- Email: [email protected]
Analytical study of third grade fluid over a rotating vertical cone in the presence of nanoparticles
International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Analytical study of third grade fluid over a rotating vertical cone in the presence of nanoparticles S. Nadeem a, S. Saleem a,b,⇑ a
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, COMSATS Institute of Information Technology, Attock 43600, Pakistan
b
a r t i c l e
i n f o
Article history: Received 18 December 2014 Received in revised form 3 February 2015 Accepted 3 February 2015 Available online 10 March 2015 Keywords: Non-Newtonian fluid Brownian motion Thermophoresis Rotating cone
a b s t r a c t Present article deals with the study of third grade fluid flow over a rotating vertical cone in the presence of nanoparticles i.e. thermophoresis and Brownian motion. Solutions for the boundary layer momentum, energy and diffusion equations are carried out by a well-known analytical technique namely Homotopy analysis method. The interesting findings for essential physical parameters are demonstrated in the form of graphs and numerical tables. Also, a suitable comparison has been made with the prior results in the literature as a limiting case of the considered problem. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In recent years the study of non-Newtonian fluid flows have attained noticeable importance due to its massive applications in many engineering and industrial processes. Non-Newtonian fluids with heat and mass transfer are important in processing of food, making of paper, lubrications with heavy oils and greases. Due to the practical significance of non-Newtonian fluids, several researchers have presented various non-Newtonian fluid models [1–3]. The third grade fluid model is one of the most substantial fluid models that display all the properties of shear thinning and shear thickening fluids. Ellahi et al. [4] studied generalized couette flow of a third-grade fluid with slip: the exact solutions. The influence of variable viscosity and viscous dissipation on the nonNewtonian flow was explored by Hayat et al. [5]. Effects of variable viscosity in a third grade fluid with porous medium: An analytic solution was discussed by Ellahi and Afzal [6]. When the forced and free convection differences are of harmonious order phenomena mixed convection occurs. It has vital appearance in atmospheric boundary layer flows, heat exchangers, solar collectors, nuclear reactors and in electronic equipment’s. These physical processes occurs in the situation where the impacts of buoyancy forces in forced convection or the influences of forced flow in natural convection become much more dominant. The interaction of ⇑ Corresponding author at: Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan. Tel.: +92 03445510959. E-mail address: [email protected] (S. Saleem). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.02.007 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
forced and natural convection is particularly noticeable physically where the forced convection flow has low velocity or moderate and large temperature differences. In the concerned article, a rotating vertical cone is positioned in a non-Newtonian nanofluid with the axis of the cone being in line with the external flow is inspected. Mixed convectional flow with heat and mass transfer problems over cones are widely finds its application in automobile and chemical industries. Some of the applications are design of canisters for nuclear waste disposal, nuclear reactor cooling system, etc. It is found that the unsteady mixed convective flows do not particularly gives similarity solution and for the few years later, several flow problems have been studied, where the non-similarity solutions are discussed. The velocities at edge of boundary layer, the body curvature, the surface mass transfer are responsible for unsteadiness and non-similarity in such kind of fluid flows. Hering and Grosh [7] have obtained a number of similarity solutions for cones with prescribed wall temperature being a power function of the distance from the apex along the generator. In recent studies, Anilkumar and Roy [8] obtained the self-similar solutions of unsteady mixed convection flow from a rotating cone in a rotating fluid. They found that the self-similar solutions are only possible, if the angular velocity at the edge and the angular velocity at the wall of cone vary inversely as a linear function of time. Alamgir [9] presented the overall heat transfer in laminar natural convection flow from vertical cones by using the integral method. The steady free convection boundary layer over a vertical cone embedded in a porous medium filled with a non-Newtonian fluid with an exponential decaying internal heat generation is
1042
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
studied by Rashidi and Rastegari [10]. In many cases, the flow can be unsteady due to the time dependent free stream velocity and there are several transport processes with surface mass transfer (suction/injection) where the buoyancy forces arise from the thermal and mass diffusion caused by the temperature and concentration gradients. Therefore, as a step towards the consequent development of investigation on combined convection flows, it is very important and useful to study the unsteady mixed convection boundary layer flow over a vertical cone with the thermal and mass diffusion, when the free stream velocity varies arbitrarily with time [11–13]. The study of convective transport of nanofluids is of superb importance due to its feature to increase the thermal conductivity of fluid as associated to base fluid. The term nanofluid is associated to such type of fluids where the suspension of nano-scale particles and the base fluid is being merged. Choi [14] was the first who used this concept. He exposed that by addition of a small amount of nanoparticles into conventional heat transfer liquids enhanced the thermal conductivity of the fluid approximately two times. A recent application of nanofluid flow is nano-drug delivery [15]. Suspension of metal nanoparticles is also being developed for other purposes, such as medical applications including cancer therapy. Also the nanofluids are regularly used as coolants, lubricants and micro-channel heat sinks. Typically nanofluids consist of metals, oxides or carbon nanotubes. Buongiorno [16] introduced seven slip mechanisms between nanoparticles and the base fluid. He revealed that the Brownian motion and thermophoresis have noteworthy effects in the laminar forced convection of nanofluids. Based on such observations, he established non-homogeneous two-component equations in nanofluids. Thermal Performance of Ethylene Glycol Based Nanofluids in an Electronic Heat Sink was analyzed by Selvakumar and Suresh. [17]. Further Akbar et al. [18] investigated interaction of nano particles for the peristaltic flow in an asymmetric channel with the induced magnetic field. In real situations in nanofluids, the base fluid do not fulfill the properties of Newtonian fluids, hereafter it is more reasonable to consider them as viscoelastic fluids. In the present paper, the base fluid is taken as third grade fluid. Rahmani et al. [19] deliberated the study of thermal and fluid effects of nonNewtonian water-based nanofluids on the free convection flow between two vertical planes. Some experimental and theoretical works related to nanofluids are given in references [14–23]. The core determination of the present work is to analyze the effects of Brownian motion and thermophoresis on mixed convection flow of a third grade fluid on a rotating vertical cone. The nonlinear partial differential equations of third grade nanofluid are initially reduce to system of nonlinear ordinary differential equations with a set of similarity transformations and the solutions are carried out by using homotopy analysis method (HAM) [24–30]. The effects of physical parameters on velocities, surface friction coefficients, temperature and nanoparticle volume fraction are calculated and discussed graphically and numerical tables. Also the results are recovered for the case where nanoparticles are not present. 2. Mathematical analysis The effects of thermophoresis and Brownian motion on the unsteady mixed convection flow on a rotating cone in a rotating third grade fluid with time dependent angular velocity has been investigated. The flow is assumed to be incompressible, axisymmetric and non-dissipative. The rotation of the cone and the fluid with the axis of cone either in same or in inverse direction causes unsteadiness in the fluid flow. Rectangular curvilinear fixed coordinate system is used to elaborate the geometry of the flow in which x, y and z and are taken along tangential, azimuthal direction and normal directions respectively. Here u, v and w are the components of velocity in x, y and z directions, respectively (see Fig. 1).
Fig. 1. Geometry of the problem.
The temperature T w and concentration C w at the wall are supposed to be linear functions of distance x. The differences in temperature and concentration fields produce the buoyancy forces in the flow. With the help of Boussinesq approximations, the boundary layer flow, temperature and concentration equations for third grade nanofluid are set as
@ðxuÞ @ðxwÞ þ ¼ 0; @x @z
ð1Þ
@u @u @u v 2 v 2 @2u þu þw ¼ e þt 2 @t @x @z @z x x ( ) a1 @ 3 u @u @ 2 u @ 2 v @ v @ v @ 2 v þ þ þ þ q @z2 @t @x @z2 @x@z @z @x @z2 ( ) a2 1 @u 2 @ 2 u @u 1 @ v 2 @u @ 2 w @w @ 2 u þ þ þ þ @x@z @z x @z @z @z2 @z @z2 q x @z ( ða1 þ a2 Þ @ 2 u @u @u @ 2 u @ 2 v @ v @ v @ 2 v v @ 2 v 3 þ2 þ þ þ @x@z @z @x @z2 @x@z @z @x @z2 x @z2 q ) " ( 2 2 2 1 @v @u @ 2 w @w @ 2 u b3 @2u @u @v þ þ þ 2 2 3 þ 2 2 x @z @z @z @z @z @z q @z @z )# 2 @u @ v @ v þ gncosa ðT 0 T 1 Þ þ gn cosa ðC C 1 Þ; ð2Þ þ4 @z @z @z2 ( ) @v @v @ v uv @ v e @ 2 v a1 @ 3 v v @ 2 u u @ 2 v þu þw þ ¼ þt 2 þ þ @t @x @z x @t @z q @z2 @t x @z2 x @z2 ( ) a2 @u @ 2 v @ v @ 2 u @ v @ 2 w @w @ 2 v ð a1 þ a2 Þ þ þ þ þ þ q @z @x@z @z @x@z @z @z2 @z @z2 q ( ) 2 2 2 2 @ v @u @ v @ u v @ u 2u @ v 1 @u @ v @ v @ 2 w @w @ 2 v þ þ þ þ þ @x@z @z @x @z2 x @z2 x @z2 x @z @z @z @z2 @z @z2 " ( )# 2 2 b @2v @u @v @u @ v @ 2 u þ3 þ4 ; ð3Þ þ 3 2 2 @z @z @z @z2 q @z @z
ðqcÞf
( 2 ) @T @T @T @2T @C @T DT @T ; ð4Þ ¼ j 2 þðqcÞp DB þu þw þ @t @x @z @z @z @z T 1 @z
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
@C @C @C @ 2 C DT @ 2 T þu þw ¼ DB 2 þ : @t @x @z @z T 1 @z2
ð5Þ
T is the temperature, C is the concentration, g is the acceleration due to gravity, j is the thermal diffusivity and D represents mass diffusivity, a is the semi-vertical angle of the cone, t is the kinematic viscosity, q is the density, n and n are the volumetric co-efficient of expansion for temperature and concentration respectively, ai ði ¼ 1; 2Þ are the second grade fluid parameters and bi ði ¼ 1; 2; 3Þ are the material moduli for the third grade fluid, ðqcÞp is the nanoparticle heat capacity, ðqcÞf is the base fluid heat capacity, DB is the Brownian diffusion coefficient and DT is the thermophoretic diffusion coefficient. It is interesting fact that the nonlinear partial differential equations can be reduced to a system of coupled ordinary differential equations by taking the velocity at wall and the free stream as an inverse function of time. So we involve the following suitable similarity and non-dimensional quantities [8]
te ¼ X2 xsina ð1 st Þ1 ; g ¼ t ¼ ðXsina Þt;
Xsina
12
t
1
ð1 st Þ 2 z;
c¼
X1 ; X
1
Gr1¼ gbcosa ðT 0 T 1 Þ
L3
t
2
L3
;
0
f ð0Þ ¼ 0 ¼ f ð0Þ;
; k2 ¼ 2
t
L2
t
;
k1 ¼
Gr1 Re2L
0
gð0Þ ¼ c;
hð0Þ ¼ /ð0Þ ¼ 1;
00
¼ 0;
ð11Þ
ðqcÞp DB ðC w C 1 Þ ; tðqcÞf
1
Nt ¼
ðqcÞp DT ðT w T 1 Þ tðqcÞf T 1
The tangential and azimuthal skin friction coefficients are
½2sxz z¼0
C fx ¼
q½Xxsina ð1 st Þ1
2
;
ð12Þ
2
;
ð13Þ
½2syz z¼0
C fy ¼
q½Xxsina ð1 st Þ1
Where
;
sxz
k2 t t ; N ¼ ; Pr ¼ ; Sc ¼ ; 2 k D j 1 B ReL
Gr2
3 a1 Xsina ð1 st Þ1 a2 Xsina ð1 st Þ1 b ðXsina Þ ð1 st Þ3 ; b¼ ; d¼ 3 ; l l tl
ð6Þ
Here s is the unsteady parameter, flow field is accelerating for s > 0 and vice versa. X1 and X2 are the angular velocities of the cone and the free stream fluid respectively, X ¼ X1 þ X2 is the composite angular velocity, c is the ratio of angular velocity of the cone to the angular velocity of fluid and c ¼ 0 implies that the fluid is rotating and the cone is at rest; besides, the fluid and the cone are rotating with equal angular velocity in the same direction for c ¼ 0:5. For c ¼ 1, the fluid is at rest and the cone is in rotation. k1 is the buoyancy force parameter, N is the ratio of the buoyancy forces and it measures the strength of thermal and chemical diffusion due to the buoyancy force which drives the flow. It is zero, infinity, positive and negative for no chemical diffusion, no thermal diffusion, when the buoyancy forces due to the combined effects of temperature and concentration differences acts in the same direction and vice versa, respectively and a; b and d is the non-Newtonian flow parameters. The continuity Eq. (1) is trivially satisfied and Eqs. (2)–(5) takes the form
n o 1 1 0 000 00 0 ð1 þ asÞf f þ sg f þ f s f 2 g 2 ð1 cÞ2 2 2 1 000 002 0 000 2k1 ðh þ N/Þ a þ b df f ða 2bÞf f 4 1 000 1 00 2 a b df g 02 2 a df gg 00 ¼ 0; 4 4
ð10Þ
f ð1Þ ¼ 0 ¼ f ð1Þ; gð1Þ ¼ 1 c; g 0 ð1Þ ¼ 0; hð1Þ ¼ /ð1Þ
wðt;x;zÞ ¼ ðtXxsina Þ2 ð1 st Þ 2 f ðgÞ;
ReL ¼ Xxsina
ð9Þ
The assumed boundary conditions for the non-Newtonian fluid flow problem in non-dimensional form are specified as
Nb ¼
x Cðt; x;zÞ C 1 ¼ ðC w C 1 Þ/ðgÞ; ðC w C 1 Þ ¼ ðC 0 C 1 Þ ð1 st Þ2 ; L
a¼
Nt 1 00 0/ s 2/ þ 21 g/0 þ / f /0 f h00 ¼ 0: Sc 2 Nb
0
x Tðt; x;zÞ T 1 ¼ ðT w T 1 ÞhðgÞ; ðT w T 1 Þ ¼ ðT 0 T 1 Þ ð1 st Þ2 ; L
Gr2¼ gbcosa ðC 0 C 1 Þ
1 00 0 h s 2h þ 21 gh0 þ Nbh0 /0 þ Nth02 ¼ 0; h f h0 f Pr 2
Where the Brownian motion parameter Nb and the thermophoresis parameter Nt are defined as
uðt; x; zÞ ¼ 21 Xxsina ð1 st Þ1 f ðgÞ;
v ðt;x;zÞ ¼ Xxsina ð1 st Þ1 gðgÞ;
1043
1 1 0 0 ð1 aÞg 00 ðfg gf Þ þ s 1 c g gg 0 asgg 000 2 2 n o 3 1 000 0 00 00 2 þ a þ b f g a þ b f g d ðf Þ þ 6ðg 0 Þ2 g 00 ¼ 0; ð8Þ 2 2
! @u @ 2 u @u @u @ v @ v @u @w þ a2 ¼l þ a1 þ þ @z @z@t @x @z @x @z @z @z ( 2 2 2 2 @w @u @u @ v @u @u @u @ v þ b3 6 þ2 þ4 þ2 @x @z @z @z @z @x @z @x ) 2 v @u @ v @u v 2 @u @w þ2 þ4 4 ; @z x @z @z x @z @x ! @v @ 2 v v @u u @ v @ v @w þ a2 þ a1 þ @z @z@t x @z x @z @z @z ( 2 2 @ v @u @v @v v @v @v @u @ v @w þ2 4 þ4 þ b3 4 @z @z @x @z @x @z @x x @z @x ) 2 @ v v 2 @ v u2 @ v @w ; þ4 þ4 þ2 @z x @z x @z @z
syz ¼ l
or
1 3 00 1 0 00 g 000 00 0 00 02 00 bf f 6df f C fx Re2x ¼ f þ a 2gg 0 sf þ f f s f ; 2 2 2 g¼0 1 3 1 0 1 00 g 0 02 C fy Re2x ¼ g 0 a sg 0 f g 0 þ f g þ s g 00 bf g 0 df g 0 : 2 2 2 2 g¼0 ð14Þ x2 Xsina ð1st Þ1
where Rex ¼ is the rotational Reynolds number. t The non-dimensional heat and mass transfer rates are listed as
ð7Þ
1
NuRex 2 ¼ h0 ð0Þ;
1044
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048 1
ShRex 2 ¼ /0 ð0Þ:
ð15Þ
3. Solution technique Since we are interested in the analytical solutions of Eqs. (7)–(10), for this purpose we have employed homotopy analysis method (HAM) which is a famous mathematical technique established by Liao [24]. The corresponding initial guesses and the linear operators for the velocities f ðgÞ; gðgÞ; the temperature field hðgÞ and the concentration field /ðgÞ are stated below, respectively.
f 0 ðgÞ ¼ 0;
ð16Þ
g 0 ðgÞ ¼ ð1 cÞ þ ð2c 1Þ expðgÞ;
ð17Þ
h0 ðgÞ ¼ expðgÞ;
ð18Þ
/0 ðgÞ ¼ expðgÞ;
ð19Þ
3
ff ðgÞ ¼
d f df ; dg3 dg
fg ðgÞ ¼
d g g; dg2
Fig. 3. h-curves for h0 (0) and u0 (0) at 15th approximation.
ð20Þ
2
ð21Þ
Table 1 Convergence table for the flow problem. Order of convergence
f ð0Þ
g 0 ð0Þ
h0 ð0Þ
/0 ð0Þ
1 5 10 15 20
0.24053 0.25188 0.25168 0.25168 0.25168
0.23918 0.24783 0.24938 0.24934 0.24934
1.6044 1.60443 1.60425 1.60425 1.60425
2.42222 2.07728 2.07855 2.07855 2.07855
2
fh ðgÞ ¼
d h h; dg2
f/ ðgÞ ¼
d / /; dg2
ð22Þ
2
ð23Þ
4. Convergence of the analytical solutions Obviously the series solutions achieved by homotopy analysis h. This paramethod contain the convergence control parameter meter controls the convergence region and the rate of approximation of the HAM solution. To check the convergence of the solutions in the satisfactory range of the values of the auxiliary parameters g ; hf ; h hh and h/ ; h-curve for 15th-order approximations have been presented. Figs. 2 and 3 show that the admissible range of values g ; hf ; h hh and h/ are 1:3 6 hf 6 0:4; 1:2 6 hg 6 0:4; 1:3 6 of
Fig. 2. h-curve of f00 (0) and g0 (0) at 15th approximation.
00
h 6 0:3 and 1:2 6 h h/ 6 0:4. The convergence Table 1 made for each of the function up to 25th order of approximations. 5. Results and discussion This section is dedicated to observe and discuss the analytical results for a different range of various physical parameters like, ratio of angular velocities c, ratio of the buoyancy forces N, second grade parameter a and b, Third grade parameter d, Brownian
0
Fig. 4. Effects of c and k1 on tangential velocity f ðgÞ.
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
1045
0
Fig. 5. Effects of a on tangential velocity f ðgÞ.
Fig. 8. Effects of c and k1 on azimuthal velocity gðgÞ.
0
Fig. 6. Effects of b on tangential velocity f ðgÞ. Fig. 9. Effects of a on azimuthal velocity gðgÞ.
0
Fig. 7. Effects of d on tangential velocity f ðgÞ. Fig. 10. Effects of b on azimuthal velocity gðgÞ.
motion parameter Nb, Thermophoresis parameter Nt, Schmidt number Sc, Prandtl number Pr and unsteadiness parameter s. 0 Figs. 4–7 refer to the variation of tangential velocity f ðgÞ for c; a; b and d respectively. It is establish from Fig. 4 that when c ¼ 0:5 the fluid and the cone are in rotation with compatible
angular velocity in the similar direction and the flow is only due to the favorable pressure gradient i.e. k1 = 1. For c > 0:5, the 0 magnitude of velocity f ðgÞ increases on the other hand the variation reduces for c < 0:5. It is found that for c < 0 the velocity 0 field f ðgÞ reaches asymptotically at the edge of the boundary
1046
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
Fig. 11. Effects of d on azimuthal velocity gðgÞ. Fig. 14. Effects of Nb on temperature hðgÞ and nanoparticle volume fraction /ðgÞ.
1
Fig. 12. Effects of N on surface skin friction coefficient C fx Re2x .
Fig. 15. Effects of Nt on temperature hðgÞ and nanoparticle volume fraction /ðgÞ.
1
Fig. 13. Effects of N on surface skin friction coefficient C fy Re2x .
layer in an oscillatory style. Actually such oscillations occur due to the surplus convection of angular momentum seems in the region 0 of boundary layer. Figs. 5 and 6 clarifies that the velocity f ðgÞ follows the reducing behavior for second grade parameter a, 0 whereas increase in magnitude is found for b. The velocity f ðgÞ
Fig. 16. Effects of Pr and Sc temperature hðgÞ and nanoparticle volume fraction /ðgÞ.
1047
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048 Table 3 Values of surface Skin friction coefficients for some relevant parameters.
a
b
1
d
0.0 1.0 2.0 0.0 0.2 0.4 0.0 1.0 2.0
1
C fx Re2x
C fy Re2x
0.00226 0.18848 1.32178 0.01533 0.02063 0.02597 0.07423 0.07930 0.08546
0.24422 0.48514 0.57451 0.26805 0.26777 0.26704 0.26635 0.28235 0.30028
Table 4 Values of heat transfer and mass transfer rates for some relevant parameters. Pr Fig. 17. Effects of Nb and Nt on surface heat transfer rate h0 ð0Þ.
Sc
N
1
3.0 8.0 13.0 0.5 1.0 1.5 0.0 0.5 1.0
Fig. 18. Effects of Nb and Nt on surface mass transfer rate /0 ð0Þ.
varies inversely with d (see Fig. 7). The influences of c; a; b and d on azimuthal velocity gðgÞ is presented in Figs. 8–11. It is depicted from Fig. 8 that the velocity gðgÞ reduces for c > 0:5 but the behavior is converse when c < 0:5. It is demonstrated in Figs. 9 and 10 that the velocity gðgÞ increases by increasing a and b respectively. Fig. 11 is plotted to show that velocity gðgÞ increases upon decreasing d. Figs. 12 and 13 are keen to notice the variations of skin friction coefficients in tangential and azimuthal directions for various values of N, respectively. Both the skin friction coefficients increase with an increase in N, respectively. It /ðgÞ is due to the fact that wall temperature at the cone boundaries is somewhat greater than
1
NuRex2
ShRex2
2.84871 3.45558 3.62775 1.57685 1.55824 1.54708 0.82487 0.84184 0.85625
0.01595 0.37508 0.47130 0.55885 1.25681 1.76238 0.52556 0.53311 0.54066
the temperature of the fluid which eventually increases the Gr 2 as compared to Gr 1 , thus higher values of N provide the larger values of skin friction coefficients. Besides it is found that as s increases from 0 to 0.5, tangential coefficient of skin friction reduces but the deviations are opposed for azimuthal coefficient of skin friction. Both the temperature hðgÞ and nanoparticle volume fraction /ðgÞ have opposite influences against different values of Brownian motion parameter Nb (see Fig. 14). The effects of Thermophoresis parameter Nt on temperature hðgÞ and nanoparticle volume fraction /ðgÞ is seen in Fig. 15. The figure shows that both temperature hðgÞ and nanoparticle volume fraction /ðgÞ enhances their variation for greater values of Nt. Fig. 16 is sketched for temperature hðgÞ and nanoparticle volume fraction /ðgÞ against Pr and Sc, respectively. It is interesting to know that both temperature hðgÞ and nanoparticle volume fraction reduces for Pr and Sc. The Nusselt number and the Sherwood number shows an increasing behavior for increasing values of N and s (see Figs. 15 and 16). Figs. 17 and 18 displays the effects of Nb and Nt on heat transfer rate h0 ð0Þ and mass transfer rate /0 ð0Þ respectively. It is revealed in Fig. 17 and 18 the change in the dimensionless heat transfer rates is seen to be greater for smaller values of the parameter Nb and this change reduces with the rise in Nt. Table 2 is testified to authorize the exactness of our analytical results. It is noticed from the Table 2 that present HAM findings and numerical results [8] follows the satisfactory criteria. The mathematical values of Skin friction coefficients in both directions
Table 2 Comparison table for important physical quantities for special case. k1
c
Present results 1 2
Anilkumar and Roy results [8] 1 2
1 2
1 2
1
1
1
1
C fx Rex
C fy Rex
NuRex
ShRex
C fx Re2x
C fy Re2x
NuRex2
ShRex2
1.0
0.0 0.25 0.50
0.63238 1.31328 1.84791
0.63940 0.22758 0.19800
0.81920 0.89010 0.93702
0.95060 1.02817 1.07979
0.63241 1.31339 1.84798
0.63949 0.22765 0.19806
0.81922 0.89011 0.93700
0.95065 1.02812 1.07977
3.0
0.0 0.25 0.50
3.79518 4.31850 4.73958
0.59658 0.13687 0.33554
1.02862 1.06525 1.09111
0.18648 1.22640 1.25442
3.79522 4.31854 4.73958
0.59651 0.13691 0.33552
1.02869 1.06539 1.09111
0.18645 1.22639 1.25444
1048
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
for different evolving constraints such as a; b and d are demonstrated in Table 3. The impact of a; b and d is to reduce the tangential skin friction coefficient, while the azimuthal skin friction coefficient is a direct function of a and d, but varies in inverse way for b: Table 4 indicates that the heat transfer and mass transfer rates shows opposite behaviors for increasing values of Pr and Sc, respectively. The ratio of buoyancy forces N increases the magnitude of both heat transfer and mass transfer rates gradually. 6. Conclusions We have theoretically examined the unsteady boundary layer flow of third grade nanofluid over a rotating cone. The model used for the nanofluid includes the effects of Brownian motion and thermophoresis. The transformed ordinary differential equations are elucidated by a strong mathematical method recognized as homotopy analysis method (HAM). The analytical results are agreed to be in conventional settlement with the previous results in literature The Nusselt number and the Sherwood number shows an increasing behavior for increasing values of N and s. The results of [8] can be recovered in the absence of nanoparticles. The Brownian motion parameter Nb increases the temperature and nanoparticle volume fraction, but the effects are just opposite for thermophoresis parameter Nt. The acquired results have promising applications in engineering and will now be available for experimental verification to give confidence for the well-posedness of this nonlinear boundary value problem. Conflict of interest It is stated that all the authors have seen the article and agree to submit it in International Journal of Heat and Mass Transfer. References [1] O.D. Makinde, Irreversibility analysis for gravity driven non-Newtonian liquid film along an inclined isothermal plate, Phys. Scr. 74 (2006) 642–645. [2] M. Qasim, Soret and Dufour effects on the flow of an Eyring–Powell fluid over a flat plate with convective boundary condition, Eur. Phys. J. Plus 129 (2014) 24. [3] Noor Fadiya Mohd Noor, Analysis for MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction, World Acad. Sci., Eng. Technol. 64 (2012) 1019–1023. [4] R. Ellahi, T Hayat, F.M. Mahomed, Generalized couette flow of a third-grade fluid with slip: the exact solutions, Z. Naturforsch. – Sect. A 65 (2010) 1071– 1076. [5] T. Hayat, R. Ellahi, S. Asghar, The influence of variable viscosity and viscous dissipation on the non-Newtonian flow: an analytical solution, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 300–331. [6] R. Ellahi, S. Afzal, Effects of variable viscosity in a third grade fluid with porous medium: an analytic solution, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 2056–2072.
[7] R.G. Hering, R.J. Grosh, Laminar combined convection from a rotating cone, ASME J. Heat Transfer 85 (1963) 29–34. [8] D. Anilkumar, S. Roy, Unsteady mixed convection flow on a rotating cone in a rotating fluid, Appl. Math. Comput. 155 (2004) 545–561. [9] M. Alamgir, Over all heat transfer from vertical cones in laminar free convection, an approximate method, ASME J. Heat Transfer 101 (1989) 174. [10] M.M. Rashidi, M.T. Rastegari, Analytical solution for free convection boundarylayer over a vertical cone in a non-Newtonian fluid saturated porous medium with internal heat generation, World Appl. Sci. J. 16 (2012) 64–74. [11] S. Saleem, S. Nadeem, R. Ul Haq, Buoyancy and metallic particle effects on an unsteady water-based fluid flow along a vertically rotating cone, Eur. Phys. J. Plus 129 (2014) 213–220. [12] C.Y. Wang, Boundary layers on rotating cones, discs and axisymmetric surfaces with a concentrated heat source, Acta Mech. 81 (1990) 245–251. [13] M.C. Ece, Free-convection flow about a cone under mixed thermal boundary conditions and a magnetic field, Appl. Math. Model. 29 (2005) 1121–1134. [14] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, Dev. Appl. Non-Newtonian Flows 66 (1995) 99–105. [15] C. Kleinstreuer, J. Li, J. Koo, Microfluidics of nano-drug delivery, Int. J. Heat Mass Transfer 51 (2008) 5590–5597. [16] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [17] P. Selvakumar, S. Suresh, Thermal performance of ethylene glycol based nanofluids in an electronic heat sink, J. Nanosci. Nanotechnol. 14 (2014) 2325– 2333. [18] Noreen Sher Akbar, M. Raza, R. Ellahi, Interaction of nano particles for the peristaltic flow in an asymmetric channel with the induced magnetic field, Eur. Phys. J. – Plus 129 (2014) 155–167. [19] Y. Rahmani, R. Yousefi, S.E. Ghasemi, D.D. Ganji, Thermal and fluid effects of non-Newtonian water-based nanofluids on the free convection flow between two vertical planes, Phys. Rev. Res. Int. 3 (4) (2013) 688–701. [20] R. Ellahi, M. Hassan, A. Zeeshan, Shape effects of nanosize particles in Cu–H2O nanofluid on entropy generation, Int. J. Heat Mass Transfer 81 (2015) 449–456. [21] O.D. Makinde, W.A. Khan, Z.H. Khan, Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet, Int. J. Heat Mass Transfer 62 (2013) 526–533. [22] A. Dominic, J. Sarangan, S. Suresh, Monica Sai, Heat transfer performance of Al2O3/water nanofluids in a mini channel heat sink, J. Nanosci. Nanotechnol. 14 (2014) 2368–2376. [23] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions, Appl. Math. Model. 37 (3) (2013) 1451–1467. [24] S.J. Liao, ‘‘Beyond Perturbation’’: a review on the homotopy analysis method and its applications, Adv. Mech. 153 (2008) 1–34. [25] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A 360 (2006) 109–113. [26] S. Nadeem, S. Saleem, Analytical treatment of unsteady mixed convection MHD flow on a rotating cone in a rotating frame, J. Taiwan Inst. Chem. Eng. 44 (4) (2013) 596–604. [27] S. Nadeem, S. Saleem, Unsteady mixed convection flow of nanofluid on a rotating cone with magnetic field, Appl. Nanosci. 4 (2013) 405–414. [28] N.F.M. Noor, O. Abdulaziz, I. Hashim, MHD flow and heat transfer in a thin liquid film on an unsteady stretching sheet by the homotopy analysis method, Int. J Numer. Method Fluid 63 (2010) 357–373. [29] R. Ellahi, A study on the convergence of series solution of non-Newtonian third grade fluid with variable viscosity: by means of homotopy analysis method, Adv. Math. Phys. 2012 (2012) 11. Article ID 634925. [30] S. Nadeem, S.T. Hussain, C. Lee, Flow of a Williamson fluid over a stretching sheet, Braz. J. Chem. Eng. 30 (2013) 619–625.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Analytical study of third grade fluid over a rotating vertical cone in the presence of nanoparticles S. Nadeem a, S. Saleem a,b,⇑ a
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, COMSATS Institute of Information Technology, Attock 43600, Pakistan
b
a r t i c l e
i n f o
Article history: Received 18 December 2014 Received in revised form 3 February 2015 Accepted 3 February 2015 Available online 10 March 2015 Keywords: Non-Newtonian fluid Brownian motion Thermophoresis Rotating cone
a b s t r a c t Present article deals with the study of third grade fluid flow over a rotating vertical cone in the presence of nanoparticles i.e. thermophoresis and Brownian motion. Solutions for the boundary layer momentum, energy and diffusion equations are carried out by a well-known analytical technique namely Homotopy analysis method. The interesting findings for essential physical parameters are demonstrated in the form of graphs and numerical tables. Also, a suitable comparison has been made with the prior results in the literature as a limiting case of the considered problem. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In recent years the study of non-Newtonian fluid flows have attained noticeable importance due to its massive applications in many engineering and industrial processes. Non-Newtonian fluids with heat and mass transfer are important in processing of food, making of paper, lubrications with heavy oils and greases. Due to the practical significance of non-Newtonian fluids, several researchers have presented various non-Newtonian fluid models [1–3]. The third grade fluid model is one of the most substantial fluid models that display all the properties of shear thinning and shear thickening fluids. Ellahi et al. [4] studied generalized couette flow of a third-grade fluid with slip: the exact solutions. The influence of variable viscosity and viscous dissipation on the nonNewtonian flow was explored by Hayat et al. [5]. Effects of variable viscosity in a third grade fluid with porous medium: An analytic solution was discussed by Ellahi and Afzal [6]. When the forced and free convection differences are of harmonious order phenomena mixed convection occurs. It has vital appearance in atmospheric boundary layer flows, heat exchangers, solar collectors, nuclear reactors and in electronic equipment’s. These physical processes occurs in the situation where the impacts of buoyancy forces in forced convection or the influences of forced flow in natural convection become much more dominant. The interaction of ⇑ Corresponding author at: Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan. Tel.: +92 03445510959. E-mail address: [email protected] (S. Saleem). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.02.007 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
forced and natural convection is particularly noticeable physically where the forced convection flow has low velocity or moderate and large temperature differences. In the concerned article, a rotating vertical cone is positioned in a non-Newtonian nanofluid with the axis of the cone being in line with the external flow is inspected. Mixed convectional flow with heat and mass transfer problems over cones are widely finds its application in automobile and chemical industries. Some of the applications are design of canisters for nuclear waste disposal, nuclear reactor cooling system, etc. It is found that the unsteady mixed convective flows do not particularly gives similarity solution and for the few years later, several flow problems have been studied, where the non-similarity solutions are discussed. The velocities at edge of boundary layer, the body curvature, the surface mass transfer are responsible for unsteadiness and non-similarity in such kind of fluid flows. Hering and Grosh [7] have obtained a number of similarity solutions for cones with prescribed wall temperature being a power function of the distance from the apex along the generator. In recent studies, Anilkumar and Roy [8] obtained the self-similar solutions of unsteady mixed convection flow from a rotating cone in a rotating fluid. They found that the self-similar solutions are only possible, if the angular velocity at the edge and the angular velocity at the wall of cone vary inversely as a linear function of time. Alamgir [9] presented the overall heat transfer in laminar natural convection flow from vertical cones by using the integral method. The steady free convection boundary layer over a vertical cone embedded in a porous medium filled with a non-Newtonian fluid with an exponential decaying internal heat generation is
1042
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
studied by Rashidi and Rastegari [10]. In many cases, the flow can be unsteady due to the time dependent free stream velocity and there are several transport processes with surface mass transfer (suction/injection) where the buoyancy forces arise from the thermal and mass diffusion caused by the temperature and concentration gradients. Therefore, as a step towards the consequent development of investigation on combined convection flows, it is very important and useful to study the unsteady mixed convection boundary layer flow over a vertical cone with the thermal and mass diffusion, when the free stream velocity varies arbitrarily with time [11–13]. The study of convective transport of nanofluids is of superb importance due to its feature to increase the thermal conductivity of fluid as associated to base fluid. The term nanofluid is associated to such type of fluids where the suspension of nano-scale particles and the base fluid is being merged. Choi [14] was the first who used this concept. He exposed that by addition of a small amount of nanoparticles into conventional heat transfer liquids enhanced the thermal conductivity of the fluid approximately two times. A recent application of nanofluid flow is nano-drug delivery [15]. Suspension of metal nanoparticles is also being developed for other purposes, such as medical applications including cancer therapy. Also the nanofluids are regularly used as coolants, lubricants and micro-channel heat sinks. Typically nanofluids consist of metals, oxides or carbon nanotubes. Buongiorno [16] introduced seven slip mechanisms between nanoparticles and the base fluid. He revealed that the Brownian motion and thermophoresis have noteworthy effects in the laminar forced convection of nanofluids. Based on such observations, he established non-homogeneous two-component equations in nanofluids. Thermal Performance of Ethylene Glycol Based Nanofluids in an Electronic Heat Sink was analyzed by Selvakumar and Suresh. [17]. Further Akbar et al. [18] investigated interaction of nano particles for the peristaltic flow in an asymmetric channel with the induced magnetic field. In real situations in nanofluids, the base fluid do not fulfill the properties of Newtonian fluids, hereafter it is more reasonable to consider them as viscoelastic fluids. In the present paper, the base fluid is taken as third grade fluid. Rahmani et al. [19] deliberated the study of thermal and fluid effects of nonNewtonian water-based nanofluids on the free convection flow between two vertical planes. Some experimental and theoretical works related to nanofluids are given in references [14–23]. The core determination of the present work is to analyze the effects of Brownian motion and thermophoresis on mixed convection flow of a third grade fluid on a rotating vertical cone. The nonlinear partial differential equations of third grade nanofluid are initially reduce to system of nonlinear ordinary differential equations with a set of similarity transformations and the solutions are carried out by using homotopy analysis method (HAM) [24–30]. The effects of physical parameters on velocities, surface friction coefficients, temperature and nanoparticle volume fraction are calculated and discussed graphically and numerical tables. Also the results are recovered for the case where nanoparticles are not present. 2. Mathematical analysis The effects of thermophoresis and Brownian motion on the unsteady mixed convection flow on a rotating cone in a rotating third grade fluid with time dependent angular velocity has been investigated. The flow is assumed to be incompressible, axisymmetric and non-dissipative. The rotation of the cone and the fluid with the axis of cone either in same or in inverse direction causes unsteadiness in the fluid flow. Rectangular curvilinear fixed coordinate system is used to elaborate the geometry of the flow in which x, y and z and are taken along tangential, azimuthal direction and normal directions respectively. Here u, v and w are the components of velocity in x, y and z directions, respectively (see Fig. 1).
Fig. 1. Geometry of the problem.
The temperature T w and concentration C w at the wall are supposed to be linear functions of distance x. The differences in temperature and concentration fields produce the buoyancy forces in the flow. With the help of Boussinesq approximations, the boundary layer flow, temperature and concentration equations for third grade nanofluid are set as
@ðxuÞ @ðxwÞ þ ¼ 0; @x @z
ð1Þ
@u @u @u v 2 v 2 @2u þu þw ¼ e þt 2 @t @x @z @z x x ( ) a1 @ 3 u @u @ 2 u @ 2 v @ v @ v @ 2 v þ þ þ þ q @z2 @t @x @z2 @x@z @z @x @z2 ( ) a2 1 @u 2 @ 2 u @u 1 @ v 2 @u @ 2 w @w @ 2 u þ þ þ þ @x@z @z x @z @z @z2 @z @z2 q x @z ( ða1 þ a2 Þ @ 2 u @u @u @ 2 u @ 2 v @ v @ v @ 2 v v @ 2 v 3 þ2 þ þ þ @x@z @z @x @z2 @x@z @z @x @z2 x @z2 q ) " ( 2 2 2 1 @v @u @ 2 w @w @ 2 u b3 @2u @u @v þ þ þ 2 2 3 þ 2 2 x @z @z @z @z @z @z q @z @z )# 2 @u @ v @ v þ gncosa ðT 0 T 1 Þ þ gn cosa ðC C 1 Þ; ð2Þ þ4 @z @z @z2 ( ) @v @v @ v uv @ v e @ 2 v a1 @ 3 v v @ 2 u u @ 2 v þu þw þ ¼ þt 2 þ þ @t @x @z x @t @z q @z2 @t x @z2 x @z2 ( ) a2 @u @ 2 v @ v @ 2 u @ v @ 2 w @w @ 2 v ð a1 þ a2 Þ þ þ þ þ þ q @z @x@z @z @x@z @z @z2 @z @z2 q ( ) 2 2 2 2 @ v @u @ v @ u v @ u 2u @ v 1 @u @ v @ v @ 2 w @w @ 2 v þ þ þ þ þ @x@z @z @x @z2 x @z2 x @z2 x @z @z @z @z2 @z @z2 " ( )# 2 2 b @2v @u @v @u @ v @ 2 u þ3 þ4 ; ð3Þ þ 3 2 2 @z @z @z @z2 q @z @z
ðqcÞf
( 2 ) @T @T @T @2T @C @T DT @T ; ð4Þ ¼ j 2 þðqcÞp DB þu þw þ @t @x @z @z @z @z T 1 @z
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
@C @C @C @ 2 C DT @ 2 T þu þw ¼ DB 2 þ : @t @x @z @z T 1 @z2
ð5Þ
T is the temperature, C is the concentration, g is the acceleration due to gravity, j is the thermal diffusivity and D represents mass diffusivity, a is the semi-vertical angle of the cone, t is the kinematic viscosity, q is the density, n and n are the volumetric co-efficient of expansion for temperature and concentration respectively, ai ði ¼ 1; 2Þ are the second grade fluid parameters and bi ði ¼ 1; 2; 3Þ are the material moduli for the third grade fluid, ðqcÞp is the nanoparticle heat capacity, ðqcÞf is the base fluid heat capacity, DB is the Brownian diffusion coefficient and DT is the thermophoretic diffusion coefficient. It is interesting fact that the nonlinear partial differential equations can be reduced to a system of coupled ordinary differential equations by taking the velocity at wall and the free stream as an inverse function of time. So we involve the following suitable similarity and non-dimensional quantities [8]
te ¼ X2 xsina ð1 st Þ1 ; g ¼ t ¼ ðXsina Þt;
Xsina
12
t
1
ð1 st Þ 2 z;
c¼
X1 ; X
1
Gr1¼ gbcosa ðT 0 T 1 Þ
L3
t
2
L3
;
0
f ð0Þ ¼ 0 ¼ f ð0Þ;
; k2 ¼ 2
t
L2
t
;
k1 ¼
Gr1 Re2L
0
gð0Þ ¼ c;
hð0Þ ¼ /ð0Þ ¼ 1;
00
¼ 0;
ð11Þ
ðqcÞp DB ðC w C 1 Þ ; tðqcÞf
1
Nt ¼
ðqcÞp DT ðT w T 1 Þ tðqcÞf T 1
The tangential and azimuthal skin friction coefficients are
½2sxz z¼0
C fx ¼
q½Xxsina ð1 st Þ1
2
;
ð12Þ
2
;
ð13Þ
½2syz z¼0
C fy ¼
q½Xxsina ð1 st Þ1
Where
;
sxz
k2 t t ; N ¼ ; Pr ¼ ; Sc ¼ ; 2 k D j 1 B ReL
Gr2
3 a1 Xsina ð1 st Þ1 a2 Xsina ð1 st Þ1 b ðXsina Þ ð1 st Þ3 ; b¼ ; d¼ 3 ; l l tl
ð6Þ
Here s is the unsteady parameter, flow field is accelerating for s > 0 and vice versa. X1 and X2 are the angular velocities of the cone and the free stream fluid respectively, X ¼ X1 þ X2 is the composite angular velocity, c is the ratio of angular velocity of the cone to the angular velocity of fluid and c ¼ 0 implies that the fluid is rotating and the cone is at rest; besides, the fluid and the cone are rotating with equal angular velocity in the same direction for c ¼ 0:5. For c ¼ 1, the fluid is at rest and the cone is in rotation. k1 is the buoyancy force parameter, N is the ratio of the buoyancy forces and it measures the strength of thermal and chemical diffusion due to the buoyancy force which drives the flow. It is zero, infinity, positive and negative for no chemical diffusion, no thermal diffusion, when the buoyancy forces due to the combined effects of temperature and concentration differences acts in the same direction and vice versa, respectively and a; b and d is the non-Newtonian flow parameters. The continuity Eq. (1) is trivially satisfied and Eqs. (2)–(5) takes the form
n o 1 1 0 000 00 0 ð1 þ asÞf f þ sg f þ f s f 2 g 2 ð1 cÞ2 2 2 1 000 002 0 000 2k1 ðh þ N/Þ a þ b df f ða 2bÞf f 4 1 000 1 00 2 a b df g 02 2 a df gg 00 ¼ 0; 4 4
ð10Þ
f ð1Þ ¼ 0 ¼ f ð1Þ; gð1Þ ¼ 1 c; g 0 ð1Þ ¼ 0; hð1Þ ¼ /ð1Þ
wðt;x;zÞ ¼ ðtXxsina Þ2 ð1 st Þ 2 f ðgÞ;
ReL ¼ Xxsina
ð9Þ
The assumed boundary conditions for the non-Newtonian fluid flow problem in non-dimensional form are specified as
Nb ¼
x Cðt; x;zÞ C 1 ¼ ðC w C 1 Þ/ðgÞ; ðC w C 1 Þ ¼ ðC 0 C 1 Þ ð1 st Þ2 ; L
a¼
Nt 1 00 0/ s 2/ þ 21 g/0 þ / f /0 f h00 ¼ 0: Sc 2 Nb
0
x Tðt; x;zÞ T 1 ¼ ðT w T 1 ÞhðgÞ; ðT w T 1 Þ ¼ ðT 0 T 1 Þ ð1 st Þ2 ; L
Gr2¼ gbcosa ðC 0 C 1 Þ
1 00 0 h s 2h þ 21 gh0 þ Nbh0 /0 þ Nth02 ¼ 0; h f h0 f Pr 2
Where the Brownian motion parameter Nb and the thermophoresis parameter Nt are defined as
uðt; x; zÞ ¼ 21 Xxsina ð1 st Þ1 f ðgÞ;
v ðt;x;zÞ ¼ Xxsina ð1 st Þ1 gðgÞ;
1043
1 1 0 0 ð1 aÞg 00 ðfg gf Þ þ s 1 c g gg 0 asgg 000 2 2 n o 3 1 000 0 00 00 2 þ a þ b f g a þ b f g d ðf Þ þ 6ðg 0 Þ2 g 00 ¼ 0; ð8Þ 2 2
! @u @ 2 u @u @u @ v @ v @u @w þ a2 ¼l þ a1 þ þ @z @z@t @x @z @x @z @z @z ( 2 2 2 2 @w @u @u @ v @u @u @u @ v þ b3 6 þ2 þ4 þ2 @x @z @z @z @z @x @z @x ) 2 v @u @ v @u v 2 @u @w þ2 þ4 4 ; @z x @z @z x @z @x ! @v @ 2 v v @u u @ v @ v @w þ a2 þ a1 þ @z @z@t x @z x @z @z @z ( 2 2 @ v @u @v @v v @v @v @u @ v @w þ2 4 þ4 þ b3 4 @z @z @x @z @x @z @x x @z @x ) 2 @ v v 2 @ v u2 @ v @w ; þ4 þ4 þ2 @z x @z x @z @z
syz ¼ l
or
1 3 00 1 0 00 g 000 00 0 00 02 00 bf f 6df f C fx Re2x ¼ f þ a 2gg 0 sf þ f f s f ; 2 2 2 g¼0 1 3 1 0 1 00 g 0 02 C fy Re2x ¼ g 0 a sg 0 f g 0 þ f g þ s g 00 bf g 0 df g 0 : 2 2 2 2 g¼0 ð14Þ x2 Xsina ð1st Þ1
where Rex ¼ is the rotational Reynolds number. t The non-dimensional heat and mass transfer rates are listed as
ð7Þ
1
NuRex 2 ¼ h0 ð0Þ;
1044
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048 1
ShRex 2 ¼ /0 ð0Þ:
ð15Þ
3. Solution technique Since we are interested in the analytical solutions of Eqs. (7)–(10), for this purpose we have employed homotopy analysis method (HAM) which is a famous mathematical technique established by Liao [24]. The corresponding initial guesses and the linear operators for the velocities f ðgÞ; gðgÞ; the temperature field hðgÞ and the concentration field /ðgÞ are stated below, respectively.
f 0 ðgÞ ¼ 0;
ð16Þ
g 0 ðgÞ ¼ ð1 cÞ þ ð2c 1Þ expðgÞ;
ð17Þ
h0 ðgÞ ¼ expðgÞ;
ð18Þ
/0 ðgÞ ¼ expðgÞ;
ð19Þ
3
ff ðgÞ ¼
d f df ; dg3 dg
fg ðgÞ ¼
d g g; dg2
Fig. 3. h-curves for h0 (0) and u0 (0) at 15th approximation.
ð20Þ
2
ð21Þ
Table 1 Convergence table for the flow problem. Order of convergence
f ð0Þ
g 0 ð0Þ
h0 ð0Þ
/0 ð0Þ
1 5 10 15 20
0.24053 0.25188 0.25168 0.25168 0.25168
0.23918 0.24783 0.24938 0.24934 0.24934
1.6044 1.60443 1.60425 1.60425 1.60425
2.42222 2.07728 2.07855 2.07855 2.07855
2
fh ðgÞ ¼
d h h; dg2
f/ ðgÞ ¼
d / /; dg2
ð22Þ
2
ð23Þ
4. Convergence of the analytical solutions Obviously the series solutions achieved by homotopy analysis h. This paramethod contain the convergence control parameter meter controls the convergence region and the rate of approximation of the HAM solution. To check the convergence of the solutions in the satisfactory range of the values of the auxiliary parameters g ; hf ; h hh and h/ ; h-curve for 15th-order approximations have been presented. Figs. 2 and 3 show that the admissible range of values g ; hf ; h hh and h/ are 1:3 6 hf 6 0:4; 1:2 6 hg 6 0:4; 1:3 6 of
Fig. 2. h-curve of f00 (0) and g0 (0) at 15th approximation.
00
h 6 0:3 and 1:2 6 h h/ 6 0:4. The convergence Table 1 made for each of the function up to 25th order of approximations. 5. Results and discussion This section is dedicated to observe and discuss the analytical results for a different range of various physical parameters like, ratio of angular velocities c, ratio of the buoyancy forces N, second grade parameter a and b, Third grade parameter d, Brownian
0
Fig. 4. Effects of c and k1 on tangential velocity f ðgÞ.
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
1045
0
Fig. 5. Effects of a on tangential velocity f ðgÞ.
Fig. 8. Effects of c and k1 on azimuthal velocity gðgÞ.
0
Fig. 6. Effects of b on tangential velocity f ðgÞ. Fig. 9. Effects of a on azimuthal velocity gðgÞ.
0
Fig. 7. Effects of d on tangential velocity f ðgÞ. Fig. 10. Effects of b on azimuthal velocity gðgÞ.
motion parameter Nb, Thermophoresis parameter Nt, Schmidt number Sc, Prandtl number Pr and unsteadiness parameter s. 0 Figs. 4–7 refer to the variation of tangential velocity f ðgÞ for c; a; b and d respectively. It is establish from Fig. 4 that when c ¼ 0:5 the fluid and the cone are in rotation with compatible
angular velocity in the similar direction and the flow is only due to the favorable pressure gradient i.e. k1 = 1. For c > 0:5, the 0 magnitude of velocity f ðgÞ increases on the other hand the variation reduces for c < 0:5. It is found that for c < 0 the velocity 0 field f ðgÞ reaches asymptotically at the edge of the boundary
1046
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
Fig. 11. Effects of d on azimuthal velocity gðgÞ. Fig. 14. Effects of Nb on temperature hðgÞ and nanoparticle volume fraction /ðgÞ.
1
Fig. 12. Effects of N on surface skin friction coefficient C fx Re2x .
Fig. 15. Effects of Nt on temperature hðgÞ and nanoparticle volume fraction /ðgÞ.
1
Fig. 13. Effects of N on surface skin friction coefficient C fy Re2x .
layer in an oscillatory style. Actually such oscillations occur due to the surplus convection of angular momentum seems in the region 0 of boundary layer. Figs. 5 and 6 clarifies that the velocity f ðgÞ follows the reducing behavior for second grade parameter a, 0 whereas increase in magnitude is found for b. The velocity f ðgÞ
Fig. 16. Effects of Pr and Sc temperature hðgÞ and nanoparticle volume fraction /ðgÞ.
1047
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048 Table 3 Values of surface Skin friction coefficients for some relevant parameters.
a
b
1
d
0.0 1.0 2.0 0.0 0.2 0.4 0.0 1.0 2.0
1
C fx Re2x
C fy Re2x
0.00226 0.18848 1.32178 0.01533 0.02063 0.02597 0.07423 0.07930 0.08546
0.24422 0.48514 0.57451 0.26805 0.26777 0.26704 0.26635 0.28235 0.30028
Table 4 Values of heat transfer and mass transfer rates for some relevant parameters. Pr Fig. 17. Effects of Nb and Nt on surface heat transfer rate h0 ð0Þ.
Sc
N
1
3.0 8.0 13.0 0.5 1.0 1.5 0.0 0.5 1.0
Fig. 18. Effects of Nb and Nt on surface mass transfer rate /0 ð0Þ.
varies inversely with d (see Fig. 7). The influences of c; a; b and d on azimuthal velocity gðgÞ is presented in Figs. 8–11. It is depicted from Fig. 8 that the velocity gðgÞ reduces for c > 0:5 but the behavior is converse when c < 0:5. It is demonstrated in Figs. 9 and 10 that the velocity gðgÞ increases by increasing a and b respectively. Fig. 11 is plotted to show that velocity gðgÞ increases upon decreasing d. Figs. 12 and 13 are keen to notice the variations of skin friction coefficients in tangential and azimuthal directions for various values of N, respectively. Both the skin friction coefficients increase with an increase in N, respectively. It /ðgÞ is due to the fact that wall temperature at the cone boundaries is somewhat greater than
1
NuRex2
ShRex2
2.84871 3.45558 3.62775 1.57685 1.55824 1.54708 0.82487 0.84184 0.85625
0.01595 0.37508 0.47130 0.55885 1.25681 1.76238 0.52556 0.53311 0.54066
the temperature of the fluid which eventually increases the Gr 2 as compared to Gr 1 , thus higher values of N provide the larger values of skin friction coefficients. Besides it is found that as s increases from 0 to 0.5, tangential coefficient of skin friction reduces but the deviations are opposed for azimuthal coefficient of skin friction. Both the temperature hðgÞ and nanoparticle volume fraction /ðgÞ have opposite influences against different values of Brownian motion parameter Nb (see Fig. 14). The effects of Thermophoresis parameter Nt on temperature hðgÞ and nanoparticle volume fraction /ðgÞ is seen in Fig. 15. The figure shows that both temperature hðgÞ and nanoparticle volume fraction /ðgÞ enhances their variation for greater values of Nt. Fig. 16 is sketched for temperature hðgÞ and nanoparticle volume fraction /ðgÞ against Pr and Sc, respectively. It is interesting to know that both temperature hðgÞ and nanoparticle volume fraction reduces for Pr and Sc. The Nusselt number and the Sherwood number shows an increasing behavior for increasing values of N and s (see Figs. 15 and 16). Figs. 17 and 18 displays the effects of Nb and Nt on heat transfer rate h0 ð0Þ and mass transfer rate /0 ð0Þ respectively. It is revealed in Fig. 17 and 18 the change in the dimensionless heat transfer rates is seen to be greater for smaller values of the parameter Nb and this change reduces with the rise in Nt. Table 2 is testified to authorize the exactness of our analytical results. It is noticed from the Table 2 that present HAM findings and numerical results [8] follows the satisfactory criteria. The mathematical values of Skin friction coefficients in both directions
Table 2 Comparison table for important physical quantities for special case. k1
c
Present results 1 2
Anilkumar and Roy results [8] 1 2
1 2
1 2
1
1
1
1
C fx Rex
C fy Rex
NuRex
ShRex
C fx Re2x
C fy Re2x
NuRex2
ShRex2
1.0
0.0 0.25 0.50
0.63238 1.31328 1.84791
0.63940 0.22758 0.19800
0.81920 0.89010 0.93702
0.95060 1.02817 1.07979
0.63241 1.31339 1.84798
0.63949 0.22765 0.19806
0.81922 0.89011 0.93700
0.95065 1.02812 1.07977
3.0
0.0 0.25 0.50
3.79518 4.31850 4.73958
0.59658 0.13687 0.33554
1.02862 1.06525 1.09111
0.18648 1.22640 1.25442
3.79522 4.31854 4.73958
0.59651 0.13691 0.33552
1.02869 1.06539 1.09111
0.18645 1.22639 1.25444
1048
S. Nadeem, S. Saleem / International Journal of Heat and Mass Transfer 85 (2015) 1041–1048
for different evolving constraints such as a; b and d are demonstrated in Table 3. The impact of a; b and d is to reduce the tangential skin friction coefficient, while the azimuthal skin friction coefficient is a direct function of a and d, but varies in inverse way for b: Table 4 indicates that the heat transfer and mass transfer rates shows opposite behaviors for increasing values of Pr and Sc, respectively. The ratio of buoyancy forces N increases the magnitude of both heat transfer and mass transfer rates gradually. 6. Conclusions We have theoretically examined the unsteady boundary layer flow of third grade nanofluid over a rotating cone. The model used for the nanofluid includes the effects of Brownian motion and thermophoresis. The transformed ordinary differential equations are elucidated by a strong mathematical method recognized as homotopy analysis method (HAM). The analytical results are agreed to be in conventional settlement with the previous results in literature The Nusselt number and the Sherwood number shows an increasing behavior for increasing values of N and s. The results of [8] can be recovered in the absence of nanoparticles. The Brownian motion parameter Nb increases the temperature and nanoparticle volume fraction, but the effects are just opposite for thermophoresis parameter Nt. The acquired results have promising applications in engineering and will now be available for experimental verification to give confidence for the well-posedness of this nonlinear boundary value problem. Conflict of interest It is stated that all the authors have seen the article and agree to submit it in International Journal of Heat and Mass Transfer. References [1] O.D. Makinde, Irreversibility analysis for gravity driven non-Newtonian liquid film along an inclined isothermal plate, Phys. Scr. 74 (2006) 642–645. [2] M. Qasim, Soret and Dufour effects on the flow of an Eyring–Powell fluid over a flat plate with convective boundary condition, Eur. Phys. J. Plus 129 (2014) 24. [3] Noor Fadiya Mohd Noor, Analysis for MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction, World Acad. Sci., Eng. Technol. 64 (2012) 1019–1023. [4] R. Ellahi, T Hayat, F.M. Mahomed, Generalized couette flow of a third-grade fluid with slip: the exact solutions, Z. Naturforsch. – Sect. A 65 (2010) 1071– 1076. [5] T. Hayat, R. Ellahi, S. Asghar, The influence of variable viscosity and viscous dissipation on the non-Newtonian flow: an analytical solution, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 300–331. [6] R. Ellahi, S. Afzal, Effects of variable viscosity in a third grade fluid with porous medium: an analytic solution, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 2056–2072.
[7] R.G. Hering, R.J. Grosh, Laminar combined convection from a rotating cone, ASME J. Heat Transfer 85 (1963) 29–34. [8] D. Anilkumar, S. Roy, Unsteady mixed convection flow on a rotating cone in a rotating fluid, Appl. Math. Comput. 155 (2004) 545–561. [9] M. Alamgir, Over all heat transfer from vertical cones in laminar free convection, an approximate method, ASME J. Heat Transfer 101 (1989) 174. [10] M.M. Rashidi, M.T. Rastegari, Analytical solution for free convection boundarylayer over a vertical cone in a non-Newtonian fluid saturated porous medium with internal heat generation, World Appl. Sci. J. 16 (2012) 64–74. [11] S. Saleem, S. Nadeem, R. Ul Haq, Buoyancy and metallic particle effects on an unsteady water-based fluid flow along a vertically rotating cone, Eur. Phys. J. Plus 129 (2014) 213–220. [12] C.Y. Wang, Boundary layers on rotating cones, discs and axisymmetric surfaces with a concentrated heat source, Acta Mech. 81 (1990) 245–251. [13] M.C. Ece, Free-convection flow about a cone under mixed thermal boundary conditions and a magnetic field, Appl. Math. Model. 29 (2005) 1121–1134. [14] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, Dev. Appl. Non-Newtonian Flows 66 (1995) 99–105. [15] C. Kleinstreuer, J. Li, J. Koo, Microfluidics of nano-drug delivery, Int. J. Heat Mass Transfer 51 (2008) 5590–5597. [16] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [17] P. Selvakumar, S. Suresh, Thermal performance of ethylene glycol based nanofluids in an electronic heat sink, J. Nanosci. Nanotechnol. 14 (2014) 2325– 2333. [18] Noreen Sher Akbar, M. Raza, R. Ellahi, Interaction of nano particles for the peristaltic flow in an asymmetric channel with the induced magnetic field, Eur. Phys. J. – Plus 129 (2014) 155–167. [19] Y. Rahmani, R. Yousefi, S.E. Ghasemi, D.D. Ganji, Thermal and fluid effects of non-Newtonian water-based nanofluids on the free convection flow between two vertical planes, Phys. Rev. Res. Int. 3 (4) (2013) 688–701. [20] R. Ellahi, M. Hassan, A. Zeeshan, Shape effects of nanosize particles in Cu–H2O nanofluid on entropy generation, Int. J. Heat Mass Transfer 81 (2015) 449–456. [21] O.D. Makinde, W.A. Khan, Z.H. Khan, Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet, Int. J. Heat Mass Transfer 62 (2013) 526–533. [22] A. Dominic, J. Sarangan, S. Suresh, Monica Sai, Heat transfer performance of Al2O3/water nanofluids in a mini channel heat sink, J. Nanosci. Nanotechnol. 14 (2014) 2368–2376. [23] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions, Appl. Math. Model. 37 (3) (2013) 1451–1467. [24] S.J. Liao, ‘‘Beyond Perturbation’’: a review on the homotopy analysis method and its applications, Adv. Mech. 153 (2008) 1–34. [25] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A 360 (2006) 109–113. [26] S. Nadeem, S. Saleem, Analytical treatment of unsteady mixed convection MHD flow on a rotating cone in a rotating frame, J. Taiwan Inst. Chem. Eng. 44 (4) (2013) 596–604. [27] S. Nadeem, S. Saleem, Unsteady mixed convection flow of nanofluid on a rotating cone with magnetic field, Appl. Nanosci. 4 (2013) 405–414. [28] N.F.M. Noor, O. Abdulaziz, I. Hashim, MHD flow and heat transfer in a thin liquid film on an unsteady stretching sheet by the homotopy analysis method, Int. J Numer. Method Fluid 63 (2010) 357–373. [29] R. Ellahi, A study on the convergence of series solution of non-Newtonian third grade fluid with variable viscosity: by means of homotopy analysis method, Adv. Math. Phys. 2012 (2012) 11. Article ID 634925. [30] S. Nadeem, S.T. Hussain, C. Lee, Flow of a Williamson fluid over a stretching sheet, Braz. J. Chem. Eng. 30 (2013) 619–625.