Scholarly article on topic 'Effects of buoyancy and thermal radiation on MHD flow over a stretching porous sheet using homotopy analysis method'

Effects of buoyancy and thermal radiation on MHD flow over a stretching porous sheet using homotopy analysis method Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — Yahaya Shagaiya Daniel, Simon K. Daniel

Abstract This paper investigates the theoretical influence of buoyancy and thermal radiation on MHD flow over a stretching porous sheet. The model which constituted highly nonlinear governing equations is transformed using similarity solution and then solved using homotopy analysis method (HAM). The analysis is carried out up to the 5th order of approximation and the influences of different physical parameters such as Prandtl number, Grashof number, suction/injection parameter, thermal radiation parameter and heat generation/absorption coefficient and also Hartman number on dimensionless velocity, temperature and the rate of heat transfer are investigated and discussed quantitatively with the aid of graphs. Numerical results obtained are compared with the previous results published in the literature and are found to be in good agreement. It was found that when the buoyancy parameter and the fluid velocity increase, the thermal boundary layer decreases. In case of the thermal radiation, increasing the thermal radiation parameter produces significant increases in the thermal conditions of the fluid temperature which cause more fluid in the boundary layer due to buoyancy effect, causing the velocity in the fluid to increase. The hydrodynamic boundary layer and thermal boundary layer thickness increase as a result of increase in radiation.

Academic research paper on topic "Effects of buoyancy and thermal radiation on MHD flow over a stretching porous sheet using homotopy analysis method"

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Alexandria Engineering Journal (2015) xxx, xxx-xxx

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Effects of buoyancy and thermal radiation on MHD flow over a stretching porous sheet using homotopy analysis method

Yahaya Shagaiya Daniel *, Simon K. Daniel 1

Department of Mathematics, Kaduna State University, Nigeria

Received 2 December 2014; revised 13 March 2015; accepted 31 March 2015

KEYWORDS MHD;

Heat transfer; Stretching sheet; Homotopy analysis method

Abstract This paper investigates the theoretical influence of buoyancy and thermal radiation on MHD flow over a stretching porous sheet. The model which constituted highly nonlinear governing equations is transformed using similarity solution and then solved using homotopy analysis method (HAM). The analysis is carried out up to the 5th order of approximation and the influences of different physical parameters such as Prandtl number, Grashof number, suction/injection parameter, thermal radiation parameter and heat generation/absorption coefficient and also Hartman number on dimensionless velocity, temperature and the rate of heat transfer are investigated and discussed quantitatively with the aid of graphs. Numerical results obtained are compared with the previous results published in the literature and are found to be in good agreement. It was found that when the buoyancy parameter and the fluid velocity increase, the thermal boundary layer decreases. In case of the thermal radiation, increasing the thermal radiation parameter produces significant increases in the thermal conditions of the fluid temperature which cause more fluid in the boundary layer due to buoyancy effect, causing the velocity in the fluid to increase. The hydrodynamic boundary layer and thermal boundary layer thickness increase as a result of increase in radiation. © 2015 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

Stretching sheet is essential in industrial processes. This process is usually following with heat and mass transfer aspects. The influence of radiation on MHD flow and heat transfer

* Corresponding author. Tel.: +234 8175678448.

E-mail addresses: shagaiya12@gmail.com (Y.S. Daniel), simondanie-

l40@yahoo.com (S.K. Daniel).

1 Tel.: +234 8025479979. Peer review under responsibility of Faculty of Engineering, Alexandria University.

in industrial and technological areas occurs at high temperatures and knowledge of radiation heat transfer becomes very vital for design of pertinent equipment. For production of plastic sheets, gas turbines, missiles, space vehicles aircraft, nuclear power plants, satellites and foils see [1-5]. The influence of variable thermal conductivity and radiation on the flow and heat transfer was carried out by Mahmoud [6]. Pal and Mondal [7] analyse the effect of variable viscosity on MHD non-Darcy boundary layer flow and heat transfer features in an incompressible electrically conducting fluid. Magnetohydrodynamic (MHD) based Nanofluids with Natural convection through porous sheet were presented by

http://dx.doi.org/10.1016/j.aej.2015.03.029

1110-0168 © 2015 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

a constant u, v fluid velocity components along the x and y direc-

B(x) magnetic field tions, respectively

Bo applied magnetic induction U(x) velocity of the stretching sheet

f dimensionless stream function x, y vertical and horizontal directions

g acceleration due to gravity

Gr Grashof number Greek Symbols

k fluid thermal conductivity a thermal diffusivity

m velocity exponent parameter ß volumetric expansion coefficient

M magnetic parameter g similarity variable

n temperature exponent parameter k buoyancy or mixed convection parameter

cp specific heat at constant pressure h dimensionless temperature

Pr Prandtl number l dynamic viscosity

qr thermal radiation V kinematic viscosity

Rex local Reynolds number P fluid density

T fluid temperature r electrical conductivity

Tw{x) temperature of the stretching sheet sw skin friction

T ± i free stream temperature w stream function

Zeeshan et al. [8]. Also, El-Aziz [9] studied influence of thermal radiation on the flow over an unsteady stretching surface and Pal and Mondal [10] extended the work to ohmic dissipation and thermal effects. Pal and Chatterjee [11] work on micropolar fluid over a stretching sheet in a non-Darcian porous medium on MHD boundary layer flow. Sheikholeslami et al. [12] consider Lorentz force to investigate CuO-water nanofluid flow and convective heat transfer. An extension was made by Sheikholeslami et al. [13] to thermal radiation on nanofluid flow using two phase model. Rashidi et al. [14] consider stream wise transverse magnetic fluid flow in porous medium with heat transfer. Important analysis concerning the Buoyancy and MHD boundary layer flow over stretching sheet was carried out by Makinde et al. [15] and Nadeen et al. [16]. To study the stability of the problem understudy see Hajmohammadi and Nourazar [17] by using a semi-analytical method and the new ''gradient energy method'' has been used by Hajmohammadi and Nourazar [17] and Hajmohammedi et al. [18]. In view of this some other researchers use semi-analytical approach on the problems of variable thermal conductivity such as the work of Khan et al. [19], Gul et al. [20] and Hajmohammadi and Nourazar [21].

The effect of radiation on MHD steady asymmetric flow was considered by Makinde [22] of an electrically conducting fluid over a stretching porous sheet. Khan et al. [23] study Oldroyd fluid with inclined magnetic field in the presence of heat transfer. The flow and heat transfer phenomenon in a power law fluid over a porous stretching sheet with effect of magnetic field was considered by researchers such as in [2427]. MHD flow of an incompressible viscous and electrically conducting fluid over a vertical stretching sheet embedded in a porous medium was considered by Pal and Mondal [3], Mohammed [4], Shateyi et al. [28]. The flow of variable electric conductivity inclined impermeable flat plate subject to a uniform surface heat flux boundary condition was carried out by Rahman et al. [29]. Looking at the effects of viscous dissipation are the works of Hajmohammadi and Nourazar [30] and Hajmohammadi et al. [31]. For effect of the buoyancy

force and thermal radiation in MHD boundary layer visco-elastic fluid over continuously moving stretching surface in a porous surface see ([32-36]). Seddeck [37] analyses thermal-diffusion and the diffusion-thermo effects on the mixed free-forced convective and mass transfer steady laminar boundary layer flow over an accelerating surface with a heat source in the presence of suction and blowing. The influence of MHD and temperature dependent of non-Newtonian nanofluid was studied by Ellahi [38]. Mahmoud [6] studied flow and heat transfer of an incompressible viscous electrically conducting fluid over a continuously moving vertical infinite plate with suction and heat flux in the presence of radiation. An extension to thermal radiation combined buoyancy and suction/blowing was carried out by Sheteyi [39].

In the present paper, the object of this study was to present a unified approach to solving the MHD flow due to influence of buoyancy and thermal radiation over a stretching porous sheet using homotopy analysis method (HAM). There have been several approaches on theoretical models developed to describe buoyancy and thermal radiation on hydrodynamic magnetic field flow over a stretching sheet. However, to the best of our knowledge, no investigation has been made yet to analyse the effects of buoyancy and thermal radiation on MHD flow over a stretching porous sheet using HAM. The non-linear differential equations are solved using HAM proposed by Liao [40] and Liao and Tan [41].

2. Mathematical analysis

In our present study on effect of thermal radiation, we assumed the flow to be laminar and stable. Consider a steady two-dimensional laminar flow of a viscous, incompressible and electrically conducting fluid past a stretching sheet. The stretching sheet is assumed to be permeable in order to give way for possible wall fluid suction/injection. By using two equal and opposite forces along the horizontal direction, with the influence of uniform magnetic field normal to the plate, the uniform magnetic field as result of velocity of the electrically

Figure 1 Schematic representation of the physical model and coordinates system.

conducting fluid is negligible. The physical model of the present study is shown in Fig.1 above. With this assumption the magnetic Reynolds number is small (valid) so that the induced magnetic field is also neglected. Assume the temperature-dependent heat source (or sink) to be in velocity with presence of buoyancy and thermal radiation. It is assumed also that the external electric field is zero and pressure gradient, and magnetic dissipation with viscous are neglected. With the exception of density in the body force known as the balance of linear momentum equation which is approximated using the Boussinesq's approximation, all the fluid properties are assuming to be constant.

Under the abovementioned assumptions, the constituted boundary layer form of the governing equations, describing the flow can be written as (see Kays and Crawford [42] and Chamkha [35])

du dv o

dx dy ' du du du B)(x) , n/rr rj, x

u^T + v^- = VTT1 ~ r —-u + gß(T- Ti),

dx dy dy2 p dT dT d2T Qo

u^T + = ^^"T H--(T -

dx dy dy2

— (Ti- T) Pcp

3pCpk* 1 dy2

(1) (2)

magnetic field strength was used by Chamkha [35]. In terms

of the stream function the velocity components are

u — ; u —

Using the transformations

g = y^V, W — paxf(g), h =

Eqs. (1)-(4) reduce to f" + ff + f2 + Grh — M2f Nr + 1 h" + fh + Ah — (1 + dx)f h

with the boundary conditions

f(o) — R, f(o) — 1, f(i) — 1 0(0) = 1, h(i) — o

(9) (10)

where R — vw/^/va is the dimensionless suction/injection velocity see Seddeek [33]. And

M —I —

Gr — gß'-

(T— T

Pr = -,

16r*Tj 3k*k

dx — £x, D — PQa

pCp pcpa

are the Hartmann number, Grashof number, Prandtl number, thermal radiation parameter and heat generation/absorption coefficients respectively.

3. The homotopy analysis method (HAM)

In this section, we will apply HAM an idea from Liao [40] to solve Eqs. (7)-(10). We choose the set of base functions (e-ng; n P 0 is an integer} to approximate the unknown functions f(g) and h(g) respectively, as

f(g) — fo(g) + Yf' (g), h(g) — ho (g) + Eh(g) i—1 i—1

subjected to the following boundary conditions:

u — ax, v — vw, T — Tw(x) — T1 + A0x at y — 0 u ! 0, T ! Tœ as y (4)

where Tw(x) is as the wall temperature, a is the stretching rate which is constant and vw is the wall suction when (vw < 0) and injection when (vw > 0). Also u, v and T are the velocity and Temperature components along the x and y axes respectively, g is the acceleration due to gravity, cp is the specific heat at constant pressure, v is the kinematics viscosity, a is the thermal diffusivity of the fluid, v is the kinematic viscosity, b is the coefficient of thermal expansion, q is the fluid density, B0 is the applied magnetic induction, a is stretching rate (a constant) and vw is the suction/injection velocity. And p*u(T1 — T) n

and Q0 (T — Tœ) are heat generated or absorbed per unit vol- N2 /(g; q), 6(g; q)

ho(g) — e-

/0(g)— R + b (1 - e-b

are taken to be the initial guess approximations, with the property

L:[c1g2 + C2g + C3] — 0 (13)

L2[c4g + C5] — 0 (14)

Here c¡{where i — 1,2,3,4, 5) are the constants. Based on Eqs. (11) and (12), the non-linear operators will be

f(g; q)]

+ f(g;q) ^

df(g; q)

- M2 + Grh(g; q) —o,

_Nr + 1 d2h(g; q)

ume (Q0 and b* are constants), where r* is termed as Stefan-Boltzmann constant and k* is as the mean absorption coefficient. To obtain the similarity solutions of Eqs. (1)-(3) subjected to the boundary conditions (4), the uniform steady

+f(g; q)

dh (g; q) dg

(1+ dx) fgp h(g; q)+Ah(g; q) —o

where q e[0,1] is an embedding parameter, and /(g; q) and 0(g; q) are kind of mapping functions for /(g) and 0(g) respectively. From the operators, we can construct the zeroth-order deformation equations as

(1 - q)L,[/(g; q)-/„(x)] — qhN, /(x; q), 0(g; q)] (17)

(1 - q)L2 [0(g; q) - 0„(x)] — qhN2 /(x; q), 0(g; q)] (18)

where h is an auxiliary non-zero parameter. The boundary conditions for Eqs. (17) and (18) are presented as

f(0; q)= R, f(g; q) = 1, /(i q) = 0 0'(0; q) = 0, 0(i; q) = 0,

Clearly, when q — 0 and q — 1, the above zeroth-order deformation equations have the following solutions:

f(g;0)= /0(g), h(g; 0) = 00(g) f(g; 1)= f(g), Â(g; 1) = 0(g)

(21) (22)

When q increases from 0 to 1, /(g; q) and °(g; q) vary from /0(g) and 00(g) to /(g) and 0(g). From Taylor's theorem and Eqs. (21) and (22), we obtained

f(g; 0)= Ä0(g) + ^/„(g)qm,

0(g; 0) = 00(g) + £0m(g)qm

fm(x) =

@"/"(x; q) m! dqm

0 1 @"0(x; q) :

q=0, 0m(x) = m! dqm 'q=°

The convergence of the series solutions (23) and (24) depends upon the choice of auxiliary parameter h. Assume that h is chosen such that the series solutions (23) and (24) are convergent at q — 1, then due to Eqs. (21) and (22). The mth-order deformation equations, we differentiate Eqs. (17) and (18) up to m times with respect to q and divide by m! and then set q — 0. The resulting deformation equations at the mth-order are

Ll[/m(g) - Xjm-l(g)] = hRl,m(g) L2[0m(g) - Vmhm-l(g)] = hR2,m(g)

with the following boundary conditions

fm (0) = 0, /„(0)= 0, /„ (1)= 0

0„(O) = 0, 0„(l)= 0

(28) (29)

i— 1 i—\ k

R1,m(g) = f—,(g) + YÄi—1(g/k(g) ^E fi—1—k(gfk—i(g))

k=0 k=0 j=0

— Mf—,(g) + Gr0i—1(g)

R2,m(g) =

L0i— 1(g) + /i—1(g)0i—1(g)

— (1 + dx)f—1(g)0i— 1(g) + D0(g)

200 100 0 -100 -200 -300 -400

Figure 2 h - curve for the velocity profile at 5th order approximation.

-1.0 -0.5 0.0 0.5 1.0

Figure 3 h - curve for the temperature profile at 5th order approximation.

1, m > 1

Assume fm(g) and 0*m(g) represent the particular solutions of Eq. (30), one after the other as i — 1, 2,3,... the computation, can easily be computed by symbolic computation software such as Mathematica, Maple, and Matlab.

4. Results and discussion

In this section we will discuss the convergence of our solutions. The analytic solution contains the auxiliary parameter h, which influences the convergence region and rate of approximation for the Homotopy Analysis Method (HAM) solution. This is called h curves approach. The constant h curve is quite rational, and whenever convergence takes place at p — 1, the quantities of exact solution should be free of the parameter h. The analytical expressions (17), (18) and (26), (27) contain the auxiliary parameter h. As pointed out by Liao [41], the convergence region and rate of approximations given by HAM are strongly dependent upon h. Figs. 2 and 3 show the h-curves for R — -0.3, M — 10, b — 14, to find the range of h for velocity profile which is -0.4 6 h 6 0.4 and temperature profile which is -1.0 6 h 6 1.0. It is apparent from Figs. 2 and 3 that the

0, m 6 1

Table 1 Comparison of wall temperature gradient — h'(0) for different values of the suction or injection parameter R and the heat generation or absorption parameter

R = 0.45 dx = 0.5 R = 0.45 dx = 1.0 R = 0 dx = 0.5 R = 0 dx = 1.0 R = -1.5 dx = 0.5 R = -1.5 dx = 1.0

Present study 0.82396 0.96190 0.94765 1.07895 1.57077 1.66182

Chamkha [21] 0.82397 0.96191 0.94769 1.07996 1.57077 1.66184

Acharya et al. [13] 0.82250 0.96180 0.94620 1.07890 1.56960 1.66030

Figure 4 Effect of Gr for the velocity profile.

Figure 5 Effect of Gr for the temperature profile.

range for admissible values for h is —0.1 6 h 6 0.1 for the velocity profile and —0.4 6 h 6 0.4 for the temperature profile.

Table 1 presents a Validation of results of wall heat transfer Q — — h'(0) for different values of the suction or injection parameter R and the heat generation or absorption parameter dx using homotopy analysis method with the result obtained by Chamkha [35] and Acharya et al. [25] by taking Gr — D — M — Nr — 0, Pr — 0.71. Thus it is seen from the table that the numerical results are in close agreement with those published previously.

The basic parameters that governed the flow are the Grashof number Gr, the Hartmann number M, thermal radiation parameter NR, suction/injection parameter R, and heat absorption or generation (heat source (D > 0) or heat sink

Figure 6 Effect of M for the velocity profile.

Figure 7 Effect of M for the temperature profile.

(D < 0)). In order to study their effects, a MATHEMATICA programme is written to compute and generate the graphs for the velocity and temperature for different values of these parameters. Some representative results are presented in Figs. 4-13 to interpret the effects of these parameters. Figs. 4 and 5 show the effects of the buoyancy force to the viscous forces of a typical velocity and temperature profiles in the boundary layer. We fixed Pr — 0.71 and all other parameters to be zero in order to depict the effects of different parameters under investigation. In Fig. 4, it is evident that the momentum boundary layer thickness increases with increasing values of Gr enabling more flow. In Fig. 5, this plot indicates increasing value of Gr results in thinning of the thermal boundary layer associated with an increase in the wall temperature gradient and hence produces an increase in the heat transfer rate.

-R = -1

■ R = -05

- R = 0

- R = 0.5

-R = 1

Figure 11 Effect of R for the temperature profile.

Figure 12 Effect of D for the velocity profile.

Figure 8 Effect of Nr for the velocity profile.

0.6 0.5 0.4 0.3 0.2 0.1 0.0

Figure 9 Effect of Nr for the temperature profile.

Figure 10 Effect of R for the velocity profile.

The influence of Hartmann number is seen in Figs. 6 and 7. We observe in Fig. 6 that increasing values of the Hartmann number results into a decrease in the velocity. This is due to the retarding nature of the Lorentz force which slows down the motion of the fluid in the boundary layer and to increase its temperature. Figs. 6 and 7 indicate decreases in the velocity distribution and increases in the temperature distribution due to effect of transverse magnetic field to an electrically conducting fluid.

Figs. 8 and 9 show the effect of thermal radiation parameter NR, on the velocity and temperature profiles with fixed

Figure 13 Effect of D for the temperature profile.

buoyancy effect (Gr — 1). In Fig. 9, the influence of NR is to increase the thermal boundary layer thickness. Increasing the radiation parameter causes increases in the velocity profile see Fig. 8. We can see in Fig. 9 that the surface temperature gradient is reduced with an increase in radiation parameter; thereby, the heat transfer rate from the surface decreases with increasing NR. The hydrodynamic boundary layer thickness increases with increasing radiation parameter.

Effect of suction/injection parameter R is shown in Figs. 10 and 11. The influence of R on the flow velocity in the boundary layer is displayed in Fig.10. The imposition of wall fluid injection increases the hydrodynamic boundary layer which shows

an increase in the fluid velocity. However, the exact opposite behaviour is produced by imposition of wall fluid suction. In Fig. 11, we observed that as injection rate increases the thermal boundary layer thickness increases. And as suction parameter value increases, the temperature profile decreases.

Figs. 12 and 13 show the effect of heat source (D > 0) or a heat sink (D < 0) in the boundary layer on the velocity and temperature profiles. For fixed buoyancy effect Gr — 1 on velocity and temperature profiles, increasing the heat generation or absorption parameter D. has the tendency to increase the thermal state of the electrical conducting fluid. This increase in the fluid temperature causes more induced flow towards the plate through the thermal buoyancy effect.

Influence of heat sink in the boundary layer absorbs energy which causes the temperature of the fluid to decrease. The effect of decrease in the temperature causes a reduction in the flow velocity in the boundary layer which is the result of buoyancy influence on both the flow and thermal problems.

5. Conclusion

Using homotopy analysis method on the motion of steady two-dimensional viscous, incompressible and electrically conducting fluid over a stretching porous sheet, one can solve the transform governing equations for buoyancy and thermal radiation on MHD flow. We have considered the effects of flow parameters such as Grashof number Gr (buoyancy effects), Hartmann number M(magnetic field parameter), thermal radiation parameter NR, suction/injection parameter R, and heat source (D > 0) or a heat sink (D < 0) in the boundary layer and the flow behaviour on the velocity and temperature fields. We have extended the scope and applicability of earlier results by Chamkha [35], Abel [32], Makinde and Ogulu [34], Shateyi [28] and Seddeek [37] to porous sheet and using homo-topy analysis method (HAM) to solve the problem using two axillary parameters by Alizadeh-Pahlavan et al. [43]. In our results, it was found that when the buoyancy parameter increases, the fluid velocity increases and the thermal boundary layer decreases. In case of the thermal radiation, we observed that increasing the thermal radiation parameter produces significant increases in the thermal conditions of the fluid temperature which cause more fluid in the boundary layer due to buoyancy effect, causing the velocity in the fluid to increase. The hydrodynamic boundary layer and thermal boundary layer thicknesses were observed to increase as a result of increasing radiation. The influence of suction and injection parameter increases as the values of both the velocity and temperature profiles decrease for a fixed buoyancy effect (Gr — 1).

Acknowledgement

The authors thank the reviewers for their interests and time as well as constructive suggestions, recommendation and comments for improving the paper.

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