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Hydromagnetic nonlinear thermally radiative nanoliquid flow with Newtonian heat and mass conditions
Accepted Manuscript Hydromagnetic nonlinear thermally radiative nanoliquid flow with Newtonian heat and mass conditions Muhammad Ijaz Khan, Ahmed Alsaedi, Sabir Ali Shehzad, Tasawar Hayat PII: DOI: Reference:
S2211-3797(17)30830-6 http://dx.doi.org/10.1016/j.rinp.2017.06.035 RINP 751
To appear in:
Results in Physics
Received Date: Accepted Date:
15 May 2017 22 June 2017
Please cite this article as: Khan, M.I., Alsaedi, A., Shehzad, S.A., Hayat, T., Hydromagnetic nonlinear thermally radiative nanoliquid flow with Newtonian heat and mass conditions, Results in Physics (2017), doi: http://dx.doi.org/ 10.1016/j.rinp.2017.06.035
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Hydromagnetic nonlinear thermally radiative nanoliquid flow with Newtonian heat and mass conditions Muhammad Ijaz Khand>1 > Ahmed Alsaedie , Sabir Ali Shehzadf and Tasawar Hayatd>e d
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan e
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 f
Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal 57000, Pakistan
Abstract: This paper communicates the analysis of MHD three-dimensional flow of Je rey nanoliquid over a stretchable surface. Flow due to a bidirectional surface is considered. Heat and mass transfer subject to volume fraction of nanoparticles, heat generation and nonlinear solar radiation are examined. Newtonian heat and mass transportation conditions are employed at surface. Concept of boundary layer is utilized to developed the mathematical problem. The boundary value problem is dictated by ten physical parameters: Deborah number, Hartman number, ratio of stretching rates, thermophoretic parameter, Brownian motion parameter, Prandtl number, temperature ratio parameter, conjugate heat and mass parameters and Lewis number. convergent solutions are obtained using homotopic procedure. Convergence zone for obtained results is explicitly identified. The obtained solutions are interpreted physically.
Key-words: Hydromagnetic flow; Viscoelastic nanofluid; Thermophoretic and Brownian 1
Corresponding author. email address: [email protected] (Ijaz Khan)
1
moment; Nonlinear thermal radiation; Heat generation.
Fig. 1: Geometry of flow problem.
1
Introduction
Nanoliquid is a suspension of nanometer-sized particles in a base liquid i.e., in water, oil and ethanol fuel. The heat transfer feature of nanoliquid depend on the thermophysical properties of nanoparticles, nanoparticle volume fraction and thermophysical characteristics of base fluid. Since nanoparticles improve the transport properties and increases the heat transfer characteristics of nanofluid. Boundary layer analysis in the presence of MHD flow of nanofluid has wide range of applications in industrial and engineering problems, for example in hot rolling, polymer processing and metal extrusion, geothermal energy extraction, fiber glass, oil exploration and continuous stretching of plastic films. Boundary layer radiative heat transportation is requisite part in producing high quality of the final commodity, which based on heat transfer rate between the stretched surface and nearby liquid particles. Nanoliquid flow involving porous medium carry great potential in increasing thermal conductivity of base fluid. Hsiao [1] studied the energy conversion problem of viscous fluid past over a nonlinear stretching sheet in presence of convective heat transfer and thermal radiation. Jang and Choi [2] numerically anayzed a nanoliquid cooled MCHS and experimentally prove that 2
water-diamond nanoliquid rise the performance by 10% in comparison to MCHS with distilled water. Impact of chemical reaction and non-uniform heat sink and source on stagnation flow of viscoelastic nanoliquid towards stretched of surface is studied by Sandeep et al. [3]. Hayat et al. [4] worked on MHD thixotropic liquid with suspension of nanomaterial embedded in medium of stratification with magnetic field. Further Hayat et al. [5] discussed ferromagnetic mixed convection flow with homogeneous-heterogeneous reaction considering magnetic dipole over a stretched surface. Few more attempts on behvior of nanoliquid are refereed in works [6-15]. The study of transportation of heat and mass in non-Newtonian liquid flow by movement of surface obtained the importance in the recent years due to a large number of industrial applications like extrusion of metals and polymers, wire and fiber coating, continuous casting, foodstu processing, exchangers, drawing of plastic sheets, and chemical processing apparatus etc. Ojjela et al. [16] presented chemically reaction flow of Je rey liquid with mixed convection and periodic suction/injection between two parallel porous plates. Flow and heat transport is discussed under the influence of induced magnetic field. Consequences of thermal radiation and heat generation/absorption in flow of Je rey liquid by semi infinite plate is studied by Ga ar et al. [17]. Three-dimensional hydromagnetic flow of non-Newtonian liquid accounting solar radiation and internal heat over a stretchable sheet is analyzed by Shehzad et al. [18]. Computational analysis in MHD flow of Je rey liquid with chemical reaction and radiation by a stretched surface is reported by Narayana and Babu [19]. Magnetohydrodynamic flow of Je rey fluid due to nolinear radially stretched surface with Newtonian heating is considered by Hayat et al. [20]. Magneto-hydrodynamic boundary-layer phenomenon of electrically conducting nonNewtonian liquids over stretched surfaces has promising application in mechanical device 3
and industrial engineering, for instance, paper production, manufacture of foods, crystal growing, extraction of polymer and drawing of plastic films. Ijaz et al. [21] studied MHD flow of Casson liquid with the reactions of homogeneous and heterogeneous. Hsiao [22] discussed transportation of heat and mass in rate type liquid accounting thermal radiation and viscous dissipation. Ramesh et al. [23] examined convective three dimemsional flow of rate type liquid in presence of thermal radiation past a porous stretched surface. Khan et al. [24] scrutinized thermally stratified slip flow of viscous fluid via thermal radiation and suction/blowing. Makinde and Animasaun [25] inspected paraboloid of revolution MHD bioconvection nanoliquid flow subject to nonlinear thermal radiation past an upper surface. Kumaran and Sandeep [26] considered MHD flow of Casson and Williamson fluids with thermophoresis and Brownian moment. The theme of present attempt is to report the hydromagnetic nonlinear thermally radiative three-dimensional Je rey nanoliquid flow through Newtonian heat and mass conditions. Flow due to a bidirectional surface is considered. Heat and mass transfer characteristics are examined accounting volume fraction of nanoparticles and heat generation/absorption. Boundary layer concept and assumption of small magnetic Reynold number are addressed. Homotopy approach [27-37] is utilized to solve the nonlinear governing systems. Aspects of several significant variables on velocity (i 0 ()) and (j 0 ()) > temperature (()) > nanoparticles volume fraction (!()) > surface drag force and heat transfer rate are addressed. The main points are concluded.
2
Formulation
Three dimensional hydromagnetic flow of Je rey nanoliquid with nonlinear thermal radiation over a stretched surface is considered. The schematic diagram, corresponding components 4
of velocity x, y, and z and flow configuration is presented in Fig. 1. Temperature and nanoparticle concentration at the surface of sheet are
CW C}
= kv W and
CF C}
= kf F and
ambient temperature and concentration due to nanoparticles refereed to be W" and F" . Induced magnetic field is neglected due to low magnetic Reynold number. Further e ects of electric field is also neglected. Under the aforestated assumptions and boundary layer approximation resulting constitutive equations are [18]: Cx Cy Cz + + = 0> C{ C| C} 3 3
2 E Cx Cx Cx E E C x + K2 E x +y +z = C C{ C| C} 1 + K1 C C} 2
3
3
E C2y Cy Cy E Cy E E +y +z = + K x 2 C C{ C| C} 1 + K1 C C} 2
C2x
Cx C} C{C}
+
C3x +x C{C} 2 Cx C 2 y C} C{C}
C2x
Cy C} C|C}
+
+
C3y +x C{C} 2
(1)
C3x y C|C} 2
Cy C 2 y C} C|C}
+
+
Cz C} C} 2
+
+
C3y y C|C} 2
C2x
3 z CC}x3
Cz C 2 y C} C} 2
+
3 z CC}y3
44
(2)
44
(3)
FF h E02 FF x> DD
FF h E02 FF y= DD
The above expressions have x> y and z which are components of velocity in the directions of {> | and }> h electrical conductivity, = (@ ) kinematic viscosity, E0 strength of magnetic field, density of fluid, dynamic viscosity, K2 retardation time and K1 ratio of relaxation and retardation times. The energy and mass species equations after the boundary layer approximation are à µ ¶2 ! 2 CW C W 1 CF Ct CW CW CW CW G W u x +y +z = W 2 + GE + +T(W W" )> (4) C{ C| C} C} ( fs )i C| C} C} W" C} x
CF CF CF C 2 F GW C 2 W +y +z = GE 2 + > C{ C| C} C} W" C} 2
i fluid density, W thermal di usivity, =
( f)s ( f)i
(5)
ratio of nanoparticle heat capacity and base
fluid heat capacity, T heat source/sink, tu radiative heat flux, GW thermophoretic di usion coe!cient and GE Brownian di usion coe!cient. The radiative heat flux (tu ) is tu =
4 W CW 4 16 W 3 CW = W > 3" C} 3" C} 5
(6)
where W and " depict to constant of Stefan-Boltzman and coe!cient of mean absorption. Then Eq. (4) takes the form CW CW C CW +z = x +y C{ C| C} C}
ÃÃ
16 W W3 W + 3" ( fs )i
!
CW C}
!
Ã
CF CW GW + GE C} C} W"
+
µ
CW C}
¶2 !
= (7)
The conditions for the present flow phenomenon are [18] x = xz ({) = d{> y = yz (|) = e|> z = 0>
CF CW = kv W> = kF F at } = 0> C} C}
x $ 0> y $ 0> W $ W" > F $ F" as } $ 4>
(8) (9)
Applying s x = d{i 0 ()> y = d|j0 ()> z = d(i() + j())> r d F F" W W" > > !() = > =} () = W" F"
(10) (11)
we get the following system of equations i 000 + (1 + K1 )((i + j)i 00 i 02 ) + 1 (i 002 (i + j)i 0000 j 0 i 000 ) (1 + K1 )Pj i 0 = 0>
(12)
j 000 + (1 + K1 )((i + j)j00 j02 ) + 1 (j 002 (i + j)j 0000 i 0 j000 ) (1 + K1 )Pj j 0 = 0>
(13)
¢0 1 ¡ 1 + Ug (1 + (z 1))3 0 + (i + j)0 + QE 0 !0 + QW 02 + Kj = 0> Pr !00 + S uOh (i + j)!0 + (QW @QE ) 00 = 0>
i = 0> j = 0> i 0 = 1> j0 = D> 0 = 1 (1 + (0))> !0 = 2 (1 + !(0)) at = 0> i 0 $ 0> j 0 $ 0> $ 0> ! $ 0 as $ 4> where 1 = K2 d as the Deborah number> Pj = rameter), =
e d
ratio of stretching rates, QW =
generation parameter, Oh =
W > GE
q
W E02 d
GW W"
(15) (16) (17)
Hartman number (magnetic pa-
thermophoretic parameter, Kj heat
Lewis number, Qe =
6
(14)
GE
Brownian motion parameter,
Su =
W
Prandtl number and 1 = kv
p
d
> 2 = kF
p
d
conjugate heat and mass transfer
parameters. The results of viscous fluid are obtained when 1 = 0= Dimensionless expression of local Nusselt (Qx{ ) and Sherwood (Vk{ ) numbers are Qx{ Uh31@2 {
µ ¶ 3 ¡ 3¢ = 1 + Ug 1 + (z 1) ( (0)) 0 (0)> Uh31@2 Vk{ = !(0)> { 4
(18)
in which Uh{ = xz ({){@ is the local Reynolds number.
3
Homotopic solutions
It is well known that Liao [27] developed homotopy analysis procedure. This procedure is especially useful for the solutions development of nonlinear expressions. Here the initial guesses in such procedure are i0 () = 1 exp()> j0 () = (1 exp()) > 0 () =
exp() 1 exp() > !0 () = 2 > 1 + 1 1 + 2
(19)
Li = i 000 i 0 > Lj = j 000 j 0 > L = 00 > L! = !00 !>
(20)
with Li (F1W + F2W h + F3W h3 ) = 0> Lj (F4W + F5W h + F6W h3 ) = 0> W 3 h ) = 0= L (F7W h + F8W h3 ) = 0> L! (F9W h + F10
Here FlW (l = 1 10) are the arbitrary constants. 7
(21)
General solutions can be written as W ip () = ip () + F1W + F2W h + F3W h3 >
(22)
W jp () = jp () + F4W + F5W h + F6W h3 >
(23)
p () = Wp () + F7W h + F8W h3 >
(24)
W 3 !p () = !Wp () + F9W h + F10 h >
(25)
W W ()> jp ()> Wp () and !Wp () represents the special function. where ip
3.1
Convergence study
The method of homotopy analysis is adopted to report the solutions of governing nonlinear expressions. This technique involves the convergence parameters which are known as auxiliary parameters. The proper and suitable choice of such parameters is very necessary for the development of convergent solutions. To select the appropriate region of convergence, the curves are plotted for velocity, temperature and nanoparticles volume friction in Fig. 2. Fig. 2 depict that the range of adjusted values of ~i > ~j > ~ and ~! are 0=95 ~i 0=15> 0=95 ~j 0=15> 0=76 ~ 0=13 and 0=70 ~! 0=20= In addition, Table 1 gives the witness of convergent solutions of momentum, energy and nanoparticle volume friction. This Table also elucidates the required number of deformations for the convergent solutions. It is evaluated that computations upto 20th order give the convergent solutions of momentum, energy and nanoparticle volume friction.
8
I
I
+ /J
+ / T + /I +/
T
J
I
³ f ³ g³T ³I
Fig. 2: Combined ~curves for i 00 > j00 > 0 and !0 = Table 1: Convergence of HAM solutions for di erent order of approximation when 1 = 1 = 2 = 0=3> P = = 0=5> 1 = Ug = 0=4> Qw = Qe = 0=2> Pr = 1=2> Oh = 1=0 and z = 0=1= Order of approximation i 00 (0) j00 (0) 0 (0) !0 (0)
4
1
1.2567
0.6225
0.3604
0.2047
5
1.2578
0.6616
0.3783
0.1700
8
1.2574
0.6624
0.3818
0.1652
9
1.2576
0.6622
0.3817
0.1652
10
1.2576
0.6623
0.3814
0.1654
18
1.2575
0.6624
0.3806
0.1661
20
1.2575
0.6624
0.3806
0.1661
25
1.2575
0.6624
0.3806
0.1661
28
1.2575
0.6624
0.3806
0.1661
30
1.2575
0.6624
0.3806
0.1661
Discussion
The current study explores the salient features of Deborah number ( 1 )> Hartman number (Pj ), Brownian motion parameter (Qe )> ratio of stretching rates (D)> thermophoretic para9
meter (QW ), temperature ratio parameter (z ) Prandtl number (Pr), Lewis number (Oh ) and conjugate heat and mass transfer parameters ( 1 , 2 )= Mostly we investigate their combined e ects on the velocities (i 0 ()), (j0 ()), temperature ( ()) and nanoparticles volume fraction (! ()) = All the graphical results are accomplished utilizing homotopy analysis method in mathematica. Variation of velocity field (i 0 ()) and (j 0 ()) against for di erent estimation of ( 1 ) is shown in Figs. 3 and 4. Figs. 3 and 4 depict that when ( 1 ) increases then the velocity of fluid enhances. Physically for higher estimation of ( 1 ) the elasticity of material increases and therefore the velocity of fluid tends to enhance. It can be noticed from Fig. 5 and 6 that when (Pj ) increases, then the velocity of fluid decays. It is due to the fact that when magnetic field is applied to liquid then there originated an opposite force, called Lorentz force which tends to reduce the velocity. To investigate the influence of (S u) on ( ()) the Fig. 7 is displayed. This Fig. confirms that larger (S u) decay the temperature field. Physically larger (Pr) has smaller thermal di usivity which causes the reduction in ( ()) = It can be noticed from Fig. 8 that higher estimation of (Ug ) causes a significant enhancement in ( ()) and thermal layer thickness. Physically the fluid particles gain more heat from the surface when (Ug ) enhances and thus ( ()) decays. Influence of temperature ratio parameter (z ) on thermal field is plotted in Fig. 9. It is noted that ( ()) and its associated thermal layer are increased for higher estimation of (z ) = To further investigate the influence of Lewis number (Oh ), Brownian motion (Qe ) and thermophoretic parameter (QW ) on nanoparticle concentration (! ()) > the Figs. 10 12 are shown= It can be noticed from Fig. 10 that for higher estimation of (Oh ) there is a significant reduction in the nanoparticles concentration (! ()) = Clearly lower (Oh ) gives stronger mole10
cular di usivity. Therefore thinner layer thickness is observed in Fig. 10. Figs. 11 and 12 displayed the variation of nanoparticle concentration profile (! ()) against di erent values of Brownian moment (Qe ) and thermophoretic parameter (QW )= Nanoparticles concentration (! ()) gradually decays for larger values of (Qe ) (see Fig. 11)= Further nanoparticles concentration (! ()) enhances for larger estimation of (QW )= Fig. 13 displays the heat transfer rate variation for di erent values of (S u) and (Ug ). It is noticed that an increase in (Ug ) 31
leads to decrease in Nusselt number. Fig. 14 depicts the e ect of (Qe ) and ( 2 ) on Re{ 2 Vk{ . Sherwood number is an increasing function of (Qe )=
E E
E E
E E
E E
K
Fig. 3: 1 variation for i 0 ()=
K
M g M g
Mg
Mg Mg Mg Mg
J +K /
Mg
Fig. 4: 1 variation for j 0 ()=
I +K /
J +K /
I +K /
K
Fig. 5: Pj variation for i 0 ()=
K
Fig. 6: Pj variation for j 0 ()=
11
5G 5G
3U
5G
3U
5G
T+K/
T+K/
3U 3U
K
Fig. 7: Pr variation for ()=
K
T w Tw
/H /H
Tw
/H
Tw
/H
K
Fig. 9: z variation for ()=
K
Fig. 10: Oh variation for !()=
1E 1E 1E
1E
I +K/
I +K/
Fig. 8:Rd variation for ()=
I +K/
T+K/
1W 1W
1W
1W
K
Fig. 11: Qe variation for !()=
K
Fig. 12: QW variation for !()=
12
Rd Rd Rd
Nb Nb
Nb Nb
5H x 6K
5H x 1Xx
Rd
3U
J
31
Fig. 13. E ects of Pr and Rg on Re{ 2 Qx{ =
5
31
Fig. 14. E ects of Ne and 2 on Re{ 2 Vk=
conclusions
Nonlinear thermal radiation and heat generation/absorption in flow of Je rey nanofluid are examined. The following results are worthmentioning. • Momentum layer thickness is increasing function of ( 1 ). • (Pj ) decelerates the momentum boundary layer. • Thermal boundary layer reduces with increasing (S u). • Heat transfer decays for larger estimation of (Ug )= • Mass transfer boots for higher (Qe ).
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16
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17
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18
S2211-3797(17)30830-6 http://dx.doi.org/10.1016/j.rinp.2017.06.035 RINP 751
To appear in:
Results in Physics
Received Date: Accepted Date:
15 May 2017 22 June 2017
Please cite this article as: Khan, M.I., Alsaedi, A., Shehzad, S.A., Hayat, T., Hydromagnetic nonlinear thermally radiative nanoliquid flow with Newtonian heat and mass conditions, Results in Physics (2017), doi: http://dx.doi.org/ 10.1016/j.rinp.2017.06.035
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Hydromagnetic nonlinear thermally radiative nanoliquid flow with Newtonian heat and mass conditions Muhammad Ijaz Khand>1 > Ahmed Alsaedie , Sabir Ali Shehzadf and Tasawar Hayatd>e d
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan e
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 f
Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal 57000, Pakistan
Abstract: This paper communicates the analysis of MHD three-dimensional flow of Je rey nanoliquid over a stretchable surface. Flow due to a bidirectional surface is considered. Heat and mass transfer subject to volume fraction of nanoparticles, heat generation and nonlinear solar radiation are examined. Newtonian heat and mass transportation conditions are employed at surface. Concept of boundary layer is utilized to developed the mathematical problem. The boundary value problem is dictated by ten physical parameters: Deborah number, Hartman number, ratio of stretching rates, thermophoretic parameter, Brownian motion parameter, Prandtl number, temperature ratio parameter, conjugate heat and mass parameters and Lewis number. convergent solutions are obtained using homotopic procedure. Convergence zone for obtained results is explicitly identified. The obtained solutions are interpreted physically.
Key-words: Hydromagnetic flow; Viscoelastic nanofluid; Thermophoretic and Brownian 1
Corresponding author. email address: [email protected] (Ijaz Khan)
1
moment; Nonlinear thermal radiation; Heat generation.
Fig. 1: Geometry of flow problem.
1
Introduction
Nanoliquid is a suspension of nanometer-sized particles in a base liquid i.e., in water, oil and ethanol fuel. The heat transfer feature of nanoliquid depend on the thermophysical properties of nanoparticles, nanoparticle volume fraction and thermophysical characteristics of base fluid. Since nanoparticles improve the transport properties and increases the heat transfer characteristics of nanofluid. Boundary layer analysis in the presence of MHD flow of nanofluid has wide range of applications in industrial and engineering problems, for example in hot rolling, polymer processing and metal extrusion, geothermal energy extraction, fiber glass, oil exploration and continuous stretching of plastic films. Boundary layer radiative heat transportation is requisite part in producing high quality of the final commodity, which based on heat transfer rate between the stretched surface and nearby liquid particles. Nanoliquid flow involving porous medium carry great potential in increasing thermal conductivity of base fluid. Hsiao [1] studied the energy conversion problem of viscous fluid past over a nonlinear stretching sheet in presence of convective heat transfer and thermal radiation. Jang and Choi [2] numerically anayzed a nanoliquid cooled MCHS and experimentally prove that 2
water-diamond nanoliquid rise the performance by 10% in comparison to MCHS with distilled water. Impact of chemical reaction and non-uniform heat sink and source on stagnation flow of viscoelastic nanoliquid towards stretched of surface is studied by Sandeep et al. [3]. Hayat et al. [4] worked on MHD thixotropic liquid with suspension of nanomaterial embedded in medium of stratification with magnetic field. Further Hayat et al. [5] discussed ferromagnetic mixed convection flow with homogeneous-heterogeneous reaction considering magnetic dipole over a stretched surface. Few more attempts on behvior of nanoliquid are refereed in works [6-15]. The study of transportation of heat and mass in non-Newtonian liquid flow by movement of surface obtained the importance in the recent years due to a large number of industrial applications like extrusion of metals and polymers, wire and fiber coating, continuous casting, foodstu processing, exchangers, drawing of plastic sheets, and chemical processing apparatus etc. Ojjela et al. [16] presented chemically reaction flow of Je rey liquid with mixed convection and periodic suction/injection between two parallel porous plates. Flow and heat transport is discussed under the influence of induced magnetic field. Consequences of thermal radiation and heat generation/absorption in flow of Je rey liquid by semi infinite plate is studied by Ga ar et al. [17]. Three-dimensional hydromagnetic flow of non-Newtonian liquid accounting solar radiation and internal heat over a stretchable sheet is analyzed by Shehzad et al. [18]. Computational analysis in MHD flow of Je rey liquid with chemical reaction and radiation by a stretched surface is reported by Narayana and Babu [19]. Magnetohydrodynamic flow of Je rey fluid due to nolinear radially stretched surface with Newtonian heating is considered by Hayat et al. [20]. Magneto-hydrodynamic boundary-layer phenomenon of electrically conducting nonNewtonian liquids over stretched surfaces has promising application in mechanical device 3
and industrial engineering, for instance, paper production, manufacture of foods, crystal growing, extraction of polymer and drawing of plastic films. Ijaz et al. [21] studied MHD flow of Casson liquid with the reactions of homogeneous and heterogeneous. Hsiao [22] discussed transportation of heat and mass in rate type liquid accounting thermal radiation and viscous dissipation. Ramesh et al. [23] examined convective three dimemsional flow of rate type liquid in presence of thermal radiation past a porous stretched surface. Khan et al. [24] scrutinized thermally stratified slip flow of viscous fluid via thermal radiation and suction/blowing. Makinde and Animasaun [25] inspected paraboloid of revolution MHD bioconvection nanoliquid flow subject to nonlinear thermal radiation past an upper surface. Kumaran and Sandeep [26] considered MHD flow of Casson and Williamson fluids with thermophoresis and Brownian moment. The theme of present attempt is to report the hydromagnetic nonlinear thermally radiative three-dimensional Je rey nanoliquid flow through Newtonian heat and mass conditions. Flow due to a bidirectional surface is considered. Heat and mass transfer characteristics are examined accounting volume fraction of nanoparticles and heat generation/absorption. Boundary layer concept and assumption of small magnetic Reynold number are addressed. Homotopy approach [27-37] is utilized to solve the nonlinear governing systems. Aspects of several significant variables on velocity (i 0 ()) and (j 0 ()) > temperature (()) > nanoparticles volume fraction (!()) > surface drag force and heat transfer rate are addressed. The main points are concluded.
2
Formulation
Three dimensional hydromagnetic flow of Je rey nanoliquid with nonlinear thermal radiation over a stretched surface is considered. The schematic diagram, corresponding components 4
of velocity x, y, and z and flow configuration is presented in Fig. 1. Temperature and nanoparticle concentration at the surface of sheet are
CW C}
= kv W and
CF C}
= kf F and
ambient temperature and concentration due to nanoparticles refereed to be W" and F" . Induced magnetic field is neglected due to low magnetic Reynold number. Further e ects of electric field is also neglected. Under the aforestated assumptions and boundary layer approximation resulting constitutive equations are [18]: Cx Cy Cz + + = 0> C{ C| C} 3 3
2 E Cx Cx Cx E E C x + K2 E x +y +z = C C{ C| C} 1 + K1 C C} 2
3
3
E C2y Cy Cy E Cy E E +y +z = + K x 2 C C{ C| C} 1 + K1 C C} 2
C2x
Cx C} C{C}
+
C3x +x C{C} 2 Cx C 2 y C} C{C}
C2x
Cy C} C|C}
+
+
C3y +x C{C} 2
(1)
C3x y C|C} 2
Cy C 2 y C} C|C}
+
+
Cz C} C} 2
+
+
C3y y C|C} 2
C2x
3 z CC}x3
Cz C 2 y C} C} 2
+
3 z CC}y3
44
(2)
44
(3)
FF h E02 FF x> DD
FF h E02 FF y= DD
The above expressions have x> y and z which are components of velocity in the directions of {> | and }> h electrical conductivity, = (@ ) kinematic viscosity, E0 strength of magnetic field, density of fluid, dynamic viscosity, K2 retardation time and K1 ratio of relaxation and retardation times. The energy and mass species equations after the boundary layer approximation are à µ ¶2 ! 2 CW C W 1 CF Ct CW CW CW CW G W u x +y +z = W 2 + GE + +T(W W" )> (4) C{ C| C} C} ( fs )i C| C} C} W" C} x
CF CF CF C 2 F GW C 2 W +y +z = GE 2 + > C{ C| C} C} W" C} 2
i fluid density, W thermal di usivity, =
( f)s ( f)i
(5)
ratio of nanoparticle heat capacity and base
fluid heat capacity, T heat source/sink, tu radiative heat flux, GW thermophoretic di usion coe!cient and GE Brownian di usion coe!cient. The radiative heat flux (tu ) is tu =
4 W CW 4 16 W 3 CW = W > 3" C} 3" C} 5
(6)
where W and " depict to constant of Stefan-Boltzman and coe!cient of mean absorption. Then Eq. (4) takes the form CW CW C CW +z = x +y C{ C| C} C}
ÃÃ
16 W W3 W + 3" ( fs )i
!
CW C}
!
Ã
CF CW GW + GE C} C} W"
+
µ
CW C}
¶2 !
= (7)
The conditions for the present flow phenomenon are [18] x = xz ({) = d{> y = yz (|) = e|> z = 0>
CF CW = kv W> = kF F at } = 0> C} C}
x $ 0> y $ 0> W $ W" > F $ F" as } $ 4>
(8) (9)
Applying s x = d{i 0 ()> y = d|j0 ()> z = d(i() + j())> r d F F" W W" > > !() = > =} () = W" F"
(10) (11)
we get the following system of equations i 000 + (1 + K1 )((i + j)i 00 i 02 ) + 1 (i 002 (i + j)i 0000 j 0 i 000 ) (1 + K1 )Pj i 0 = 0>
(12)
j 000 + (1 + K1 )((i + j)j00 j02 ) + 1 (j 002 (i + j)j 0000 i 0 j000 ) (1 + K1 )Pj j 0 = 0>
(13)
¢0 1 ¡ 1 + Ug (1 + (z 1))3 0 + (i + j)0 + QE 0 !0 + QW 02 + Kj = 0> Pr !00 + S uOh (i + j)!0 + (QW @QE ) 00 = 0>
i = 0> j = 0> i 0 = 1> j0 = D> 0 = 1 (1 + (0))> !0 = 2 (1 + !(0)) at = 0> i 0 $ 0> j 0 $ 0> $ 0> ! $ 0 as $ 4> where 1 = K2 d as the Deborah number> Pj = rameter), =
e d
ratio of stretching rates, QW =
generation parameter, Oh =
W > GE
q
W E02 d
GW W"
(15) (16) (17)
Hartman number (magnetic pa-
thermophoretic parameter, Kj heat
Lewis number, Qe =
6
(14)
GE
Brownian motion parameter,
Su =
W
Prandtl number and 1 = kv
p
d
> 2 = kF
p
d
conjugate heat and mass transfer
parameters. The results of viscous fluid are obtained when 1 = 0= Dimensionless expression of local Nusselt (Qx{ ) and Sherwood (Vk{ ) numbers are Qx{ Uh31@2 {
µ ¶ 3 ¡ 3¢ = 1 + Ug 1 + (z 1) ( (0)) 0 (0)> Uh31@2 Vk{ = !(0)> { 4
(18)
in which Uh{ = xz ({){@ is the local Reynolds number.
3
Homotopic solutions
It is well known that Liao [27] developed homotopy analysis procedure. This procedure is especially useful for the solutions development of nonlinear expressions. Here the initial guesses in such procedure are i0 () = 1 exp()> j0 () = (1 exp()) > 0 () =
exp() 1 exp() > !0 () = 2 > 1 + 1 1 + 2
(19)
Li = i 000 i 0 > Lj = j 000 j 0 > L = 00 > L! = !00 !>
(20)
with Li (F1W + F2W h + F3W h3 ) = 0> Lj (F4W + F5W h + F6W h3 ) = 0> W 3 h ) = 0= L (F7W h + F8W h3 ) = 0> L! (F9W h + F10
Here FlW (l = 1 10) are the arbitrary constants. 7
(21)
General solutions can be written as W ip () = ip () + F1W + F2W h + F3W h3 >
(22)
W jp () = jp () + F4W + F5W h + F6W h3 >
(23)
p () = Wp () + F7W h + F8W h3 >
(24)
W 3 !p () = !Wp () + F9W h + F10 h >
(25)
W W ()> jp ()> Wp () and !Wp () represents the special function. where ip
3.1
Convergence study
The method of homotopy analysis is adopted to report the solutions of governing nonlinear expressions. This technique involves the convergence parameters which are known as auxiliary parameters. The proper and suitable choice of such parameters is very necessary for the development of convergent solutions. To select the appropriate region of convergence, the curves are plotted for velocity, temperature and nanoparticles volume friction in Fig. 2. Fig. 2 depict that the range of adjusted values of ~i > ~j > ~ and ~! are 0=95 ~i 0=15> 0=95 ~j 0=15> 0=76 ~ 0=13 and 0=70 ~! 0=20= In addition, Table 1 gives the witness of convergent solutions of momentum, energy and nanoparticle volume friction. This Table also elucidates the required number of deformations for the convergent solutions. It is evaluated that computations upto 20th order give the convergent solutions of momentum, energy and nanoparticle volume friction.
8
I
I
+ /J
+ / T + /I +/
T
J
I
³ f ³ g³T ³I
Fig. 2: Combined ~curves for i 00 > j00 > 0 and !0 = Table 1: Convergence of HAM solutions for di erent order of approximation when 1 = 1 = 2 = 0=3> P = = 0=5> 1 = Ug = 0=4> Qw = Qe = 0=2> Pr = 1=2> Oh = 1=0 and z = 0=1= Order of approximation i 00 (0) j00 (0) 0 (0) !0 (0)
4
1
1.2567
0.6225
0.3604
0.2047
5
1.2578
0.6616
0.3783
0.1700
8
1.2574
0.6624
0.3818
0.1652
9
1.2576
0.6622
0.3817
0.1652
10
1.2576
0.6623
0.3814
0.1654
18
1.2575
0.6624
0.3806
0.1661
20
1.2575
0.6624
0.3806
0.1661
25
1.2575
0.6624
0.3806
0.1661
28
1.2575
0.6624
0.3806
0.1661
30
1.2575
0.6624
0.3806
0.1661
Discussion
The current study explores the salient features of Deborah number ( 1 )> Hartman number (Pj ), Brownian motion parameter (Qe )> ratio of stretching rates (D)> thermophoretic para9
meter (QW ), temperature ratio parameter (z ) Prandtl number (Pr), Lewis number (Oh ) and conjugate heat and mass transfer parameters ( 1 , 2 )= Mostly we investigate their combined e ects on the velocities (i 0 ()), (j0 ()), temperature ( ()) and nanoparticles volume fraction (! ()) = All the graphical results are accomplished utilizing homotopy analysis method in mathematica. Variation of velocity field (i 0 ()) and (j 0 ()) against for di erent estimation of ( 1 ) is shown in Figs. 3 and 4. Figs. 3 and 4 depict that when ( 1 ) increases then the velocity of fluid enhances. Physically for higher estimation of ( 1 ) the elasticity of material increases and therefore the velocity of fluid tends to enhance. It can be noticed from Fig. 5 and 6 that when (Pj ) increases, then the velocity of fluid decays. It is due to the fact that when magnetic field is applied to liquid then there originated an opposite force, called Lorentz force which tends to reduce the velocity. To investigate the influence of (S u) on ( ()) the Fig. 7 is displayed. This Fig. confirms that larger (S u) decay the temperature field. Physically larger (Pr) has smaller thermal di usivity which causes the reduction in ( ()) = It can be noticed from Fig. 8 that higher estimation of (Ug ) causes a significant enhancement in ( ()) and thermal layer thickness. Physically the fluid particles gain more heat from the surface when (Ug ) enhances and thus ( ()) decays. Influence of temperature ratio parameter (z ) on thermal field is plotted in Fig. 9. It is noted that ( ()) and its associated thermal layer are increased for higher estimation of (z ) = To further investigate the influence of Lewis number (Oh ), Brownian motion (Qe ) and thermophoretic parameter (QW ) on nanoparticle concentration (! ()) > the Figs. 10 12 are shown= It can be noticed from Fig. 10 that for higher estimation of (Oh ) there is a significant reduction in the nanoparticles concentration (! ()) = Clearly lower (Oh ) gives stronger mole10
cular di usivity. Therefore thinner layer thickness is observed in Fig. 10. Figs. 11 and 12 displayed the variation of nanoparticle concentration profile (! ()) against di erent values of Brownian moment (Qe ) and thermophoretic parameter (QW )= Nanoparticles concentration (! ()) gradually decays for larger values of (Qe ) (see Fig. 11)= Further nanoparticles concentration (! ()) enhances for larger estimation of (QW )= Fig. 13 displays the heat transfer rate variation for di erent values of (S u) and (Ug ). It is noticed that an increase in (Ug ) 31
leads to decrease in Nusselt number. Fig. 14 depicts the e ect of (Qe ) and ( 2 ) on Re{ 2 Vk{ . Sherwood number is an increasing function of (Qe )=
E E
E E
E E
E E
K
Fig. 3: 1 variation for i 0 ()=
K
M g M g
Mg
Mg Mg Mg Mg
J +K /
Mg
Fig. 4: 1 variation for j 0 ()=
I +K /
J +K /
I +K /
K
Fig. 5: Pj variation for i 0 ()=
K
Fig. 6: Pj variation for j 0 ()=
11
5G 5G
3U
5G
3U
5G
T+K/
T+K/
3U 3U
K
Fig. 7: Pr variation for ()=
K
T w Tw
/H /H
Tw
/H
Tw
/H
K
Fig. 9: z variation for ()=
K
Fig. 10: Oh variation for !()=
1E 1E 1E
1E
I +K/
I +K/
Fig. 8:Rd variation for ()=
I +K/
T+K/
1W 1W
1W
1W
K
Fig. 11: Qe variation for !()=
K
Fig. 12: QW variation for !()=
12
Rd Rd Rd
Nb Nb
Nb Nb
5H x 6K
5H x 1Xx
Rd
3U
J
31
Fig. 13. E ects of Pr and Rg on Re{ 2 Qx{ =
5
31
Fig. 14. E ects of Ne and 2 on Re{ 2 Vk=
conclusions
Nonlinear thermal radiation and heat generation/absorption in flow of Je rey nanofluid are examined. The following results are worthmentioning. • Momentum layer thickness is increasing function of ( 1 ). • (Pj ) decelerates the momentum boundary layer. • Thermal boundary layer reduces with increasing (S u). • Heat transfer decays for larger estimation of (Ug )= • Mass transfer boots for higher (Qe ).
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18