Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition

International Journal of Heat and Mass Transfer 102 (2016) 766–772

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Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition Muhammad Waqas a, Muhammad Farooq b, Muhammad Ijaz Khan a, Ahmed Alsaedi c, Tasawar Hayat a,c, Tabassum Yasmeen d,e,⇑ a

Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia d Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK e Department of Mechanical Engineering, University of Engineering & Technology Peshawar, Pakistan b c

a r t i c l e

i n f o

Article history: Received 14 March 2016 Received in revised form 10 May 2016 Accepted 17 May 2016

Keywords: MHD Micropolar material Viscous dissipation Convective condition Mixed convection

a b s t r a c t The present paper addresses magnetohydrodynamics (MHD) flow of micropolar liquid towards nonlinear stretched surface. Analysis is presented with viscous dissipation, Joule heating and convective boundary condition. Characteristics of heat transfer are analyzed with mixed convection phenomenon. Dimensional nonlinear equations are converted into dimensionless expressions by employing suitable transformations. Homotopic procedure is implemented to solve the governing dimensionless problems. Behaviors of several sundry variables on the flow and heat transfer is scrutinized. Skin friction coefficient and local Nusselt number are presented and evaluated. Obtained results are also compared with the available data in the limiting case and good agreement is noted. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Recently the investigation of micropolar liquids has fascinated the consideration of recent investigators. Such consideration is because of the reality that conventional Newtonian fluids cannot accurately depict the features of fluid flow in various industrial applications and biology. Eringen [1] established the theory taking into account the local properties emerging from the intrinsic motion and the microstructure of the fluid components. The theory is likely to deliver a exact model for liquid characteristics noticed in definite synthetic materials including lubricants, animal blood, polymers, paints and colloidal solutions etc. The existence of smoke or dust, especially in a gas might likewise be demonstrated utilizing micropolar liquid model. Besides this the analysis of micropolar liquid has ample significance in liquid crystal solidification. The theory of thermo-micropolar fluid is modified by Eringen [2]. He defined relations of liquids with microstructures. The stiff particles in a little volume element revolve about the centroid of the volume part in micropolar liquids. Peddieson and McNitt [3] ⇑ Corresponding author at: Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK. E-mail addresses: [email protected] (M. Waqas), [email protected]. pk (M.I. Khan), [email protected] (T. Yasmeen). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.05.142 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

extended the analysis of Eringen [1] for the theory of boundary layer. Turkyilmazoglu [4] explored the heat transfer features in flow of micropolar liquid by a porous shrinking surface. Impact of double stratifications in flow of micropolar liquid with chemical reaction is studied by Rashad et al. [5]. Sheikholeslami et al. [6] scrutinized the impact of heat transfer in micropolar liquid in a permeable channel. Analysis of micropolar fluid via a porous enhanced channel is addressed by Cao et al. [7]. Characteristics of micropolar liquid through a bidirectional stretched surface is analyzed by Mehmood et al. [8]. Analysis of flows due to stretched surface through heat transfer has acknowledged ample consideration owing to their possible demands in several industrial procedures for instance in metal extrusion, continuous casting, hot rolling, drawing of plastic films etc. Particularly in polymer industry aerodynamic extrusion of plastic sheets is very important. This procedure includes the heat transfer between the surrounding and surface fluid. Moreover the rate of stretching in a hot/cold fluids greatly depends upon the quality of the material with desired properties. In such process heat transfer has important role in controlling the cooling rate. Makinde [9] presented computational modeling of nanofluid flow due to convectively heated unsteady stretching sheet. Hydromagnetic viscous liquid with slip condition and homogeneousheterogeneous reactions is analyzed by Abbas et al. [10]. Mabood

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et al. [11] considered MHD flow of viscous liquid in presence of transpiration. They also reported the interaction of chemical reaction in this attempt. Malvandi et al. [12] explored the unsteady stagnation point flow of a nanofluid over a stretched sheet with slip effects. Zheng at al. [13] investigated the characteristics of velocity slip in flow of nanofluid over a stretched surface. Investigation of stagnation point flow due to shrinking/stretching surface with heat transfer is reported by Bhattacharyya [14]. Lin et al. [15] studied radiation effects on Marangoni convection flow of pseudoplastic nanofluid with variable thermal conductivity. Recently impact of variable thermal properties in flow of Maxwell fluid due to variable thicked sheet is presented by Hayat et al. [16]. Characteristics of magnetic field effects has tremendous implementations in engineering, medicine and physics. Several industrial equipments for instance bearings, pumps, MHD generators and boundary layer control are influenced by the communication between the electrically conducting liquid and a magnetic field [17–24]. Further it is also analyzed that magnetohydrodynamic flow becomes more significant with the heat transfer phenomenon. In this direction several attempts have been implemented to evaluate the combined impacts of the magnetic field and heat transfer. For instance slip effects in MHD flow due to stretched cylinder is addressed by Mukhopadhyay [25]. Farooq et al. [26] studied the Newtonian heating effects in MHD flow of Jeffrey fluid. Impact of convective heat transfer in MHD flow of Jeffrey liquid over a stretched sheet is reported by Hayat et al. [27]. Mustafa et al. [28] scrutinized the MHD flow of Maxwell fluid with convective heat transfer. Further mixed convection flows of nonlinear liquids has ample demands in numerous engineering and industrial process. Several attempts have been reported in this direction. Analysis of viscoelastic fluid with variable thermal conductivity and mixed convection is reported by Hayat et al. [29]. Abbasi et al. [30] examined double stratification and mixed convection effects in magneto Maxwell nanofluid. MHD slip flow of micropolar liquid due to vertical shrinking surface is demonstrated by Das [31]. Hayat et al. [32] explored convective flow of Maxwell liquid with thermal radiation and mixed convection. The present investigation looks at the nonlinear stretched flow of MHD micropolar liquid with mixed convection, viscous dissipation, Joule heating and convective condition. Flow caused is due to a nonlinear stretched sheet. Analytic solutions are achieved by utilizing homotopic procedure [33–49]. Numerical values are presented to demonstrate the convergence of the derived series solutions. Moreover graphical results are demonstrated and analyzed.

2. Formulation Consider the MHD mixed convection flow of an incompressible micropolar fluid induced by a stretching surface at y = 0. The

temperature at the plate is passively adjusted through heated fluid of temperature Tf below the surface of the wall. Let T1 be the temperature outside the thermal boundary layer. Further we assumed a general power-law surface velocity distribution uw = cxn with c > 0. Here n = 1 and n > 1 correspond to the linear and nonlinear stretching cases respectively. The flow is subjected to a non-uniform applied magnetic field B(x)=B0xn  1/2 (for detail see Fig. 1). The electric field is absent whereas the induced magnetic field is neglected by assuming low magnetic Reynolds number. The governing problems under the boundary layer approximation are expressed as follows:

@u @ v þ ¼ 0; @x @y

ð1Þ




@ 2 u k @N rB2 ðxÞ þ u; þ gbT ðT  T 1 Þ  2 @y q @y q

u

@u @u þv ¼ @x @y

u


 @N @N c @ 2 N k @u þv ¼ ;  2N þ @x @y qj @y2 qj @y

u

@T @T @2T þv ¼a 2þ @x @y @y

mþ

k

q




l þ k1 qcp

ð2Þ

ð3Þ


2 @u rB2 ðxÞ 2 þ u ; @y qcp

ð4Þ

with the following boundary conditions

@u @T ; k1 ¼ hf ðT f  TÞ at @y @y as y ! 1:

u ¼ uw ¼ cxn ; v ¼ 0; N ¼ m0 y ¼ 0; u ! 0; N ! 0; T ! T 1

ð5Þ

Here u and v are the velocity components parallel to the x- and y-axis respectively, k the vortex viscosity, q the fluid density, l the dynamic viscosity, g the gravitational acceleration, bT the thermal   expansion coefficient, m the kinematic viscosity, c ¼ l þ 2k j the spin gradient viscosity, a = k1/qcp the thermal diffusivity, N the microrotation velocity, j = m/c the microinertia, Tf the convective fluid temperature, T the fluid temperature, T1 the ambient fluid temperature, cp the specific heat at constant pressure, k1 the thermal conductivity, m0 the boundary parameter and hf the convective heat transfer coefficient. Here viscous case is recovered by putting k = 0. Using the transformations

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðn þ 1Þ n1 cðn þ 1Þ n1 n 2 g¼ x y; N ¼ cx x 2 gðgÞ; 2m 2m T  T1 0 hðgÞ ¼ ; u ¼ cxn f ðgÞ; Tf  T1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   cmðn þ 1Þ n1 n1 0 x 2 f ðgÞ þ v¼ gf ðgÞ ; 2 nþ1

ð6Þ

Eqs. (2)–(5) are reduced as 000

00

ð1 þ KÞ f þ ff 

2n 02 2 2 0 0 f þ kg þ kh  Ha2 f ¼ 0; nþ1 nþ1 nþ1


 K 00 3n  1 0 2K 0 00 g þ fg  f g ð2g þ f Þ ¼ 0; 1þ 2 nþ1 nþ1 h00 þ Pr f h0 þ Pr Ecð1 þ KÞ f 0

002

0 2

þ PrEcHa2 ðf Þ ¼ 0;

ð7Þ ð8Þ ð9Þ

00

f ð0Þ ¼ 0; f ð0Þ ¼ 1; gð0Þ ¼ m0 f ð0Þ; h0 ð0Þ 0

¼ cð1  hð0ÞÞ; f ðgÞ ! 0; gðgÞ ! 0; hðgÞ ! 0 as g ! 1;

Fig. 1. Schematic diagram.

ð10Þ

where primes indicates differentiation with respect to g, K the micropolar parameter, k the buoyancy or mixed convection parameter, Grx the thermal buoyancy parameter, Rex the local Reynolds number, Ha the Hartman number, Pr the Prandtl number, Ec the

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Eckert number and c the Biot number. These parameters are defined as follows:

K¼

k

l

Ha2 ¼

;k¼

Grx Re2x

; Grx ¼

gbT ðT f  T 1 Þ x3

m2

; Rex ¼

uw x

v

;

rffiffiffi

hf m ðn1Þ rB2 ðxÞ m u2w ; Pr ¼ ; Ec ¼ ;c¼ x 2 : cqxn1 a cp ðT f  T 1 Þ k1 c

ð11Þ

The skin friction coefficient Cf with surface shear stress sw and local Nusselt number Nux with surface heat flux qw are given by

Cf ¼




2sw ; qu2w

Nux ¼

sw ¼ ðl þ kÞ

@u þ kN @y

 ;

ð12Þ

y¼0


 xqw @T : ; qw ¼ k1 @y y¼0 k1 ðT f  T 1 Þ

ð13Þ

The dimensionless skin friction coefficient and local Nusselt number are

C f Rex1=2

rffiffiffiffiffiffiffiffiffiffiffiffi nþ1 00 ð1 þ ð1  m0 ÞKÞ f ð0Þ; ¼ 2

Fig. 2. ⁄-curves for the functions f, g and h for linear stretching case.

ð14Þ

rffiffiffiffiffiffiffiffiffiffiffiffi nþ1 0 h ð0Þ; 2

Nux Rex1=2 ¼ 

ð15Þ

where Rex = uwx/v is the local Reynolds number. From Eq. (14) it should be noted that when m0 = 0 (called strong concentration). Moreover m0 = 0 in Eq. (5) implies N = 0 near the wall. This represents the concentrated particle flows in which the microelements close to the wall surface are unable to rotate. Eq. (5) has further two cases. (i) m0 = 1/2 corresponds to the vanishing of antisymmetric part of the stress tensor and it shows weak concentration of microelements. (ii) m0 = 1.0 is used for the modeling of turbulent boundary layer flow. Fig. 3. ⁄-curves for the functions f, g and h for nonlinear stretching case.

3. Series expressions Here initial approximations and auxiliary linear operators required for HAM solutions are presented below:

f 0 ðgÞ ¼ ð1  eg Þ; g 0 ðgÞ ¼ m0 eg ; h0 ðgÞ ¼ 000

c

1þc

expðgÞ;

0

Lf ¼ f  f ; Lg ¼ g 00  g 0 ; Lh ¼ h00  h;

ð16Þ ð17Þ

Eq. (17) satisfies the following mentioned properties:

Lf ðC 1 þ C 2 eg þ C 3 eg Þ ¼ 0; Lg ðC 4 eg þ C 5 eg Þ ¼ 0; Lh ðC 6 eg þ C 7 eg Þ ¼ 0

ð18Þ

where Ci (i = 1–7) indicate the arbitrary constants. 4. Convergence of the derived series solutions 0

Fig. 4. Impact of K on f .

It is well known that the derived series solutions contain controlling convergence parameters ⁄f, ⁄g and ⁄h. These parameters are beneficial in controlling and adjusting the convergence of the homotopic solutions. Therefore, the ⁄-curves have been plotted in Figs. 2 and 3. Here ⁄ -curves in Fig. 2 are sketched for the case of linearly stretching sheet (i.e. for n = 1.0) while ⁄-curves shown in Fig. 3 are plotted for non-linearly stretching sheet (i.e. for n = 1.5). It is noted from Fig. 2 that the permissible values of ⁄f, ⁄g  g 6 0:25 and and ⁄h are 1:35 6  hf 6 0:2, 1:45 6 h 1:45 6  hh 6 0:30: However the permissible values of ⁄f, ⁄g and hf 6 0:2, 1:45 6  hg 6 0:25 and ⁄h in Fig. 3 are 1:35 6  1:45 6  hh 6 0:30.

5. Discussion Our aim here is to discuss the velocity, micro-rotation velocity, temperature, skin friction and Nusselt number graphically. Fig. 4 describes the behavior of K on the velocity distribution. This figure illustrates that velocity in the x- direction is increased for larger K. In fact higher values of material parameter corresponds to the low viscosity fluid. Thus velocity distribution enhances. Features of k on velocity distribution is shown in Fig. 5. Larger values of k enhance

M. Waqas et al. / International Journal of Heat and Mass Transfer 102 (2016) 766–772

769

0

Fig. 5. Impact of k on f .

0

Fig. 6. Impact of Ha on f .

0

Fig. 8. Impact of K on g.

Fig. 9. Impact of m0 on g.

Fig. 7. Impact of n on f .

Fig. 10. Impact of n on g.

the buoyancy force which boosts the velocity distribution. Behavior of Hartman number Ha on velocity distribution is reported in Fig. 6. We noted that magnetic parameter Ha corresponds to smal0 ler velocity f (g). Here hydrodynamic case is stronger in comparison to hydromagnetic situation. The Lorentz force appeared in magnetic parameter becomes stronger with an increase in Ha. In 0 fact the stronger Lorentz force decays the velocity f (g). Fig. 7 shows the effects of power index on velocity distribution. Here velocity distribution and its related boundary layer thickness reduces via larger n. Figs. 8–10 are sketched for the variations of micropolar parameter K, boundary parameter m0 and power index n on the microrotation velocity distribution g(g). Fig. 8 portrays the behavior of K on

g(g). For larger K, it is detected that initially g reduces by enhancing K. For K > 0 micro-rotation velocity g is greater for K = 0. Fig. 9 portrays the influence of boundary parameter m0 on g(g). Here g(g) is increased for larger m0. For m0 = 0 micro-rotation velocity g(g) is zero. Impact of power index n on microrotation velocity g is illustrated in Fig. 10. Here microrotation velocity reduces via larger n. Characteristics of Prandtl number Pr, Biot number c and Eckert number Ec on temperature distribution are shown in the Figs. 11–13. Fig. 11 plots Pr variation on temperature distribution. It is noted that temperature distribution and associated thermal boundary layer thickness are reduced through larger Pr. Obviously Prandtl number and thermal diffusivity have inverse relationship. Due to this reason, one anticipates a thinner thermal boundary

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M. Waqas et al. / International Journal of Heat and Mass Transfer 102 (2016) 766–772

Fig. 11. Impact of Pr on h.

Fig. 14. Impacts of K and n on C f Re1=2 x .

Fig. 15. Impacts of k and Ha on C f Re1=2 x . Fig. 12. Impact of c on h.

Fig. 16. Impacts of c and Pr on Nux Re1=2 . x

Fig. 13. Impact of Ec on h.

layer in higher Prandtl number fluid. Fig. 12 reports the behavior of c on temperature distribution. Larger values of c boost the temperature. Since Biot number c contains the heat transfer coefficient which enhances for higher values of c. Hence temperature enhances. For c = 0 there is no heat transfer at the wall (insulated wall) and for c ? 1 one recovers the case of prescribed surface temperature. Effects of Ec on temperature distribution are plotted in Fig. 13. Here temperature enhances for larger Ec. Heat energy in the fluid is stored for larger Ec. It is due to friction forces that ultimate enhances the temperature profile. Figs. 14–17 are prepared to analyze the impacts of different emerging parameters on skin friction and Nusselt number

Fig. 17. Impacts of Ec and Ha on Nux Re1=2 . x

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M. Waqas et al. / International Journal of Heat and Mass Transfer 102 (2016) 766–772 Table 1 Convergence of series solutions for different order of approximations when K ¼ k ¼ 0:2, Ha ¼ Ec ¼ 0:1; c ¼ 0:3, Pr = 0.7, m0 = 0.5, n = 1.5 and ⁄f = ⁄g = ⁄h =  0.7. 0

00

0

Table 4 Comparative study of present K ¼ 0 ¼ Ha ¼ k ¼ m0 :

results

00

of f ð0Þ

with

[50]

via

n

when

Order of approximations

f ð0Þ

g (0)

h (0)

n

Cortell [50]

Present results

1 10 15 20 25 27 30 40

0.99578 0.96785 0.95801 0.95542 0.95478 0.95468 0.95468 0.95468

0.52706 0.52486 0.52130 0.52020 0.52009 0.52003 0.52003 0.52003

0.21192 0.18233 0.17353 0.17098 0.17070 0.17061 0.17061 0.17061

0.0 0.2 0.5 0.75 1.0 1.5 3.0 7.0 10.0 20.0 100.0

0.627547 0.766758 0.889477 0.953786 1.000000 1.061587 1.148588 1.216847 1.234875 1.257418 1.276768

0.6276 0.7669 0.8895 0.9539 1.0000 1.0616 1.1486 1.2169 1.2349 1.2575 1.2768

Table 2 Skin friction coefficient C f Re1=2 via various values of K, k, Ha, c, Ec and Pr when n = 1.5. x K

k

c

Ha

Ec

Pr

m0 = 0.0

m0 = 0.5

Re1=2 x Cf 0.0 0.3 0.6 0.2

0.2

0.0 0.3 0.6 0.2

0.1

0.3

0.0 0.2 0.4 0.1

0.1

0.5 0.7 0.9 0.3

0.0 0.4 0.8 0.1

0.7

1.0 1.1 1.2

0.9984 1.1400 1.2585 1.1651 1.0640 0.9750 1.0910 1.1111 1.1694 1.0786 1.0672 1.0591 1.1041 1.0732 1.0455 1.1103 1.1134 1.1164

0.9984 1.0751 1.1463 1.1183 1.0190 0.9326 1.0456 1.0645 1.1194 1.0335 1.0225 1.0147 1.0583 1.0274 0.9998 1.0642 1.0674 1.0702

Table 3 Local Nusselt number Nux Re1=2 via various values of K, k, Ha, c, Ec and Pr when n = 1.5 x and m0 = 0.5. K 0.0 0.3 0.6 0.2

k

0.0 0.4 0.8 0.2

Ha

0.0 0.2 0.5 0.1

c

0.5 0.6 0.7 0.3

Pr

1.0 1.1 1.2 1.0

Ec

Nux Re1=2 x

0.0 0.2 0.4

0.1711 0.1706 0.1700 0.1668 0.1736 0.1779 0.1711 0.1698 0.1632 0.2273 0.2479 0.2651 0.1837 0.1869 0.1898 0.1833 0.1585 0.1351

respectively. Influences of K and n on skin friction are disclosed through Fig. 14. Here skin friction coefficient enhances for larger K and n. Fig. 15 depicts the behaviors of k and Ha on skin friction. It is observed that higher values of k reduces the skin friction however opposite behavior is examined via larger Ha. Features of c and Pr on Nusselt number are analyzed through Fig. 16. Clearly Nusselt number is increasing function of c and Pr. Fig. 17 illustrates the impacts of Ec and Ha on Nusselt number. It is inspected that Nusselt number reduces via larger Ec and Ha.

Table 5 0 Comparative study of present results of h (0) with [50] via n and Ec when Pr = 1.0 and c ? 1. Ec

n

Cortell [50]

Present results

0.0

0.2 0.5 1.5 3.0 10.0 0.2 0.5 1.5 3.0 10.0

0.610262 0.595277 0.574537 0.564472 0.554960 0.574985 0.556623 0.530966 0.517977 0.505121

0.6102 0.5952 0.5748 0.5648 0.5550 0.5752 0.5568 0.5310 0.5181 0.5055

0.1

Convergence of the derived solutions is studied through Table 1. Clearly velocity, micro-rotation and temperature converges at 30th order of approximations respectively. Tables 2 and 3 includes numerical values of skin friction and local Numerical number. Here skin friction coefficient enhances for larger K, Ha and Pr however it reduces when k; c and Ec are enhanced. For m0 = 0, skin friction coefficient is higher than m0 = 0.5. Larger values of k; c and Pr consequences in the enhancement of Nusselt number however it reduces for higher values of K, Ha and Ec. A comparative study in the special situation is made with the work done by Cortell [50]. Here good agreement is noted (see Tables 4 and 5). 6. Conclusions We have examined the MHD mixed convection flow of micropolar liquid by nonlinear stretching surface with viscous dissipation, Joule heating and convective boundary condition. Major points are mentioned below. 0

 Impacts of K and k on f are opposite to that of Ha and n.  Temperature and thermal boundary layer thickness decrease with an increase in Prandtl number.  Microrotation velocity profile increases for larger m0.  Skin-friction coefficient for m0 = 0 are more when compared with m0 = 0.5.  Local Nusselt number enhances for k; c and Pr however it reduces via larger K, Ha and Ec.

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