In this paper, we consider the electroosmotic flow (EOF) of a viscoplastic fluid within a slit nanochannel modulated by periodically arranged uncharged slipping surfaces and no-slip charged surfaces embedded on the channel walls. The objective of the present study is to achieve an enhanced EOF of a non-Newtonian yield stress fluid. The Herschel-Bulkley model is adopted to describe the transport of the non-Newtonian electrolyte, which is coupled with the ion transport equations governed by the Nernst-Planck equations and the Poisson equation for electric field. A pressure-correction-based control volume approach is adopted for the numerical computation of the governing nonlinear equations. We have derived an analytic solution for the power-law fluid when the periodic length is much higher than channel height with uncharged free-slip patches. An agreement of our numerical results under limiting conditions with this analytic model is encouraging. A significant EOF enhancement and current density in this modulated channel are achieved when the Debye length is in the order of the nanochannel height. Flow enhancement in the modulated channel is higher for the yield stress fluid compared with the power-law fluid. Unyielded region develops adjacent to the uncharged slipping patches, and this region expands as slip length is increased. The impact of the boundary slip is significant for the shear thinning fluid. The results indicate that the channel can be cation selective and nonselective based on the Debye layer thickness, flow behavior index, yield stress, and planform length of the slip stripes.

1.
F.
Kamişli
, “
Flow analysis of a power-law fluid confined in an extrusion die
,”
Int. J. Eng. Sci.
41
,
1059
(
2003
).
2.
C.
Qi
and
C.-O.
Ng
, “
Rotating electroosmotic flow of viscoplastic material between two parallel plates
,”
Colloids Surf., A
513
,
355
(
2017
).
3.
C.
Canpolat
,
S.
Qian
, and
A.
Beskok
, “
Induced-charge electro-osmosis of polymercontaining fluid around a metallic rod
,”
Microfluid. Nanofluid.
16
,
247
(
2014
).
4.
T.-A.
Lee
,
W.-H.
Liao
,
Y.-F.
Wu
,
Y.-L.
Chen
, and
Y.-C.
Tung
, “
Electrofluidic circuit-based microfluidic viscometer for analysis of Newtonian and non-Newtonian liquids under different temperatures
,”
Anal. Chem.
90
,
2317
(
2018
).
5.
E. C.
Bingham
,
Fluidity and Plasticity
(
McGraw-Hill
,
1922
), Vol. 2.
6.
W. H.
Herschel
and
R.
Bulkley
, “
Measurement of consistency as applied to rubber-benzene solutions
,”
Proc. Am. Soc. Test Mater.
26
,
621
633
(
1926
).
7.
N.
Casson
, “
A flow equation for pigment-oil suspensions of the printing ink type
,” in
Rheology of Disperse Systems
, edited by
C. C.
Mill
(
Pergamon Press
,
London
,
1959
), pp.
84
104
.
8.
T. C.
Papanastasiou
, “
Flows of materials with yield
,”
J. Rheol.
31
,
385
(
1987
).
9.
E.
Lauga
and
H. A.
Stone
, “
Effective slip in pressure-driven Stokes flow
,”
J. Fluid Mech.
489
,
55
(
2003
).
10.
M.
Cloitre
and
R. T.
Bonnecaze
, “
A review on wall slip in high solid dispersions
,”
Rheol. Acta
56
,
283
(
2017
).
11.
P.
Panaseti
,
A.-L.
Vayssade
,
G. C.
Georgiou
, and
M.
Cloitre
, “
Confined viscoplastic flows with heterogeneous wall slip
,”
Rheol. Acta
56
,
539
(
2017
).
12.
M. M.
Denn
, “
Extrusion instabilities and wall slip
,”
Annu. Rev. Fluid. Mech.
33
,
265
(
2001
).
13.
H.
Walls
,
S. B.
Caines
,
A. M.
Sanchez
, and
S. A.
Khan
, “
Yield stress and wall slip phenomena in colloidal silica gels
,”
J. Rheol.
47
,
847
(
2003
).
14.
S. G.
Hatzikiriakos
, “
Slip mechanisms in complex fluid flows
,”
Soft Matter
11
,
7851
(
2015
).
15.
A. S.
Haase
,
J. A.
Wood
,
L. M.
Sprakel
, and
R. G.
Lammertink
, “
Inelastic non-Newtonian flow over heterogeneously slippery surfaces
,”
Phys. Rev. E
95
,
023105
(
2017
).
16.
A. Y.
Malkin
, “
The state of the art in the rheology of polymers: Achievements and challenges
,”
J. Polym. Sci. A
51
,
80
(
2009
).
17.
T.
Jiang
,
A.
Young
, and
A.
Metzner
, “
The rheological characterization of HPG gels: Measurement of slip velocities in capillary tubes
,”
Rheol. Acta
25
,
397
(
1986
).
18.
P.
Ballesta
,
G.
Petekidis
,
L.
Isa
,
W.
Poon
, and
R.
Besseling
, “
Wall slip and flow of concentrated hard-sphere colloidal suspensions
,”
J. Rheol.
56
,
1005
(
2012
).
19.
U.
Yilmazer
and
D. M.
Kalyon
, “
Slip effects in capillary and parallel disk torsional flows of highly filled suspensions
,”
J. Rheol.
33
,
1197
(
1989
).
20.
A.-L.
Vayssade
,
C.
Lee
,
E.
Terriac
,
F.
Monti
,
M.
Cloitre
, and
P.
Tabeling
, “
Dynamical role of slip heterogeneities in confined flows
,”
Phys. Rev. E
89
,
052309
(
2014
).
21.
J. M.
Piau
, “
Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges: Meso-and macroscopic properties, constitutive equations and scaling laws
,”
J. Non-Newtonian Fluid Mech.
144
,
1
(
2007
).
22.
A.
Ramamurthy
, “
Wall slip in viscous fluids and influence of materials of construction
,”
J. Rheol.
30
,
337
(
1986
).
23.
A.
Fortin
,
D.
Côté
, and
P.
Tanguy
, “
On the imposition of friction boundary conditions for the numerical simulation of Bingham fluid flows
,”
Comput. Meth. Appl. Mech. Eng.
88
,
97
(
1991
).
24.
D. M.
Kalyon
, “
Apparent slip and viscoplasticity of concentrated suspensions
,”
J. Rheol.
49
,
621
(
2005
).
25.
Y.
Damianou
,
G. C.
Georgiou
, and
I.
Moulitsas
, “
Combined effects of compressibility and slip in flows of a Herschel-Bulkley fluid
,”
J. Non-Newtonian Fluid Mech.
193
,
89
(
2013
).
26.
Y.
Damianou
and
G. C.
Georgiou
, “
Viscoplastic Poiseuille flow in a rectangular duct with wall slip
,”
J. Non-Newtonian Fluid Mech.
214
,
88
(
2014
).
27.
P.
Panaseti
and
G. C.
Georgiou
, “
Viscoplastic flow development in a channel with slip along one wall
,”
J. Non-Newtonian Fluid Mech.
248
,
8
(
2017
).
28.
L.
Ferrás
,
J.
Nóbrega
, and
F.
Pinho
, “
Analytical solutions for Newtonian and inelastic non-Newtonian flows with wall slip
,”
J. Non-Newtonian Fluid Mech.
175
,
76
(
2012
).
29.
H. A.
Stone
,
A. D.
Stroock
, and
A.
Ajdari
, “
Engineering flows in small devices: Microfluidics toward a lab-on-a-chip
,”
Annu. Rev. Fluid. Mech.
36
,
381
(
2004
).
30.
X.
Wang
,
C.
Cheng
,
S.
Wang
, and
S.
Liu
, “
Electroosmotic pumps and their applications in microfluidic systems
,”
Microfluid. Nanofluid.
6
,
145
(
2009
).
31.
C.-O.
Ng
and
C.
Qi
, “
Electroosmotic flow of a viscoplastic material through a slit channel with walls of arbitrary zeta potential
,”
Phys. Fluids
25
,
103102
(
2013
).
32.
X.
Zhang
,
Y.
Shi
,
S.
Kuang
,
W.
Zhu
,
Q.
Cai
,
Y.
Wang
,
X.
Wu
, and
T.
Jin
, “
Microscale effects of Bingham-plastic liquid behavior considering electroviscous effects in nano-or microsized circular tubes
,”
Phys. Fluids
31
,
022001
(
2019
).
33.
N.
Bag
and
S.
Bhattacharyya
, “
Electroosmotic flow of a non-Newtonian fluid in a microchannel with heterogeneous surface potential
,”
J. Non-Newtonian Fluid Mech.
259
,
48
(
2018
).
34.
T. M.
Squires
, “
Electrokinetic flows over inhomogeneously slipping surfaces
,”
Phys. Fluids
20
,
092105
(
2008
).
35.
S. S.
Bahga
,
O. I.
Vinogradova
, and
M. Z.
Bazant
, “
Anisotropic electro-osmotic flow over super-hydrophobic surfaces
,”
J. Fluid Mech.
644
,
245
(
2010
).
36.
C.-O.
Ng
and
H. C.
Chu
, “
Electrokinetic flows through a parallel-plate channel with slipping stripes on walls
,”
Phys. Fluids
23
,
102002
(
2011
).
37.
P.
Papadopoulos
,
X.
Deng
,
D.
Vollmer
, and
H.-J.
Butt
, “
Electrokinetics on superhydrophobic surfaces
,”
J. Phys.: Condens. Matter
24
,
464110
(
2012
).
38.
S.
De
,
S.
Bhattacharyya
, and
S.
Hardt
, “
Electroosmotic flow in a slit nanochannel with superhydrophobic walls
,”
Microfluid. Nanofluid.
19
,
1465
(
2015
).
39.
S.
Bhattacharyya
and
S.
Pal
, “
Enhanced electroosmotic flow in a nano-channel patterned with curved hydrophobic strips
,”
Appl. Math. Model.
54
,
567
(
2018
).
40.
A.
Afonso
,
L.
Ferrás
,
J.
Nóbrega
,
M.
Alves
, and
F.
Pinho
, “
Pressure-driven electrokinetic slip flows of viscoelastic fluids in hydrophobic microchannels
,”
Microfluid. Nanofluid.
16
,
1131
(
2014
).
41.
M. H.
Matin
and
W. A.
Khan
, “
Electrokinetic effects on pressure driven flow of viscoelastic fluids in nanofluidic channels with Navier slip condition
,”
J. Mol. Liq.
215
,
472
(
2016
).
42.
S.
Mukherjee
,
P.
Goswami
,
J.
Dhar
,
S.
Dasgupta
, and
S.
Chakraborty
, “
Ion-size dependent electroosmosis of viscoelastic fluids in microfluidic channels with interfacial slip
,”
Phys. Fluids
29
,
072002
(
2017
).
43.
C. L.
Berli
, “
The apparent hydrodynamic slip of polymer solutions and its implications in electrokinetics
,”
Electrophoresis
34
,
622
(
2013
).
44.
E.
Mitsoulis
and
J.
Tsamopoulos
, “
Numerical simulations of complex yield-stress fluid flows
,”
Rheol. Acta
56
,
231
(
2017
).
45.
R. B.
Bird
,
W. E.
Stewart
, and
E. N.
Lightfoot
,
Transport Phenomena
(
John Wiley & Sons
,
2007
).
46.
W. M.
Deen
,
Analysis of Transport Phenomena
(
Oxford University Press
,
New York
,
1998
), Vol. 3.
47.
P.
Panaseti
,
Y.
Damianou
,
G. C.
Georgiou
, and
K. D.
Housiadas
, “
Pressure-driven flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters
,”
Phys. Fluids
30
,
030701
(
2018
).
48.
M.
Philippou
,
Z.
Kountouriotis
, and
G. C.
Georgiou
, “
Viscoplastic flow development in tubes and channels with wall slip
,”
J. Non-Newtonian Fluid Mech.
234
,
69
(
2016
).
49.
R.
Huilgol
, “
Variational inequalities in the flows of yield stress fluids including inertia: Theory and applications
,”
Phys. Fluids
14
,
1269
(
2002
).
50.
L.-H.
Yeh
and
J.-P.
Hsu
, “
Effects of double-layer polarization and counterion condensation on the electrophoresis of polyelectrolytes
,”
Soft Matter
7
,
396
(
2011
).
51.
L.-H.
Yeh
,
M.
Zhang
,
S.
Qian
, and
J.-P.
Hsu
, “
Regulating DNA translocation through functionalized soft nanopores
,”
Nanoscale
4
,
2685
(
2012
).
52.
C. A.
Fletcher
,
Computational Techniques for Fluid Dynamics
, 2nd ed. (
Springer
,
Berlin
,
1991
), Vol. 2.
53.
B. P.
Leonard
, “
A stable and accurate convective modelling procedure based on quadratic upstream interpolation
,”
Comput. Meth. Appl. Mech. Eng.
19
,
59
(
1979
).
54.
S.
Patankar
,
Numerical Heat Transfer and Fluid Flow
(
CRC Press
,
1980
).
55.
C.
Zhao
,
E.
Zholkovskij
,
J. H.
Masliyah
, and
C.
Yang
, “
Analysis of electroosmotic flow of power-law fluids in a slit microchannel
,”
J. Colloid Interface Sci.
326
,
503
(
2008
).
56.
K.
Ellwood
,
G.
Georgiou
,
T.
Papanastasiou
, and
J.
Wilkes
, “
Laminar jets of Bingham plastic liquids
,”
J. Rheol.
34
,
787
(
1990
).
57.
M.
Chatzimina
,
G. C.
Georgiou
,
I.
Argyropaidas
,
E.
Mitsoulis
, and
R.
Huilgol
, “
Cessation of Couette and Poiseuille flows of a Bingham plastic and finite stopping times
,”
J. Non-Newtonian Fluid Mech.
129
,
117
(
2005
).
58.
R. L.
Whitmore
,
Rheology of the Circulation
(
Pergamon
,
1968
).
59.
C. R.
Ethier
and
C. A.
Simmons
,
Introductory Biomechanics: From Cells to Organisms
(
Cambridge University Press
,
2007
).
60.
R. L.
Fournier
,
Basic Transport Phenomena in Biomedical Engineering
(
CRC Press
,
2017
).
61.
J.
Philip
and
R.
Wooding
, “
Solution of the Poisson-Boltzmann equation about a cylindrical particle
,”
J. Chem. Phys.
52
,
953
(
1970
).
62.
A.
Syrakos
,
G. C.
Georgiou
, and
A. N.
Alexandrou
, “
Performance of the finite volume method in solving regularised Bingham flows: Inertia effects in the lid-driven cavity flow
,”
J. Non-Newtonian Fluid Mech.
208
,
88
(
2014
).
You do not currently have access to this content.