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Complex Flows of Immiscible Microfluids and Nanofluids with Velocity Slip Boundary Conditions

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Numerous experiments on the laminar flows of the micro- and nanofluids in the tubes and channels revealed noticeable discrepancies between the measured flow parameters and those predicted by the Navier–Stokes equations due to the tangential slip of the particles at the wall roughness. A brief literature review on the velocity slip and temperature jump boundary conditions at the solid walls for the microfluidic and nanofluidic flows is given. The analytical results on the Poiseuille and Couette flows between the parallel plates and coaxial cylinders, through the tubes and channels of different geometry at the second-order velocity slip boundary conditions, are given. It is shown that due to proper choice of the wall materials and roughness, the efficiency of the microfluidic and nanofluidic devices could be increased by significant decrease of the wall shear stress and apparent viscosity of the complex flow. Laminar shear-driven flow of three immiscible fluids between parallel plates is considered. The velocity field profiles, shear stress, and apparent viscosity are computed. Influence of the model parameters on the computed values is discussed in connection to the Fahraeus–Lindquist effect in micro- and nanofluids.
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Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
COMPLEX FLOWS OF IMMISCIBLE MICROFLUIDS AND
NANOFLUIDS WITH VELOCITY SLIP BOUNARY CONDITIONS1
1Vitalina Cherevko, 2,3Natalya Kizilova
1Kharkov National University, Svobody sq., 4, Kharkov 61022, Ukraine
2Warsaw University of Technology, ul. Nowowiejska 24, Warsaw 00-665
Poland
3Vilnius Gediminas Technical University, J. Basanavičiaus st. 28, Vilnius
03224 Lithuania
Abstract Numerous experiments on the laminar flows of the micro and nano flu-
ids in the tubes and channels revealed noticeable discrepancies between the meas-
ured flow parameters and those predicted by the Navier-Stokes equations due to
the tangential slip of the particles at the wall roughness. A brief literature review
on the velocity slip and temperature jump boundary conditions at the solid walls
for the microfluidic and nanofluidic flows is given. The analytical results on the
Poiseuille and Couette flows between the parallel plates and coaxial cylinders,
through the tubes and channels of different geometry at the second order velocity
slip boundary conditions are given. It is shown due to proper choice of the wall
materials and roughness the efficiency of the microfluidic and nanofluidic devices
could be increased by significant decrease of the wall shear stress and apparent
viscosity of the complex flow. Laminar shear-driven flow of three immiscible flu-
ids between parallel plates is considered. The velocity field profiles, shear stress
and apparent viscosity are computed. Influence of the model parameters on the
computed values is discussed in connection to the Fahreus-Lindquist effect in mi-
cro and nano fluids.
1 Introduction
During the last decade the microfluids (suspensions of microparticles with
d=10-100 μm) and nanofluids (suspensions of nanoparticles with diameters d=10-
100 νm) have become important components of different microunits for mixing
and purification of the microscopic volumes of technical and biological fluids,
biochemical analysis and medical diagnostics in the lab-on-a-chip flow systems,
as well as efficient nanofluid-based microcoolers/heaters and other devices [1-3].
1 Cherevko V., Kizilova N. Complex flows of immiscible microfluids and nanofluids with velocity
slip bounary conditions. In: Nanophysics, Nanomaterials, Interface Studies, and Applications, Springer
Proceedings in Physics , vol. 183, O. Fesenko, L. Yatsenko (eds.). – 2017. – P. 207–230.
https://link.springer.com/chapter/10.1007%2F978-3-319-56422-7_15
207
V. Cerevko and N. Kizilova
Numerous experiments with flows of micro and nano fluids through the micro-
tubes, ducts and channels revealed that the measured pressures, velocites, volu-
metric flows and shear rates had not corresponded to those values computed from
classical Poiseuille and Couette flow solutions in the corresponding geometry with
no slip boundary conditions (BC) [4]. The most essential differences had been
found in the flow patterns, pressure drops, early transition to turbulence and higher
friction losses that could be explained by the bigger influence of the wall rough-
ness on the micro and nano fluid flows [4,5]. Therefore, reformulation of the BC
problem is needed for better understanding of the differences between the flows of
macroscopic fluids and micro/nano fluids.
Applicability of the slip boundary conditions have already been studied for
the Newtonian fluid flows [6], macroscopic liquid flows near biological surfaces
and interfaces [7], walls with special slide coating [8], penetrable walls [9], in
polymer melts [10], and in turbulent flows with boundary slip [11]. The first for-
mulation of the nonlinear slip BC has been proposed by Navier in 1873 in the
form
ˆ
(Tn) v 0 on


where is the boundary of the flow domain
, v
and are the fluid veloc-
ity and stress tensor, and
ˆ
T
n
are normal and tangential unit vectors to the sur-
face, is the ‘friction’ coefficient. Validity of the Navier BC for the fluid flows
through rigid microtubes has been studied in [12].
Nanofluids as suspensions of nanoparticles and polymer molecules ex-
hibit high thermal and electric conductivity, low specific heat, and unique elec-
tromagnetic properties due to high strength, thermal and electric conductivity of
the nanoparticles and their magnetic properties [13]. Classical fluid dynamics and
thermomechanical theories developed for the macroscopic systems are not fully
applicable to the suspensions of nanoparticles as well as to uniform fluids at the
micro and nano scales. Velocity slip, viscous dissipation, thermal creep and non-
continuum effects like scattering at the wall, adhesion and changes in conforma-
tions must be taken into account [14-16] as well as electrokinetic phenomena [17].
For the solid nanoparticles in the concentrated (C>5%) nanofluids the shear-
thickening behavior may also lead to the high pressure gradients for the steady
fluid flow than those predicted by the Poiseuille law [18]. Gas microflows in
MEMS and microfluidic devices can be used for extracting biological samples,
cooling integrated circuits and active control over the aerodynamic forces [19,20].
The first experimental study conducted for the gas flow in the rectangular
glass channels with hydraulic radius =45.5–83.1
h
D m
and silicon channels
with = 55.8–72.4
h
D m
of microminiature Joule–Thomson refrigerators re-
vealed the friction coefficient about 10–30% higher in silica channels and in 3–5
times high in glass channels than those predicted by the Moody chart for the fric-
tion factor against the Reynolds number at different relative roughness h
/D
,
where
is the roughness height [21]. In the micro and nano channels due to the
208
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
tremendous increase in the surface-to-volume ratio the relative roughness becomes
the most influencing factor, which must be accounted for in the BC via complex
geometry of the wall with roughness, as well as complex interaction of the flow
and nanoparticles with the wall.
Experimental study of the fluid flow (1-, 2-propanol and 1-,3-pentanol)
through silicon microchannels with =5;12;25
h
D m
also demonstrated an in-
crease in the friction coefficient by 5–30% depending on the temperature within
the limits compared to the classical computations on the Navier-
Stokes equations with no-slip boundary conditions [22]. Water flow through rec-
tangular stainless steel microchannels with =133-367
T085С
h
D m
and width to height
ratios W/H=0.333-1 has been studied in [23]. The friction factors for both laminar
and turbulent flows have been found deviated from the classical predictions, and
the geometry factor W/H was found to have important effects on the flow. The
laminar to turbulence transition occurred at the critical Reynolds numbers
Re*=200–700 depending on and W/H. The value Re* becomes lower as the
size of the microchannel decreases. Water flow through the stainless steel and
fused silica circular microtubes with diameters D=50–254
h
D
m
and /D
=0.69–
3.5% at Re=100-2000 also exhibited higher friction than those predicted by the
classical fluid dynamics [15]. The difference increased with the decreasing D and
increasing Re values. The flow transition has been observed at Re*=300-900 de-
pending on the microtube diameter D=50–150 m
. For the fluid flows in rectan-
gular metallic channels with widths W=150-600 m
and heights H=22.7-
26.3 m
an approximate 20% increase over the classical theory prediction in the
friction factor at low H/W ratios has been revealed [24]. The water flow through
trapezoidal silicon microchannels with =51.3–168.9
h
D m
and /D
=1.76-
2.85% at Re<1500 demonstrated the friction factor by 8–38% higher than the clas-
sical theory prediction for laminar flows [25].
A good review of literature published between 1983 and 2005 on the ex-
perimental studies of the friction coefficient and laminar to turbulence transition
Re* values in the microchannels and tubes of different geometry and materials is
given in [16]. It is shown when h
/D
<1% the classical computations for the cor-
responding Poiseuille and Couette laminar flows remain still valid. A positive de-
viation of the friction factor from the conventional theory is observed due to the
high roughness and compressibility effects. For instance, smaller friction factors
detected in gas flows through fused silica microtubes with D=10-20 m
are pro-
duced by rarefaction effect.
The results of the abovementioned experimental studies confirmed that
the flow resistances are about 10–90% higher than the theoretical values for the
corresponding geometry and material parameters, and some of them even by
350% over the theoretical predictions [16,21,26].
209
V. Cerevko and N. Kizilova
For lower Re numbers, the required pressure drop is roughly the same as
for the Poiseuille flow, but at bigger Re the pressure gradient becomes higher than
those compared to the predicted by the Poiseuille law due to higher friction at the
wall or/and development of turbulence [15]. The dependence of the flow behav-
iour on the wall material has been shown for the metal, glass and silica tubes. At
the same flow rate and tube diameter the fluid flow through the fused silica micro-
tube requires a higher pressure gradient than through the stainless steel microtube.
2. Problem formulations with slip boundary conditions.
The wall roughness may be taken into account by modeling the fluid
layer that is in contact with the wall as flow in the porous medium [27]. This ap-
proach was applied to the macroflows in [28].
The modified roughness-viscosity model has been proposed in [29] for
circular tubes with axisymmetric roughness. Instead of the conventional dynamic
viscosity
in the Navier-Stokes equations the viscosity R
where
2
R
Re
r
ARe 1exp Re


r


(1)
is the so-called roughness viscosity, r is the radial coordinate, Re uD /
, u is
the average flow velocity,
is the fluid density, Re U /

, U is the velocity
at the top of the roughness element, A is the material-dependent constant. Solution
of the steady incompressible Navier-Stokes equations with modified viscosity in
the form (1) has been found numerically for the no-slip boundary conditions and
compared to the experimental data in [15].
Another effective viscosity model for concentrated nanofluids has been
proposed in the form [30]:
bf
eff 0.3 1.03
pf
1(d/d)C
(2)
where
b
f
is the viscosity of the base fluid, is the diameter of nanoparticles,
p
d
1/3
fAbf
N

fb
d6M is the base fluid equivalent diameter, C is the nanopar-
ticle volume fraction,
b
f
and
b
f
M is the density at temperature T=293 K and
molecular weight of the base fluid, and A
N
is the Avogadro number.
For the concentrated nanofluids the nonlinear approximation
2
eff bf 1 2
(1 k C k C )

 (3)
has also been used. For instance, in [31] 12
k 39.1, k 533.9
values have been
accepted.
The effective density of the suspension can be introduced in a usual form
accepted for mixtures [32]
eff p bf
C(1

 C)
(4)
210
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
where p
is the density of particle material: eq.(4) may also bee generalized for
the set of particles of different densities and concentrations.
Then expressions for the effective heat conductivity k and specific heat
of the nanofluid have been derived in the form [33]
p
c
pbf bfp
b
fB
pbf bfp
4B
Brownian bf bf
p
(k 2k ) 2C(k k )
kk k ,
(k 2k ) C(k k )
kT
k510Ccf(T,C),
d





rownian
(5)
pp bfbf
eff
pbf
cC c (1 C)
cC(1C)

, (6)
where and are heat conductivity and specific heat of the particles
and base fluid, is the Boltzmann constant,
pbf
k,k pbf
c,c
B
k
is the constant dependent on the
material, concentration of the particles and temperature, f( is the function
determined from experiments.
T,C)
The heat transfer by the steady laminar nanofluid flow in the 2d micro-
tube ( d50m
, L 250 m
) has been studied experimentally for the Al2O3,
CuO, SiO2 and ZnO nanoparticles with =25, 45, 65,80 nm and concentrations
C=1-4% in ethylene glycol as a base fluid [34]. The corresponding steady non-
linear incompressible 2d Navier-Stokes equations and heat equation have been
solved by finite volume method. It was found the nanofluid with SiO2 particles
had the highest Nusselt number, followed then by ZnO, CuO, Al2O3, and lastly by
pure ethylene glycol. The Nusselt number for all cases increased with the Rey-
nolds number, volume fraction C of the particles and with decrease of the particle
diameter .
p
d
p
d
Steady 2d Navier-Stokes equations with no-slip boundary conditions are
considered in the most papers on the flows in the microchannels. The roughness is
introduced as small amplitude wall waviness and the solution is sought as power
expansions over the small parameter /D
[26]. The same approach applied to the
macroflows revealed appearance of the slip flow over the wall with regular [35]
and randomly rough [36] walls. The slip coefficient has been derived by using
conformal mapping techniques [37] and method of matched asymptotic [38]. The
numerical results show that microflows are more complex than macroflows [26]
and the pressure drops are up to 65% higher than those in the classical Poiseuille
flow when the roughness /D
rises to 5%. As it was previously shown in the
classic experiments by J. Nikuradze on rough pipes [39], in the macroflows the
same roughness ~5% had no influence on the friction factor [40]. In the micro-
flows apparent fluctuations in flow fields have been found, and influence of the
roughness on the flow profile and the friction factor were different [26].
211
V. Cerevko and N. Kizilova
The Maxwell first order slip model [41] of diffusive reflection of the par-
ticles from the rough surface of the isothermal wall within the so-called Knudsen
layer is
w
2u
uu Kn 0
n


, (7)
where is the wall velocity,
w
uKn / L
is the Knudsen number,
is the mean
free path, L is the characteristic length,
is the tangential momentum accommo-
dation factor (or friction), and
=1 for purely diffuse reflection, has been used
for Lattice-Boltzmann simulation of the microchannel flow [14]. In gases the
mean free pass is determined by the diameters of particles and their numerical
concentration N (number per unite volume) as follows 2
p
d)
1
(2N
[42].
The thickness of the Knudsen layer in the discrete hard sphere model can be
computed as [43]
B
2
p
kT
dp
, (8)
where p is the pressure, while in the continuous media
2p

. (9)
A modification of (7) has been proposed in [44] in the form
w
2Knu
uu 0
1Knf(Kn)n



, (10)
where f( is an empirical parameter to be determined; |fKn) (Kn)| 1
.
The model (7) can be considered at 0.01 Kn 0.1
because at
the no-slip boundary conditions give good results [42,44]. For the flow
regimes with the second-order slip boundary condition is used in the
form proposed by J.C. Maxwell
Kn 0.01
0.1 Kn 1
2
w2
2uKnu
uu Kn 0
n2
n



(11)
In that way, the boundary condition (7) is used for the microchannel
flows, while the condition (11) is more exact for the nanochannel flows, as well as
flows of concentrated nanofluids of nanoparticles or fullerenes (C60 or others).
Then the diffusive reflection and free path may be related to the particles. In the
general form the second order boundary condition (11) has been proposed in the
form [45]
2
2
w1 2 2
uu
uu CKn CKn 0
nn


. (12)
212
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
The term in (12) is the constant slip length [19] that equals to the mean free
pass of the fluid particles. A good review on the experimental data of the coeffi-
cients is given in [44, page 74]. Summarizing the table, we may accept
, for further computations.
1
CKn
1,2
15] 2
C
C
1;1.
1
C [[0.5;1.31]
Molecular dynamics simulations revealed that the velocity slip at the wall
decreases with increase of the ratio /
Kn 0
for both regular and stochastic rough-
ness [46]. In the transition regime the constitutive laws for the stress
tensor, heat flux vector and other parameters break down requiring higher-order
correcting terms [47]. When the coupled heat and mass transfer is studied, the Na-
vier-Stokes and heat balance equations are solved with the modified second order
boundary conditions in the form
.1
2
w
w
2u3(1)KnReT
uu Kn 0,
n2 Ecs
2(2 )Kn T
TT 0,
(1)Prn











(13)
where T and Tw are temperatures in the flow and at the wall, s is the tangential di-
rection (coordinate),
is the energy accommodation coefficient, pV
C/C
is
the ratio of specific heats, Re= uD /
p
C T
is the Reynolds number, is
the Prandtl number, Ec
p
Pr C / k
2
u /
is the Eckert number, k is the heat conductiv-
ity, is the specified temperature difference in the domain. Perfect energy ex-
change is also correspond to
T
1
when the energy of the reflected (scattered)
particles corresponds to the wall temperature. Thermal and tangential momentum
accommodation coefficients have been measured for different typical gases and
surfaces and was shown to be strongly dependent on the material and surface state
[48]. Their values can be reduced by applying suitable surface preparation tech-
niques.
For the second-order slip conditions the effective viscosity the following
expression has been proposed [44] and studied for the Navier-Stokes tube flow
[49]
bf 0
eff
1
2
,
1Kn tan( Kn )



, (14)
where 14
, 0.4
, 064 / 3 (1 4 / b)
,
b
1
.
As it was shown in [50,51], the experimental data on the isothermal
Poiseuille flows of dilute compressible gases in the microchannels better corre-
spond to the following velocity slip condition
w
uu ln 0
s

, (15)
213
V. Cerevko and N. Kizilova
than to the Maxwell condition (13), because the compressible fluid always slips
along the wall in the direction of diminishing density and, hence, for the case of
isothermal flows, in the direction of decreasing pressure [52].
Kinematics of the nanotube’s nanostructure can be modelled by Euler–
Bernoulli plug flow beam theory (EBPF) which was found applicable to the pipe
flows when L/D>10 [53]. Oscillations of the microtubes conveying nanofluids has
been studied using the EBPF theory with no-slip velocity profiles [54].
Therefore the value /
can be considered as a criterion for the no-slip
boundary conditions acceptance. When
the no-slip condition is satisfied,
otherwise significant slip at the wall is present. As it was shown by comparative
numerical simulations of the micro and nano channel flows in diffrent geometries
conducted by molecular dynamics simulations, numerical solutions of the
Boltzmann equation as well as direct computations on the Navier-Stokes equa-
tions, the slip-flow approach is remarkably robust in the meaning that it is
qualitatively accurate and physically relevant [19]. The classical pressure-
driven and shear-driven channels flows being generalized for the velocity slip
and thermal jump BC have the analytical solutions [19,44,55]. It is convenient
for the comparative studies by both experiments and numerical simulations
with determination of the model coefficients in the BC formulations for di-
luted and concentrated micro and nano suspensions.
3. Analytical solutions for the classical Poiseuille and Couette laminar flows
with 2-nd order velocity slip boundary conditions.
The analytical solutions for the Poiseuille flows in the circular pipes and be-
tween the parallel plates and coaxial cylinders [55] and for the Couette flows be-
tween the parallel plates [44] have been obtained for the surfaces with the same
roughness. In this chapter similar solutions as well as a set of new ones are ob-
tained and analyzed for different surfaces, so the BC (12) are accepted in the form
2
wj j
2
j
uu
uu 0,
nn



(16)
where j=1,2, .
2
j 1jjj 2jj
CKn, C Kn


3.1. Couette flow between the parallel plates. The shear-driven laminar
flow between the plates and
y0yh
moved with velocities 11
V(V,0,0)
and accordingly is described by the velocity distribution
with the volumetric rate
and constant shear stress
22
V(V,0,0
12
V (V V )
h
ouette 0
QSv
uette 2 1
(V V
)
1
v(y) y /
C(y)
Co ) / h
h
1
dy(V
2
V)h/2
 , where
is the fluid viscosity.
For the micro and nano flows, the coefficients in the same linear velocity dis-
tribution determined from (16) that give the following results
214
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
slip 12112 21
Couette 21 21
2
slip 12 2112
Couette 21
slip 21
Couette 21
Vh V V V V
v(y)hh
(V V )h 2h( V V )
Q,
2(h )
VV
.
h

 






 



y,
(17)
When 12
0

0
, (17) have the form of classic Couette flow solution; at
12
 (17) gives the expressions presented in [44]; due to the linear velocity
field the 2-nd order term is not influenced the solution. If we assume for
the sake of definiteness then (17) gives
2
VV1
slip slip
12 1 22 1
12
Couette Couette
21 21
(V V) (V V)
v(0)V ,v(h)V
hh



 
(18)
and both differences in (18) are positive when 12
h
, i.e. the difference in
the constant slip lengths of the two plates must exceed the distance between the
plates. This condition is valid for the sufficiently high Knudsen numbers. In con-
nection with development of rheology of the micro and nano fluids, the correction
(18) must be accounted for in the viscosity estimations by the rotational visco-
simeters.
Comparison of the micro (slip BC) with macro (no-slip BC) flow gives
slip 1221
Couette
Couette 21
()(VV)h
QQ 2(h ) 0




when 12
h
, and
slip 2112
Couette
Couette 21
(V V )( ) 0
h(h )
 

 when 12
or 12
h
.
In that way, when 12
h
 the slip BC flow experiences lower shear rate
and lower apparent viscosity with higher flow rate than the no-slip BC flow.
3.2. Couette flow between coaxial rotating cylinders. Two coaxial cylinders
with axis 0z and radiuses and rotated at the angular velocity
1
R2
R1
and
(let’s assume ) are considered (Fig.1). Then the knows solution of the
Navier-Stokes equation for the angular velocity
2
2
1
v
of the fluid at the slip BC is
22 22
22 11 2 1 12
22 22
21 21
22 22
22 11 2 1 1 2
rot 22 22 2
21 21
RR( )RR
1
v(r) r ,
r
RR RR
RR( )RR
1.
RR RR r








(19)
215
V. Cerevko and N. Kizilova
Fig.1. Couette flow between two rotating coaxial cylinders.
For the micro/nano fluid flow the same solution with the BC (16) gives
44
slip 221 11 2
33
22 21 11 12
34 43
21 2 1 1 11 2 2 2
33
22 21 11 12
44
slip 221 11 2
rot 33
22 21 11 12
34 43
21 2 1 1 11 2 2 2
3
2
RA RA
v(r) r
R(R )A R(R )A
RR (R ) R R (R )
1,
r
R(R )A R(R )A
RA RA
(r)
R (R )A R (R )A
RR (R ) RR (R )
R














32
2211112
1
(R )A R (R )A r


, (20)
where 2
1,2 1 .2 1, 2 1, 2 1,2
AR R

 .
At some sets of the parameters 1212
,,,

the values
3222 22
1,2 1,2 2,1 2 1 2 ,1 2 ,1 1 2
slip
1,2 33
22 21 11 12
24
2,1 1,2 2,1 1,2 2,1 1,2 1, 2 1,2
33
22 21 11 12
R(R(R R) R(R R)
v(R)
R (R )A R (R )A
2R)2 RR(R )
R (R )A R (R )A









will be bigger ( ) and the wall shear stress
slip
1,2 1, 2 1, 2
v(R) R
44234 43
221 11 2 1,2 212 1 1 11 2 2 2
slip
1,2
rot 332
22 21 11 121,2
(RA RA)R RR(R ) RR(R )
(R )
(R (R )A R (R )A )R




1
will be smaller than in the corresponding no-slip BC flow. Using this regularity,
the low and high shear stress micro/nano fluid devices could be designed for the
separated layered or intensive mixing flows between the rotating cylinders when
the condition is not valid.
21
RR R
216
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
3.3. Laminar flow in an inclined duct. Laminar x
vv(y)
Couette flow in
a rectangular duct with sufficiently big length L and width inclined at the
angle
y0
to the horizon (Fig.2) is considered.
Fig.2. Schema of the laminar flow in an inclined duct.
At the free surface the kinematic BC is taken in the form
yh
dv(y) 0
dy
. (21)
The velocity, flow rate and shear stress are determined by the well-known ex-
pressions
incl
3
incl
incl b
gsin
v(y) y(2hy),
2
gsin
Qh,
3
y
gsin (h y) 1 ,
h





(22)
where bgh si n

is the shear stress at the bottom of the duct.
For the same solution with velocity slip BC one can obtain from (16), (21) the
following expressions

slip
incl
2
slip
incl
slip
b
incl
gsin
v (y) 2( h ) y(2h y) ,
2
gsin h
Qh(h),
3
y
1,
h








(23)
where ,
are the slip coefficients for the bottom.
Comparing (22) and (23), one may conclude that
217
V. Cerevko and N. Kizilova
slip slip 2incl
incl incl
slip slip
incl incl
incl incl
gsin gsin
v(0) (h )0,v(h) 2(h )h v(h)
2
QQ,
 
 




,
for positive ,
values. Since in some papers the negative coefficient in (12)
has been reported, the comparative results are definitely valid for .
2
C
12
CCKn/h
3.4. Poiseuille flow between the parallel plates. Laminar flow be-
tween the plates (Fig.3) driven by the pressure drop
x
vv(y)
yh
p
pp

be-
tween the inlet x0
p
p
and outlet xL
p
p
at the no-slip BC has the follow-
ing distributions of the flow profiles, volumetric flow rate and shear stress
Fig.3. Poiseuille flow between the parallel plates.
2
23w
paral paral paral
ph y 2 ph
v(y) 1 ,Q , y,
2L h 3 L h





(24)
where w
p
h/L
is the wall shear stress.
The same parabolic velocity profile at the slip BC (16) and different slip
coefficients of the upper and lower plates has the form
2
2
slip 12 21 1 2 12 1 2
paral 12
2
12 12
12
32
slip 12 12 12 1221
paral 12
slip w12
paral
h( 2 ) h ( )
ph
v(y)L2 2h( )
h( ) y
y,
2h ( ) 2
2h 2h ( ) 3h( 2 ) 3( )
2ph
Q,
3L 2h( )
h(
h
 

 













12
12
)y.
2h ( )






(25),
Comparative study of (24) and (25) revealed that the differences in the shear
rates at no-slip and slip BC is constant and becomes negative
218
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
slip 12 12
paral
paral 12
h( )
p0
L2h( )

 



when
12
12 12
2h,
h( )




or (26)
12
12 12
2h,
h( ).




With the constraint (26) the corresponding difference in the flow rates
2
slip 12 21 1 2 12 1 2
paral
paral 12
h( 2 ) h ( )
ph
QQ 2
L2h()



 
could be either positive or negative, but a wide set of the parameters
1212
,,,

providing the condition could be chosen among
them. In that way, the shear rate and apparent viscosity could be decreased by the
wall slip BC providing bigger flow rate in the channel flow at the same pressure
drop
slip paral
paral
QQ
p
. In [44,55] the solution of the problem is given for the plates with the
same slip coefficients.
3.5. Poiseuille flow in the circular tubes. The pressure-driven (
p
pp

)
axisymmetric ( 0
 ) laminar flow x
vv(r)
through the tube with axis, ra-
dius R and length L is described by the following velocity profile, flow rate and
shear stress
24
pois max pois pois wall
2
rp
vv1 ,Q R,
8L R
R



r
,
(27)
where ,
2
max
vpR/(4

L)
wall
p
R/2L
.
For the micro and nano fluids with BC (16) the same solution has the form
[19,44,55]

2
2
slip
pois 2
slip 22
pois
slip
pois
pois
pR r R
v(r) 1 2 ,
4L R R
p
QRR4(R),
8L
,







(28)
where ,
are the slip coefficients for the circular wall.
Comparison of (27) and (28) demonstrates that when
slip pois
pois
QQ/R
1,2
C
or at the high Knudsen numbers Kn . According to the data for
presented in [44], the nanofluid flows benefits of higher efficiency due to bigger
flow rates at the same pressure drop driving the flow. Note that for the microfluids
(
1
R C / C 2
0
) the slip BC flow rate is always lower than the classical flow.
219
V. Cerevko and N. Kizilova
3.6. Poiseuille flow in the tubes of elliptic cross-section. Similar pressure-
driven laminar flow through the tube with axis 0z and elliptic cross-section
22
22
xy1
ab

with semi-major axis a and b is described by the following distribu-
tions
22 33
ell max ell
22 22
22
xz yz
22 22
xy abp
v(x,y)V 1 ,Q ,
ab 4L(ab)
pbx pay
,,
LL
ab ab










(29)
where
22
max 22
p
ab
V2L
ab
.
In this case the solution of the Navier-Stokes equations with the slip BC (16)
has the form
22
slip
max
ell 22
slip
ell
ell 22
x2(a )y2(b )
v(x,y)V 1
ab
ab
QQ1 ,
ab
 
 

 









,
}
(30)
while the shear stress field keeps the same form (29).
Similar shift of the velocity profile and increase in the volumetric rate will be
obtained when /max{a,b
or .
12
Kn max{a, b} C / C
3.7. Poiseuille flow between two coaxial circular tubes. The same pressure-
driven flow through the annulus between the concentric tubes with long axis 0x
and radiuses and is considered (Fig4).
1
R2
R
Fig.4. Schema of the Poiseuille flow between coaxial cylinders.
Classical Navier-Stoke solution for the no-slip boundary conditions at both cir-
cular walls is
220
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
22
22 21
annul 1
21 1
222
44 21
annul 2 1
21
22
21
annul
21
RR
Pr
v(r) Rr ln
4L ln(R/R) R
(R R )
p
QRR ,
8L ln(R /R)
RR
P1
2r ,
4L ln(R / R ) r

,






(31)
The velocity in (31) is maximal and shear stress is zero
22 2
221 21
annul max 1 annul
21 21
RR (R/R)1
P
vu R 1ln ,
4L 2ln(R /R) ln(R/R)




0,
at
22
21
21
RR
rr* 2ln(R / R )
 (Fig.4).
When the flow is subjected to the slip BC (16) with different slip coeffi-
cients at the inner and outer tubes, the solution has the form
slip 22
11 11 1 2 2 2 2 2 22 2
annul
22 2
11121 11222112
slip 22
11 11 1 2 2 2 2 2 22 2
annul
111
P
v(r) R(R2R2)(RlnR )R(R2R2)
4LZ
(R ln R )(R R 2( R R ))R R ln(r) r ,
p
Q 2(R (R 2 R 2 )(R ln R ) R (R 2 R 2 )
8LZ
(R lnR ))


 





22 22
12
21 21 112221
222244
22 1121 21
slip 22
12 21 112221
annul
RR
(R R ) (R R 2( R R ))
2
2R ln R 2R ln R R R (R R ) ,
RR
P(R R 2( R R )) 2r ,
4LZ r


 





(32)
where 12 2 1 12 21
ZRRln(R/R) R R
.
In this case, reaches its maximum at
slip
annul
v
22
** 21 112221
21 112 2
RR2(R R )
rr 2(ln(R / R ) / R / R )



  and or de-
pending on the slip coefficients
** * ** *
rrrr
1,2 1,2
,

and radii . Like in the circular
tubes (eqs.(27), (28)), one can reach by a proper choice of the
walls’ (i.e. slip) parameters
1,2
R
slip
annu
Q
1212
,,,
annul
lQ

depending on the radii . In [55]
the solution (32) is presented in the simplified form at
12
RR,
12
and 12

.
221
V. Cerevko and N. Kizilova
4. Layered laminar flow of immiscible micro/nano fluids between moving
plates.
Shear-driven laminar flow of three liquids with different viscosities 1, 2 , 3
between two parallel plates moved with velocities and (Fig.5) is
considered. The thickness of the layers ,
1
U
1
h
2
UU1
1
h2
h
and h2
h
are assumed to
be constant. The differences in the viscosities can be produced by different con-
centrations of the micro/nano particles in the uniform suspension cased by interac-
tion with walls. Then the layers with 1
and 3
are thin in comparison to the
innermost layer of the fluid with viscosity 2
and can be considered as boundary
layers.
Fig. 5. Laminar flow of three fluids between moving plates.
The solution of the Navier-Stokes equations in this case is given by linear func-
tion (see chapter 3.1)
(j) (j)
(j)
1
uCyC
2
, (33)
where , are constants determined from the slip conditions at the walls
j1,2,3(j)
1,2
C
(1) 2 (1)
(1) 11
2
y0
(3) 2 (3)
(3) 22
2
yh
du d u
uU
dy dy
du d u
uU
dy dy
 

 

1
2
,
,
(34)
and velocity and shear stress continuity conditions at the interfaces


1
1
2
2
(1) (2 )
(1) (2 ) 12
yh yh
(2) (3)
(2) (3) 23
yh yh
du du
u u 0, 0,
dy dy
du du
uu 0, dy dy




0.
(35)
222
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
Substitution of (33) into (34) and (35) gives the expressions for ,
and finally the velocities in the non-dimensional form are
(1,2, 3)
1,2
C



13212
c
23211
c
332112
c
1
V(u
1
Vuu(u
1
Vu(u
  
  
  
1)Y,
1)Y,
1)Y,
(36)
where ,
(1,2, 3)
1,2,3 1
Vu/UYy/h
are dimensionless coordinates,
,
c1 21
l( ) a)Kn
2 1 21 2
1 )l(1 1 12
(a  1,2 1,2
lh/h
, 1,2 1,2 3
/

,
, ,
1,2 1,2
 21
/U 11
a/h uU 1 2 2
1 )l(
  , 221 1
l(1 )
  ,
31
(1 2 21
a Kn ) u a
 .
From (36) one may compute the flow rate
12
12
hh h
(1) (2 ) (3 )
0h h
Q u (y)dy u (y)dy u (y)dy

.
and obtain for the non-dimensional flow rate the following expression
1
qQ/(Uh)

21 1112 22
12112
1
q (1 ) 2 u (1 u)l l (1 ) 2 u (1 u )l l
2
(1 u ) 2 ( a u a )Kn .
  
 
(37)
When , ,
1
l02
l112
1
 , we have the uniform flow and (36), (37)
coinside with (17).
The laminar flow between the parallel plates is the basic model for the
rotational viscosimeters. Let us assume one plate corresponds to a moving wall
() while another experiences the viscous shear stress
1
U3
. Then the apparent
viscosity can be computed as 3
/
app hU
1
which gives after the substitu-
tion of (37) the following dimensional expression
123
app
12 3 1 21 3 2 13 12 21 3
h
h( )h( )h ( )Kn

 
 . (38)
5. Results and discussions.
The velocity profiles are presented schematically in Fig.5. Numerical calcula-
tions of the flow rate have been carried out at the following values of the parame-
ters ,
1,2
l 0.1, 0.2,0.31,2
a 0.5,1,1.5
, 1, 2 0.25,0.5,1, 2, 4
, Kn 0.02,0.05,0.08,
and flow regimes u10.2,0.5,0.8 ;2;5;8
. The dependences of the non-dimensional
flow rate on one of the parameters
q1,2 1,2 1, 2
, ,lu, a
while others are kept constant.
Here correspond to the microfluids and
Kn 0.02,0.05,0.08Kn 0.2,0.5,0.8
for
nanofluids.
223
V. Cerevko and N. Kizilova
When the difference between the velocities of the two plates increases,
the flow rate increases proportionally to the Kn number that determines the tan-
gential slip of the particles (Fig.6a). When 21
/UU 8
, the maximal difference
in ~2 times is observed for Kn 0.2
and Kn 0.8
. When the slip coefficient
increases, the flow rate becomes bigger due to accumulation of the slip at lower,
upper or both plates (Fig.6b).
2
a
The
dependence (Fig.6c) has been computed at
1
q( )21
, while the de-
pendence 2
q( )
(Fig.6d) has been computed at 11
. In both cases the two-layer
flow with one boundary layer has been considered. When the viscosity is bigger
in the layer which is in contact with the faster moving plate, the flow rate de-
creases. In that way the layer with decreased viscosity serves as a lubrication that
decreases the resistivity to the flow and, therefore increases the flow rate. Any-
way, in the nanofluid flows the opposite effect is observed (Fig.6c,d at )
because the terms with Kn in (37) become significant. At different sets of the val-
ues the sign
Kn 0.8
1,2
u,l 1, 2 1, 2
, , 1, 2
q/
 could be either positive or negative.
a b
c d
224
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
e f
Fig.6. Dependences on (a), (b),
qu2
a1
(c), 2
(d), (e), (f) for different val-
ues
1
l
,0.8
2
l
Kn , 0 0.08 0.50.02 .05, ,0.2,
.
The dependencies have been computed at
1,2
q(l ) 10.25
, 22
. When the
thickness of the faster layer with lower viscosity increases, the flow rate also in-
creases (Fig.6e). When the thickness of the slower layer with high viscosity in-
creases, the flow rate varies insignificantly (Fig.6f).
The computed apparent viscosity (38) differs from the viscosity 2
of
the core fluid app 2
when
12 3 1 21 3 2 12 21 3
h( )h( )( )Kn0
 
 and vise versa.
For instance, when 231

and 12 21

, the accepted for-
mula for the apparent viscosity will give the overestimated viscosity values; when
231

 and 12 21

, the computed values will be underestimated.
In the case presented in Fig.5, the Magnus forces will decrease/increase concentra-
tion of particles in the lower/upper layer. In the case 123

correspon-
dent to the velocity distributions in Fig.5, the relationship between app
and 2
will depend on the whole set of the model parameters. Some computational results
are given in Fig.7.
The measured apparent viscosity of the fluid will be overestimated for the mi-
crofluids with at different velocity slip coefficients (curves 1-3 in
Fig.7a) and underestimated in the nanofluids with (curves 4-6 in
Fig.7a). When
0.7Kn
11
1
a
0.2Kn
(or 13
), the apparent viscosity is underestimated,
while when 13
, the viscosimeter gives correct values 2app
for the
microfluids (curves 1-3 in Fig.7b). The obtained results allow correction of the
viscosity measurements of the nanofluids and microfluids in the viscosimeters.
225
V. Cerevko and N. Kizilova
a b
Fig.7. Dependencies app 2 1
/(a)
at ,
20.1a20.25
(a) and app 2 1
/()

at
,
1,2 0.5a21
(b) when l1,2 0.1
; the curves 1-6 correspond to
Kn 0.02,0.05,0.08,0.2,0.5,0.8
.
6. Conclusions.
A review of the analytical solutions for the pressure-driven Poiseuille flow
between the parallel plates, through the circular and elliptic tubes, and the annulus
between two coaxial circular tubes, as well as the shear-driven Couette flows be-
tween the moving parallel plates, rotating coaxial cylinders and inclined duct are
generalized for the 1-st and 2-nd order velocity slip BC and different slip coeffi-
cients at the opposite walls. It was found that by a proper choice of the wall coef-
ficients one can reach lower shear stress and bigger flow rates at the same pressure
drop or shear rate at the expense of tangential velocity slip in the micro and nano-
channels.
Experimental measurements of viscosities of the microfluids and nanofluids in
viscosimeters are based on accepted formulae that may be erroneous due to veloc-
ity slip BC and concentration phenomena. The Magnus forces may appear due to
rotation of the moving particles as well as their interaction with the walls. Then
the concentrations of the micro or nanoparticles in the layers which are in contact
with the walls may be bigger or lower than in the core of the flow. As it was
shown, the apparent viscosity computed for the rotational viscosimeter accounting
to the concentration effects could give overestimated and underestimated values of
the viscosity. Using the obtained regularities, the proper conditions of the experi-
ments can be developed to obtain the values 2app
. The corresponding de-
pendencies for the capillary viscosimeters can also be computed in the same way
for the layered laminar Poiseuille flows in the tubes, channels and annulus given
in the chapter 3.
226
Complex Flows of Immiscible Microfluids and Nanofluids with Velocity …
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... CNTs have a thermal conductivity λ>3000 W/mK, carbon nanofibers λ~10−1000 W/mK, and copper nanotubes λ~400 W/mK [11]. Addition of NPs to basic technical and biological fluids enhances their thermal properties as coolants [1,12], especially CNTs and cellulose nanofibers [11,13]. Thermal parameters of CNTs and CNT-based nanomaterials are reviewed in [14]. ...
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