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ORIGINAL ARTICLE
Unsteady MHD boundary-layer flow and heat transfer of nanofluid over a permeable shrinking sheet in the presence of thermal radiation
Samir Kumar Nandy a,% Sumanta Sidui b, Tapas Ray Mahapatra c
a Department of Mathematics, A.K.P.C Mahavidyalaya, Bengal, Hooghly 712 611, India b Department of Mathematics, Ajhapur High School, Ajhapur, Burdwan 713 401, India c Department of Mathematics, Visva-Bharati, Santiniketan 731 235, India
Received 3 July 2014; revised 3 September 2014; accepted 7 September 2014
KEYWORDS
Unsteady flow; Heat transfer; Magnetic field; Thermal radiation; Shrinking sheet
Abstract Forced convection in unsteady boundary layer flow of nanofluid over a permeable shrinking sheet in the presence of thermal radiation is studied. A variable magnetic field is applied normal to the sheet. The nanofluid model includes Brownian motion and thermophoresis effects. The governing momentum, energy and nanofluid solid volume fraction equations are solved numerically using fourth order Runge-Kutta method with shooting technique. The effects of various physical parameters on dimensionless velocity, temperature, nanoparticle volume fraction, as well as the skin friction, local Nusselt and local Sherwood numbers are analyzed. The numerical results indicate that dual solutions exist for certain values of the magnetic parameter (M), wall mass suction (s) and unsteadiness parameter (A). It is found that both magnetic field and wall mass suction widen the range of unsteadiness parameter for which the solution exists. The skin friction coefficient, local Nusselt and Sherwood numbers increase for the first solution and decrease for the second solution with the increase in M.
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1. Introduction
The boundary layer flow of an electrically conducting fluid in the presence of magnetic field has wide applications in many engineering problems such as MHD generator, plasma studies,
* Corresponding author. Tel.: +91 9434667221; fax: +91 3211 246772.
E-mail address: nandysamir@yahoo.com (S.K. Nandy).
Peer review under responsibility of Faculty of Engineering, Alexandria
University.
nuclear reactors, geothermal energy extraction, and oil exploration. Also, radiative heat transfer in the boundary layer flow is very important from application point of view, because the quality of the final product is very much dependent on the rate of heat transfer of the ambient fluid particles. Again unsteady flows, such as start-up process and periodic fluid motion, are very much important in engineering practices. In many engineering problems, such as helicopter rotor, the ship propeller, the cascades of blades of turbo-machinery unsteady environment occurs. Hence it is very much important to
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Nomenclature
A nondimensional unsteadiness parameter Rex local Reynolds number
B variable magnetic field s mass suction parameter
Bo uniform magnetic field Shx local Sherwood number
c constant proportional to the shrinking velocity T temperature of the fluid
cf skin friction coefficient T T w temperature of the fluid at the sheet
cP specific heat at constant pressure T1 temperature of the fluid in the free stream
C nanoparticle volume fraction u, v velocity components along x and y axes, respec-
C ^w nanoparticle volume fraction at the surface tively
C1 nanoparticle volume fraction in the free stream uw velocity of the shrinking sheet
Db Brownian diffusion coefficient vw velocity of mass transfer
DT thermophoresis diffusion coefficient
f(n) dimensionless stream function Greek symbols
Ki mean absorption coefficient am thermal diffusivity
Le Lewis number /(n) rescaled nanoparticle volume fraction
M dimensionless magnetic parameter n similarity variable
Nb dimensionless Brownian motion parameter h dimensionless temperature of the fluid
Nt dimensionless Thermophhoresis motion parameter V kinematic viscosity of the fluid
Nux local Nusselt number k dimensional unsteadiness parameter
Pr Prandtl number Pf density of the base fluid
Preff effective Prandtl number s ratio of the effective heat capacity of the nanopar
qm wall mass flux ticle material and the ordinary fluid
qw wall heat flux w stream function
R thermal radiation parameter
investigate the simultaneous effects of thermal radiation, magnetic field and unsteadiness.
Various industries such as chemical production, power generation in a power plant, production of microelectronics, advanced nuclear systems demand an efficient heat exchanger device. Classical heat transfer fluids such as water, ethylene glycol, engine oil have a limitation in heat transfer efficiency. In contrast, metals are very good conductors. To overcome this disadvantage of fluids, it would be desired to combine metals and fluids to produce a heat transfer medium that behaves like a fluid, but has the thermal conductivity of a metal. The necessity for improving the thermal conductivity and enhancing the heat transfer has led to the utilization of nanoparticles in the fluid.
A liquid containing suspension of a metallic (such as Cu, Al, Fe, Hg and Ti) or a non-metallic (such as metal oxide) nanometer-sized (generally less than 100 nm) solid particles or fibers is termed as nanofluid. The main characteristics of the nanofluid are the enhancement of the thermal conductivity of the base fluid, minimal clogging in few passage, long term stability and homogeneity (see Choi [1], Choi et al. [2]). Because of the wide range of applications of nanofluids, significant research has been carried out in recent years by the researchers to study the heat transfer characteristics of these fluids.
Buongiorno [3] was the first who formulated the nanofluid model taking into account the effects of Brownian motion and thermophoresis. In his work, he has pointed that although there are some elements those affect nanofluid flow such as inertia, Brownian diffusion, thermophoresis, Magnus effect, fluid drainage, diffusiophoresis and gravity; only Brownian diffusion and thermophoresis are important mechanisms in nanofluid. It should be mentioned that a lot of researchers
(see [4-6]) have used the mathematical nanofluid model proposed by Buongiorno [3] to study different problems under different physical situations.
Bachok et al. [7] investigated the forced convective boundary layer flow of nanofluid over a moving plate and they concluded that dual solutions exist when the plate and free stream move in opposite directions. Khan and Pop [8] analyzed the boundary layer flow and heat transfer of nanofluid over a stretching surface. The effect of magnetic field on free convection flow of nanofluid past a vertical flat plate was considered by Hamad [9]. The boundary layer flow and heat transfer in a viscous fluid containing metallic nanoparticles over a nonlinear stretching sheet are investigated by Hamad and Ferdows [10]. They studied different types of nanoparticles and found that the behavior of the fluid flow changes with the change of type of the nanoparticles. Numerous recent studies on nanofluids can be found in [11-15].
Recently, the boundary layer flow near a shrinking sheet gets attention due to increasing engineering applications. The existence and uniqueness of steady viscous flow due to a shrinking sheet was established by Miklavcic and Wang [16] and they concluded that for some specific value of suction at the sheet, dual solutions exist and also in certain range of value of suction, no boundary layer solution exists. On the other hand, Wang [17] investigated the stagnation point flow toward a shrinking sheet and obtained dual solutions for some values of the ratio of shrinking and stagnation flow rates. Following these works, some very important investigations in this direction were made (see [18-23]).
The problem of boundary layer flow over a shrinking sheet with mass transfer is investigated by Fang et al. [24]. In that paper, they reported that boundary layer solution exists only for mass suction parameter s P 2 and for the unsteadiness
parameter b 6 0. Later, Bachok et al. [25] analyzed the unsteady three dimensional boundary layer flow due to a permeable shrinking sheet. In another paper, Bachok et al. [26] studied the unsteady boundary layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet and they concluded that the unsteadiness parameter as well as the mass suction parameter widen the range of stretching/ shrinking parameter for which the boundary layer solution exists. Very recently, the unsteady flow and heat transfer over an unsteady shrinking sheet with suction in a nanofluid was analyzed by Rohni et al. [27] and it was found that dual solutions exist for a certain range of wall mass suction, unsteadiness and nanofluid parameters.
The aim of the present paper (which is an extension of Rohni et al. [27]) is to study the simultaneous effect of the magnetic field and thermal radiation on the flow and heat transfer due to the unsteady, two-dimensional laminar flow of a viscous nanofluid caused by a permeable shrinking sheet. In this paper, a similarity analysis is performed to reduce the governing equations to ordinary differential equations which are subsequently solved numerically using fourth order Runga-Kutta method with shooting technique. Results presented focus on how the magnetic field, surface mass transfer, Brownian motion, thermophoresis and thermal radiation affect the heat and mass transfer characteristics of the flow. The present study is of immediate interest to all those which are highly affected with heat enhancement concept.
2. Flow analysis
Consider unsteady two-dimensional laminar boundary-layer flow of incompressible electrically conducting viscous nano-fluid past a permeable shrinking sheet. The flow is subjected to a transverse magnetic field of strength B which is assumed to be applied in the positive y-direction, normal to the surface. It is assumed that the velocity of the shrinking sheet is uw(x, t) and the velocity of the mass transfer is vw(x, t), where x is the coordinate measured along the shrinking sheet and t is the time. It is also assumed that the constant surface temperature and concentration of the sheet are Tw and Cw, while the uniform temperature and concentration far from the sheet are Tx and CM, respectively.
Under these assumptions, the unsteady boundary-layer equations governing the flow, heat and mass transfer are
du du du d2u rB2
dt dx dy dy2
dT dT dT _ dt dx dy m dy2 pfCp dy
dC dT Dt
dy dy Ti
—+u—+vdC-D @2c+DLdT
dt dx dy B dy2 Tœ dy2 '
(1) (2)
where u and v are the velocity components in the x and y-directions respectively, v is the kinematic viscosity, r is the electrical conductivity (assumed constant), pf is the density
of the base fluid, am is the thermal diffusivity, DB is the Brownian diffusion coefficient, DT is the thermophoresis diffusion coefficient and cp is the specific heat at constant pressure. Here s is the ratio of the effective heat capacity of the nanopar-ticle material and the heat capacity of the ordinary fluid, T is the fluid temperature and C is the nanoparticle volume fraction.
The term r®!u in the R.H.S. of Eq. (2) is the Lorentz force which arises due to the interaction of the fluid velocity and the applied magnetic field. In writing Eq. (2), we have neglected the induced magnetic field since the magnetic Reynolds number for the flow is assumed to be very small. This assumption is justified for flow of electrically conductive fluids such as liquid metals e.g. mercury, liquid sodium, etc. (see Shercliff [28]). Eq. (3) depicts that heat can be transported in a nano-fluid by convection, by conduction and also by virtue of nano-particle diffusion and radiation. The term u + v fy is the heat
convection, the term am fyT is the heat conduction; the term sDB fy @y is the thermal energy transport due to Brownian diffusion, the term sD- (jyj is the energy transport due to ther-mophoretic effect and — is the nanoparticle heat diffusion
r Pfcp @y r
by radiation. Eq. (4) shows that the nanoparticles can move homogeneously within the fluid (by the term u @C + v fy), but they also possess a slip velocity relative to the fluid due to Brownian diffusion DB and the thermophoresis T^ fyT .
Here the boundary conditions are
u — uw(x, t) — —
Cx 1- kt '
— vw(x' t); T — Tw; C — Cw at y — 0'
u ! 0, T! T^, C! C^ as y n, The wall mass transfer velocity then becomes
vw(x, t) = -y*!
cm - kt
where s is the constant wall mass transfer parameter with s > 0 for suction and s < 0 for injection, respectively. Using Rosseland's approximation for radiation (see Brewster [29]), we can write
4r dT4
"3K "dy '
where rj is the Stefan-Boltzmann constant and K is the mean absorption coefficient. Assuming the temperature difference within the flow is such that T4 may be expanded in a Taylor series about T1 and neglecting higher order terms we get T4 ~ 4TlT — 3T1. Hence from Eq. (7), using the above result, we have
16r 1 ti @2t
3K dy2
To attain the similarity solutions of the Eqs. (1)-(4) with the boundary conditions (5), we take the transverse unsteady magnetic field strength applied to the sheet is of the form B — B0/V1 — kt, where B0 is constant. This form of B0(t) has also been considered by Vajravelu et al. [30] while analyzing the MHD flow and heat transfer over an unsteady stretching sheet. The stream function and dimensionless variable can be taken as
0(g)—-T
- kt ' C — C C - C '
v(l - kt)'
where the stream function W is defined in the usual way u — dW/dy and v — —dW/dx. Substituting (9) into Eqs. (1)-(4), we obtain the following ordinary differential equations
/''' + //'' -/'2 - a (/' +1/") - Mf — 0'
-1-0" +/0' - Agh' + Nèh'/' + Nt002 -
Pre// 2
/'' + Le(/- Ag/' + N0'' = 0'
(l0) (ll) (l2)
„ J?2 k M — —0 ; A — -; Pre// —
Pr 4r1 73, v
- R —-1 1 Pr —-
1 + 4R/3 ' Kl amp/Cp' am '
v = I Nb = tDb(C» - Ci), Nt = sDT(Tw - Ti), Le = JL. (13) Pf v vTi Db
Here M is the dimensionless magnetic parameter, A is the unsteadiness parameter, Preff is the effective Prandtl number, R is the thermal radiation parameter, Pr is the Prandtl number, v is the kinematic viscosity of the fluid, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter and Le is the Lewis number.
It is worth noting that the temperature actually does not depend on Prandtl number (Pr) and the thermal radiation parameter (R) independently, but depends only on a combination of them termed as effective Prandtl number Preff which is directly proportional to the Prandtl number and inversely proportional to the thermal radiation parameter. The corresponding boundary conditions are
f(0) = s, f (0) = -1, 0(0) = 1, /(0) = 1 f (1)= 0, 0(i) = 0, /(1) = 0. (14)
The physical quantities of interest are the skin friction coefficient Cf, the local Nusselt number Nux and the local Sherwood number Shx which are defined as
'/ — 2 ' P/uW
Nux = Xqw
k(Tw - Ti)
DB(Cw - C1) '
where sw is the shear stress at the stretching surface, qw and qm are the wall heat and mass fluxes, respectively. Hence using Eq. (10) we get
RelJ2Cf = /'(0), NUxRe-1'2 = -0' (0), ShxRe'1'2
= -/' (0) (16)
where Rex — Uw (x, t)x/v is the local Reynolds number based on the stretching velocity uw(x, t).
3. Numerical solution
By using a similarity transformation, the governing equations of the problem are reduced to a system of coupled, non-linear ordinary differential equations, which are solved numerically by fourth order Runga-Kutta method with shooting technique (see for details Gladwell and Sayers [31]). For the sake of
brevity, further details of the solution process are not presented here. Since the physical domain in this problem is unbounded, whereas the computational domain has to be finite, we apply the far field boundary conditions for the similarity variable g at a finite value denoted by gmax. In all our computations we have used the value of gmax — 12 which is sufficient to satisfy the far field boundary conditions asymptotically for all values of the physical parameters considered in this problem. For these numerical computations, a uniform step size Dg — 0.01 is taken and the solutions are obtained with an absolute error tolerance of 10 6 in all cases.
4. Results and discussion
The analysis of the results obtained by the applied numerical scheme explores the condition for which the boundary layer solutions are possible. Following Fang et al. [24] and Rohni et al. [27], we have considered the decelerating shrinking sheet only (i.e., A 6 0). To assess the accuracy of the present results, comparison of the critical values of A (i.e Ac, beyond which the boundary layer solution is not possible) between the present results and the previously published results are made for special cases (in the absence of magnetic field and thermal radiation) and found excellent agreement. This gives us confidence of our numerical results (see Table 1).
Numerical computations are carried out for several sets of values of the governing parameters, namely, unsteadiness parameter (A), magnetic parameter (M), thermal radiation parameter (R), effective Prandtl number (Preff), Brownian motion parameter (Nb), thermophoresis parameter (Nt), Lewis number (Le) and suction parameter (s) using the numerical procedure discussed in the previous section. In order to illustrate the salient features of the model, the numerical results are presented in Figs. 1-15.
Fig. 1 shows that the dual solutions of skin friction coeffi-cient/"(0) with the unsteadiness parameter A(< 0) for several values of the magnetic parameter M. It reveals that it is possible to get dual solutions of the boundary layer equations for A < 0. These dual solutions are in the range Ac 6 A < 0 and no solution exists for A < Ac < 0, where Ac is the minimum value of A for which the solution exists. The value of Ac for different values of M are shown in figure. It is observed that the value of |Ac| increases as the strength of the magnetic parameter increases. This shows that as the value of M increases the range of the values of A for which the solution of the boundary layer equations exists, also increases. Also, for a certain mass suction parameter (s — 2.1), in the case of upper solution branch, /"(0) increases with the increase of M. However, for the lower solution branch, opposite trend is observed. The variation of/"(0) with A(< 0) for different values of mass suction parameter s are depicted in Fig. 2. The fig-
Table 1 Comparison of the critical values of unsteadiness parameter Ac for several values of suction parameter s with M = 0.
s Present study Rohni et al. [27]
2.10 -1.654850 -1.6550
2.15 -3.860346 -3.8605
2.20 -8.349103 -8.3488
Figure 1 Dual solutions of skin friction coefficient /'(0) with
A(< 0) for several values of M when s = 2.1 . Figure 4 Variation of -/'(0) with A for several values of M
when s = 2.1, Le = 2.0, Nb = 0.5, Nt = 0.5 and Pre// = 1.0.
Ac1 =-<>■! >62989 .............
A,^»- 2.333451 ^^ ^ - s
' ' ' '
A^ = - 5.064349 .....
M = 0.1
- first solution
--second solution
-6.0 -4.5 -3.0 -1.5 0.0
Figure 2 Dual solutions of skin friction coefficient /"(0) with A(< 0) for several values of s when M = 0.1 .
Figure 5 Variation off (g) for several values of M when s = 2.1, A = -1.0, Le = 2.0, Nb = 0.5, Nt = 0.5 and Preff = 0.71 .
M = 0.2 M = 0.1
S = 2.1 n!\.M=0'0 V N ^^ /1 N S -OVl
first solution
--second solution N \
Figure 3 Variation of — h(0) with A for several values of M when s = 2.1, Le = 2.0, Nb = 0.5, Nt = 0.5 and Preff = 1.0.
M = 0.2 first solution second solution
Figure 6 Variation of f (g) for several values of A when s = 2.1, M = 0.2, Le = 2.0, Nb = 0.5, Nt = 0.5 and Preff = 0.71.
Figure 7 Variation of/(g) for several values of s when A — — 1.0, M — 0.2, Le — 2.0, Nb — 0.5, Nt — 0.5 and Pre// — 0.71.
Figure 10 Variation of h(g) for several values of s when M — 1.0, A — —2.0, Le — 2.0, Nb — 0.5, Nt — 0.5 and Pre// — 0.71.
Figure 8 Variation of h(g) for several values of M when s — 2.0, A — —1.0, Le — 2.0, Nb — 0.5, Nt — 0.5 and Pre// — 0.71.
Figure 11 Variation of h(g) for several values of Nt when s — 2.0, M — 1.0, A — —2.0, Le — 2.0, Nb — 0.5, and Pre/f — 0.71.
Figure 9 Variation of h(g) for several values of A when s — 2.1, M — 0.2, Le — 2.0, Nb — 0.5, Nt — 0.5 and Pre/f — 0.71.
Figure 12 Variation of h(g) for several values of Nb when s — 2.1, A — —1.0, M — 0.5, Le — 2.0, Nt — 0.5 and Pre/f — 0.71.
Figure 13 Variation of h(g) for several values of Pre// when s — 2.0, M — 1.0, A — —2.0, Le — 2.0, Nb — 0.5, and Nt — 0.5.
0.6--V
0.20.00 0
Figure 14 Variation of /(g) for several values of Le when s — 2.0, M — 1.0, A — —2.0, Nt — 0.5, Nb — 0.5, and Pre/f — 0.71.
0.40.20.0 -
0 12 3 4
Figure 15 Variation of /(g) for several values of M when s — 2.0, A — —2.0, Le — 2.0, Nt — 0.5, Nb — 0.5, and Preff — 0.71.
ure reveals that as the wall mass suction parameter increases, the solution domain expands with the critical value of A moving to the left.
The numerical result reveals that dual solutions exist for certain values of the physical parameters. In order to ascertain which of the two solutions is expected to appear physically, the stability analysis of the dual solutions is required. Based on the stability analysis, only the first solution (upper branch) is stable and corresponds to the physically meaningful solution. On the other hand, the second solution (lower branch) is unstable and not physically meaningful. The relevant procedure can be found in studies reported by Merkin [32], Weidman et al. [33], Mahapatra et al. [34] and very recently by Rosca and Pop [35]. The novel result that emerges from the analysis is that, though two solutions exist mathematically, only the stable solution is physically meaningful and can be realized physically.
The variation of the heat transfer rate —h' (0) and the concentration rate —/'(0) with A(< 0) for several values of the magnetic parameter M with fixed values of the other parameters is shown in Figs. 3 and 4 respectively. The dual nature of the solutions are also seen in these figures. From these figures, we can see that for the upper solution branch, due to an increment in value of M, both the heat transfer and the concentration rates increase. However for the lower solution branch, the opposite trend is observed.
Figs. 5-7 are drawn to analyze the influence of magnetic parameter (M), unsteadiness parameter (A) and wall mass suction parameter (s) on the horizontal velocity component f (g). Fig. 5 displays the variation off (g) with g for several values of M. It is seen that horizontal velocity at a point increases with increase in M for the first solution and the opposite is true for the second solution. In fig. 6, the effect of unsteadiness parameter on the velocity profiles is shown for both the solution branches with fixed values of s(= 2.1) and M(= 0.2). The figure reveals that for both the solution branches, the velocity profiles are not monotonic. For the first solution branch, as the magnitude of the unsteadiness parameter (i.e, |A|) increases, f(g)| decreases up to a point near the sheet but beyond this point opposite trend is observed. Fig. 7 illustrates the effect of the mass transfer parameter s on the horizontal velocity profile f(g) for two solution branches with A — —1.0 and M — 0.2. The figure reveals that the velocity penetration into the fluid becomes shorter with the increase of s for the upper solution branch and the velocity profiles penetrate deeper for a higher mass suction parameter.
The variation of h(g) with g is shown in Fig. 8 for several values of M with fixed values of the other parameters. It is observed that temperature at a point decreases with increase in M for the first solution and the temperature increases with increase in M for the second solution. This is a consequence of the fact that the temperature field is influenced by the advec-tion of the fluid velocity above the sheet. Fig. 9 shows the variation of temperature h(g) with g for several values of the unsteadiness parameter A(< 0). The figure reveals that the temperature at a point decreases as the magnitude of the unsteadiness parameter increases. This is due to the fact that the heat transfer rate increases with the increase in unsteadiness parameter which in turn reduces the temperature of the fluid. The temperature profiles for various mass suction parameter (s) are shown in Fig. 10. From the figure it is seen that the thermal boundary layer thickness decreases with increasing s. Physically this can be explained as follows; Due
to mass suction, the fluid is brought closer to the surface and consequently it prevents the vorticity diffusion. Also it is observed that, the thermal boundary layer thickness for the first solution is always thinner than that of the second solution.
The effects of thermophoresis parameter (Nt) and Brownian motion parameter (Nb) on the temperature profiles for selected values of parameters are shown in Figs. 11 and 12, respectively. As the parameters Nt and Nb increase, the temperature at a point increases. As a consequence, the thickness of the thermal boundary layer increases with the increase of Nt and Nb. The thermophoresis phenomenon describes the fact that small micron sized particles suspended in a non-isothermal gas will acquire a velocity in the direction of decreasing temperature. An increase in Nt results in an increase of the temperature difference between the sheet and the ambient fluid and consequently the thermal boundary layer thickness increases. Fig. 13 represents the influence of the effective Prandtl number (Preff) on the dimensionless temperature profile. The temperature and the thermal boundary layer thickness reduce with Preff. The physical reason is that, due to increase in Prandtl number (Pr), the thermal conductivity of the fluid reduces and consequently thermal boundary layer thickness decreases.
The effect of Lewis number (Le) on the dimensionless rescaled nanoparticle volume fraction /(g) is shown in Fig. 14. It is observed that the concentration boundary layer thickness decreases with increasing Le. This is due to the decrease in mass diffusivity or the Brownian motion of the nanoparticle. Fig. 15 shows the variation of the dimensionless rescaled nanoparticle volume fraction /(g) with magnetic parameter M in the concentration boundary layer. It is seen that the concentration boundary layer thickness decreases with magnetic parameter for the first solution and the opposite trend is observed for the second solution.
5. Concluding remark
The effects of magnetic, suction, thermal radiation, thermo-phoresis, Brownian motion and Lewis number parameters on the boundary layer flow and heat transfer of nanofluid over a shrinking sheet are investigated numerically using fourth order Runge-Kutta method with shooting technique. The effects of these parameters on dimensionless velocity, temperature, rescaled nanoparticle volume fraction, skin friction, rescaled Nusselt and Sherwood numbers can be summarized as follows:
1. The dual solutions are found to exist for the flow over a shrinking sheet.
2. The range of the existence of the solutions to the problem increases with increasing magnetic and suction parameters.
3. The dimensionless velocity increases with increasing magnetic and suction parameters for the first solution and decreases for the second solution.
4. The dimensionless temperature and the rescaled nanoparti-cle volume fraction both decrease with increasing M for the first solution and increase for the second solution.
5. The dimensionless temperature decreases with increasing values of unsteadiness, suction and effective Prandtl parameters but increases with increasing values of Nt and Nb.
6. The dimensionless rescaled nanoparticle volume fraction decreases with increasing Lewis number.
7. The magnitude of skin friction increases with increasing magnetic and suction parameters for the first solution and opppsite is true for the second solution.
8. The local Nusselt and Sherwood numbers increase with increase magnetic parameter for the first solution and decrease for the second solution.
Acknowledgements
We thank the reviewers for their helpful suggestions and comments, which enable to improve the presentation of the paper.
One of the author S.K. Nandy is grateful to the University
Grants Commission, New Delhi, India, for providing financial
support under a Minor Research Project Grant No. F. PSW-
002/13-14 to carry out the work.
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