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Influence of Induced Magnetic Field on Free Convection of Nanofluid Considering Koo-Kleinstreuer-Li (KKL) Correlation

Influence of Induced Magnetic Field on Free Convection of Nanofluid Considering... applied sciences Article Influence of Induced Magnetic Field on Free Convection of Nanofluid Considering Koo-Kleinstreuer-Li (KKL) Correlation 1 2 3 , 4 , M. Sheikholeslami , Q. M. Zaigham Zia and R. Ellahi * Department of Mechanical Engineering, Babol University of Technology, Babol 484, Iran; mohsen.sheikholeslami@yahoo.com Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 45550, Pakistan; zaighum_zia@comsats.edu.pk Department of Mathematics & Statistics, FBAS, IIUI, H-10 Sector, Islamabad 44000, Pakistan Mathematics Faculty of Science Taibah University, Madinah 41411, Munawwarah, Saudi Arabia * Correspondence: rellahi@engr.ucr.edu or rahmatellahi@yahoo.com Academic Editor: Fan-Gang Tseng Received: 3 September 2016; Accepted: 24 October 2016; Published: 2 November 2016 Abstract: In this paper, the influence of induced magnetic field on free convection of Al O -water 2 3 nanofluid on permeable plate by means of Koo-Kleinstreuer-Li (KKL) model is reported. Impact of Brownian motion, along with the properties of nanofluid, are also taken into account. The resulting equations are solved utilizing Runge-Kutta integration method. Obtained results are examined for innumerable energetic parameters, namely Al O volume fraction, suction parameter, and Hartmann 2 3 and magnetic Prandtl numbers. Results indicate that the velocity profile reduces with rise of the suction parameter and magnetic Prandtl and Hartmann numbers but it increases with addition of nanoparticles. Shear stress enhances with rise of suction parameter, magnetic Prandtl and Hartmann numbers. Temperature gradient improves with augment of suction parameter. Keywords: induced magnetic field; nanofluid; free convection; permeable plate; current density 1. Introduction Magnetohydrodynamic (MHD) free convection has several applications, such as combustion modeling, geophysics, fire engineering, etc. In recent decades, nanotechnology has been presented as a new passive technique for heat transfer improvement. MHD nanofluid natural convection in a tilted wavy cavity has been presented by Sheremet et al. [1]. They illustrated that a change of titled angle causes convective heat transfer to be enhanced. 3D MHD free convective heat transfer was examined by Sheikholeslami and Ellahi [2] using Lattice Boltzmann method (LBM). Their results showed that Lorentz forces cause the temperature gradient to reduce. Ismael et al. [3] investigated the influence of Lorentz forces on nanofluid flow in an enclosure with moving walls. Their outputs indicated that the impact of Lorentz forces reduces with direction of magnetic field. Sheikholeslami and Ellahi [4] utilized LBM to study Fe O -water flow, with the aim of drug delivery. They concluded that the 3 4 velocity gradient reduces with the rise of magnetic number. The influence of non-uniform Lorentz forces on nanofluid flow style has been studied by Sheikholeslami Kandelousi [5]. He concluded that improvement in heat transfer reduces with rise of Kelvin forces. A new model for nanofluid on peristaltic flow was presented by Tripathi and Beg [6]. They reported different behavior for nanofluid temperature profiles with changing temperature. Kouloulias et al. [7] presented an experimental analysis for free convection of nanofluid. They showed that greater nanoparticle volume fraction leads to higher Rayleigh numbers. Appl. Sci. 2016, 6, 324; doi:10.3390/app6110324 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 324 2 of 13 The influence of thermal radiation on magnetohydrodynamic nanofluid motion has been reported by Sheikholeslami et al. [8]. They concluded that the nanofluid concentration gradient augments with the rise of the radiation parameter. Mineral oil-based nanofluids have been utilized in natural convection by Peña et al. [9]. MHD Fe O -water flow in a wavy cavity with moving wall has been 3 4 investigated by Sheikholeslami and Chamkha [10]. The influence of Lorentz forces on forced convective heat transfer has been examined by Sheikholeslami et al. [11]. They illustrated that a greater Reynolds number has a more sensible effect on Kelvin forces. Akbar and Khan [12] investigated the impact of magnetic field on nanofluid motion in an asymmetric channel. Hakeem et al. [13] studied the influence of Lorentz forces on various nanofluids by means of second order slip flow mode. They showed that a unique solution exists for this problem for high Hartman number values. Several researchers have investigated about this subject [14–22]. In almost all the previous papers, the authors neglected the induced magnetic field. However, in various physical states it is necessary to consider this effect in governing equations. This assumption is considered in order to simplify the mathematical analysis of the problem. Furthermore, the induced magnetic field produces its own magnetic field in the fluid; therefore, it can amend the original magnetic field. Also, nanofluid motion in the magnetic field produces mechanical forces which change the motion of motion. Ghosh et al. [23] reported the impact of induced Lorentz forces on temperature profile. Unsteady magnetohydrodynamic flow on a cone has been investigated by Vanita and Kumar [24]. Beg et al. [25] examined the impact of induced magnetic field on boundary layer flow. The influence of atherosclerosis on hemodynamics of stenosis has been forecasted by Nadeem and Ijaz [26]. They showed that the velocity gradient on the wall of titled arteries reduces with augment of Strommers number. The chief end of this paper is to illustrate the influence of induced magnetic field on nanofluid hydrothermal treatment between two vertical plates. To obtain outputs, Runge-Kutta method is selected. The impacts of the suction parameter, magnetic Prandtl and Hartmann numbers, volume fraction of nanofluid on temperature, and induced magnetic, velocity and current density profiles are examined. 2. Problem Statement Al O -water fluid through two vertical permeable sheets is investigated as illustrated in Figure 1. 2 3 The boundary conditions are clear in this figure. The variables are only the function of y because plates are infinite. Velocity and magnetic field vectors are considered as v = [u, v , 0] and b = [b , b , 0] 0 0 respectively. The governing equations and boundary conditions can be obtained as follows: d u  b db du e 0 x + + g (T T ) + v = 0 (1) n f n f 0 0 dy  dy dy n f 1 d b du db x x + b + v = 0 (2) 0 0 dy dy dy e n f d T dT n f + v = 0 (3) dy dy n f dT q b 0 = 0, u 0 = 0, 0 = (4) ( ) ( ) ( ) dy k db (h) = 0, u (h) = 0, T (h) = T (5) dy Appl. Sci. 2016, 6, 324 3 of 13 () , C , ( ) and  can be introduced as [3]: n f n f n f n f 3  / 1 ( p ) n f f = 1 + , ( ) = (1 ) ( ) + ( ) , n f f p ( / +2)( / 1) p f p f (6) ( C ) = ( C ) + ( C ) (1 ),  =  +  (1 ) p p p n f p f n f p f k and  are obtained according to Koo-Kleinstreuer-Li (KKL) model [27]: n f n f 3 k /k 1 ( ) p f  T 4 0 k = + 1 + 5 10 c g (d , T, ) n f p, f f k /k 1 + k /k +2 p p ( p ) ( p ) f f g d , T,  = (7) a + a Ln d + a Ln d + a Ln () + a ln d Ln () Ln (T) + p p p 1 2 5 3 4 a + a Ln d + a Ln d + a Ln () + a ln d Ln () 6 7 p p 8 9 p 8 2 R = d 1/k 1/k , R = 4 10 km /W p p f p,e f f f f k f Brownian = +  (8) n f 2.5 k Pr (1 ) f All needed coefficients and properties are illustrated in Tables 1 and 2 [27]. Dimensionless parameters are presented as: e 2 1 U = , B =  b g h DT ,  = DT (T T ) , DT = qh/k , Y = 2 f x f 0 f g h DT f q (9) f B h  v h 0 e 0 Pr = C , Pm =    , Ha = , V = p 0 e f f f k f f f Finally, the dimensionless governing equations are d U Ha dB A A A dU 1 6 1 + +  + V = 0 (10) A dY A A dY dY 2 2 2 d B dB dU + A V Pm + A HaPm = 0 (11) 5 0 5 dY dY dY d  A d + V Pr = 0 (12) dY A dY B (0) = 0, U (0) = 0, (0) = 1 (13) dY dB (1) = 0, U (1) = 0,  (1) = 0 (14) dY Induced current density can be defined: J = dB/dY (15) C and Nu can be expressed as: C = U (0), Nu = A /(0). (16) f 4 1 Appl. Sci. 2016, 6, 324 4 of 13 Appl. Sci. 2016, 6, 324   4 of 13  Figure 1. Geometry of the problem. Figure 1. Geometry of the problem.  Table 1. Constants of Al O Water [27]. Table1. Constants of  Al O2 3Water   [27].  Al O  Water Coefficient Coef Va ficient lues Values Al O Water 2 3 a   a 552.813 2.813  a 6.115 a   2 6.115  a 0.695 0.695 2   a 4.1  10 −2 a 0.176 4.1 × 10   4 5 a 298.198 0.176  a 34.532 −298.198  a 3.922 a a 0.235 9 −34.532  a 0.999 −3.922  −0.235  Table 2. Properties of water and Al O [27]. 2 3 −0.999  3 5 1 1 Material r (kg/m ) C (j/kgk) k (W/mk) b  10 (K ) d (nm) s (Wm) p p Table 2. Properties of water and  Al O   [27]. Pure water 997.1 4179 0.613 21 - 0.05 Al O 3970 765 25 0.85 47 1  10 2 3 3 51  1   d(nm)   C(j/ kgk) k(W / m k)  Material  (kg / m ) 10 (K )   m p p Pure water  997.1  4179  0.613  21 ‐  0.05  10 Al O   3970  765  25  0.85  47  11  0   3. Runge‐Kutta Method  In  Runge‐Kutta  method,  at  first  the  following  definitions  are  applied:  x Ux,,Y xU ,  21 3 xB,, x B x,x . The final system and initial conditions are:  45 6 7 Appl. Sci. 2016, 6, 324 5 of 13 3. Runge-Kutta Method In Runge-Kutta method, at first the following definitions are applied: x = U, x = Y, x = U , 2 1 3 x = B, x = B , x = , x =  . The final system and initial conditions are: 4 5 6 7 0 1 0 1 B C B C x B 3 C B C B C B C B [Hax A A x A V x ] C 5 1 6 6 1 0 3 B 3 C A B C B C B C B x C = (17) B C B C B C B C x A V x Pm A x HaPm 5 B 5 0 5 5 3 C B C B C @ A x @ A 6 7 0 3 x V x Pr 7 0 7 0 1 0 1 0 x B C B C 0 x B C B C B C B C u x B 1 C B 3 C B C B C B 0 C = B x C (18) B C B C B C B C u x 2 5 B C B C @ A @ A u x 3 6 1 x Equations (17) and (18) are solved utilizing fourth order Runge-Kutta method. According to U (1) = 0, B (1) = 0,  (1) = 0, unknown initial conditions can be obtained by Newton’s method. 4. Results and Discussion Steady two-dimensional nanofluid hydrothermal treatment between two parallel vertical permeable plates is studied considering induced magnetic field. The Runge-Kutta integration scheme is utilized to solve this problem. MAPLE code has been validated by comparison with a previously published paper [28]. Table 3 indicates good accuracy of present code. The influences of important parameters such as magnetic Prandtl number (Pm), Hartmann number (Ha), suction parameter (V ) and nanoparticle volume fraction () on flow style are examined. Table 3. Comparison of skin friction tension over the upper wall between the present results and previous work. H , Hartmann number; V , suction parameter; Pr, Prandtl number. a 0 V Pr Pr Ha Sarveshanand and Singh [21] Present Work 0 m 0.5 0.7 0.5 5 0.016 0.015 0.75 0.7 0.5 5 0.011 0.011 1 0.7 1 5 0.018 0.018 1 0.015 0.5 0.5 2.695 2.700 Impact of  on induced magnetic field, current density, temperature and velocity distributions is shown in Figure 2. As volume fraction of nanofluid augments, nanofluid velocity and temperature are enhanced due to an increase in fluid motion by adding nanoparticles. Induced current density increases with an augment of  while the opposite behavior is shown for induced magnetic field. Influence of suction parameter on hydrothermal behavior is depicted in Figure 3. Velocity, temperature and induced current density decreases, with an augment of suction parameter while induced magnetic field enhances with rise of V . Therefore, this parameter can be considered as control parameter for engineering designs. Appl. Sci. 2016, 6, 324 6 of 13 Appl. Sci. 2016, 6, 324   6 of 13  0.002 Appl. Sci. 2016, 6, 324   6 of 13    0.002    -0.001 0.0015         -0.001 0.0015    -0.002 0.001 -0.002 -0.003 0.001 -0.003 0.0005 -0.004 0.0005 -0.004 -0.005 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.005 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a)  (b) 0.16 0.015 (a)  (b) 0.16 0.015       0.12     0.12 0.01   0.01 0.08 0.08 0.005 0.04 0.005 0.04 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (c)  (d) (c)  (d) Figure 2. (a)Effect of nanoparticle volume fraction on velocity, (b) induced magnetic field; (c) induced  Figure 2. Effect of nanoparticle volume fraction on velocity (U); induced magnetic field (B); induced Figure 2. (a)Effect of nanoparticle volume fraction on velocity, (b) induced magnetic field; (c) induced  current  density;  (d)  temperature  field  distributions  when  VH  1, a 5,Pm 1, Pr 6.8 .   curr current ent density   densit (Jy );and   (d)te  mperatur temperature e field   field ( ) distribu distributions tions  when when VVH =1, 1, Haa =55, ,PPmm = 1, P 1, r Pr 6.= 8 6.8 .   . Magnetic Prandtl number: Pm; nanoparticle volume fraction:   .  Magnetic Prandtl number: Pm; nanoparticle volume fraction: . Magnetic Prandtl number: Pm; nanoparticle volume fraction:   .  0.02 0.03 0.02 0.03 V= 0.2 V= 0.2 0 0 V= 0.2 V= 0.2 0 0 V= 0.4 V= 0.4 V= 0.4 0.015 V= 00.4 0 0 0.015 V= 0.6 V= 0.6 V= 0.6 V= 0.6 0 0 0.015 0 0 0.015 V= 1.0 V= 1.0 0V= 1.0 V= 1.0 0 0 0 0.01 0.01 -0.015 -0.015 -0.03 -0.03 0.005 0.005 -0.045 -0.045 -0.06 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.06 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Y Y Y Y (a)  (b) (a)  (b) Figure 3. Cont. U J B B B Appl. Sci. 2016, 6, 324 7 of 13 Appl. Sci. 2016, 6, 324   7 of 13  Appl. Sci. 2016, 6, 324   7 of 13  0.1 0.6 0.6 0.1 V= 0.2 V= 0.2 0 V= 0.2 V= 0.2 0 0 V= 0.4 V= 0.4 0 0.5 V= 00.4 V= 0.4 0 0 0.5 0.08 0.08 V= 0.6 V= 0.6 V= 0.6 V= 0.6 0 0 0 V= 1.0 V= V= 1.0 1.0 V= 1.0 0 0 0 0 0.0.44 0.06 0.06 0.3 0.3 0.04 0.04 0.2 0.2 0.02 0.1 0.02 0.1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 Y 0.6 0.8 1 Y (c)  (d) (c)  (d) Figure 3. (a) Effect of suction parameter on velocity; (b) induced magnetic field; (c) induced current  Figure Figure 3. 3. (aEf ) Ef fect fect of of suction  suction parameter  parameter on on velocity  velocity; (U );(binduced ) induced magnetic  magnetifield c fiel(B d; );(cinduced ) induced curr  current ent   density; (d) induced temperature field distributions when Ha  5, Pm 1,  0.04, Pr 6.8 .  density (J) and induced temperature field () distributions when Ha = 5, Pm = 1,  = 0.04, Pr = 6.8. density; (d) induced temperature field distributions when Ha  5, Pm 1,  0.04, Pr 6.8 .  Figure  4  depicts  the  impacts  of  Lorentz  forces  on  induced  magnetic  field,  induced  current  density Figure an  4d  dep velocity icts  the distr  imp ibutions. acts  of As  Lorent  Lorentz z  fo force rces s  augm on  indu ents,ced  the  ma back gn flow etic  fi appea eld, rindu s andced  in  turn curre  nt  Figure 4 depicts the impacts of Lorentz forces on induced magnetic field, induced current density velocity of nanofluid decreases. In addition, it can be seen that the maximum velocity point shifts to  density and velocity  and velocity distributions.  distributions. As Lorentz  As for Lorentz ces augments,  forces augm the back ents, flow  the appears back flow and appea in turn rs velocity and in turn of   the  hot  wall.  Induced  magnetic  field  decreases  with  rise  of  magnetic  field  strength  but  induced  velocity nanofluid  of nanofluid decreases. decre In addition, ases. In it addit can beion, seen it ca that n be the seen maximum  that the vel ma ocity ximum point velocity shifts to point the hot shi wall. fts to  current density is enhanced with the rise of Lorentz forces. Influence of  Pm   on induced magnetic  Induced magnetic field decreases with rise of magnetic field strength but induced current density is the  hot  wall.  Induced  magnetic  field  decreases  with  rise  of  magnetic  field  strength  but  induced  field, induced current density and velocity profiles is depicted in Figure 5. Without the magnetic  enhanced with the rise of Lorentz forces. Influence of Pm on induced magnetic field, induced current current density is enhanced with the rise of Lorentz forces. Influence of  Pm   on induced magnetic  field, the shape of the velocity profiles is parabolic but in the existence of the magnetic field its shape  density and velocity profiles is depicted in Figure 5. Without the magnetic field, the shape of the field, induced current density and velocity profiles is depicted in Figure 5. Without the magnetic  changes  to  being  flattened.  The  nanofluid  motion  and  induced  magnetic  field  reduces  with  an  velocity profiles is parabolic but in the existence of the magnetic field its shape changes to being field, the shape of the velocity profiles is parabolic but in the existence of the magnetic field its shape  augment of  Pm . Induced current density rises with augment of Pm.  flattened. The nanofluid motion and induced magnetic field reduces with an augment of Pm. Induced changes  to  being  flattened.  The  nanofluid  motion  and  induced  magnetic  field  reduces  with  an  current density rises with augment of Pm. augment of 0.00 4 Pm . Induced current density rises with aug 0.001ment of Pm.  Ha = 1 Ha = 1 0.004 Ha = 5 0.001 Ha = 5 Ha = 10 Ha = 10 0.003 Ha = 1 Ha = 1 Ha = 20 Ha = 20 Ha = 5 Ha = 5 -0.001 Ha = 10 Ha = 10 0.003 Ha = 20 Ha = 20 0.002 -0.002 -0.001 0.002 -0.003 -0.002 0.001 -0.004 -0.003 0.001 -0.005 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.004 (a)  (b) -0.005 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a)  (b) Figure 4. Cont.  Appl. Sci. 2016, 6, 324 8 of 13 Appl. Sci. 2016, 6, 324   8 of 13  0.012 Appl. Sci. 2016, 6, 324   8 of 13  Ha = 1 Ha = 5 0.01 0.012 Ha = 10 Ha = 1 Ha = 20 Ha = 5 0.008 0.01 Ha = 10 Ha = 20 0.008 0.006 0.006 0.004 0.004 0.002 0.002 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (c) Y (c) Figure 4. (a) Effect of Hartmann number (Ha) on velocity; (b), induced magnetic field; (c)induced  Figure 4. Effect of Hartmann number (Ha) on velocity (U); induced magnetic field (B) and induced Figure 4. (a) Effect of Hartmann number (Ha) on velocity; (b), induced magnetic field; (c)induced  current density distributions whenVP  1, m 1,  0.04, Pr 6.8 .  current density (J) distributions when V = 1, Pm = 1,  = 0.04, Pr = 6.8. current density distributions whenVP  1, m 1,  0.04, Pr 6.8 .  0.002 0.005 0.002 0.005 Pm = 0.05 Pm = 0.05 Pm = 0.05 Pm = 0.1 Pm = 0.05 Pm = 0.1 Pm = 0.1 Pm = 0.5 Pm = 0.1 Pm = 0.5 0.004 0.004 Pm = 0.5 Pm = 1 Pm = 0.5 Pm = 1 Pm = 1 Pm = 1 0.003 0.003 -0.002 -0.002 0.002 0.002 -0.004 -0.004 0.001 0.001 -0.006 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.006 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a)  (b) 0.012 (a)  (b) Pm = 0.05 0.012 Pm = 0.1 Pm = 0.05 0.01 Pm = 0.5 Pm = 0.1 Pm = 1 0.01 Pm = 0.5 0.008 Pm = 1 0.008 0.006 0.006 0.004 0.004 0.002 0.002 0 0.2 0.4 0.6 0.8 1 (c) 0 0.2 0.4 0.6 0.8 1 Figure  5.  (a)  Effect  of  magnetic  Prandtl  number  (Pm)  on  velocity;  (b)  induced  magnetic  field;   Figure 5. Effect of magnetic Prandtl number (Pm) on velocity (U); induced magnetic field (B) (c) induced current density distributions when VH (c) 1, a 5,  0.04, Pr 6.8 .  and induced current density (J) distributions when V = 1, Ha = 5,  = 0.04, Pr = 6.8. Figure  5.  (a)  Effect  of  magnetic  Prandtl  number  (Pm)  on  velocity;  (b)  induced  magnetic  field;   Influences  of  magnetic  Prandtl  number  Pm ,  Hartmann  number  Ha ,  suction  parameter      (c) induced current density distributions when VH  1, a 5,  0.04, Pr 6.8 .  V   and nanoparticle volume fraction     on skin friction coefficient are depicted in Figures 6 and      7. According to these data, a correlation is presented for skin friction coefficient as follows:  Influences  of  magnetic  Prandtl  number  Pm ,  Hartmann  number  Ha ,  suction  parameter       V   and nanoparticle volume fraction    on skin friction coefficient are depicted in Figures 6 and  7. According to these data, a correlation is presented for skin friction coefficient as follows:  B Appl. Sci. 2016, 6, 324 9 of 13 Appl. Sci. 2016, 6, 324   9 of 13  Influences of magnetic Prandtl number Pm , Hartmann number Ha , suction parameter V ( ) ( ) ( ) and nanoparticle volume fraction () on skin friction coefficient are depicted in Figures 6 and 7. CV  0.25335 0.34173 0.011146Ha 0.12279 f 0 According to these data, a correlation is presented for skin friction coefficient as follows:  0.0076966VHa 0.079691V Pm 2.54355V 00 0 (19)  C = 0.25335 0.34173V 0.011146Ha 0.12279 f 0  0.00154943Ha Pm 0.0424 Ha1.15754 Pm +0.0076966V22 Ha + 0.079691V Pm 2.543552 V  2 0 0 0 0.11737VH 0.0001179a 0.033354Pm 13.69282 (19) +0.00154943Ha Pm 0.0424Ha  1.15754Pm 2 2 2 2 +0.11737V + 0.0001179Ha + 0.033354Pm + 13.69282 0.05 0.05    0.04  0.04   0.03 0.03 0.02 0.02 0.01 0.01 1 5 10 15 20 0.05 0.2 0.4 0.6 0.8 1 Ha Pm (a)  (b) 0.2   0.15  0.1 0.05 0.2 0.4 0.6 0.8 1 (c) Figure  6.  Influences  of  magnetic  Prandtl  number  Pm ,  Hartmann  number  Ha ,  suction      Figure 6. Influences of magnetic Prandtl number Pm , Hartmann number Ha , suction parameter ( ) ( ) (V ) and nanoparticle volume fraction () on skin friction coefficient C when Pr = 6.8. parameter  V   and  nanoparticle  volume  fraction     on  skin  friction  coefficient  C   when 0     f   0 f (a) V = 1, Pm = 1; (b) V = 1, Ha = 5; (c) Ha = 5, Pm = 1. 0 0 Pr  6.8 . (a) VP 1, m 1; (b) VH 1, a 5 ; (c) Ha  5, Pm 1 .  0 0 f Appl. Sci. 2016, 6, 324 10 of 13 Appl. Sci. 2016, 6, 324   10 of 13  (a)  (b) (c)  (d) (e)  (f) Figure  7.  3D  surface  plots  for  skin  friction  coefficient.  (a)  Pm 0.55,  0.02 ;  (b)  Figure 7. 3D surface plots for skin friction coefficient. (a) Pm = 0.55,  = 0.02 ; (b) Ha = 12.5,  = 0.02; Ha 12.5,  0.02 ;  (c)  Ha 12.5, Pm 0.55 ;  (d)  V 0.6,  0.02 ;  (e) VP 0.6,m 0.55 ;   0 0 (c) Ha = 12.5, Pm = 0.55; (d) V = 0.6,  = 0.02; (e) V = 0.6, Pm = 0.55; (f) V = 0.6, Ha = 12.5. 0 0 0 (f) VH 0.6, a 12.5 .  It can be concluded that C has reverse relationship with all active parameters except for . It can be concluded that  C   has reverse relationship with all active parameters except for   .  Figure 8 shows the influence of V and  on Nusselt number. In addition, a good correlation has been Figure 8 shows the influence of  V   and    on Nusselt number. In addition, a good correlation has  presented for the Nusselt number as follows: been presented for the Nusselt number as follows:  Nu = 0.80342 + 4.92333V + 1.72177 3.50195V 0 0 Nu 0.80342 4.92333V 1.72177 3.50195V (20) 2 2 (20)  +1.822V 0.75963  1.822V 0.75963 As suction parameter (V ) and nanoparticle volume fraction () increase, temperature gradient As suction parameter  V   and nanoparticle volume fraction     increase, temperature gradient    increases. Therefore, Nu is enhanced with enhancement of V , . increases. Therefore, Nu is enhanced with enhancement of  V , .  Appl. Sci. 2016, 6, 324 11 of 13 Appl. Sci. 2016, 6, 324   11 of 13     2.5   1.5 0.05 0.1 0.2 0.3 0.4 (a) (b) Figure 8. Influence of nanofluid volume fraction ( ) and suction parameter (V ) on Nusselt number  Figure 8. Influence of nanofluid volume fraction () and suction parameter (V ) on Nusselt number (Nu) when  Ha 5, Pm 1, Pr 6.8 .  (Nu) when Ha = 5, Pm = 1, Pr = 6.8. 5. Conclusions  5. Conclusions The influence of induced magnetic field on nanofluid motion and forced convection between  The influence of induced magnetic field on nanofluid motion and forced convection between two vertical permeable plates is investigated. To solve coupled equations, Runge‐Kutta method is  two vertical permeable plates is investigated. To solve coupled equations, Runge-Kutta method is utilized. The influence of different dimensionless parameters on induced magnetic field, velocity and  utilized. The influence of different dimensionless parameters on induced magnetic field, velocity and temperature distributions are considered. Results illustrate that current density augments with a rise  temperature distributions are considered. Results illustrate that current density augments with a rise of volume fraction of nanofluid and Hartmann and magnetic Prandtl numbers, while it is reduced  of volume fraction of nanofluid and Hartmann and magnetic Prandtl numbers, while it is reduced with a rise in the suction parameter. As Lorentz force increases, velocity and induced magnetic field  with a rise in the suction parameter. As Lorentz force increases, velocity and induced magnetic field are reduced and maximum velocity point shifts to the left side.  are reduced and maximum velocity point shifts to the left side. Acknowledgments:   R. Ellahi is grateful to Prof. Sultan Z Alamri, Dean Faculty of Science and Prof. Yousef  Acknowledgments: R. Ellahi is grateful to Sultan Z Alamri, Dean Faculty of Science and Yousef Alharbi, Chairman Mathematics Department, Taibah University, Madinah Munawwarah, Saudi Arabia for their kind cooperation. Alharbi, Chairman Mathematics Department, Taibah University, Madinah Munawwarah, Saudi Arabia for their  R. Ellahi is also thankful to PCST to honed him with 7th top most Productive Scientist Award in category A and kind cooperation. R. Ellahi is also thankful to PCST to honed him with 7th top most Productive Scientist Award  Thomson Reuters to rank him among top 1% highly cited researchers on Web of Science in 2015–2016. in category A and Thomson Reuters to rank him among top 1% highly cited researchers on Web of Science in  Author Contributions: This paper is contributed in all respect by M. Sheikholeslami, Q. M. Zaigham Zia and 2015–2016.  R. Ellahi equally. Author Contributions:   This paper is contributed in all respect by Sheikholeslmai, Zaigham and Ellahi equally.  Conflicts of Interest:   The authors declare no conflict of interest  Nu Appl. Sci. 2016, 6, 324 12 of 13 Conflicts of Interest: The authors declare no conflict of interest. Nomenclature B Dimensionless induced horizontal magnetic field v Velocity vector Vector of magnetic field k Thermal conductivity c Specific heat J Induced current density T Temperature Ha Hartmann number Pr Prandtl number V Suction parameter Pm Magnetic Prandtl number U Dimensionless horizontal velocity Greek Symbols Dimensionless distance Coefficient of thermal expansion Electrical conductivity Dynamic viscosity of nanofluid Dimensionless temperature Nanofluid volume fraction Density Subscripts p Solid f Base fluid References 1. Sheremet, M.A.; Oztop, H.F.; Pop, I. MHD natural convection in an inclined wavy cavity with corner heater filled with a nanofluid. J. Magn. Magn. Mater. 2016, 416, 37–47. [CrossRef] 2. Sheikholeslami, M.; Ellahi, R. Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid. Int. J. Heat Mass Transf. 2015, 89, 799–808. [CrossRef] 3. Ismael, M.A.; Mansour, M.A.; Chamkha, A.J.; Rashad, A.M. Mixed convection in a nanofluid filled-cavity with partial slip subjected to constant heat flux and inclined magnetic field. J. Magn. Magn. Mater. 2016, 416, 25–36. [CrossRef] 4. Sheikholeslami, M.; Ellahi, R. Simulation of ferrofluid flow for magnetic drug targeting using Lattice Boltzmann method. J. Z. Naturforschung A 2015, 70, 115–124. 5. Kandelousi, M.S. Effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. Eur. Phys. J. Plus 2014, 129, 248. [CrossRef] 6. Tripathi, D.; Beg, O.A. A study on peristaltic flow of nanofluids: Application in drug delivery systems. Int. J. Heat Mass Transf. 2014, 70, 61–70. [CrossRef] 7. Kouloulias, K.; Sergis, A.; Hardalupas, Y. Sedimentation in nanofluids during a natural convection experiment. Int. J. Heat Mass Transf. 2016, 101, 1193–1203. [CrossRef] 8. Sheikholeslami, M.; Ganji, D.D.; Younus Javed, M.; Ellahi, R. 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Heat Mass Transf. 2016, 102, 544–554. [CrossRef] 17. Ellahi, R.; Bhatti, M.M.; Riaz, A.; Sheikholeslami, M. Effects of magnetohydrodynamics on peristaltic flow of jeffrey fluid in a rectangular duct through a porous medium. J. Porous Media 2014, 17, 143–157. [CrossRef] 18. Ellahi, R.; Hassan, M.; Zeeshan, A. Aggregation effects on water base Al O nanofluid over permeable 2 3 wedge in mixed convection. Asia-Pac. J. Chem. Eng. 2016, 11, 179–186. [CrossRef] 19. Akbar, N.S. Ferromagnetic CNT suspended H O+Cu nanofluid analysis through composite stenosed arteries with permeable wall. Phys. E Low-Dimens. Syst. Nanostruct. 2015, 72, 70–76. [CrossRef] 20. Afsar Khan, A.; Ellahi, R.; Mudassar Gulzar, M.; Sheikholeslami, M. Effects of heat transfer on peristaltic motion of Oldroyd fluid in the presence of inclined magnetic field. J. Magn. Magn. Mater. 2014, 372, 97–106. [CrossRef] 21. Mahian, O.; Kianifar, A.; Kalogirou, S.A.; Pop, I.; Wongwises, S. A review of the applications of nanofluids in solar energy. Int. J. Heat Mass Transf. 2013, 57, 582–594. [CrossRef] 22. Ahmad, S.; Rohni, A.M.; Pop, I. Blasius and sakiadis problems in nanofluids. Acta Mech. 2011, 218, 195–204. [CrossRef] 23. Ghosh, S.K.; Beg, O.A.; Zueco, J. Hydromagnetic free convection flow with induced magnetic field effects. Meccanica 2010, 14, 175–185. [CrossRef] 24. Anand Kumar, V. Numerical study of effect of induced magnetic field on transient natural convection over a vertical cone. Alex. Eng. J. 2016, 55, 1211–1223. 25. Beg, O.A.; Bekier, A.Y.; Prasad, V.R.; Zueco, J.; Ghosh, S.K. Non-similar, laminar, steady, electrically-conducting forced convection liquid metal boundary layer flow with induced magnetic field effects. Int. J. Therm. Sci. 2009, 48, 1596–1606. [CrossRef] 26. Nadeem, S.; Ijaz, S. Impulsion of nanoparticles as a drug carrier for the theoretical investigation of stenosed arteries with induced magnetic effects. J. Magn. Magn. Mater. 2016, 410, 230–241. [CrossRef] 27. Kandelousi, M.S. KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel. Phys. Lett. A 2014, 378, 3331–3339. [CrossRef] 28. Sarveshanand; Singh, A.K. Magnetohydrodynamic free convection between vertical parallel porous plates in the presence of induced magnetic field. Springer Plus 2015, 4, 333. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Influence of Induced Magnetic Field on Free Convection of Nanofluid Considering Koo-Kleinstreuer-Li (KKL) Correlation

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applied sciences Article Influence of Induced Magnetic Field on Free Convection of Nanofluid Considering Koo-Kleinstreuer-Li (KKL) Correlation 1 2 3 , 4 , M. Sheikholeslami , Q. M. Zaigham Zia and R. Ellahi * Department of Mechanical Engineering, Babol University of Technology, Babol 484, Iran; mohsen.sheikholeslami@yahoo.com Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 45550, Pakistan; zaighum_zia@comsats.edu.pk Department of Mathematics & Statistics, FBAS, IIUI, H-10 Sector, Islamabad 44000, Pakistan Mathematics Faculty of Science Taibah University, Madinah 41411, Munawwarah, Saudi Arabia * Correspondence: rellahi@engr.ucr.edu or rahmatellahi@yahoo.com Academic Editor: Fan-Gang Tseng Received: 3 September 2016; Accepted: 24 October 2016; Published: 2 November 2016 Abstract: In this paper, the influence of induced magnetic field on free convection of Al O -water 2 3 nanofluid on permeable plate by means of Koo-Kleinstreuer-Li (KKL) model is reported. Impact of Brownian motion, along with the properties of nanofluid, are also taken into account. The resulting equations are solved utilizing Runge-Kutta integration method. Obtained results are examined for innumerable energetic parameters, namely Al O volume fraction, suction parameter, and Hartmann 2 3 and magnetic Prandtl numbers. Results indicate that the velocity profile reduces with rise of the suction parameter and magnetic Prandtl and Hartmann numbers but it increases with addition of nanoparticles. Shear stress enhances with rise of suction parameter, magnetic Prandtl and Hartmann numbers. Temperature gradient improves with augment of suction parameter. Keywords: induced magnetic field; nanofluid; free convection; permeable plate; current density 1. Introduction Magnetohydrodynamic (MHD) free convection has several applications, such as combustion modeling, geophysics, fire engineering, etc. In recent decades, nanotechnology has been presented as a new passive technique for heat transfer improvement. MHD nanofluid natural convection in a tilted wavy cavity has been presented by Sheremet et al. [1]. They illustrated that a change of titled angle causes convective heat transfer to be enhanced. 3D MHD free convective heat transfer was examined by Sheikholeslami and Ellahi [2] using Lattice Boltzmann method (LBM). Their results showed that Lorentz forces cause the temperature gradient to reduce. Ismael et al. [3] investigated the influence of Lorentz forces on nanofluid flow in an enclosure with moving walls. Their outputs indicated that the impact of Lorentz forces reduces with direction of magnetic field. Sheikholeslami and Ellahi [4] utilized LBM to study Fe O -water flow, with the aim of drug delivery. They concluded that the 3 4 velocity gradient reduces with the rise of magnetic number. The influence of non-uniform Lorentz forces on nanofluid flow style has been studied by Sheikholeslami Kandelousi [5]. He concluded that improvement in heat transfer reduces with rise of Kelvin forces. A new model for nanofluid on peristaltic flow was presented by Tripathi and Beg [6]. They reported different behavior for nanofluid temperature profiles with changing temperature. Kouloulias et al. [7] presented an experimental analysis for free convection of nanofluid. They showed that greater nanoparticle volume fraction leads to higher Rayleigh numbers. Appl. Sci. 2016, 6, 324; doi:10.3390/app6110324 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 324 2 of 13 The influence of thermal radiation on magnetohydrodynamic nanofluid motion has been reported by Sheikholeslami et al. [8]. They concluded that the nanofluid concentration gradient augments with the rise of the radiation parameter. Mineral oil-based nanofluids have been utilized in natural convection by Peña et al. [9]. MHD Fe O -water flow in a wavy cavity with moving wall has been 3 4 investigated by Sheikholeslami and Chamkha [10]. The influence of Lorentz forces on forced convective heat transfer has been examined by Sheikholeslami et al. [11]. They illustrated that a greater Reynolds number has a more sensible effect on Kelvin forces. Akbar and Khan [12] investigated the impact of magnetic field on nanofluid motion in an asymmetric channel. Hakeem et al. [13] studied the influence of Lorentz forces on various nanofluids by means of second order slip flow mode. They showed that a unique solution exists for this problem for high Hartman number values. Several researchers have investigated about this subject [14–22]. In almost all the previous papers, the authors neglected the induced magnetic field. However, in various physical states it is necessary to consider this effect in governing equations. This assumption is considered in order to simplify the mathematical analysis of the problem. Furthermore, the induced magnetic field produces its own magnetic field in the fluid; therefore, it can amend the original magnetic field. Also, nanofluid motion in the magnetic field produces mechanical forces which change the motion of motion. Ghosh et al. [23] reported the impact of induced Lorentz forces on temperature profile. Unsteady magnetohydrodynamic flow on a cone has been investigated by Vanita and Kumar [24]. Beg et al. [25] examined the impact of induced magnetic field on boundary layer flow. The influence of atherosclerosis on hemodynamics of stenosis has been forecasted by Nadeem and Ijaz [26]. They showed that the velocity gradient on the wall of titled arteries reduces with augment of Strommers number. The chief end of this paper is to illustrate the influence of induced magnetic field on nanofluid hydrothermal treatment between two vertical plates. To obtain outputs, Runge-Kutta method is selected. The impacts of the suction parameter, magnetic Prandtl and Hartmann numbers, volume fraction of nanofluid on temperature, and induced magnetic, velocity and current density profiles are examined. 2. Problem Statement Al O -water fluid through two vertical permeable sheets is investigated as illustrated in Figure 1. 2 3 The boundary conditions are clear in this figure. The variables are only the function of y because plates are infinite. Velocity and magnetic field vectors are considered as v = [u, v , 0] and b = [b , b , 0] 0 0 respectively. The governing equations and boundary conditions can be obtained as follows: d u  b db du e 0 x + + g (T T ) + v = 0 (1) n f n f 0 0 dy  dy dy n f 1 d b du db x x + b + v = 0 (2) 0 0 dy dy dy e n f d T dT n f + v = 0 (3) dy dy n f dT q b 0 = 0, u 0 = 0, 0 = (4) ( ) ( ) ( ) dy k db (h) = 0, u (h) = 0, T (h) = T (5) dy Appl. Sci. 2016, 6, 324 3 of 13 () , C , ( ) and  can be introduced as [3]: n f n f n f n f 3  / 1 ( p ) n f f = 1 + , ( ) = (1 ) ( ) + ( ) , n f f p ( / +2)( / 1) p f p f (6) ( C ) = ( C ) + ( C ) (1 ),  =  +  (1 ) p p p n f p f n f p f k and  are obtained according to Koo-Kleinstreuer-Li (KKL) model [27]: n f n f 3 k /k 1 ( ) p f  T 4 0 k = + 1 + 5 10 c g (d , T, ) n f p, f f k /k 1 + k /k +2 p p ( p ) ( p ) f f g d , T,  = (7) a + a Ln d + a Ln d + a Ln () + a ln d Ln () Ln (T) + p p p 1 2 5 3 4 a + a Ln d + a Ln d + a Ln () + a ln d Ln () 6 7 p p 8 9 p 8 2 R = d 1/k 1/k , R = 4 10 km /W p p f p,e f f f f k f Brownian = +  (8) n f 2.5 k Pr (1 ) f All needed coefficients and properties are illustrated in Tables 1 and 2 [27]. Dimensionless parameters are presented as: e 2 1 U = , B =  b g h DT ,  = DT (T T ) , DT = qh/k , Y = 2 f x f 0 f g h DT f q (9) f B h  v h 0 e 0 Pr = C , Pm =    , Ha = , V = p 0 e f f f k f f f Finally, the dimensionless governing equations are d U Ha dB A A A dU 1 6 1 + +  + V = 0 (10) A dY A A dY dY 2 2 2 d B dB dU + A V Pm + A HaPm = 0 (11) 5 0 5 dY dY dY d  A d + V Pr = 0 (12) dY A dY B (0) = 0, U (0) = 0, (0) = 1 (13) dY dB (1) = 0, U (1) = 0,  (1) = 0 (14) dY Induced current density can be defined: J = dB/dY (15) C and Nu can be expressed as: C = U (0), Nu = A /(0). (16) f 4 1 Appl. Sci. 2016, 6, 324 4 of 13 Appl. Sci. 2016, 6, 324   4 of 13  Figure 1. Geometry of the problem. Figure 1. Geometry of the problem.  Table 1. Constants of Al O Water [27]. Table1. Constants of  Al O2 3Water   [27].  Al O  Water Coefficient Coef Va ficient lues Values Al O Water 2 3 a   a 552.813 2.813  a 6.115 a   2 6.115  a 0.695 0.695 2   a 4.1  10 −2 a 0.176 4.1 × 10   4 5 a 298.198 0.176  a 34.532 −298.198  a 3.922 a a 0.235 9 −34.532  a 0.999 −3.922  −0.235  Table 2. Properties of water and Al O [27]. 2 3 −0.999  3 5 1 1 Material r (kg/m ) C (j/kgk) k (W/mk) b  10 (K ) d (nm) s (Wm) p p Table 2. Properties of water and  Al O   [27]. Pure water 997.1 4179 0.613 21 - 0.05 Al O 3970 765 25 0.85 47 1  10 2 3 3 51  1   d(nm)   C(j/ kgk) k(W / m k)  Material  (kg / m ) 10 (K )   m p p Pure water  997.1  4179  0.613  21 ‐  0.05  10 Al O   3970  765  25  0.85  47  11  0   3. Runge‐Kutta Method  In  Runge‐Kutta  method,  at  first  the  following  definitions  are  applied:  x Ux,,Y xU ,  21 3 xB,, x B x,x . The final system and initial conditions are:  45 6 7 Appl. Sci. 2016, 6, 324 5 of 13 3. Runge-Kutta Method In Runge-Kutta method, at first the following definitions are applied: x = U, x = Y, x = U , 2 1 3 x = B, x = B , x = , x =  . The final system and initial conditions are: 4 5 6 7 0 1 0 1 B C B C x B 3 C B C B C B C B [Hax A A x A V x ] C 5 1 6 6 1 0 3 B 3 C A B C B C B C B x C = (17) B C B C B C B C x A V x Pm A x HaPm 5 B 5 0 5 5 3 C B C B C @ A x @ A 6 7 0 3 x V x Pr 7 0 7 0 1 0 1 0 x B C B C 0 x B C B C B C B C u x B 1 C B 3 C B C B C B 0 C = B x C (18) B C B C B C B C u x 2 5 B C B C @ A @ A u x 3 6 1 x Equations (17) and (18) are solved utilizing fourth order Runge-Kutta method. According to U (1) = 0, B (1) = 0,  (1) = 0, unknown initial conditions can be obtained by Newton’s method. 4. Results and Discussion Steady two-dimensional nanofluid hydrothermal treatment between two parallel vertical permeable plates is studied considering induced magnetic field. The Runge-Kutta integration scheme is utilized to solve this problem. MAPLE code has been validated by comparison with a previously published paper [28]. Table 3 indicates good accuracy of present code. The influences of important parameters such as magnetic Prandtl number (Pm), Hartmann number (Ha), suction parameter (V ) and nanoparticle volume fraction () on flow style are examined. Table 3. Comparison of skin friction tension over the upper wall between the present results and previous work. H , Hartmann number; V , suction parameter; Pr, Prandtl number. a 0 V Pr Pr Ha Sarveshanand and Singh [21] Present Work 0 m 0.5 0.7 0.5 5 0.016 0.015 0.75 0.7 0.5 5 0.011 0.011 1 0.7 1 5 0.018 0.018 1 0.015 0.5 0.5 2.695 2.700 Impact of  on induced magnetic field, current density, temperature and velocity distributions is shown in Figure 2. As volume fraction of nanofluid augments, nanofluid velocity and temperature are enhanced due to an increase in fluid motion by adding nanoparticles. Induced current density increases with an augment of  while the opposite behavior is shown for induced magnetic field. Influence of suction parameter on hydrothermal behavior is depicted in Figure 3. Velocity, temperature and induced current density decreases, with an augment of suction parameter while induced magnetic field enhances with rise of V . Therefore, this parameter can be considered as control parameter for engineering designs. Appl. Sci. 2016, 6, 324 6 of 13 Appl. Sci. 2016, 6, 324   6 of 13  0.002 Appl. Sci. 2016, 6, 324   6 of 13    0.002    -0.001 0.0015         -0.001 0.0015    -0.002 0.001 -0.002 -0.003 0.001 -0.003 0.0005 -0.004 0.0005 -0.004 -0.005 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.005 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a)  (b) 0.16 0.015 (a)  (b) 0.16 0.015       0.12     0.12 0.01   0.01 0.08 0.08 0.005 0.04 0.005 0.04 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (c)  (d) (c)  (d) Figure 2. (a)Effect of nanoparticle volume fraction on velocity, (b) induced magnetic field; (c) induced  Figure 2. Effect of nanoparticle volume fraction on velocity (U); induced magnetic field (B); induced Figure 2. (a)Effect of nanoparticle volume fraction on velocity, (b) induced magnetic field; (c) induced  current  density;  (d)  temperature  field  distributions  when  VH  1, a 5,Pm 1, Pr 6.8 .   curr current ent density   densit (Jy );and   (d)te  mperatur temperature e field   field ( ) distribu distributions tions  when when VVH =1, 1, Haa =55, ,PPmm = 1, P 1, r Pr 6.= 8 6.8 .   . Magnetic Prandtl number: Pm; nanoparticle volume fraction:   .  Magnetic Prandtl number: Pm; nanoparticle volume fraction: . Magnetic Prandtl number: Pm; nanoparticle volume fraction:   .  0.02 0.03 0.02 0.03 V= 0.2 V= 0.2 0 0 V= 0.2 V= 0.2 0 0 V= 0.4 V= 0.4 V= 0.4 0.015 V= 00.4 0 0 0.015 V= 0.6 V= 0.6 V= 0.6 V= 0.6 0 0 0.015 0 0 0.015 V= 1.0 V= 1.0 0V= 1.0 V= 1.0 0 0 0 0.01 0.01 -0.015 -0.015 -0.03 -0.03 0.005 0.005 -0.045 -0.045 -0.06 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.06 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Y Y Y Y (a)  (b) (a)  (b) Figure 3. Cont. U J B B B Appl. Sci. 2016, 6, 324 7 of 13 Appl. Sci. 2016, 6, 324   7 of 13  Appl. Sci. 2016, 6, 324   7 of 13  0.1 0.6 0.6 0.1 V= 0.2 V= 0.2 0 V= 0.2 V= 0.2 0 0 V= 0.4 V= 0.4 0 0.5 V= 00.4 V= 0.4 0 0 0.5 0.08 0.08 V= 0.6 V= 0.6 V= 0.6 V= 0.6 0 0 0 V= 1.0 V= V= 1.0 1.0 V= 1.0 0 0 0 0 0.0.44 0.06 0.06 0.3 0.3 0.04 0.04 0.2 0.2 0.02 0.1 0.02 0.1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 Y 0.6 0.8 1 Y (c)  (d) (c)  (d) Figure 3. (a) Effect of suction parameter on velocity; (b) induced magnetic field; (c) induced current  Figure Figure 3. 3. (aEf ) Ef fect fect of of suction  suction parameter  parameter on on velocity  velocity; (U );(binduced ) induced magnetic  magnetifield c fiel(B d; );(cinduced ) induced curr  current ent   density; (d) induced temperature field distributions when Ha  5, Pm 1,  0.04, Pr 6.8 .  density (J) and induced temperature field () distributions when Ha = 5, Pm = 1,  = 0.04, Pr = 6.8. density; (d) induced temperature field distributions when Ha  5, Pm 1,  0.04, Pr 6.8 .  Figure  4  depicts  the  impacts  of  Lorentz  forces  on  induced  magnetic  field,  induced  current  density Figure an  4d  dep velocity icts  the distr  imp ibutions. acts  of As  Lorent  Lorentz z  fo force rces s  augm on  indu ents,ced  the  ma back gn flow etic  fi appea eld, rindu s andced  in  turn curre  nt  Figure 4 depicts the impacts of Lorentz forces on induced magnetic field, induced current density velocity of nanofluid decreases. In addition, it can be seen that the maximum velocity point shifts to  density and velocity  and velocity distributions.  distributions. As Lorentz  As for Lorentz ces augments,  forces augm the back ents, flow  the appears back flow and appea in turn rs velocity and in turn of   the  hot  wall.  Induced  magnetic  field  decreases  with  rise  of  magnetic  field  strength  but  induced  velocity nanofluid  of nanofluid decreases. decre In addition, ases. In it addit can beion, seen it ca that n be the seen maximum  that the vel ma ocity ximum point velocity shifts to point the hot shi wall. fts to  current density is enhanced with the rise of Lorentz forces. Influence of  Pm   on induced magnetic  Induced magnetic field decreases with rise of magnetic field strength but induced current density is the  hot  wall.  Induced  magnetic  field  decreases  with  rise  of  magnetic  field  strength  but  induced  field, induced current density and velocity profiles is depicted in Figure 5. Without the magnetic  enhanced with the rise of Lorentz forces. Influence of Pm on induced magnetic field, induced current current density is enhanced with the rise of Lorentz forces. Influence of  Pm   on induced magnetic  field, the shape of the velocity profiles is parabolic but in the existence of the magnetic field its shape  density and velocity profiles is depicted in Figure 5. Without the magnetic field, the shape of the field, induced current density and velocity profiles is depicted in Figure 5. Without the magnetic  changes  to  being  flattened.  The  nanofluid  motion  and  induced  magnetic  field  reduces  with  an  velocity profiles is parabolic but in the existence of the magnetic field its shape changes to being field, the shape of the velocity profiles is parabolic but in the existence of the magnetic field its shape  augment of  Pm . Induced current density rises with augment of Pm.  flattened. The nanofluid motion and induced magnetic field reduces with an augment of Pm. Induced changes  to  being  flattened.  The  nanofluid  motion  and  induced  magnetic  field  reduces  with  an  current density rises with augment of Pm. augment of 0.00 4 Pm . Induced current density rises with aug 0.001ment of Pm.  Ha = 1 Ha = 1 0.004 Ha = 5 0.001 Ha = 5 Ha = 10 Ha = 10 0.003 Ha = 1 Ha = 1 Ha = 20 Ha = 20 Ha = 5 Ha = 5 -0.001 Ha = 10 Ha = 10 0.003 Ha = 20 Ha = 20 0.002 -0.002 -0.001 0.002 -0.003 -0.002 0.001 -0.004 -0.003 0.001 -0.005 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.004 (a)  (b) -0.005 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a)  (b) Figure 4. Cont.  Appl. Sci. 2016, 6, 324 8 of 13 Appl. Sci. 2016, 6, 324   8 of 13  0.012 Appl. Sci. 2016, 6, 324   8 of 13  Ha = 1 Ha = 5 0.01 0.012 Ha = 10 Ha = 1 Ha = 20 Ha = 5 0.008 0.01 Ha = 10 Ha = 20 0.008 0.006 0.006 0.004 0.004 0.002 0.002 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (c) Y (c) Figure 4. (a) Effect of Hartmann number (Ha) on velocity; (b), induced magnetic field; (c)induced  Figure 4. Effect of Hartmann number (Ha) on velocity (U); induced magnetic field (B) and induced Figure 4. (a) Effect of Hartmann number (Ha) on velocity; (b), induced magnetic field; (c)induced  current density distributions whenVP  1, m 1,  0.04, Pr 6.8 .  current density (J) distributions when V = 1, Pm = 1,  = 0.04, Pr = 6.8. current density distributions whenVP  1, m 1,  0.04, Pr 6.8 .  0.002 0.005 0.002 0.005 Pm = 0.05 Pm = 0.05 Pm = 0.05 Pm = 0.1 Pm = 0.05 Pm = 0.1 Pm = 0.1 Pm = 0.5 Pm = 0.1 Pm = 0.5 0.004 0.004 Pm = 0.5 Pm = 1 Pm = 0.5 Pm = 1 Pm = 1 Pm = 1 0.003 0.003 -0.002 -0.002 0.002 0.002 -0.004 -0.004 0.001 0.001 -0.006 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.006 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a)  (b) 0.012 (a)  (b) Pm = 0.05 0.012 Pm = 0.1 Pm = 0.05 0.01 Pm = 0.5 Pm = 0.1 Pm = 1 0.01 Pm = 0.5 0.008 Pm = 1 0.008 0.006 0.006 0.004 0.004 0.002 0.002 0 0.2 0.4 0.6 0.8 1 (c) 0 0.2 0.4 0.6 0.8 1 Figure  5.  (a)  Effect  of  magnetic  Prandtl  number  (Pm)  on  velocity;  (b)  induced  magnetic  field;   Figure 5. Effect of magnetic Prandtl number (Pm) on velocity (U); induced magnetic field (B) (c) induced current density distributions when VH (c) 1, a 5,  0.04, Pr 6.8 .  and induced current density (J) distributions when V = 1, Ha = 5,  = 0.04, Pr = 6.8. Figure  5.  (a)  Effect  of  magnetic  Prandtl  number  (Pm)  on  velocity;  (b)  induced  magnetic  field;   Influences  of  magnetic  Prandtl  number  Pm ,  Hartmann  number  Ha ,  suction  parameter      (c) induced current density distributions when VH  1, a 5,  0.04, Pr 6.8 .  V   and nanoparticle volume fraction     on skin friction coefficient are depicted in Figures 6 and      7. According to these data, a correlation is presented for skin friction coefficient as follows:  Influences  of  magnetic  Prandtl  number  Pm ,  Hartmann  number  Ha ,  suction  parameter       V   and nanoparticle volume fraction    on skin friction coefficient are depicted in Figures 6 and  7. According to these data, a correlation is presented for skin friction coefficient as follows:  B Appl. Sci. 2016, 6, 324 9 of 13 Appl. Sci. 2016, 6, 324   9 of 13  Influences of magnetic Prandtl number Pm , Hartmann number Ha , suction parameter V ( ) ( ) ( ) and nanoparticle volume fraction () on skin friction coefficient are depicted in Figures 6 and 7. CV  0.25335 0.34173 0.011146Ha 0.12279 f 0 According to these data, a correlation is presented for skin friction coefficient as follows:  0.0076966VHa 0.079691V Pm 2.54355V 00 0 (19)  C = 0.25335 0.34173V 0.011146Ha 0.12279 f 0  0.00154943Ha Pm 0.0424 Ha1.15754 Pm +0.0076966V22 Ha + 0.079691V Pm 2.543552 V  2 0 0 0 0.11737VH 0.0001179a 0.033354Pm 13.69282 (19) +0.00154943Ha Pm 0.0424Ha  1.15754Pm 2 2 2 2 +0.11737V + 0.0001179Ha + 0.033354Pm + 13.69282 0.05 0.05    0.04  0.04   0.03 0.03 0.02 0.02 0.01 0.01 1 5 10 15 20 0.05 0.2 0.4 0.6 0.8 1 Ha Pm (a)  (b) 0.2   0.15  0.1 0.05 0.2 0.4 0.6 0.8 1 (c) Figure  6.  Influences  of  magnetic  Prandtl  number  Pm ,  Hartmann  number  Ha ,  suction      Figure 6. Influences of magnetic Prandtl number Pm , Hartmann number Ha , suction parameter ( ) ( ) (V ) and nanoparticle volume fraction () on skin friction coefficient C when Pr = 6.8. parameter  V   and  nanoparticle  volume  fraction     on  skin  friction  coefficient  C   when 0     f   0 f (a) V = 1, Pm = 1; (b) V = 1, Ha = 5; (c) Ha = 5, Pm = 1. 0 0 Pr  6.8 . (a) VP 1, m 1; (b) VH 1, a 5 ; (c) Ha  5, Pm 1 .  0 0 f Appl. Sci. 2016, 6, 324 10 of 13 Appl. Sci. 2016, 6, 324   10 of 13  (a)  (b) (c)  (d) (e)  (f) Figure  7.  3D  surface  plots  for  skin  friction  coefficient.  (a)  Pm 0.55,  0.02 ;  (b)  Figure 7. 3D surface plots for skin friction coefficient. (a) Pm = 0.55,  = 0.02 ; (b) Ha = 12.5,  = 0.02; Ha 12.5,  0.02 ;  (c)  Ha 12.5, Pm 0.55 ;  (d)  V 0.6,  0.02 ;  (e) VP 0.6,m 0.55 ;   0 0 (c) Ha = 12.5, Pm = 0.55; (d) V = 0.6,  = 0.02; (e) V = 0.6, Pm = 0.55; (f) V = 0.6, Ha = 12.5. 0 0 0 (f) VH 0.6, a 12.5 .  It can be concluded that C has reverse relationship with all active parameters except for . It can be concluded that  C   has reverse relationship with all active parameters except for   .  Figure 8 shows the influence of V and  on Nusselt number. In addition, a good correlation has been Figure 8 shows the influence of  V   and    on Nusselt number. In addition, a good correlation has  presented for the Nusselt number as follows: been presented for the Nusselt number as follows:  Nu = 0.80342 + 4.92333V + 1.72177 3.50195V 0 0 Nu 0.80342 4.92333V 1.72177 3.50195V (20) 2 2 (20)  +1.822V 0.75963  1.822V 0.75963 As suction parameter (V ) and nanoparticle volume fraction () increase, temperature gradient As suction parameter  V   and nanoparticle volume fraction     increase, temperature gradient    increases. Therefore, Nu is enhanced with enhancement of V , . increases. Therefore, Nu is enhanced with enhancement of  V , .  Appl. Sci. 2016, 6, 324 11 of 13 Appl. Sci. 2016, 6, 324   11 of 13     2.5   1.5 0.05 0.1 0.2 0.3 0.4 (a) (b) Figure 8. Influence of nanofluid volume fraction ( ) and suction parameter (V ) on Nusselt number  Figure 8. Influence of nanofluid volume fraction () and suction parameter (V ) on Nusselt number (Nu) when  Ha 5, Pm 1, Pr 6.8 .  (Nu) when Ha = 5, Pm = 1, Pr = 6.8. 5. Conclusions  5. Conclusions The influence of induced magnetic field on nanofluid motion and forced convection between  The influence of induced magnetic field on nanofluid motion and forced convection between two vertical permeable plates is investigated. To solve coupled equations, Runge‐Kutta method is  two vertical permeable plates is investigated. To solve coupled equations, Runge-Kutta method is utilized. The influence of different dimensionless parameters on induced magnetic field, velocity and  utilized. The influence of different dimensionless parameters on induced magnetic field, velocity and temperature distributions are considered. Results illustrate that current density augments with a rise  temperature distributions are considered. Results illustrate that current density augments with a rise of volume fraction of nanofluid and Hartmann and magnetic Prandtl numbers, while it is reduced  of volume fraction of nanofluid and Hartmann and magnetic Prandtl numbers, while it is reduced with a rise in the suction parameter. As Lorentz force increases, velocity and induced magnetic field  with a rise in the suction parameter. As Lorentz force increases, velocity and induced magnetic field are reduced and maximum velocity point shifts to the left side.  are reduced and maximum velocity point shifts to the left side. Acknowledgments:   R. Ellahi is grateful to Prof. Sultan Z Alamri, Dean Faculty of Science and Prof. Yousef  Acknowledgments: R. Ellahi is grateful to Sultan Z Alamri, Dean Faculty of Science and Yousef Alharbi, Chairman Mathematics Department, Taibah University, Madinah Munawwarah, Saudi Arabia for their kind cooperation. Alharbi, Chairman Mathematics Department, Taibah University, Madinah Munawwarah, Saudi Arabia for their  R. Ellahi is also thankful to PCST to honed him with 7th top most Productive Scientist Award in category A and kind cooperation. R. Ellahi is also thankful to PCST to honed him with 7th top most Productive Scientist Award  Thomson Reuters to rank him among top 1% highly cited researchers on Web of Science in 2015–2016. in category A and Thomson Reuters to rank him among top 1% highly cited researchers on Web of Science in  Author Contributions: This paper is contributed in all respect by M. Sheikholeslami, Q. M. Zaigham Zia and 2015–2016.  R. Ellahi equally. Author Contributions:   This paper is contributed in all respect by Sheikholeslmai, Zaigham and Ellahi equally.  Conflicts of Interest:   The authors declare no conflict of interest  Nu Appl. Sci. 2016, 6, 324 12 of 13 Conflicts of Interest: The authors declare no conflict of interest. Nomenclature B Dimensionless induced horizontal magnetic field v Velocity vector Vector of magnetic field k Thermal conductivity c Specific heat J Induced current density T Temperature Ha Hartmann number Pr Prandtl number V Suction parameter Pm Magnetic Prandtl number U Dimensionless horizontal velocity Greek Symbols Dimensionless distance Coefficient of thermal expansion Electrical conductivity Dynamic viscosity of nanofluid Dimensionless temperature Nanofluid volume fraction Density Subscripts p Solid f Base fluid References 1. Sheremet, M.A.; Oztop, H.F.; Pop, I. MHD natural convection in an inclined wavy cavity with corner heater filled with a nanofluid. J. Magn. Magn. Mater. 2016, 416, 37–47. [CrossRef] 2. Sheikholeslami, M.; Ellahi, R. Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid. Int. J. Heat Mass Transf. 2015, 89, 799–808. [CrossRef] 3. Ismael, M.A.; Mansour, M.A.; Chamkha, A.J.; Rashad, A.M. 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Applied SciencesMultidisciplinary Digital Publishing Institute

Published: Nov 2, 2016

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