Mixed convection MHD flow of nanofluid over a non-linear stretching sheet with effects of viscous dissipation and variable magnetic field/Misrios konvekcijos nanoskyscio MHD tekejimas pro netiesini pailginta kolektoriu esant klampiajai sklaida ir kintamam magnetiniam laukui.
Matin, M. Habibi ; Dehsara, M. ; Abbassi, A. 等
1. Introduction
Many recent studies have been focused on the problem of magnetic
field effect on laminar mixed convection boundary layer flow over a
vertical non-linear stretching sheet [1-3]. Some industrial examples of
the problem are extrusion processes, cooling of nuclear reactors, glass
fiber production and crystal growing. Malarvizhi et al. [4], have
investigated free and mixed convection flow over a vertical plate with
prescribed temperature and heat flux. Kayhani, Khaje and Sadi [5]
studied the natural convection boundary layer along impermeable inclined
surfaces embedded in porous medium. Mohebujjaman et al. [6], studied
magneto hydrodynamics (MHD) heat transfer mixed convection flow along a
vertical stretching sheet in the presence of magnetic field with heat
generation. Also, Kumaran et al. [7], studied transition of MHD boundary
layer flow past a stretching sheet. Fadzilah et al. [8] have
investigated numerically free convection boundary layer in a Viscous
Fluid. Salleh et al. [9] studied forced boundary layer Flow at a Forward
Stagnation Point. The study by Prasad [10], has taken into account the
effects of temperature dependent properties on the MHD forced convection
over a non-linear stretching plate. In recent years, convective heat
transfer from nanofluids has been noticeable. Conventional fluids, such
as water, ethylene glycol mixture and some types of oil have low heat
transfer coefficient, the reason for which might be related to the low
conduction coefficient of these fluids. Choi [11], was the first person
who utilizes nanofluid. Choi et al. [12] affirmed that the addition of a
one percent by volume of nanoparticles to usual fluids increases the
thermal conductivity of the fluid up to approximately two times.
Recently several modeling of the natural or mixed convection of
nanofluids have been investigated numerically. Ho et al. [13] studied
the effect of natural convection of nanofluid in an enclosure due to
uncertainties in viscosity and thermal conductivity. Ghasemi and
Aminossadati [14] presented the numerical solution of natural convection
in an inclined enclosure filled with a water-CuO nanofluid. Maiga et al.
[15] studied the effect of nanofluid on forced convection heat transfer
enhancement. Wang and Mujumdar [16-18] reported numerical
investigations, experiments and applications of nanofluids, which are
very useful and can be applied. Therefore, the mixed convection heat
transfer of nanofluid over a vertical stretching sheet in the presence
of variable magnetic field and viscous dissipation effects were not
investigated. The importance of viscous dissipation term is due to the
increase in friction coefficient, when solid particles are in contact
with the solid plate. Also, the presence of nanoparticles in the
magnetic field may have interesting results.
In the present study, mixed convection MHD flow of nanofluid along
a non-linear stretching sheet with the presence of viscous dissipation
and variable magnetic field is investigated. The nanofluid is assumed to
be homogeneous with average physical properties of basic fluid and
nanoparticles. The governing boundary layer equations are transformed
into non-linear ordinary differential equations by considering suitable
similarity variables. The resultant similarity equations are solved
using an implicit finite-difference scheme known as Keller Box method.
2. Mathematical analysis
A steady state two dimensional mixed convection boundary layer flow
of nanofluid from a vertically stretching sheet with variable magnetic
field and viscous dissipation effect is considered.
The nanofluid is assumed as a homogeneous fluid with average
physical properties of basic fluid and nanoparticles. Therefore the
nanofluid is assumed to be a single phase solution. In other words we
have just investigated the macroscopic behavior of the nanofluid in the
mixed convection boundary layer. As it can be seen in [19] and [20],
these assumptions are used for natural convection boundary layer flow of
nanofluid over a vertical plate. The nanofluid is composed of solid
particles suspended in dense fluid (e.g. aqueous suspended) i.e., we
have a mixture. Therefore the gravity force applies to the whole
nanofluid as a body force. A quiescent incompressible and electrically
conducting fluid in the presence of a magnetic field B(x) perpendicular
to the sheet is taken into account along with the dissipation effect.
Fig. 1. shows the schematic view of the physical model and coordinates
of the system.
[FIGURE 1 OMITTED]
The x-axis is assumed to be in the direction of the flow and the
y-axis to be perpendicular to it. The temperature at the sheet
([T.sub.w]) is larger than the ambient temperature ([T.sub.[infinity]]).
The base fluid is water and the considered nanoparticles include CuO,
Cu, [Al.sub.2][O.sub.3], Ti[O.sub.2], Ag and Si[O.sub.2]. For
incompressible viscous fluid flow, the governing equations based on the
Boussinesq approximation and considering constant physical properties of
the base fluid and nanoparticles can be written as
[[partial derivative]u/[partial derivative]x] + [[partial
derivative]v/[partial derivative]y = 0 (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
The boundary conditions are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where [[rho].sub.nf], [[mu].sub.nf] and [[alpha].sub.nf] are
effective density, effective dynamic viscosity and effective diffusivity
of the nanofluid, respectively [21].
[[rho].sub.nf] = (1 - [phi]) [[rho].sub.f] + [[phi][[rho].sub.s]
(5)
[[mu].sub.nf] = [[mu].sub.f]/[(1 - [phi]).sup.2.5] (6)
[([rho][beta]).sub.nf] = (1 - [phi]) [([rho][beta]).sub.f] + [phi]
[([rho][beta]).sub.s] (7)
[[alpha].sub.nf] = [k.sub.nf]/[([rho][C.sub.p]).sub.nf] (8)
here [[mu].sub.f] is the dynamic viscosity of the basic fluid;
[[rho].sub.f], [[rho].sub.s], [([C.sub.p]).sub.f] and
[([C.sub.p]).sub.s] are the density of basic fluid, the density of the
nanoparticle, heat capacity of the basic fluid and heat capacity of the
nanoparticle, respectively; [k.sub.nf] is the thermal conductivity of
the nanofluid and [([rho][C.sub.p]).sub.nf] is the heat capacitance of
the nanofluid, which are as follows
[([rho][C.sub.p]).sub.nf] = (1 - [phi])[([rho][C.sub.p]).sub.f] +
[phi] [([rho][C.sub.p]).sub.s] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
where [k.sub.f], [k.sub.s] are the thermal conductivity of the base
fluid and nanoparticles, respectively.
The functional form of magnetic field is as
B(x)= [B.sub.0][square root of ([x.sup.m-1])] [22, 23].
The following dimensionless similarity variable is used to
transform the governing equations into the ordinary differential
equations
[eta] = y/x [square root of (m+1/2)] [([Re.sub.x]).sup.1/2] (11)
where [Re.sub.x] = [[rho].sub.f][u.sub.w](x)/[[mu].sub.f] x.
The dimensionless stream function and dimensionless temperature are
f([eta]) = [psi](x,y)[([Re.sub.x]).sup.1/2]/[u.sub.w](x) (12)
[theta]([eta]) = T - [T.sub.[infinity]]/[T.sub.w] -
[T.sub.[infinity]] (13)
where the stream function [psi]/(x,y) satisfies the Eq. (1).
u = [partial derivative][psi]/[partial derivative]y, v = - [partial
derivative][psi]/[partial derivative]x (14)
By applying the similarity transformation parameters, the momentum
Eq. (2) and energy Eq. (3) can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
Therefore, the transformed boundary conditions are
[f.sub.[eta]](0) = 1, f(0) = 0, [theta](0) = 1 (17)
[f.sub.[eta]]([infinity]) = 0, [theta]([infinity]) = 0 (18)
The dimensionless parameters of Gr/[Re.sup.2], Nu and [C.sub.f] are
the Richardson number, Nusselt number and stretching sheet friction
coefficient respectively. They are defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)
3. Numerical method
Two dimensional equations of flow and energy for a vertical, non
linear stretching sheet have been considered. These equations include,
the viscous dissipation and variable (non linear) magnetic field. Then,
they were transformed into similarity form. From similarity solution,
two non linear coupled equations were derived. These two equations are
converted into five first order equations. Then the system of
first-order equations is solved numerically using an efficient implicit
finite-difference scheme known as Keller Box method. The non-linear
discretized system of the equations is linearized, using Newton's
method [24-26]. The system of obtained equations is a block-tridiagonal
which is solved using the block-tridiagonal-elimination technique. A
step size of [DELTA][eta] = 0.005 was selected to satisfy the
convergence criterion of 10-4 in all cases. In this solution,
[[eta].sub.[infinity]] = 5 is sufficient to apply perfect effect of
boundary layer.
4. Results and discussion
Here, MHD mixed convection heat transfer of nanofluid along a
vertical, non-linear stretching sheet has been considered. The effect of
volume fraction of nano particles, stretching, MHD effects and non
dimensional Eckert and Richardson numbers are considered. Also, the
effects of the type nano particles on thermal and hydrodynamic boundary
layers based on similarity variables have been shown.
Table 1 depicts a comparison between the present results and
Hamad's results [19]. As illustrated in Table 1 there is only a
small difference between the results, which confirms the validity of the
present results.
In Table 2, thermal properties of nanoparticles of the present work
are seen. Also, Tables 3 and 4 show the values of Nusselt number and
stretching sheet friction coefficient for different governed physical
parameters and different nanoparticles.
In Figures [f.sub.n] is non dimensional velocity which is 1 on the
sheet and also is zero in a distance sufficiently far away. Similarly,
[theta] implied to non dimensional temperature with the same limits of
the non dimensional velocity.
[FIGURE 2 OMITTED]
Fig. 2 illustrates the effect of magnetic parameter on the non
dimensional velocity in presence of Richardson number. As it is seen,
when Richardson number increases, the velocity boundary layer thickness
also increases which shows low shear stress on the wall. The reason is
due to the fact that the higher the Richardson, the more natural
convection, which follows the reduction in the velocity gradient. Also,
a decrease in magnetic parameter causes an increase in the boundary
layer thickness. The phenomenon is due to the effect of Lorentz force
which causes fluid momentum amplification and as a result, higher
velocity gradient is created on the sheet. Also, it can be seen that
increasing Richardson number and stable natural convection, the effect
of magnetic parameter becomes more important.
In Fig. 3 the non dimensional velocity profile has been shown for
different Richardson numbers with the variation of volume fraction of
nanoparticles. As evident, when Richardson number is not higher than
unity, within whole boundary layer thickness the increase in
nanoparticle volume fraction causes reduction in velocity boundary layer
thickness. While, at high Richardson numbers, the increase in
nanoparticles volume fraction near the wall causes an increase in
velocity profile.
[FIGURE 3 OMITTED]
As can be seen in Fig. 4, when magnetic parameter is changed
between 1 and -1 the velocity boundary layer increases. In addition, it
can be understood that the stretching parameter has more affects on
velocity profile for Richardson numbers higher than unity or equal to
unity.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
Fig. 5 shows the effect of Eckert number with Richardson number on
velocity profile. As can be seen, the effect of Eckert number on
boundary layer thickness is negligible at Richardson numbers lower than
unity. The reason which might be associated to this is that natural
convection is closer to forced convection flow and the order of
magnitude of inertia forces on velocity is more than shear forces, while
at Richardson numbers higher than unity, the increase in Ec number
causes the boundary layer thickness becomes thicker.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
Figs. 6-8 show, velocity distribution profiles for various three
types of used nanoparticles such as Cu, Ag, Si[O.sub.2] and pure water.
In these figures, it is obvious that, boundary layer thickness changes
with the change of nanoparticle type. Of course, the order of magnitude
of this variation is relatively low, and main reason of this variation
is different physical and mechanical properties of nanoparticles such as
dynamic viscosity, density and expansion coefficients.
[FIGURE 9 OMITTED]
In Fig. 9, the non dimensional temperature profiles versus
variation of magnetic parameter and Richardson numbers have been
plotted. As it is seen, an increase in magnetic parameter causes an
increase in the thermal boundary layer. This increase is due to the
Laurent forces effect. The Laurent force increases the nanofluid
resistance, because of intermediate shear layer which causes increase of
temperature. By considering the Richardson number, this condition is the
same for natural, forced and mixed convection. Also, an increase in
Richardson number causes a decrease in thermal boundary layer, which
follows the increase in temperature gradient profile for different
Eckert numbers. As is seen, the on the wall and as a result, heat
transfer increases.
[FIGURE 10 OMITTED]
Fig. 10 shows the variation of Si[O.sub.2] volume fraction on the
temperature distribution for different Richardson numbers. As can be
seen, the increase in volume fraction has caused the temperature
profiles to shift up for all different Richardson numbers. The reason
can be the increase in nanoparticles friction. Because, by increasing
the volume fraction, momentum and contact between nano solid particles
would increase. Also, more Si[O.sub.2] particles cause more contact
between solid particles and base fluid and finally the thicker thermal
boundary layer.
[FIGURE 11 OMITTED]
The non dimensional temperature profiles versus different
stretching parameter ([beta]) origin velocity and also the values of 0
and 1 show linear and constant velocity for stretching sheet
respectively. By considering the Fig. 11, it is seen that the increase
in stretching parameter has caused the thermal boundary layer.
[FIGURE 12 OMITTED]
Fig. 12 shows the variation of temperature increase in Ec number
caused the increase in fluid temperature. Because, the higher the Ec
number, the higher the viscous dissipation effect in thermal boundary
layer.
In addition, Figs. 13-15 show temperature profiles for some types
of nanoparticles with water as base fluid. It can be found that
nanoparticles cause an increase in thermal boundary layer of natural,
forced and mixed convection. As a result, temperature gradient on the
wall decreases, which is in fact what was expect. Therefore, it can be
said that the effect of existence of nano particles is to cause velocity
and temperature to shift to the upper bands. This situation recovers the
condition of shear forces and heat transfer of the sheet.
[FIGURE 13 OMITTED]
[FIGURE 14 OMITTED]
[FIGURE 15 OMITTED]
5. Conclusions
In this work, an analytical and numerical study of mixed convection
with the effects of MHD flow of nanofluid over a non-linear stretching
sheet and viscous dissipation was considered. The nanofluid was made of
such nanoparticles as Si[O.sub.2], Ti[O.sub.2], CuO and Ag with pure
water as a base fluid. The nanofluid was assumed as a homogeneous fluid
with average physical properties of basic fluid and nanoparticles. The
values of Nusselt and drag coefficients for different non dimensional
parameters of stretching and magnetic field effects, Eckert and
Richardson numbers and volume fraction of nanoparticles were given. The
results of these parameters compared to each other. The results show
that, adding nanoparticles to the base fluids in forced, natural and
mixed convection would cause a reduction in shear force and a decrease
in stretching sheet heat transfer coefficient. Also, in nanofluid the
decrease in magnetic parameter and increase in Eckert number cause
better thermal conditions. In addition, using Ag nanoparticles showed
better thermal conditions in comparison with other nanoparticles.
Nomenclature
b--stretching rate, positive constant; B(x)--magnetic field, tesla;
[B.sub.0]--magnetic rate, positive constant; [C.sub.p]--specific heat at
constant pressure, J/(kg K); Ec--Eckert number, [u.sub.w]
[(x).sup.2]/[C.sub.p] ([T.sub.w] - [T.sub.[infinity]]); f--dimensionless
velocity variable; g--gravitational acceleration, m/[s.sup.2];
[Gr.sub.x]--Grashof number, g([T.sub.w] - [T.sub.[infinity]])
[[beta].sub.f]/[v.sup.2.sub.f]; k--thermal conductivity, W/(m K);
m--index of power law velocity, positive constant; Mn--magnetic
parameter, 2[sigma][B.sup.2.sub.0]/[[rho].sub.[infinity]]b(m +1);
Pr--Prandtl number, [[mu].sub.f] ([C.sub.p])/[k.sub.f];
[Re.sub.x]--local Reynolds number,
[[rho].sub.f][u.sub.w](x)x/[[mu].sub.f]; T--temperature variable, K;
[T.sub.w]--given temperature at the sheet, K;
[T.sub.[infinity]]--temperature of the fluid far away from the sheet, K;
[increment of T]--sheet temperature, K; u--velocity in x-direction, m/s;
[u.sub.w]--velocity of the sheet, m/s; v--velocity in y-direction, m/s;
X--horizontal distance, m; Y--vertical distance, m; [psi]--stream
function, [m.sup.2]/s; v--kinematic viscosity, [m.sup.2]/s;
p--stretching parameter, 2m/m +1; [[beta].sub.f]--thermal expansion
coefficient of the basic fluid, [K.sup.-1]; [[beta].sub.s]--thermal
expansion coefficient of the basic nanoparticle, [K.sup.-1];
[[beta].sub.nf]--thermal expansion coefficient of the basic nanofluid,
[K.sup.-1]; [mu]--dynamic viscosity, kg/(m s); [rho]--density,
kg/[m.sup.3]; [sigma]--electrical conductivity, mho/s;
[theta]--dimensionless temperature variable, T -
[T.sub.[infinity]]/[T.sub.w] - [T.sub.[infinity]]; [phi]--solid volume
fraction, [([m.sup.3]).sub.s]/[m.sup.3]; [alpha]--thermal diffusivity,
[m.sup.2]/s; [rho][C.sub.p]--heat capacitance of the basic fluid,
J/([m.sup.3] K);
Subscripts: f--basic fluid; s--nanoparticle; nf--nanofluid.
Received March 31, 2011
Accepted June 28, 2012
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M. Habibi Matin, Mechanical Engineering Department, Amirkabir
University of Technology, 424 Hafez Avenue, P.O. Box, 15875-4413 Tehran,
Iran, E-mail: m.habibi@aut.ac.ir
M. Dehsara, Mechanical Engineering Department, Amirkabir University
of Technology, 424 Hafez Avenue, P.O. Box, 15875- 4413 Tehran, Iran,
E-mail: bm_dehsara@aut.ac.ir
A. Abbassi, Mechanical Engineering Department, Amirkabir University
of Technology, 424 Hafez Avenue, P.O. Box, 15875- 4413 Tehran, Iran,
E-mail: abbassi@aut.ac.ir
cross ref http://dx.doi.org/10.5755/j01.mech.18A2334
Table 1
Comparison between the present results with results by
Hamad et al. [19] for various nanoparticles in water
(Pr = 6.2) at Ec = 0, Mn = 0, Gr/[Re.sup.2] = 1, [beta] = 1
[phi] [f.sub.[eta] [eta]](0) -[[theta].sub.[eta]](0)
Hamad Present Hamad Present
et al. results et al. results
[Al.sub.2] 0.0 0.4427 0.4389 0.8995 0.8876
[O.sub.3] 0.2 0.3190 0.3208 0.7113 0.7164
Ag 0.0 0.4382 0.4396 0.9005 0.9017
0.2 0.3053 0.3084 0.7132 0.7156
Cu 0.0 0.4367 0.4401 0.9492 0.9511
052 0.3059 0.3072 0.8481 0.8510
Table 2
Thermo-physical properties of water and nanoparticles [19, 27-29]
Physical Fluid Si Cu Ag
Properties Phase [O.sub.2]
(water)
[rho], kg 997.1 3970 8933 10500
[m.sup.-3]
Cp, J 4179 765 385 235
[kg.sup.-1]
[K.sup.-1]
[beta] x 21.0 0.63 1.67 1.89
[10.sup.5],
[K.sup.-1]
k, W 0.613 36.0 401 429
[m.sup.-1]
[K.sup.-1]
Physical [Al.sub.2] Ti CuO
Properties [O.sub.3] [O.sub.2]
[rho], kg 3970 4250 6500
[m.sup.-3]
Cp, J 765 686.2 540
[kg.sup.-1]
[K.sup.-1]
[beta] x 0.85 0.90 0.85
[10.sup.5],
[K.sup.-1]
k, W 40 8.9538 18.0
[m.sup.-1]
[K.sup.-1]
Table 3
Skin friction coefficient and Nusselt number for different
values of the physical parameters. Si[O.sub.2]-Water,
Pr = 6.2
[phi] [beta] Ec Mn Gr/[Re.sup.2] << 1
[C.sub.f] Nu
0.2 1 0.02 0.50 0.0949 6.6746
0.75 0.1009 7.9201
1.0 0.1065 9.1075
[phi] [beta] Mn Ec [C.sub.f] Nu
0.2 1.0 1.0 0 0.1065 9.6732
0.10 0.1064 6.8558
0.20 0.1063 4.0472
[phi] Ec Mn P [C.sub.f] Nu
0.2 0.02 1.0 -1.0 0.0251 5.6558
0.0 0.0579 6.8273
1.0 0.1065 9.1075
[beta] Ec Mn t [C.sub.f] Nu
1 0.02 1.0 0.0 0.1065 9.1075
0.1 0.0955 7.2426
0.2 0.0831 5.8316
[phi] Gr/[Re.sup.2] = 1 Gr/[Re.sup.2] >> 1
[C.sub.f] Nu [C.sub.f] Nu
0.2 0.0696 7.4707 0.0011 8.9152
0.0776 8.8526 0.0120 10.6034
0.0849 10.1115 0.0234 12.0412
[phi] [C.sub.f] Nu [C.sub.f] Nu
0.2 0.0850 10.5207 0.0238 12.2357
0.0844 8.5015 0.0217 11.2921
0.0837 6.5427 0.0198 10.4268
[phi] [C.sub.f] Nu [C.sub.f] Nu
0.2 0.0087 6.7544 0.0485 7.9912
0.0398 7.6005 0.0119 9.0457
0.0849 10.1115 0.0234 12.0412
[beta] [C.sub.f] Nu [C.sub.f] Nu
1 0.0849 10.1115 0.0234 12.0412
0.0793 7.8262 0.0318 9.0896
0.0716 6.0776 0.0366 6.6858
Table 4
Skin friction coefficient and Nusselt number for different
values of the physical parameters and different kind of
nanoparticles. Mn = 1.0, Pr = 6.2, [beta] = 1.0,
[phi] = 0.20
Gr/[Re.sup.2] << 1 Gr/[Re.sup.2] = 1
[C.sub.f] Nu [C.sub.f] Nu
Water 0.1039 3.8975 0.0602 8.6312
Si[O.sub.2] 0.1007 2.6564 0.0661 6.2364
Cu 0.1143 1.5406 0.0793 5.3308
Ag 0.1182 1.1091 0.0828 4.9909
[Al.sub.2] 0.1007 2.6631 0.0659 6.2476
[O.sub.3]
Ti[O.sub.2] 0.1016 2.4820 0.0670 6.2252
CuO 0.1079 2.0169 0.0738 5.7444
Gr/[Re.sup.2] >> 1
[C.sub.f] Nu
Water -0.0614 11.6052
Si[O.sub.2] -0.0296 9.5413
Cu -0.0159 9.2215
Ag -0.0131 9.0091
[Al.sub.2] -0.0301 9.5323
[O.sub.3]
Ti[O.sub.2] -0.0283 9.7045
CuO -0.0196 9.5077
