


123

Qðk;nÞ ¼ wkS2ðk;nÞk þ wnS2ðk;nÞ

n ; ð26Þ

then the integral of squared sensitivities over the parameter

set X is given by the sum

W ¼ 1

4dkdn

XNk�2

i¼2

XNn�2

j¼2

Qðki;njÞ þQðkiþ1;njÞ þQðki;njþ1Þ þQðkiþ1;njþ1Þh i

: ð27Þ

In Eq. 27, Nk and Nn are the number of k and n data points,

respectively. The limits of the sum start at i = 2 and j = 2

and end at Nk - 2 and Nn - 2 because the sensitivities are

not defined for data points on the edge of X owing to the

central difference approximation used in their calculation.

The central difference derivative at points where k = 1

was computed using the Newtonian pressure solution, since

a relaxation time of zero (corresponding to the data point

preceding k = 1 with dk = 1) reduces the Carreau vis-

cosity to a constant Newtonian viscosity.

Contours of total sensitivity of the wall pressure profile

to the Carreau parameters are plotted in Fig. 7, with the

dark areas representing low sensitivity and the light areas

representing high sensitivity. The areas in which the

pressure is sensitive to changes in k and n are the floor

and ceiling just upstream of the junction, and the ceiling

immediately upstream of the corner. The areas adjacent to

the inlet and outlet show little sensitivity since the pres-

sure is constrained to be zero at the inlet and outlet

boundaries. The sensitivity grows very large at the corner

owing to the singularities in the pressure field there. We

exclude the area surrounding the corner edge where

W [ 5 from our sensitivity results, as we expect the

pressure predictions to be unreliable there. It is, however,

likely that the largest pressure variations do occur in the

region of the corner, although because the pressure jumps

from high to low across the corner (see Fig. 6), a discrete

sensor placed at the corner would most likely average out

this variation. From the sensitivity plot, it appears that the

best places to position pressure sensors would be in the

center of the channel ceiling just before the junction, and

on the area of sensitivity following the corner in the outlet

channel.

CHANNEL CEILING

x position µm

y po

sitio

n µm

100 200 300

100

200

CHANNEL FLOOR

x position µm

y po

sitio

n µm

100 200 300 400

100

200

0

1

2

3

4

5

0

1

2

3

4

5

A

B

C D E

Fig. 7 Total sensitivity of the

pressure profile on the inner

surfaces of the microchannel

T-junction to the Carreau

parameters (k, n) over the

entire range of parameter

values. The top image shows

contours of the total sensitivity

on the ceiling (region A) and on

the inlet side wall of the

microchannel (region B), while

the bottom image shows

sensitivity contours on the outlet

inside wall (region C), channel

floor (region D), and back wall

(region E). It is clear that the

region of highest sensitivity is

on the channel roof (region A)

around one channel width

upstream from the T-junction

inlet, while the pressure profile

along the end wall (region E) is

a region of low sensitivity.

Pressure sensors are best

positioned in places where the

pressure sensitivity is high, for

example on the channel ceiling


123

Interestingly, the end wall pressure profile is seen to be

less sensitive to the constitutive parameters than the floor

and ceiling profiles. In two-dimensional simulations

(Zimmerman et al. 2006), the end wall pressure was suc-

cessfully used to infer k and n. Given that there appears to

be more information present in the ceiling pressure profiles

than those of the end wall, this is a strong indication that

inference of k and n is possible using the ceiling profile

alone.

Since the microchannel is fabricated by first etching the

channel layout onto a chip and then bonding a second chip

to the first to create the ceiling, the simplest method of

fitting micropressure sensors to an experimental chip would

be to build them into the ceiling chip before the two chips

were bonded. The fact that the ceiling pressure profile is

the most sensitive of the surface profiles to the Carreau

parameters means that a chip can be fabricated with sensors

on the channel ceiling with the expectation that the sensor

readings can be used to infer the Carreau constitutive

parameters k and n from a single flow experiment.

5 Discussion

Results from three-dimensional simulations of non-New-

tonian electro-osmotic flow in a T-junction microchannel

show a significant vertical velocity component in the flow,

which is of the same order of magnitude as the main

channel flow at the channel walls near the corner, where

the EDL experiences a vertical body force owing to the

curved microchannel walls. This component contributes to

larger pressure variations over the floor and ceiling of the

microchannel compared to the side walls, and especially

the end wall. While the viscosity profile appears to be very

similar to that of equivalent two-dimensional flows (Zim-

merman et al. 2006), the three-dimensional velocity and

pressure fields render the ceiling a better place to position

pressure sensors to infer the fluid constitutive parameters

than the end wall, which we considered in our previous

work.

The increased sensitivity observed here suggests that the

channel ceiling is an excellent place to position micro-

pressure transducers for pressure measurement. The anal-

ysis of sensor positioning, Carreau parameter inference,

and solution uniqueness is too broad to include here and

will form the basis of a future study.

The ability to determine both the k and n values of an

unknown fluid from a single experimental measurement is

a key feature of our potential microrheometer design.

A wide range of shear rates, and hence viscosities, as

induced as the fluid is forced to turn the corners of the

T-junction. In a similar flow in a T-junction, the region of

detectable strain rate of the level of 200 Hz is confined to a

thin region in the immediate vicinity of the corner (Ban-

dulasena et al. 2010). This can be contrasted to the wide-

spread shear field with a modal value of 200 Hz.

Another point relating to the suitability of the micro-

channel T-junction for inferring Carreau fluid parameters is

that the magnitude of the pressure differences on the

interior channel surfaces is directly related to the magni-

tude of the zero shear viscosity of the fluid. This rela-

tionship suggests that the microrheometer is better suited to

measuring complex fluids with high zero shear viscosities,

as the resulting pressure differences will be larger and

therefore easier to measure accurately with micropressure

transducers.

In the present study, we have neglected the effects of

temperature variation in the channel owing to the mecha-

nisms of electrical resistance (joule) heating and viscous

dissipation. It has been shown (Horiuchi and Dutta 2004;

Tang et al. 2004) that the dominant heating effect in a

microchannel flow of the dimensions considered here is

joule heating from the applied electric field. Under typical

electro-osmotic flow conditions, the heating effect is

counteracted by dissipation to the channel walls to the

extent that the temperature increase across the T-junction

itself (where the variations in viscosity occur) is small, of

the order of 1 K. Localized heating occurs at the sharp

corners in the channel, where temperature increases are

much larger, causing a reduction in the fluid viscosity. As

the corners are the points in the flow where the shear rate is

largest, the viscosity there is already small, and the overall

effect is likely to be an apparent drop in the infinite shear

viscosity parameter l? owing to the temperature increase

at the corners. While this effect may affect the ranges over

which the Carreau viscosity is sensitive to changes in its

parameter values, it is unlikely to destroy the close rela-

tionship between the Carreau parameters and the flow field.

Another assumption was that there was no electropho-

resis (electric migration) of charged particles present in the

flow. While this assumption can be quite valid for simple

solvents, the presence of complex macromolecules such as

long chain polymers or even blood cells in the test fluid

may lead to aggregation of particles in the flow that in turn

may acquire net charges and experience an electrophoretic

body force. If polymers attain charges their configuration

may be affected by the electric field as well as the flow

field, which may alter their viscous response so that they no

longer follow the Carreau viscosity model. These consid-

erations are best tested through experimental work; the

numerical evidence here suggests that if these potential

problems are not severe, the electro-osmotic T-junction

flow is suitable for use as a microrheometer for Carreau

complex fluids.


123

6 Conclusions

The electro-osmotic flow of shear-thinning non-Newtonian

liquids obeying the Carreau viscosity function have been

simulated in three dimensions in a microchannel T-junction

geometry for a wide range of constitutive parameters. We

have outlined the key features for producing a novel mi-

crorheometer instrumented with cheap piezo electric

pressure sensors and have carried out a sensitivity analysis

in order to propose the most appropriate positions of these

sensors within the T-channel network. Further work on the

analysis of the related inverse problem necessary for

inferring the constitutive parameters will be addressed in a

future study.

Acknowledgments WZ acknowledges support from EPSRC Grant

Nos. GR/A01435 and GR/S83746. TC would like to thank the Uni-

versity of Sheffield for a doctoral scholarship. We acknowledge

support from the EPSRC Grant EP/E01867X/1 (Bridging the Gap

between Mathematics, ICT and Engineering Research at Sheffield).

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