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ORIGINAL ARTICLE
Heat and mass transfer effects on the mixed convective flow of chemically reacting nanofluid past a moving/stationary vertical plate
B. Mahanthesh a,b, B.J. Gireeshab,c*, Rama Subba Reddy Gorlac
a Department of Mathematics, AIMS Institute of Higher Education, Peenya, Bangalore 58, Karnataka, India b Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta-577 451, Shimoga, Karnataka, India c Department of Mechanical Engineering, Cleveland State University, Cleveland 44114, OH, USA
Received 4 July 2015; revised 21 January 2016; accepted 27 January 2016
KEYWORDS
Nanofluid;
Heat and mass transfer; Thermal radiation; Laplace transform technique; Chemical reaction; Nusselt and Sherwood number
Abstract The problem of conjugate effects of heat and mass transfer over a moving/stationary vertical plate has been studied under the influence of applied magnetic field, thermal radiation, internal heat generation/absorption and first order chemical reaction. The fluid is assumed to be electrically conducting water based Cu-nanofluid. The Tiwari and Das model is used to model the nanofluid, whereas Rosseland approximation is used for thermal radiation effect. Unified closed form solutions are obtained for the governing equations using Laplace transform method. The velocity, temperature and concentration profiles are expressed graphically for different flow pertinent parameters. The physical quantities of engineering interest such as skin friction, Nusselt number and Sherwood number are also computed. The obtained analytical solutions satisfy all imposed initial and boundary conditions and they can be reduced to known previous results in some limiting cases. It is found that, by varying nanoparticle volume fraction, the flow and heat transfer characteristics could be controlled.
© 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
* Corresponding author at: Department of Mechanical Engineering, Cleveland State University, Cleveland 44114, OH, USA, Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta-577 451, Shimoga, Karnataka, India. E-mail addresses: bmanths@gmail.com (B. Mahanthesh), g.bijjanal-jayanna@csuohio.edu (B.J. Gireesha), r.gorla@csuohio.edu (R.S.R. Gorla).
Peer review under responsibility of Faculty of Engineering, Alexandria University.
In many industrial and engineering applications, the heat and mass transfer is a consequence of buoyancy effects caused by thermal diffusion and chemical species. Therefore, the study of conjugate effects of heat and mass transfer is handy for improving many technologies such as underground energy transport, polymer and ceramics production, enhanced oil recovery, food processing, formation and dispersion of fog, the distribution of temperature and moisture over agricultural fields, and environmental pollution. The heat and mass
http://dx.doi.org/10.1016/j.aej.2016.01.022
1110-0168 © 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Nomenclature
B magnetic field uo constant
b body forces u, v, w velocity components along x, y, and z directions
Bo magnetic field strength (ms—1)
C concentration of the fluid x coordinate along the plate (m)
Cf skin friction co-efficient y coordinate normal to the plate (m)
cp specific heat coefficient of fluid (J/kg K)
D mass diffusion co-efficient Greek symbols
E electric field rp pressure gradient
f dimensionless velocity component V kinematic viscosity (m2 s—1)
g acceleration due to gravity l dynamic viscosity (kgm—1 s—1) shear rate
Grt thermal Grashof number / nanoparticle volume fraction
Grc solute Grashof number r* Stefan-Boltzmann constant
J current density r electrical conductivity
j mass flux q density (kg/m3)
k* Rosseland mean absorption co-efficient bT thermal expansion co-efficient
k thermal conductivity (W/mK) bc solutal volumetric coefficient
ki chemical reaction co-efficient k moving parameter
K chemical reaction parameter h dimensionless temperature
M2 magnetic parameter u dimensionless concentration
Nu Nusselt number g, s dimensionless variables
Pr Prandtl number
Qo uniform volumetric heat source/sink Subscripts
Q heat source/sink parameter f base fluid
q heat flux nf nanofluid
qr radiative heat flux s nanoparticles
R thermal radiation parameter w at the wall condition
Sc Schmidt number oc far from the wall
Sh Sherwood number Constants xl — x6, al — a9, bl — b2,fl — f7,gl — g7
t time
T fluid temperature (K)
transfer flow of an electrically conducting fluid in the presence of transverse magnetic field also finds a variety of applications such as MHD generators, pumps, flow meters, nuclear reactors, accelerators and in metallurgical industries. Its relevance is also seen in many practical applications in geophysical and astrophysical situations [1]. Sarpkaya [2] was the first to study the effectiveness of MHD flows in fluids. A few recent studies regarding MHD heat and mass transfer with different physical conditions are [3-12].
On the other hand, the radiative heat transfer has encountered a variety of applications such as nuclear power plants, gas turbines, and propulsion devices for space vehicles, missiles and aircraft. Additionally, heat and mass transfer under the influence of chemical reaction has also attracted considerable attention of many authors due to their wide range of applications. Keeping this in view, conjugate heat and mass transfer flow past a vertical permeable plate with thermal radiation and chemical reaction is investigated by Pal and Talukdar [13]. They found that, the thermal radiation and chemical reaction effect decrease the velocity and concentration profiles. Numerical investigation of heat and mass transfer of an electrically conducting fluid over a moving surface with first order chemical reaction effect is carried out by Chamkha [14]. Uwanta and Omokhuale [15] have analyzed the influence of radiation and chemical reaction effect on heat and mass transfer of a viscoelastic fluid in a fixed plane. Later, the combined
effects of radiation and chemical reaction on the MHD free convection flow of an electronically conducting viscous fluid over an inclined plate have been studied by Ali et al. [16] and have obtained a closed form solution by employing Laplace transform technique. Das et al. [17] came up with an analytical solution for the problem of unsteady free convection flow and mass transfer over a vertically moving plate in the presence of thermal radiation using Laplace transform method. Several attempts have been made to analyze the effect of thermal radiation and chemical reaction under various physical situations (see [18-24]).
In recent times, the heat/mass transfer analysis in nanoflu-ids has been the topic of extensive research due to intensification of thermal properties in heat transfer processes. The nanofluids can be described as a mixture of nanometer sized particles (diameter less than 100 nm) in the base fluid. The nanoparticles are prepared from copper, aluminum, titanium oxide, gold and the materials that are stable chemically. The base fluids are ordinary fluids, including water, toluene, ethy-lene glycol and mineral oils. In a wide variety of industrial and engineering applications, the transfer of heat energy is a common and necessary process, and the enrichment of heating/cooling in an industrial process may create a number of advantages such as energy saving, reduction of processing time and also the greater quality of the final product. However, the heat transfer efficiency also depends on the thermal
Fig. 1 Physical system and Geometry of the problem.
performance of working fluids. Thus, improving the thermal conductivity of working fluids becomes a major and challenging task for industrial necessity. The inspiration of suspended nanoparticles in a base fluid to increase the thermal conductivity was proposed by Choi [25] about a decade ago. Thereafter, theoretical and experimental investigations on the nanofluid heat transfer property have been conducted by Wang et al. [26], Eastman et al. [27], Buongiorno [28], etc. They have concluded that the thermal conductivity of the base fluid can be dramatically enhanced in the presence of nanoparticles. Besides, there are two models are available to incorporate nanoparticle effect on fluid flow problems namely singlephase model and two-phase model. Further, Buongiorno or Tiwari and Das model is used to model the single-phase nano-fluid. Two important mechanisms such as Brownian motion and thermophoresis are treated in Buongiorno model. However, in Tiwari and Das model, the effective fluid properties are taken into account.
The nanofluids have tremendous applications such as engine cooling, solar water heating, cooling of electronic equipments, cooling of transformer oil, cooling of heat exchanging devices, in chillers, refrigerator-freezers, nuclear reactors, and space vehicles, due to their higher thermal conductivity and convective heat transfer rates. Recently, Sheik-holeslami and Ganji [29] studied the MHD effects on nanofluid flow in a permeable channel. Kuznetsov and Nield [30] critically analyzed the natural convection flow of a nano-fluid past a vertical plate. The problem of transient MHD free convection flow of a nanofluid over a rotating vertical plate is addressed by Hamad and Pop [31]. They solved the governing equations analytically using the perturbation method and found that, the inclusion of nanoparticles into the base fluid is capable of varying the flow pattern. Turkyilmazoglu [32] reported the exact solution for heat and mass transfer of mag-netohydrodynamic flow of nanofluid. Further, Turkyilma-zoglu and Pop [33] have examined the heat and mass transfer of unsteady natural convection flow of nanofluid past
a vertical infinite plate with thermal radiation. Das [34] discussed the problem of free convection flow of nanofluid bounded by moving vertical plate with constant heat source and convective boundary condition in a rotating frame of reference. He found that, the skin friction coefficient increases with an increase in the nanoparticle volume fraction. Recently, Sheikholeslami and Ganji [35] have presented the heat and mass transfer behavior of unsteady flow of nanofluid between parallel plates in the presence of thermal radiation. They conclude that, the solutal boundary layer thickness increases with increase in radiation parameter. Freidoonimehr et al. [36] presented the analytical solution for three-dimensional squeezing nanofluid flow in a rotating channel using Tiwari and Das model. With the aid of same model Gireesha et al. [37] have studied the effect of nanoparticles on flow and heat transfer of dusty fluid. Das et al. [38] have found that the volume fraction parameter decreases the rate of heat transfer in a fully developed mixed convection flow through a vertical channel filled with water based nanofluid with the uniform transverse magnetic field. Das and Jana [39] studied the MHD flow past a moving vertical plate in the presence of thermal radiation using Laplace transform method for closed form expressions of flow fields. Also, they found that, the shear stress at the plate for Cu-water nanofluid is found to be lower. Heat transfer in flow of nanofluid with heat source/sink and an induced magnetic field is addressed by Gireesha et al. [40].
An Extension to this topic, Sheikholeslami et al. [41-44] added a new dimension to the study of nanofluid flow in channel/different geometry under diverse physical aspects by using single/two-phase model. Motivated by aforementioned studies, we intend to investigate combined effects of heat and mass transfer on natural convection flow of water based nanofluid over a stationary/moving vertical plate. The effects of heat source/sink, chemical reaction and thermal radiation are also considered. To the best of our knowledge, this study has not been considered by any authors. Hence, the novelty of this study is to obtain the closed form solution for the present problem by employing Laplace transform technique. The basic nonlinear partial differential equations are derived by using Boussinesq approximation and then are transformed to dimensionless equations using suitable non-dimensional variables. The resultant equations have been solved analytically.
2. Mathematical formulation and solution
Consider an unsteady, laminar flow of nanofluid over an infinite stationary/moving vertical flat plate. The plate is at rest initially with constant ambient temperature and concentration. The plate starts to move vertically with the velocity ku0 for t > 0. The temperature and concentration are raised or lowered to Tw and Cw correspondingly. The magnetic Reynolds number is assumed to be so small, so that the induced magnetic field and Hall current are negligible. The governing equations of conservation of mass, momentum, energy and concentration of nanofluid are given by the following:
V- U = 0, dU
Pnf=~Vp + InfV2 U + J x B + gb,
Table 1 Thermo-physical properties of water and copper
nanoparticles [39].
Physical properties Water/base fluid Cu (copper)
p (kg/m3) 997.1 8933
cp (J/kg K) 4179 385
k (W/m K) 0.613 401
ßt x 105 (K-1) 21 1.67
ßc x 106 (m2/h) 298.2 3.05
o (s/m) 5.5 x 10-6 59.6 x 106
/ » dT
{Pcp)nf -q; — -V ' q,
where U — (u, v, w) is the velocity vector, pnf the effective density of nanofluid, inf the effective dynamic viscosity of nano-fluid, Vp the pressure gradient, B the magnetic field, g the acceleration due to gravity, b the body forces, T the temperature, (cp)nf the effective specific heat, q — —knfVTthe heat flux, C the concentration, j — —DVC the mass flux, knf the effective thermal conductivity, D the diffusion co-efficient, t the time and J — rnf(E + U x B) the current density.
The x-axis is taken along the plate in the vertically upward direction and y-axis is normal to it. In the plane y — 0 the plate coincides and the flow being confined to y > 0 as shown in Fig. 1. A uniform transverse magnetic field of strength B0 and radiative heat flux qr are applied normal to the flow direction and it is assumed that there is no applied or polarization voltage exists. It is assumed that the pressure gradient is neglected and the fluid is electrically conducting, water based nanofluid embedded with copper nanoparticles. Further, it is considered as, the base fluid and suspended nanoparticles are in thermal equilibrium. In Table 1, the thermo-physical properties of nanofluid are presented. The 1st order chemical reaction between the nanofluid and the species concentration is taken into the account. The reaction is assumed to take place entirely in the stream.
Under aforementioned assumptions and usual Boussinesq's approximation, the governing equations in its component form readily read as [39];
pnf dT
f dyi " f0u + g(Pßr)nf(T - T1
+ g(pßc)nf (C - C
, » dT d2T dqr
(pc?)nf -Qt — knf @yi - -r + Ö0(T- T1
dC nd2 C -,,rr -Qt — D -y, - kl(C - Ci
where rnf is the effective electrical conductivity, pr^f the effective thermal volumetric coefficient of expansion, bc the effective solutal volumetric coefficient of expansion, k the chemical reaction coefficient and Q0 the uniform
volumetric heat source/sink. The effective properties of nanofluid are given by following co-relations (see Das and Jana [39]);
pnf — (1 - /)Pf + /ps
(Pcp)nf — (1 - /) (Pcp)f + HpCp)s;
(pßr)nf -(1 - /)(pßr)f + /(pßr),,
(pßC)nf -(1 - /)(pßC)f + /(pßa)„
rnf — Of
3(r - 1)/
o + 2 - (r - 1)/J'
ks + 2kf - 2/ ( kf - ks
Of' kf ks + 2kf + / (kf - ks
where /-the volume fraction of nanoparticles, pf and ps are density of base fluid and nanoparticles respectively, cpf and cPs-specific heat of the base fluid and nanoparticles respectively, Gf and rs are the electrical conductivity of base fluid and nanoparticles correspondingly, bT/ and are thermal volumetric coefficient of the base fluid and nanoparticles correspondingly, bC and are solutal volumetric coefficient of the base fluid and nanoparticles correspondingly, and kf and ks are thermal conductivity of base fluid and nanoparticles respectively.
The appropriate initial and boundary conditions of the present problem are
t — 0 : u — 0, T — Ti, C — Ci for all y p 0, t > 0 : u — ku0, T — Tw, C — Cw at y — 0, (2.9)
t > 0 : u -> 0,
C ! Ci as y
where k is the direction of plate movement. It is worth to mention that, k — 0 corresponds to stationary plate and k — 0 corresponds to moving plate. Now by using Rosseland approximation, the radiative heat flux qr can be expressed as (see Sheikholeslami and Ganji [24]);
4o* dT4
3k* dy
(2.10)
where r* is the Stefan Boltzmann constant and k* is the Rosseland mean absorption co-efficient. Now by assuming small temperature difference with the flow, the temperature can be expressed as a linear function of temperature, and this is by employing Taylor series to expand the r4 about rœ as follows:
T4 — TL + 3Tl(T- To) + 6Tl(T- Tx
(2.11)
Beyond the first order, neglecting the higher order terms in the above equation, one can get;
t4 ^4t! t-3tl.
(2.12)
In the view of Eqs. (2.10) and (2.12), the Eq. (2.6) will take the following form;
, » dT , d2T 16ff*Tl d2T
(pcp)nfdT — knf+ T^T1 ~d~T + Q0(T- T»).
dy2 3k* dy2
(2.13)
It is opportune to non-dimensionalize the equations by employing following variables;
Uoy g =-'
s _ up u
6 — -
T— T
■*■ x nc
C— C C — C '
M2 —
CfB20Vf Grt = gßifT — Ti)Vf Grc^gßCf(Cw — Ci)v/
'' - .3 ' ,.3 '
Pr —
(ic, x
^f R- 4r*T1 Q_ QoVf D K_k1Vf
' R _~Pn ' Q ^ ' Sc _ ' K _ 2"
f kkf u2( pcp)f Vf u2
(2.14)
On substituting these dimensionless variables into the governing equations and boundary conditions, one can get;
@f = a1 @f + Grta20 + Grca3 u — M2a4f, @g2
96 926
+ a<>6'
@s @g2
_ ± - K(p '
9s Sc 9g2 '
s _ 0 s > 0 s > 0
f _ 0 ' 6 _ 0 ' u _ 0 8g P 0 ' f _ k 6 _ 1' u _ 1 at g _ 0 ' f ! 0 ' 6 ! 0 ' u ! 0 as g !
(2.15)
(2.16)
(2.17)
(2.18)
By applying Laplace transform on both sides of Eqs. (2.15)-(2.18), we have;
d2f - -+ Grta2- + Grca3- — (M2a4 + s)f _ 0 '
a5 + (a6 — s)- _ 0 '
1 Ö —(K + s) - _ 0'
Sc @g2
With corresponding initial and boundary conditions are
/ = 0 , h = 0 , p = 0 8g P 0
/=-, h = -, p = - at g = 0
/ ! 0 , h ! 0 , p ! 0 as g !1
where/ = /0°°/(s , g)e-ssds, h = /0°° h(s , g)e-ssds, p = /0°° u(s , g) e~ssds and s > 0. Now by solving above system, one can have;
(2.19)
(2.20) (2.21)
(2.22)
- (g ' s)_ eV Sc(s + K)g '
6(g 's) _-e—/S——g ' s a5
( 2. 23 ) (2.24)
f(g 's) _se
[s — 01 Grca81
è2 I s — b2
s—a6 ' a5
— Sc(s+K)g
(2.25)
Table 2 The numerical values of skin friction co-efficient for
various values of M2 ' / ' Grt' Grc and s when
R _ 2 , Q = —1, Pr _ 6.2 ' Sc _ 0.6 and K _ 5.
M2 / Grt Grc s Cf k _ 0 Cf k - 0
0 0.2 10 10 0.5 —7.69E + 03 —7.70E + 03
1 26.7770 25.4943
1.5 4.4823 3.0559
2 3.2406 1.6775
2.5 3.2703 1.5767
1 0 4.9185 3.7519
0.05 6.1722 4.9443
0.1 7.4002 6.1350
0.15 10.0833 8.8008
0.2 26.7770 25.4943
—4 20.3474 1.91E + 01
—2 21.2659 19.9832
0 22.1844 20.9017
2 23.1029 21.8203
4 24.0214 22.7388
—4 —4.2812 —5.5639
—2 0.1557 — 1.1270
0 4.5926 3.3099
2 9.0295 7.7468
4 13.4663 12.1837
0.1 6.4963 4.1781
0.2 10.2000 8.4604
0.3 14.3132 12.8126
0.4 19.5713 18.2037
0.5 26.7770 25.4943
In order to determine, the flow fields in the time domain, by applying inverse Laplace transform on both sides of (2.23)-(2.25), we get;
f(g 's) _ kf (g 's 'aba9) -
Grta7 b1
{¿¡(g 's ' au a9 ' b1
—f3(g 's ' a5 ' 06' b1) — f1(g 's ' ab a9 ) + f4(g ^ ' a5 ' a6)g ' Grca8
1 {/5 (g ' s ' a1 ' a9 ' b2) — f6(g ' s ' Sc ' K' b2)
—f1(g's 'au a9) + f7(g's ' Sc 'K)g
6(g 's) _ ft(g 's ' a5 ' a6)'
u(g 's) _ f7(g 's ' Sc ' K)' where
(2.26)
(2.27)
(2.28)
X1 _ (1 — < X3 _(1 — /
X5 _ (1 — '
X6 _~r'-
X2 _ (1 — 1
+ ^ (pßc)f '
) + / ^ (PcWf
ks + 2kf — 2/ ( kf — ks
(pßT)/
3(r — 1)/
r + 2 — (r — !)/_
ks + 2kf + / (kf — ks
Table 3 The numerical values of Nusselt number and skin friction co-efficient for various values of Pr when R = 2, Q = -1, M2 = 1, Grt = Grc = 4, s = 0.5, Sc = 0.6 and K= 5.
Pr Ordinary fluid / = 0) Nanofluid ( / - 0) £T
Nu C Nu C
6.2 i.677i 3.75i9 i.490i 25.4943 H
io 2.i300 3.3672 i.8924 4.0267
20 3.0i22 2.857i 2.6762 2.6724
50 4.7628 2.2944 4.23i5 i.8037
ioo 6.7355 i.959i 5.9842 i.3685
200 9.5255 i.696i 8.4630 i.0469
iooo 2i.2997 i.3057 i8.9238 0.5884
Fig. 3 The effect of nanoparticle volume fraction parameter on velocity profiles for (a) moving plate and (b) stationary plate in both Grt > 0 and Grt < 0 cases.
1 Г /a9„ / g
''er/c
2^/xal
- ^/a9s ^Ja9 t
,, , ч ebi'l eV ai er/cijPaf+V^+bOï) + e
/¡(g, t, ai, 09, bi)= —^ V^sai v '
Fig. 2 The effect of magnetic parameter on velocity profiles for f3(g, s, a5, a6, bi) = —<
er/c ( ffi \J (a9 + "i)s )
eV ai V/cj^-ffi+ A/(bi -a6)^ + e
X2 X3 X4
T25-; a2 = — ; a3 = — ; a4 = — ;
if /-—g ( g f4(g,t,a5,«0 = 2\eV a er/c\2-fä,
er/c(2pfe— V(bi —a6)s)
\/-a6s
g er/c
2^/xa55
V-aes) >,
if 3 1 g a2a5
a5 =—:r- <xe + -R >, ae = —, a7 =-, ag--
x5Pr \ 4 ) x5 ai — a5 ai — Sc
/5 (g, t, ai, a>, ¿2,
eV ai "er/^2p= + V(a9 + ¿2^) + e
ly2 , aiae + M a4a5 a9 = M a4, bi =-,
ai — a5
M2a4Sc — aiK ai — Sc
k er/c{ffi \/(a9 + ¿2)t)
Red: Ordinary fluid
Blue: Nanofluid
V-.\\ V \Л Grt = -20,-10, 0, 10, 20
\4\\\ \4k.
.... Red: Ordinary fluid
II Blue: Nanofluid
X = 0 Grt = -20, -10, 0, 10, 20
Щ /--v^
ah: ' W
Fig. 4 The effect of thermal Grashof number on velocity profiles Fig. 5 The effect of solute Grashof number on velocity profiles for (a) moving plate and (b) stationary plate in both / _ 0 and for (a) moving plate and (b) stationary plate in both Grt > 0 and / - 0 cases. Grt < 0 cases.
f6(g , Sc ,KA)= —
4 e^er/c^ +j(K + + e
/7(4 , Sc ,K)= 2
ieVSaerfcl-
l ■ V:
2VScr -
2—Sc
-Л/(К+Ъ2
Knerfc
2\iScc
Here er/c() is complementary error function and is defined as er/c(x) = p2pp J1 e~'2 dt. The skin friction coefficient, Nusselt number and Sherwood number are most important physical quantities in the point of engineering interest. The skin friction
coefficient at the plate g = 0 is given by; . @/(g
cf = -
Cf = + fe - g3 - gl + g4g
+ ~Ъ2~ fg5 - g« - gl + g7 g ' here gl = (-^/fjerf—— - —_e-a9S),
(2.29)
g2 = -eblS
— _ebis
erfp ( a9 + _ 1 ) _+__
( t Ъ-—erf^/ ( - 1 - a« ) - + —1 _
lu a5 y/sa5n
-(а9+Ъ1)т
-(bi-a6)T
/-a« , ,- , 1
-erf V-a6s + e
' a5 y/sa5n
g 5 = -eb2s
— _ob2s
erfp—+—s1 \^~+cr~erfp——+—
y/sa1n 1
-(a9+b2)s
-(K+b2)i
= - ^Sc^'7_+7_e-K').
where erf(-) is the error function and is defined as erf(x) — Pp f0x e~'2dt. The Nusselt number is given by;
Fig. 6 The effect of time parameter on velocity profiles for (a) moving plate and (b) stationary plate in both Grt > 0 and Grt < 0 cases.
œ 0.4
1 Red: t = 0.1
V Blue: x= 1.0
\ \ 4> = 0.0, 0.1, 0.2
Fig. 7 The effect of nanoparticle volume fraction parameter on temperature profile.
Nu = -
dh(g, s)
Nu = , /—^erf—-affi + —ea6s, \ja5 y'sa5p
(2.30)
i Red: R = 0.0
1 Blue: R = 5.0
Q = -2,-1,0, 1,2
2 T| 3
Fig. 8 The effect of heat source/sink parameter on temperature profile.
Red: R = 0.0
% Blue: R = 0.5
'v'A T = 0.1, 0.5, 0.9
V XV'\
0.0 0.5
1.5 2.0
Fig. 9 The effect of time parameter on temperature profile
Fig. 10 The effect of radiation parameter on temperature profile.
The Sherwood number is defined as
@u(g, s )
Sh = J — erfVKx + —= e-Ks.
V Sc -JSKSC
Fig. 11 The effect of Prandtl number on temperature profile.
Fig. 14 The effect of Schmidt number on concentration profile.
Fig. 12 The effect of chemical reaction parameter on concentration profile.
Fig. 13 The effect of time parameter on concentration profile.
It is worth to mention that, by neglecting heat source/sink, chemical reaction and mass transfer effects, the solutions of the present study are identical to Das and Jana [39].
3. Result and discussion
In the present study, the Laplace transform scheme has been employed to obtain the analytical expressions for velocity, temperature and concentration profiles as well as the skin friction coefficient, Nusselt number and Sherwood number. In order to analyze the salient features of different flow fields, a numerical computation is carried out for different values of pertinent parameter. The results of numerical computation are presented through graphs 2-18 and in Tables 2 and 3. The values of thermo-physical properties of base fluid (water) and nanoparticle (copper) are taken from the data presented in Table 1. The range of nanoparticle volume fraction is chosen as 0 6 / 6 0.2. The value of Prandtl number is taken as 6.2, which physically corresponds to base fluid (water). It is worth to mention that, Gr > 0 and Gr < 0 means that the cooling and heating of the plate by free convection current correspondingly and Gr — 0 correspond to the absence of free convection current. Throughout our numerical computation, we have chosen other parameters as s — M2 — 0.5 , R — 2, Q — 1, Sc — 0.6 , Gr' — Grc — 10 and K — 0.5 unless otherwise stated.
The impact of magnetic field on the dimensionless velocity profile is depicted in Fig. 2 for both stationary and moving plate case. From Fig. 2, it is observed that, an increase in magnetic parameter leads to decrease in the velocity profile in both Gr > 0 and Gr < 0 cases. This is due to the fact that, the application of transverse magnetic field to an electrically conducting fluid gives rise to a resistive type of force called the Lorentz force. This force has the tendency to slow down the motion of the fluid; consequently, the fluid velocity decreases significantly. This outcome is consistently agreed with that of Sheik-holeslami and Ganji [44]. Fig. 3 depicts the variation of dimensionless velocity profiles / versus g for different values of the nanoparticle volume fraction parameter in both Gr > 0 and Gr < 0 cases. By increasing nanoparticle volume fraction parameter results in an increase in the friction force within the fluid. As a consequence the velocity field decreases in both moving and stationary plates. It is also observed that, the momentum of the fluid is lower in the case of Gr > 0 than that of Gr < 0.
The inclusion of buoyancy effects is illustrated in Figs. 4 and 5. The occurrence of buoyancy influences complicates
Fig. 15 The effect of heat source/sink parameter on skin friction and Nusselt number profiles.
Fig. 16 The effect of radiation parameter on skin friction and Nusselt number profiles.
Fig. 17 The effect of time parameter on skin friction and Nusselt number profiles.
the flow analysis with thermal and solutal analysis. From Fig. 4 it is noticed that, the velocity increases in the neighborhood of the plate by increasing the thermal Grashof number. But this trend is qualitatively opposite at the long distance
from the plate. Physically speaking, the thermal Grashof number signifies the relative importance of buoyancy force to the viscous hydrodynamic force. The larger Grashof number indicates lower viscous effects in momentum equation.
Accordingly, the momentum of the fluid is higher. It is also observed that, the velocity of ordinary fluid is large when compared to the nanofluid velocity. On the other hand, an expected outcome was obtained for the influence of solutal Grashof number in both cases. That is, the fluid velocity gets enhanced due to increase in the species buoyancy force, which is as shown in Fig. 5. The solutal Grashof number is described as the ratio of the species buoyancy force to the viscous hydro-dynamic force. As solutal Grashof number increases, the viscous hydrodynamic force decreases. As a result, the momentum boundary layer thickens. In Fig. 6, we can observe that an increase in time leads to an increase in the velocity profile in both the cases. Importantly, the behavior of velocity profile noticed in Figs. 2-6 is qualitatively same in both moving and stationary plates.
The effect of nanoparticle volume fraction, heat source/ sink, time parameter, thermal radiation and Prandtl number on the dimensionless temperature profile versus g has been plotted in Figs. 7-11 correspondingly. It is observed in Fig. 7 that, an increase in the volume fraction parameter leads to an intensification of the temperature profile. Strengthening of volume fraction of nanoparticles enhances the thermal conductivity of the nanofluid and as a result the thermal boundary layer thickens. The volume fraction parameter is a key parameter and it has a significant role in the improvement of heat characteristics of the fluids. Thus we can conclude that, the temperature can be controlled by varying the nanoparticle volume fraction in many industrial processes. It is also observed that, the temperature profile is higher for larger time (s = 1) than smaller time (s = 0.1). Before discussing the influence of Q, it is to be noted that Q > 0 corresponds to the internal heat source and Q < 0 corresponds to heat sink. That is the way we have chosen the values of Q as —2, —1,0,1,2. From Fig. 8, it is clear that, by increasing the values of Q > 0 the energy is released and these results in temperature increase, while, by increasing the values of Q < 0 energy is absorbed and this causes decrease in temperature field. Further, it is seen that, the temperature profile is higher in the presence of thermal radiation effect (R — 5) than its absence (R — 0). This is because, by introducing the thermal radiation provides more heat into the fluid, and as a result the temperature and the thickness of its boundary layer are intensified, which is similar result observed by Turkyilmazoglu and Pop [23]. From Fig. 9
it is observed that an increase in time parameter, results in a significant increase in the temperature profile. The similar trend is observed for the influence of thermal radiation parameter on the temperature profile as can be shown in Fig. 10. In addition, we observed in Fig. 10 that, the temperature of nano-fluid is higher than that of ordinary fluid. The fluids with large Pr have low thermal diffusivity, so it causes low heat penetration. As a consequence, the temperature profile reduces significantly, which is illustrated as in Fig. 11.
The influence of chemical reaction, time and Schmidt number on dimensionless concentration profiles has been plotted in Figs. 12-14 correspondingly. Fig. 12 shows that, an increase in the chemical reaction parameter decreases the concentration profile rapidly. The central reason is that, the number of solute molecules undergoing chemical reaction gets increased as chemical reaction parameter increases, which leads to decrease in concentration field. Thus, a destructive chemical reaction reduces the solutal boundary layer thickness significantly. This outcome is in excellent agreement with that of obtained by Chamkha [14]. From Fig. 13, it is evident that the concentration of the fluid increases with increase in time. Fig. 14 anticipates that an increase in the Schmidt number corresponds to a weaker solute diffusivity which allows a shallower penetration of solutal effect. As a consequence the concentration decreases with increase in Sc. Thus, the solute boundary layer is thicker for smaller values of Sc and vice versa. We have chosen the Sc values as Sc — 0.22, 0.30, 0.60, 0.78 which correspond to hydrogen, helium, water vapor, and ammonia respectively. Further, it is observed that the concentration of profile becomes steeper in the presence of chemical reaction.
Fig. 15 is prepared to demonstrate the influence of heat source/sink on Cf and Nu versus /. An increase in Q corresponds to increase in Cf whereas, opposite phenomenon can be seen with Nu. This is due to the fact that, for large values of heat source/sink parameter the momentum boundary layer thickens. Consequently, the skin friction co-efficient increases with heat source/sink parameter. The influence of thermal radiation and time parameter on Cf and Nu is quite same as the impact of Q, and this is as shown in Figs. 16 and 17. It is also observed from Figs. 15-17 that, both the skin friction co-efficient and Nusselt number decrease with increase in /. Finally, Fig. 18 has been plotted to perceive the influence of time parameter on Cf and Sh versus K. The skin friction
co-efficient and Nusselt number reveal an opposite behavior for the influence of both time and chemical reaction parameter. Table 2 explores that, the skin friction co-efficient is an increasing function of / Gr' Grc and s, whereas it is decreasing function of M2. It also depicts that, the drag force at the plate is lower when the plate is at stationary as compared with moving. Physically, the momentum boundary layer thickness is higher for large values of / , Gr', Grc and s. Hence, the skin friction drag is enhanced. The influence of Prandtl number on the Nu and C/ is qualitatively opposite, as can be seen in Table 3. Further, it is noted that the Nusselt number at the plate g — 0 is higher in the ordinary fluid than that of nanofluid.
4. Conclusion
An unsteady MHD free convection flow of a fluid past a vertical moving/stationary plate has been studied in the presence of nanoparticles, thermal radiation, heat source/sink and chemical reaction effects. The set of governing equations has been solved analytically using Laplace transform method. Based on the obtain results, the following conclusions are drawn;
• Intensifying magnetic field strength leads to reduction of drag force at the plate. On the other hand, the opposite trend is accounted as nanoparticle volume fraction varies.
• The effects of thermal radiation, heat source/sink and transient are qualitatively same on shear stress and Nusselt number.
• Larger values of the nanoparticle volume fraction showed a significant impact on velocity as well as temperature profile. By varying nanoparticle volume fraction, the flow and heat transfer characteristics could be controlled.
• The influence of thermal and solute Grashof number stabilizes the momentum boundary layer growth.
• As a result of thermal radiation and heat source/sink, the temperature profile enhances rapidly.
• The influence of Prandtl number and time parameter is opposite on temperature field.
• The presence of chemical reaction and heavier species reduces the concentration in the boundary layer.
• The inclusion of nanoparticles in the base fluids offers a potential in improving the heat transfer performance.
Acknowledgment
One of the authors (B.J. Gireesha) is thankful to the University Grants Commission, India, for the financial support under the scheme of Raman Fellowship for Indian Scholars in USA.
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