Propulsion and Power Research 2016;5(3):211-222

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ORIGINAL ARTICLE

MHD flow of Boungiorno model nanofluid over a vertical plate with internal heat generation/absorption

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B. Gangaa, S. Mohamed Yusuff Ansarib, N. Vishnu Ganeshc, A.K. Abdul Hakeema,n

aDepartment of Mathematics, Providence College for Women, Coonoor 643104, India Department of Mathematics, Jamal Mohamed College, Trichy 620020, India

cDepartment of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts & Science, Coimbatore 641020, India

Received 7 May 2015; accepted 10 September 2015 Available online 12 August 2016

KEYWORDS

Heat generation/ absorption; Homotopy analysis method;

Magnetohydrodynam-ics (MHD); Nanofluid; Vertical plate

Abstract A mathematical analysis has been carried out to investigate the effect of internal heat generation/absorption on steady two-dimensional radiative magnetohydrodynamics (MHD) boundary-layer flow of a viscous, incompressible nanofluid over a vertical plate. A system of governing nonlinear PDEs is converted into a set of nonlinear ODEs by suitable similarity transformations and then solved analytically using HAM and numerically by the fourth order Runge-Kutta integration scheme with shooting method. The effects of different controlling parameters on the dimensionless velocity, temperature and nanoparticle volume fraction profiles are discussed graphically. The reduced Nusslet number and the local Sherwood number are tabulated. It is found that the nanosolid volume fraction profile decreases in the presence of heat generation and increases in the case of heat absorption and a reverse trend is observed in velocity profile. An excellent agreement is observed between present analytical and numerical results. Furthermore, comparisons have been made with bench mark solutions for a special case and obtained a very good agreement.

© 2016 National Laboratory for Aeronautics and Astronautics. 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/).

"Corresponding author.

E-mail address: abdulhakeem6@gmail.com (A.K. Abdul Hakeem). Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.

1. Introduction

Nanofluids are suspensions of nanoparticles in fluids which was introduced by Choi [1] that show significant

2212-540X © 2016 National Laboratory for Aeronautics and Astronautics. 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/).

http://dx.doi.Org/10.1016/j.jppr.2016.07.003

enhancement of their properties at modest nanoparticle concentrations. Many of the publications on nanofluids are about understanding their behavior so that they can be utilized where straight heat transfer enhancement is paramount as in many industrial applications such as nuclear reactors, transportation, electronics as well as biomedicine and food. This concept attracted various researchers towards nanofluids, and various theoretical and experimental studies have been done to find the thermal properties of nanofluids. Boungiorno et al. [2] studied the thermal conductivity of nanofluids experimentally. The same author proposed an analytical model for convective transport in nanofluids taking into the account of Brownian diffusion and thermophoresis [3].

In recent year, the natural convection flow of nanofluid has been studied in the following publications [4-16]. Kuznetsov and Nield [17] investigated the natural convec-tive boundary-layer flow of a nanofluid past a vertical plate using Boungiorno model. Gorla and Chamkha [18] studied the natural convective boundary layer flow of nanofluid in a porous medium. Khan and Pop [19] studied the boundary layer flow of a nanofluid past a stretching sheet by considering the Brownian diffusion and thermophoresis effects. Khan and Aziz [20] investigated the boundary layer flow of a nanofluid past a vertical surface with a constant heat flux. Aziz and Khan [21] studied natural convective flow of a nanofluid over a convectively heated vertical plate. Recently, Rashad et al. [22] analyzed the natural convection flow of nanofluid over a vertical plate with stream wise temperature variation.

The interaction of natural convection with thermal radiation has increased greatly during the last decade due to its importance in many practical involvements. When free convection flows occur at high temperature, radiation effects on the flow become significant. Radiation effects on the free convection flow are important in context of space technology, processes in engineering areas occur at high temperature. Based on these applications, Olanrewaju et al. [23] investigated the boundary layer flow of nanofluids over a moving surface in a flowing fluid in the presence of radiation past a moving semi-infinite flat plate in a uniform free stream. Poornima et al. [24] analyzed the simultaneous effects of thermal radiation and magnetic on heat and mass transfer flow of nanofluids over a non-linear stretching sheet. Recently, Turkyilmazoglu and Pop [25] studied the thermal radiation effects on the flow of single phase nanofluid over a infinite vertical plate.

The study of heat generation or absorption effects is very important in cooling processes. Although, exact modeling of internal heat generation or absorption is quite difficult, some simple mathematical models can express its average behavior for most physical situations [26,27]. Ahmed et al. [28] investigated the effects of heat source/sink on the boundary layer flow of single phase nanofluid over a stretching tube. Very recently, Akilu and Narahari [29] studied the effects of internal heat generation/absorption on natural convection flow of a nanofluid over an inclined plate numerically.

The main goal of this paper is to analyze the effect of internal heat generation/absorption on steady radiative magnetohydrodynamic free convective boundary layer flow of an incompressible nanofluid over a vertical flat plate both analytically and numerically. The analytical solutions are obtained using HAM and the fourth order Runge-Kutta method along with shooting technique is used to find the numerical solutions for the physical problem.

2. Formulation of the problem

We consider the steady two-dimensional boundary layer flow of a nanofluid over vertical plate in the presence of magnetic field intensity, thermal radiation and volumetric rate of heat generation/absorption. We select a coordinate frame in which the x-axis is aligned vertically upwards. We consider a vertical plate at y—0. At this boundary, the temperature T and the nanoparticle volume fraction ^ take constant values Tw and respectively. The temperature T and the nanoparticle volume fraction of the nanofluid ^ take values TM and respectively as y — 1. We also consider influence of a constant magnetic field strength B0 which is applied normally to the plate. It is further assumed that the induced magnetic field is negligible in comparison to the applied magnetic field. Under the above assumptions, the boundary layer equations governing the flow, thermal and concentration fields can be written in dimensional form as Ref. [17].

? + £ —o d)

_ d2u dx ^ dy2

u 2 ■v-J -*B»u

+ [ (1 - <Pi)Pf - Tœ)- (Pp -Pf 1 g (0 - ) ]

dT dT 2 Q , u— + v— = aS T + f^~(T- Ti) dx dy \Pc)f

D d0 dT + DT ÍdT B dy dy Tœ\ dy

1 ( dq_

(Pc)f\ dy

# # / d2<A DT /32T\

uî + vî =D'W) +DZW) №

where u and v are the velocity components along the x and y directions respectively. p is the fluid pressure, pf is the density of base fluid, pp is the nanoparticle density, ^ is the absolute viscosity of the base fluid, a — -p^- is the thermal

diffusivity of the base fluid, t — jPCz is the ratio of

nanoparticle heat capacity and the base fluid heat capacity, ^ is the local solid volume fraction of the nanofluid, ¡3 is volumetric thermal expansion coefficient of the base fluid, DB is the Brownian diffusion coefficient, DT is the

thermophoretic diffusion coefficient, T is the local temperature and g is the acceleration due to gravity and Q is the heat generation/absorption coefficient. The boundary conditions are taken to be,

u — 0, v = 0, T — Tw, T — Tw, $ — (pw at y = 0 (5)

u — v — 0, T — Ti, $ — as y-i.

The radiative heat flux qr is described by Rosseland approximation [30,31] such that

4a* dT4

qr —~3ksiy (7)

where a* and ô are the Stefan-Boltzmann constant and k the mean absorption coefficient, respectively. We assume that the temperature differences within the flow are sufficiently small so that the T4 can be expressed as a linear function after using Taylor series to expand T4 about the free stream temperature T1 and neglecting higher-order terms. This result is the following approximation:

T4 ffi 4T1T- 3T1 (8)

Using Eqs. (7) and (8) in Eq. (3), we obtain

u — + v— — aV2T , ox oy (pc)f

d$ dT DT fdT

+t DB —--—- —

dy dy T i\ dy

(T - Ti)

1 /16 a*T31 d2T " (pc)^ V 3ô dy2

3. Similarity transformations

The following quantities are introduced to transform Eqs. (2), (4) and (9) into ordinary differential equations.

n = y-Ra^J4, w = aRal/4s(n),

$ - $00

0(1)— ¿-Ai,

f (n) —

$w - $o

with the local Rayleigh number which is defined as Ra = (1 - - Ti)x3

and the stream function ip(x,y) is defined such that dip d\y

dy' dx

So, the continuity equation Eq. (1) is identically satisfied. After some algebraic manipulation, the momentum, energy and the solid volume fraction equations are obtained as follows,

s''' + 4^(3ss" - 2s' 2 - 4Mv/V/) + 0 - Nrf = 0 (13) + f) 0" + 4s0' + Nbf'0 + Nt02 + Appro = 0 (14)

f +- Lesf +— 0" — 0

where primes denote differentiation with respect to n and the non-dimensional parameters, Prandtl number (Pr), buoyancy-ratio parameter (Nr), Brownian motion parameter (Nb), thermophoresis parameter (Nt), Lewis number (Le), magnetic parameter (M), radiation parameter (N) and heat generation (X>0) or absorption parameter (X<0) are defined as follows,

Pr= -, Nr —

(Pp -Pf i) ($w -

a Pf ifi(Tw - Ti) (1 - $iY

(pc)pDB{ $ w *r 1 ) Ar _ (pc)pDr (Tw - Ti)

(Pc)f a

Le — —, M — Db

4a* T 3

aB0x1=2

(pc)f aT i

Pf\j(1 - $i)gP(Tw - Ti) ' Qx1=2

and X —

(pc)f\J(1 - $i)gP(Tw - Ti)

The corresponding boundary conditions are as follows, s(rj) — s'(n) — 0, 0(n) — f(n) — 1 at n — 0. (16)

s'(n) — 0(n) — f(n) — 0 as n-i.

The quantities of practical interest are the Nusselt number Nu and Sherwood number Sh defined by

k(Tw - Ti)

Db{$w - $i

where qw and q' are the wall heat flux and mass flux. The reduced Nusselt number Nur in the presence of thermal radiation and local Sherwood number Shr, can be introduced and represented as follows,

Nur — Ra14Nu — - ^ 1 + ^ 0' (0) Shr — Ra14Sh — -f (0)

4. Analytical solution by homotopy analysis method

The Eqs. (13)-(15) are solved under the corresponding boundary conditions (16) and (17) by using HAM. For HAM solutions, we choose the initial guesses and auxiliary linear operators in the following form:

S0(n)=1 - e-n - ne-i 00(i) = e-i Mi) = e -n (20)

L1(s) — s'''- s', L2 (0) — 0'' + 0, L3f)— f' + f

Li(ci + C2en + cse - n) — L2(cien + C2e - n)

= Ls(cien + C2e - n) — 0 (22)

and c1, c2 and c3 are constants and p e [0,1] denotes the embedding parameter. h1, h2 and h3 indicate the non-zero auxiliary parameters. We then construct the following problems:

Zeroth-order deformation problems,

(1 -P)L1 p)-s0(n)] — Ph1N1 [s(np); 0(np),f(n,p)]

(1 - p)La[e(n, p)-00(n)]— ph2N2[s(t], p), 0(n, p),f(n, p)]

(1 -p)L3 f(n,p)-fo(n)] = ph3N3[s(n,p), 0(n,p),f(n,p)]

s(0, p) — 0, s'(0, p) — 0, s'(1, p) = 0.

0(0, p)—1, 0(1, p) = 0.

f (0, p)—1, f (1, p) = 0. and

N1 [s(n, p), 0(n, p),f (n, p)] = s"'(n, p)

+ 4pr (3s(n, p)s"(n, p) - 2s'2 (n, p) - 4M\J~Prs'(j], p)) +0(n, p)- Nrf (]], p)

N2[s(n,p), 0(n,p),f(n,p)]= ( 1 + — )0"(n,p)

+ 3 s(n, p)0 (n, p) + Nbf (n, p)0'(], p) +Nt0 2(n, p) + Av/v0(n, p)

N3[s(n, p), 0(n, p),f (n, p)] = f "(n, p) 3 Nb

+4 Les(n p)f' (n p) + n 0"(n p)

For p — 0 and p — 1, we have s(n, 0) — S0(n), s(n, 1) — s(n)

0(n, 0) — 00(n), 0(n, 1) — 0(n)

f(n, 0)— f0(n), f(n, 1)— f (n)

Due to Taylor's series with respect to p. We have

s(n p) — s0(n) + ^ sm (n)pm m = 1

0(n,p)— 00(n)+ £ 0m(]])pm

f(n,p)— f0(n)+Y, fm(])p

Sm(n) — 0m (n) — fm(n) —

m — 1

dm(s(n, p)) ml dpm

dm(0(n, p))

1 ^ (f(n, p))

and thus, mth-order deformation problems

L1 [sm (n)- Xmsm - 1 (n)] — h1

L2[0m(n)-Xm0m- Kn)] — h2^m(n)

L3[fm(n)-Xmf m - 1^)] — hsfn)

(25) (26) and sm (0) — s'm(0) — s"(1) — 0

(27) 0m(0) — ■ 0m (1) — 0

(28) fm (0) — where, :fm(1) : — 0

Rs -/' m m - 1 -1 + 4pr m- 1 3 sm - 1 - \ ; _ A

s - 2£ sm

i — 0

+ 0m - 1 - Nrfm - 1

Rm—(1+4N) 0m -1+4E sm -1

+NbYJf'm - 1 -A + N^ 0m - 1 -i0i

+kVPr0m- 1

" 3 m -1 Nt

Rfm — f"m - 1 + 4 LeJ2sm - 1 - f + m 0m - 1

(32) X"

(m < 1) (m> 1)

which h is chosen in such a way that these three series are convergent at p — 1, therefore we have,

(35) s(n) — s0(n)^5] sm(n)

(36) 0(n) — 00(n)^5] 0m(n)

f(n)= fM+Y. fmin)

4.1. Convergence of HAM

As pointed by Liao [32], the convergence rate of approximation for the HAM solution strongly depends on the values of auxiliary parameter. It is essential to adopt a proper value of the auxiliary parameters h1, h2 and h3 which can adjust and control the convergence of the HAM solution. The range of the h curves of the functions s"(0), 0'(0) and f (0) for 15th order of approximations is shown in Figure 1. It is found that the range of the admissible values of hj, h2 and h3 are -0.9 < h1 < 0.15, - 0.8 < h2 < 0.15 and - 0.7 < h3 < 0.15.

Ress — s'" + 4pr (3ss" - 2s'2 - 4MPrs') + 0 - Nrf (54)

Rese = + ^ + 3 s°' + Nbf 0' + Nté2 + AVFr-e

Resf = f'' + - Lesf +— 0" f 4 Nb

In order to choose the optimal value of auxiliary parameter h we have presented the average residual errors as (see Refs. [33,34] for more details)

/ \ #2

1 — As,m — — ^^

— ■ 0

Aem——X0

Ress Y sj(ikx)

Resé Í Y^ 0j(iAx)

\j = 0 )

minimum values of As,m, Ag,m and Afm corresponding to the non-linear algebraic equations:

dAs>m _ Q dAgm _ q dAf m _ Q

In the present calculation h — h1 — h2 — h3 — -0.3 for the whole region n.

yi — y2; y2 — y3;

y3 —-^Fr (3y1y3 - 2y2 - 4M\J~PryYj -y4 + Nry6; y4— ^

y5 ^ - 3 y1y5 - Nby7y5 - Nty2 - Ap/PryA 3

J6 — y?,

- 3T Nt ,

y7— — Leyiy?- Nby5 (57)

With the boundary conditions y1 (0) — y2(0) — 0 and y4(0) — y6(0) — 1 (58)

To solve Eq. (57) with Eq. (58) as an initial value problem we must need the values for y3(0) i.e. s"(0), y5(0) i.e. 0(0) and y7(0) i.e. f (0) but no such values are given. The initial guess values for s"(0), 0'(0) and f (0) are chosen and the fourth order Runge-Kutta integration scheme is applied to obtain the solution. Then we compare the calculated values of s'(n), 0(n) and f(n) at nx with the given boundary conditions s' (i^) — 0, 0 and

f(noo) — 0 and adjust the values of s"(0), 0'(0) and f'(0) using the shooting technique to give better approximation for the solution. The process is repeated until we get the results correct up to the desired accuracy of 10- 8 level, which fulfils the convergence criterion.

Af m — —J2

Resf PTf/iAx)

where Ax —10/K and K —15. For the given order of approximation m, the optimal value of h is given by the

Figure 1 s"(0), 0'(0) and f'(0) plots for determining of optimum of 'h' coefficient.

5. Results and discussion

The effects of the pertinent physical parameters on the velocity, temperature and concentration profiles of the nanofluid can be observed from the graphical illustrations (Figures 2-22). In order to verify the present analytical and numerical results, a comparison has been made with Bejan [35] and observed a good agreement which is shown in Table 1.

5.1. Velocity profile results

The effect of Prandtl number on dimensionless velocity profile along the vertical plate is shown in Figure 2. It is noted that the vertical velocity increases as Pr increases. Figure 3 demonstrates the dimensionless velocity profile in the presence of magnetic field and thermal radiation to examine the effect of Lewis number. From this figure, it is apparent that the velocity enhances as Lewis number increases. Figure 4 is depicted to expose the effect of magnetic parameter on the velocity profile.

Figure 2 Effects of Prandtl number on the dimensionless velocity Figure 5 Effects of radiation parameter on the dimensionless velocity profile for Le = 1, N=M = 0.5, i = 0.1 and Nb = Nt=Nr=0.1. profile for Pr = 1, Le = 1, M = 0.5, i = 0.1 and Nb=Nt=Nr=0.1.

Figure 6 Effects of buoyancy-ratio parameter on the dimensionless Figure 3 Effects of Lewis number on the dimensionless velocity velocity profile for Pr = 1, Le = 1, M=N = 0.5, i = 0.1 and

profile for Pr = 1, N=M = 0.5, i = 0.1 and Nb = Nt=Nr=0.1.

Nb=Nt=0.1.

Figure 4 Effects of magnetic parameter on the dimensionless velocity profile for Pr = 1, Le = 1, N = 0.5, i = 0.1 and Nb=Nt=Nr = 0.1.

Figure 7 Effects of Brownian motion parameter and thermophoresis parameter on the dimensionless velocity profile for Pr = 1, Le = 1, M=N = 0.5, i = 0.1 and Nr=0.1.

Figure 8 Effects of heat generation/absorption parameter on the Figure 11 Effects of magnetic para^CT °n the dimensionless tempera-dimensionless velocity profile for Pr = 1, Le = 1, M=N = 0.5 and ture profile for Pr =1 Le = 1, N= a5, i—0.1 and Nb =Nt=Nr =

Nb Nt Nr 0.1.

Figure 12 Effects of radiation parameter on the temperature profile Figure 9 Effects of Prandtl number on the dimensionless tempera- for Pr = 1, Le = 1, M = 0.5, i = 0.1 and Nb=Nt=Nr=0.1. ture profiles for Le = 1, M=N = 0.5, i = 0.1 and Nb=Nt=Nr=0.1.

Figure 10 Effects of Lewis number on the temperature profile for Pr = 1, M=N = 0.5, i = 0.1 and Nb=Nt=Nr=0.1.

Figure 13 Effects of buoyancy-ratio parameter on the dimensionless temperature profile for Pr = 1, Le = 1, Nb=Nt=0.1, i = 0.1 and M=N = 0.5.

Figure 17 Effects of Lewis number on the dimensionless solid

Figure 14 Effects of Brownian motion parameter and thermophor- Voiume fraction of the nanofluid profile for Pr = 1, M = N = 0.5, Я = 0.1 esis parameter on the dimensionless temperature profile for Pr = 1, and Nb—Nt—Nf—0 1 Le = 1, Nr=0.1, Я = 0.1 and M=N = 0.5. . .

Figure 15 Effects of heat generation/absorption parameter on the Figure 18 Effects of magnetic parameter on the dimensionless solid dimensionless temperature profile for Pr = 1, Le = 1, M=N = 0.5 and volume fraction of the nanofluid profile for Pr = 1, Le = 1, N = 0.5,

Nb Nt Nr 0.1.

Я = 0.1 and Nb=Nt=Nr=0.1.

Figure 16 Effects of Prandtl number on the dimensionless solid Figure 19 Effects of radiation parameter on the dimensionless solid volume fraction of the nanofluid profile for Le = 1, M=N = 0.5, Я = 0.1 volume fraction of the nanofluid profile for Pr = 1, Le = 1, M = 0.5,

and Nb=Nt=Nr=0.1.

Я = 0.1 and Nb=Nt=Nr=0.1.

0 2 4 6 8

Figure 20 Effects of buoyancy-ratio parameter on the dimensionless solid volume fraction of the nanofluid profile for Pr = 1, Le = 1, M=N = 0.5, Я = 0.1 and Nb=Nt=Nr=0.1.

_l—,—,—,—I—,—,—,—I—,—I ,—i_—,—,—,—1_

0 2 4 6 8

Figure 22 Effects of heat generation/absorption parameter on the dimensionless solid volume fraction of the nanofluid profile for Pr = 1, Le = 1, M=N= 0.5 and Nb=Nt=Nr=0.1.

Table 1 Comparison test results. Values of the reduced Nusslet number Nur=RalJ4Nu in the limiting case of a regular fluid. The present results are with Le = 10, Nr=Nb = Nt = 10-5.

Pr Bejan [35]

1 0.401 10 0.465 100 0.490

Present results (absence of magnetic, radiation parameter and heat generation/ absorption parameter)

Analytical

Numerical

0.40103 0.46496 0.49000

0.40102817 0.46496287 0.49000028

Figure 21 Effects of Brownian motion and thermophoresis parameters on the dimensionless solid volume fraction of the nanofluid profiles for Pr = 1, Le = 1, Nr=0.1, Я = 0.1 and M=N = 0.5.

It reveals that the magnetic field reduces the dimensionless velocity distribution. Due to enhancement of magnetic field strength, a resistive type force called Lorentz force associated with the magnetic field makes the boundary layer thinner. Figure 5 displays the influences of the radiation parameter on the dimensionless velocity profile in the presence of magnetic field. It is interesting to observe that there is significant enhancement in the velocity distribution, when the thermal radiation increases.

Figure 6 is presented to show the dimensionless velocity profile for different values of the buoyancy-ratio parameter. It is found that the velocity distribution decreases due to an increase in the buoyancy-ratio parameter. Figure 7 shows the effects of Brownian motion and thermophoresis parameters on the dimensionless velocity profile. It is found that the increasing values of Nb and Nt improve the velocity distribution. Figure 8 exhibits the effect of the heat generation/absorption parameter Я on the dimensionless velocity profile. It is clear that the increasing values of Я are to increase the velocity distribution.

5.2. Temperature profile results

Figure 9 illustrates the influence of the Prandtl number on the dimensionless temperature profile. It is observed that increasing values of Prandtl number are to increase the temperature distribution in the flow region. This is due to the fact the thermal boundary layer thickness increases with Prandtl number.

Figure 10 shows the influence of Lewis number on the dimensionless temperature profile of the nanofluid. It is clear that increasing values of Le increase the temperature distribution. The variations of magnetic parameter and radiation parameter on the dimensionless temperature profile along the vertical plate are shown in Figures 11 and 12. From these figures, it is found that both the magnetic field and thermal radiation enhance the temperature distribution.

Figure 13 is depicted to analyze the variations of the buoyancy-ratio parameter on the dimensionless temperature profile. The temperature is found to be increase, as the order of buoyancy ratio increases. Figure 14 is plotted to show the dimensionless temperature profile for different values of the Brownian motion and thermophoresis parameters. It is obvious that the Brownian motion and thermophoresis parameters augment the temperature distribution. The dimen-sionless temperature profile along the vertical plate is shown

in Figure 15 with the variations of heat generation/absorption parameter.

It is clear that the temperature distribution rises in the presence of heat generation and reduces in the case of heat absorption. In the presence of heat source (X>0) the additional energy is generated in the boundary layer. But in the case of heat sink (X<0) an amount of energy is observed from the boundary layer and the temperature decreases.

5.3. Nanosolid volume fraction profile results

Figure 16 is a plot of the dimensionless concentration profile to detect the influence of the Prandtl number. It is noted that there is a significant decrement in concentration distribution, when the Prandtl number increases. Figure 17 represents the effect of Lewis number on the dimensionless nanoparticle volume fraction profile in the presence of magnetic field and thermal radiation. It is found that increasing values of Le decrease the nanoparticle volume fraction within the nanoparticle volume fraction boundary layer. The dimensionless concentration profile along the vertical plate is shown in Figure 18 with the variations of magnetic field parameter. From the figure, it is noticed that the concentration distribution increases, when the magnetic parameter increases. Figure 19 exhibits the effect of radiation parameter on the dimensionless nanoparticle volume fraction within the nanoparticle volume fraction boundary layer in the presence of magnetic field. As the radiation parameter increases, the nanoparticle volume fraction decreases.

Figure 20 is a plot of the dimensionless concentration profile with the effect of the buoyancy-ratio parameter. The nanoparticle volume fraction is found to be increase as the order of buoyancy-ratio increases. Figure 21 is presented to show the effects of the Brownian motion and thermophor-esis parameters on the dimensionless nanoparticle volume fraction profile in the presence of magnetic field and thermal radiation. It is clear from the figure that both the nanofluid parameters (i.e.) thermophoresis and Brownian motion parameters help in decreasing the concentration distribution.

Figure 22 is plotted to examine the effect of the heat generation/absorption parameter on the dimensionless nano-particle volume fraction profile in the presence of magnetic field and thermal radiation. It is observed that the dimen-sionless concentration profile decreases as X increases.

5.4. Results of reduced Nusselt number and Sherwood number

Table 2 reveals the values of the reduced Nusselt number and Sherwood number and also it assures the accuracy of present analytical and numerical results.

The reduced Nusselt number increases with increasing values of N & Le and decreases with M, Pr, Nr, Nb and Nt. The Nur values rises in the case of heat absorption and reduces in heat generation case. The local Sherwood number increases with N, Pr, Le, Nb & Nt and decreases with M & Nr. The Shr values rises in the case of heat generation and reduces in heat absorption case.

Table 2 Variations of Nur and Shr.

Parameters Values Nur Shr

Analytical Numerical Analytical Numerical

M 0 0.10526 0.10525465 1.36474 1.36473456

0.5 0.05813 0.05812797 1.28582 1.28581535

1 0.01474 0.01473962 1.22737 1.22737233

N 2 0.05813 0.05812797 1.28582 1.28581535

4 0.07058 0.07058035 1.28449 1.28448996

6 0.07116 0.07115861 1.28881 1.28880763

Pr 3 0.05813 0.05812797 1.28582 1.28581535

6 0.02824 0.02823726 1.37204 1.37204119

8 0.00903 0.00902899 1.41019 1.41018869

Le 12 0.05813 0.05812797 1.28582 1.28581535

13 0.05846 0.05846360 1.32175 1.32175173

14 0.05876 0.05876141 1.35590 1.35590142

X - 0.2 0.32156 0.32155691 1.05066 1.05065676

0 0.20811 0.20811256 1.16028 1.16027713

0.2 0.05812 0.05812797 1.28582 1.28581535

Nr 0.4 0.05923 0.05923288 1.29447 1.29447419

0.5 0.05813 0.05812797 1.28582 1.28581535

0.6 0.05699 0.05699930 1.27695 1.27694506

Nb = Nt 0.4 0.06247 0.06246597 1.28249 1.28249815

0.5 0.05813 0.05812797 1.28582 1.28581535

0.6 0.05396 0.05395929 1.28899 1.28898625

Note: While studying the effect of individual parameters the following values are assumed M=0.5, N = 2, Pr=3, Le = 12, X = 0.2, Nr=0.5 and Nb=Nt = 0.5.

6. Conclusion

A two dimensional steady free convective MHD laminar incompressible boundary layer flow of an electrically conducting nanofluid past a vertical plate with thermal radiation effect in the presence of internal heat generation/ absorption is studied both analytically and numerically. The present type boundary layer problems can also be solved using some new semi analytical techniques [36,37]. The analysis shows that the velocity, temperature and the solid volume fraction of the nanofluid profiles in the respective boundary layers depend on eight dimensionless parameters, namely Prandtl number Pr, Lewis number Le, Brownian motion parameter Nb, thermophoresis parameter Nt, buoyancy ratio parameter Nr, magnetic parameter M, radiation parameter N and heat generation or absorption parameter i. The following specific conclusions are obtained:

• The dimensionless velocity profile of the nanofluid enhances with the increase of Prandtl number, Lewis number, radiation parameter, Brownian motion parameter and thermophoresis parameter and it diminishes with magnetic parameter and buoyancy-ratio parameter. The dimensionless velocity enhances in the presence of heat generation and decreases in the case of heat absorption.

• The increasing of values of Prandtl number, Lewis number, magnetic parameter, radiation parameter, buoyancy ratio parameter, Brownian motion parameter and the thermophoresis parameter lead to increase the nano-fluid temperature profile. The temperature distribution increases in the presence of heat generation and decreases in the case of heat absorption.

• The nanosolid volume fraction profile diminishes with Prandtl number, Lewis number, radiation parameter, Brownian motion parameter and thermophoresis parameter and augments with magnetic parameter and buoyancy-ratio parameter. The nanosolid volume fraction profile decreases in the presence of heat generation and increases in the case of heat absorption.

• The reduced Nusselt number rises in the case of heat absorption and reduces in heat generation case. The local Sherwood number values rises in the case of heat generation and reduces in heat absorption case.

Acknowledgments

The authors wish to express their sincere thanks to the honorable referees and the editor for their valuable comments and suggestions to improve the quality of the paper. One of the authors (N. Vishnu Ganesh) gratefully acknowledges the financial support of Rajiv Gandhi National Fellowship (RGNF), UGC, New Delhi, India.

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