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Unsteady slip flow of Carreau nanofluid over a wedge with nonlinear radiation and new mass flux condition
Results in Physics 7 (2017) 2261–2270
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Unsteady slip flow of Carreau nanofluid over a wedge with nonlinear radiation and new mass flux condition M. Khan a, M. Azam a,⇑, A.S. Alshomrani b a b
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 27 April 2017 Received in revised form 2 June 2017 Accepted 23 June 2017 Available online 4 July 2017 Keywords: Unsteady wedge flow Carreau nanofluid Non-linear radiation Velocity slip and nanoparticles mass flux conditions
a b s t r a c t This article addresses a numerical investigation for the unsteady 2D slip flow of Carreau nanofluid past a static and/or moving wedge with the nonlinear radiation. A zero nanoparticle mass flux and convective boundary conditions are implemented. Further, the most recently devised model for nanofluid is adopted that incorporates the effects of Brownian motion and thermophoresis. A set of suitable transformation is demonstrated to alter the nonlinear partial differential equations into nonlinear ordinary differential equations and then tackled numerically by employing bvp4c in Matlab package. The numerical computations for the wall heat flux (Nusselt number) and wall mass flux (Sherwood number) are also performed. Effects of several controlling parameters on the velocity, temperature and nanoparticles concentration are explored and discussed in detail. Our study reveals that the temperature and the associated thermal boundary layer thickness are enhancing function of the temperature ratio parameter for both shear thickening and shear thinning fluids. Moreover, it is noticed that the velocity in case of moving wedge is higher than static wedge. Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction From last two decades, modern evaluation in engineering has pushed the advanced thermal systems to enhance their performance. This can be achieved by the possibility of using nanofluids in such systems as the characteristic feature of nanofluids in thermal conductivity enhancement. The concept of nanofluid was devised by Choi [1] in 1995. There are two main methods used to develop nanofluids which are the single-step and the two-step method (see Akoh et al. [2] and Eastman et al. [3]). Both of these techniques possess advantages and disadvantages as reported by Wang and Mujumdar [4]. Transport properties of nanofluids can be analyzed by two different models proposed by Buongiorno [5] and Tiwari and Das [6]. Here, we are adopting Buongiorno’s model to study the mechanism of heat transfer in nanofluids. Buongiorno proposed a model that ignores the limitations of dispersion and homogeneous models. He demonstrated the seven slip mechanisms that generate a parallel velocity between the base fluid and nanoparticles. These are inertia, Magnus effect, thermophoresis, Brownian diffusion, fluid drainage, gravity and diffusiophoresis. He concluded that thermophoresis and Brownian diffusion are
⇑ Corresponding author. E-mail address: [email protected] (M. Azam).
main slip mechanisms in nanofluids. On the evidence of these facts, he demonstrated two-component four-equation nonhomogeneous equilibrium model subject to mass, momentum and heat transport in nanofluids. In view of such facts, several studies on nanofluids were presented by many authors. Makinde and Aziz [7] reported the problem of boundary layer flow of nanofluid in the presence of convective surface conditions. They observed the nanoparticle concentration is enhanced with the enhancment of the Biot number. A comparative analysis for convective heat and mass transfer in nanofluid was studied by Sandeep et al. [8]. They concluded that the rate of heat and mass transfer in Oldroyd-B nanofluid is larger when compared to Maxwell and Jeffrey nanofluids. Khan et al. [9] conducted a numerical investigation on unsteady flows of Carreau nanofluid over contracting or expanding cylinder. They observed that the rate of heat transfer increases for growing values of Biot number. Umawathi and Sheremet [10] considered a problem of nanofluid in a rectangular conduct to analyze the impact of variable thermal conductivity. They noticed that the heat transfer rate is smaller for regular fluid when compared to nanofluid. Recently, Khan and Azam [11] presented the numerical study of unsteady heat and mass transfer of Carreau nanofluid in the presence of magnetic field. They noticed that the temperature and nanoparticles concentration are higher due to the presence of magnetic field. The Carreau viscosity model can characterize the rheology of several polymeric solutions such as pure poly ethylene oxide, 1%
http://dx.doi.org/10.1016/j.rinp.2017.06.038 2211-3797/Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Nomenclature x y u v t V T C
s
p I
l l0 l1 c_ C
g
W 0
f h / We NR Pr b C fx Nux Shx Re
distance along the surface distance normal to the surface velocity component in x-direction velocity component in y-direction time velocity field temperature field concentration field Cauchy stress tensor pressure identity tensor apparent viscosity zero shear rate viscosity infinite shear rate viscosity shear rate time material constant local similarity variable stream function dimensionless velocity dimensionless temperature dimensionless concentration local Weissenberg number radiation parameter Prandtl number wedge angle parameter local skin friction coefficient local Nusselt number local Sherwood number local Reynolds number
methylcellulose tylose in glycerol solution and 0:3% hydroxyethylcellulose. These polymers are intensively used in capillary electrophoresis to enhance the resolution in the partition of proteins. It predicts a specific area where the shear rate and viscosity have linear relationship. It also predicts the power law region. In the existing data, different researchers considered the problem of Falkner-Skan for several values of wedge angle parameter b using different methods like as the classical Blasius problem for b ¼ 0 [12]. The study of Falkner-Skan flow [13] of viscous fluid past a static or moving wedge has been presented by Riley and Weidman [14]. A finite difference scheme was used by Asaithambi [15] to find the numerical solution of the Falkner-Skan problem for 0:19884 6 b 6 2. Zaturska and Banks [16] presented the analytic solutions for b > 1. Yang and Chien [17] explored the analytical solutions for the wedge angle parameter b ¼ 1. Raju and Sandeep [18] conducted a comparative study for the heat and mass transfer analysis of Falkner-Skan and Blasius flow of Casson fluid over a wedge. They noted that rate of heat and mass transfer in Falkner-Skan flow is high when compared to Blasius flow. Khan et al. [19] reported a numerical investigation on unsteady flows of MHD Carreau nanofluid past a wedge. They concluded that thermal boundary layer thickness is an enhancing function of the Biot number. Sheremet et al. [20] presented a numerical investigation for the problem of unsteady natural convection of nanofluid in wavy-walled cavity using implicit finite difference scheme. They observed that an increment in the wavy contraction ratio leads to enhance in the wave amplitude. Raju and Sandeep [21] presented a work on Falkner-Skan flow of MHD Carreau fluid over a wedge with cross diffusion effects. They predicted that improving values of wedge angle parameter grows the velocity field. Khan et al. [22] presented a numerical study to analyze the impact of heat generation or absorption and melting phenomenon on
A1 n l
c q m am
Uw Tw Cw L T1 C1 DT DB k cp k
r
a; c hw Nt Nb Sc A
sw
qw qm k
first Rivlin-Ericksen tensor power law index slip length Biot number density of fluid kinematic viscosity thermal diffusivity stretching velocity surface temperature surface nanoparticle concentration slip parameter ambient fluid temperature ambient fluid concentration thermophoresis diffusion coefficient Brownian diffusion coefficient thermal conductivity specific heat capacity mean absorption coefficient Stefan–Boltzman constant constants temperature ratio parameter thermophoresis parameter Brownian motion parameter Schmidt number unsteadiness parameter wall shear stress wall heat flux wall mass flux velocity ratio parameter
Falkner-Skan flow of Carreau nanofluid past a wedge. They noted that nanoparticle concentration distribution decrease by growing values of melting parameter. On the other hand, the radiative characteristics of non-Newtonian fluids have been paid much attention because of its variable characteristics found in base fluids. Thermal radiations have key role in controlling heat transfer mechanism in polymer processing industrial area as the quality of final product depends on the factors of heat controlling. The influences of radiation in boundary layer flow were reported by many authors. Probably, Smith [23] reported the first study on the impact of radiation involving the boundary layer flow. Viskanta and Grosh [24] explored the influences of thermal radiation on heat transfer and temperature profiles in Falkner-Skan flow considering Rosseland approximation. They noticed that thermal radiation becomes an additional part in missile reentry, gas cooled nuclear reactors etc. Cortell [25] reported a numerical study for the characteristics of nonlinear thermal radiation in boundary layer flow over a stretching sheet. He observed that radiation parameter and temperature ratio parameter have opposite behavior qualitatively in temperature profiles. Pantokratoras [26] demonstrated a numerical investigation for the natural convection flow over a vertical plate subject to linear and nonlinear thermal radiation. He explored that when the wall shear stress enhances the wall heat transfer depreciates and vice versa. Khan et al. [27] analyzed the impact of homogeneous and heterogeneous reaction on the flow of Burger fluid. Kumaran et al. [28] reported a numerical study for MHD Maxwell and Casson fluid flow with cross diffusion effects. They noted that rate of heat and mass transfer was high in Maxwell fluid when compared to Casson fluid. Turkyilmazoglu [29] considered a problem of flow and heat transfer of micropolar fluid over a porous stretching sheet. He found the analytical unique solution of the considered problem. The effects of thermophoresis and Brownian
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motion on flow of Williamson and Casson fluids were analyzed numerically by Kumaran and Sandeep [30]. They examined that the performance of Casson fluid was better as compared to Williamson fluid. Khan and Khan [31] examined the impact of thermophoresis particle deposition on three dimensional flow of Burgers fluid in the presence of thermal radiation. They concluded that concentration field depreciated for improving values of thermophoretic parameter. Ali and Sandeep [32] demonstrated a numerical study to investigate the Cattaneo-Christov model for MHD Casson fluid. Turkyilmazoglu [33] also presented the multiple solutions for slip flow over a shrinking surface considering magnetic field effects. On the basis of motivation of the aforesaid investigations, the aim of present study is to present numerical solutions on the characteristics of nonlinear thermal radiation on the unsteady twodimensional Falkner-Skan flow of Carreau nanofluid past a static or moving wedge for 0 6 b 6 2. Additionally, velocity slip and convective boundary conditions are implemented at the boundary. Furthermore, zero nanoparticles mass flux condition is also implemented which leads to a more realistic physical problem. It is important to mention that Carreau viscosity model is an important class of generalized Newtonian fluid which characterizes the shear thinning as well as shear thickening nature of the fluid. This model reduces to viscous fluid when the power law index n ¼ 1 and We ¼ 0. Suitable transformations are utilized to alter the governing nonlinear boundary-layer equations to non-linear ordinary differential equations. Finally, these resulting equations are solved numerically by adopting an effective numerical approach bvp4c Matlab package. A comparison between the current results and existing study in limiting case is conducted with very good agreement. Mathematical analysis We consider the unsteady two-dimensional Falkner-Skan flow of a non-Newtonian Carreau fluid with the insertion of nanoparticles. Heat and mass transfer properties are explored through the Brownian motion and thermophoresis effects. The fluid flow is induced by a stretching wedge subject to stretching velocity
partial slip at the surface of wedge is implemented. The flux of nanoparticle volume fraction at the surface is assumed to be zero. The surface temperature T w of the wedge is also assumed to be higher than the ambient temperature ðT w > T 1 Þ. The Cauchy stress tensor s for Carreau viscosity model [34,35] can be written as n1
s ¼ pI þ lA1 ; l ¼ l0 ½1 þ ðCc_ Þ2 2 ; c_ ¼
ð1Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 trðA21 Þ; Ri Rj c_ ij c_ ji ¼ 2 2
ð2Þ
where I is the identity tensor, p the pressure, l0 the zero shear rate viscosity, C the time material parameter, c_ the shear rate, n the power law index and A1 the first Rivlin-Ericksen tensor. For the unsteady two-dimensional laminar flow, the velocity, temperature and nanoparticles concentration fields are demonstrated as
V ¼ ½uðx; y; tÞ; v ðx; y; tÞ; 0;
T ¼ Tðx; y; tÞ;
C ¼ Cðx; y; tÞ:
ð3Þ
Utilizing the aforesaid assumptions and boundary layer approximations, the governing equations of the Carreau nanofluid model can be represented as [19]
›u ›v þ ¼ 0; ›x ›y
ð4Þ
" 2 #n1 2 @u ›u ›u @U e @U e @2u 2 @u þu þv ¼ þ Ue þm 2 1þC @t ›x ›y @y @y @t @x 2 2 @ u @u þ m ð n 1 Þ C2 2 @y @y " 2 #n3 2 @u 1 þ C2 ; @y
ð5Þ
" 2 # @T @T @T @2T @C @T DT @T 1 @qr þu þv ¼ am 2 þ s DB þ ; @t @x @y @y @y @y T 1 @y qcp @y ð6Þ
m
bx U w ðx; tÞ ¼ 1ct . It is stated here that U w ðx; tÞ > 0 indicates the stretching wedge surface velocity and U w ðx; tÞ < 0 relates to a contracting wedge surface velocity. The free stream velocity of the axm considered problem is U e ðx; tÞ ¼ 1ct where a; b; c and m are positive constants. In Fig. 1, the total angle of the wedge is defined by 2m is the wedge angle parameter. The nonlinear X ¼ bp and b ¼ mþ1
thermal radiation is also taken into account. The temperature at the surface of the wedge is controlled by a convective heating analysis which is referred to heat transfer coefficient hf . The velocity
@C @C @C @ 2 C DT @ 2 T þu þv ¼ DB 2 þ : @t @x @y @y T 1 @y2
ð7Þ
The appropriate boundary conditions in the current problem are as follows
u ¼ U w þ U slip ; DB
v ¼ 0;
k
@T ¼ hf ðT w T Þ; @y
@C DT @T þ ¼ 0 at y ¼ 0; @y T 1 @y
u ! Ue ;
T ! T1;
C ! C1
Here, s ¼ ðqcÞp =ðqcÞf
ð8Þ as y ! 1:
ð9Þ
is the ratio of the effective heat capacity
of the nanoparticle to the effective heat capacity of the base fluid, am ¼ qkcp the thermal diffusivity with cp the specific heat, q the density of fluid and k the thermal conductivity, DT the thermophoresis diffusion coefficient, DB the Brownian diffusion coefficient, qr the radiative heat flux and C 1 the ambient fluid concentration. The partial slip velocity U slip for Carreau fluid model in present problem can be stated as:
U slip Fig. 1. Physical description of flow problem.
" 2 #n1 2 @u 2 @u ¼l ; 1þC @y @y
ð10Þ
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where l is the slip length having dimension of length. Utilizing the Rosseland approximation for the radiation [23], the simplified form of radiative heat flux can be written as
qr ¼
4r @T 4 ; 3k @y
ð11Þ
where k and r indicate the mean absorption coefficient and Stefan-Boltzmann constant, respectively. It is stated that unlike classical cases the nonlinear structure of thermal radiation is considered here. Note that, the impact of thermal radiation in the linearized Rosseland approximation is simply a rescaling of the Prandtl number by a factor including the radiation parameter. Thus it is noted here that the energy equation with the addition of nonlinear thermal radiation is highly nonlinear in including the additional temperature ratio parameter. It is important to mention here that for a planer boundary layer flow over a horizontal flat surface [23], from Eq. (7), we obtain
qr ¼
16r 3 @T : T @y 3k
ð12Þ
By substituting of Eq. (12), in Eq. (6) results in the energy equation with nonlinear thermal radiation in the form.
! # 16r T 3 @T am þ 3k qcp @y " 2 # @C @T DT @T þ s DB þ : @y @y T 1 @y
@T @T @T @ þu þv ¼ @t @x @y @y
"
ð13Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 1Þue 2mxue ; Wðx; y; tÞ ¼ f ðgÞ; g¼y 2m x mþ1 T T1 C C1 hðgÞ ¼ ; /ðgÞ ¼ ; Tw T1 C1
ð14Þ
where W is the stream function, T ¼ T 1 ð1 þ ðhw 1ÞhÞ and hw ¼ TT1w > 1 the temperature ratio parameter. In perspective of Eq. (14), Eqs. (5), (7) and (13) reduce to the following ordinary equations.
n on on3 n 0 2 o 2 00 2 00 2 000 00 1 þ nWe2 ðf Þ f þ ff þ b 1 f 1 þ We2 ðf Þ n o g 00 0 A f þ f 1 ¼ 0; 2 A 2 h00 þ Pr f h0 Pr gh0 þ Nbh0 /0 þ Ntðh0 Þ 2 i 4 d h þ ð1 þ ðhw 1ÞhÞ3 h0 ¼ 0; 3NR dg A Nt 00 h ¼ 0; /00 þ Scf /0 Sc g/0 þ 2 Nb
f ð0Þ ¼0;
0
hð1Þ ! 0;
Sc ¼ hw ¼
Pr ¼
m 2c ; A¼ ; ðm þ 1Þaxm1 am
hf sDT ðT w T 1 Þ sDB C 1 ; Nb ¼ ; c¼ am T 1 am k m DB
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mx ; ð1 þ mÞue
;
Tw ; T1
NR ¼
kk 4r
T3 1
;
b¼
2m ; mþ1
b k¼ : a
ð20Þ
The physical quantities of engineering interest are the local skin friction coefficient C f , the local Nusselt number Nu and the local Sherwood number Sh. They are defined as
sw jy¼0 xqw xqm ; Nu ¼ ; ; Sh ¼ qu2e kðT w T 1 Þ DB C 1
ð21Þ
with sw ; qw and qm as the wall shear stress, wall heat flux and wall mass flux, respectively and they are given by
" 2 #n1 2 @u 2 @u ; 1þC sw ¼ l0 @y @y y¼0 @T @C þ q j ; q ¼ D : qw ¼ k B r m y¼0 @y y¼0 @y y¼0
ð22Þ
The dimensionless surface drag, heat transfer rate and mass transfer rate are thus given by
h in1 2 2 00 00 ð2 bÞ1=2 Re1=2 C f ¼ f ð0Þ 1 þ We2 ðf ð0ÞÞ ;
n o 4 ½1 þ ðhw 1Þhð0Þ3 ; ð2 bÞ1=2 Re1=2 Nu ¼ h0 ð0Þ 1 þ 3N R ð2 bÞ1=2 Re1=2 Sh ¼ /0 ð0Þ; where Re ¼
xU e
m
ð23Þ ð24Þ ð25Þ
is the local Reynolds number.
ð15Þ
Discussion of numerical results
ð16Þ
The behavior of unsteady slip flow of Carreau nanofluid in the presence of nonlinear radiation and new mass flux condition is studied numerically. The numerical computations have been performed using Matlab function bvp4c for distinct values of physical parameters namely the local Weissenberg number We, unsteadiness parameter A, power law index n, wedge angle parameter b,
ð17Þ
2
00
2
f ð0Þ ¼ k þ Lf ð0Þ 1 þ We ðf ð0ÞÞ
h0 ð0Þ ¼ cð1 hð0ÞÞ; f ð1Þ ! 1;
h
Nt ¼
Table 1 Comparison values of ð2 bÞ1=2 Re1=2 Nu for different values of Pr and b when n ¼ 1; c ¼ N R ¼ 106 ; hw ¼ 1:7 and We ¼ Nt ¼ Nb ¼ k ¼ L ¼ Sc ¼ A ¼ 0.
where prime indicates the differentiation with respect to g. The associated conditions of the present problem are the 00
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 ðm þ 1ÞU 3e ; We ¼ 2mx rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 1ÞU e L¼l ; 2mx
Cf ¼
The non-dimensional suitable variables can be demonstrated in the following way
0
the Brownian motion parameter, c the Biot number, N R the radiation parameter, k the velocity ratio parameter and Sc the Schmidt number. These parameters are stated as follows:
in1 2
Nb/0 ð0Þ þ Nth0 ð0Þ ¼ 0; /ð1Þ ! 0:
b¼0
; ð18Þ ð19Þ
In the above equations, We indicates the local Weissenberg number, Pr the Prandtl number, A the unsteadiness parameter, L the velocity slip parameter, Nt the thermophoresis parameter, Nb
b ¼ 0:3
Pr
White [36]
Present results
White [36]
Present results
0.1 0.3 0.6 0.72 1.0 2.0 6.0 10.0
0.1980 0.3037 0.3916 0.4178 0.4696 0.5972 0.8672 1.0297
0.198129 0.303718 0.391675 0.418091 0.469600 0.597234 0.867278 1.029747
0.2090 0.3278 0.4289 0.4592 0.5195 0.6690 0.9872 1.1791
0.209152 0.327829 0.428924 0.459551 0.519519 0.669045 0.987268 1.179130
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M. Khan et al. / Results in Physics 7 (2017) 2261–2270
Prandtl number Pr, radiation parameter N R , temperature ratio parameter hw , Biot number c, velocity slip parameter L, Brownian motion parameter Nb, Schmidt number Sc, velocity ratio parameter k and thermophoresis parameter Nt. For the better understanding the computed results are depicted through tables and plots and discussed in detail.
To assess the accuracy of the method which we have used, a comparison with the existing data is conducted in limiting manners. Our results of the local Nusselt number for some selected values of the Prandtl number and the wedge angle parameter are compared in limiting cases with those of White [36] (see Table 1). There is an excellent agreement between the present results and
Table 2 Numerical values of the local Nusselt number for different values of A; b; N R ; hw ; c; L; Nt; Nb and Sc when We ¼ 1:0; k ¼ 0:3 and Pr ¼ 3:0. ð2 bÞ1=2 Re1=2 Nu
Parameters A
b
NR
hw
c
L
Nt
Nb
Sc
n ¼ 0:5
n ¼ 1:5
0.0 0.2 0.3 0.2
0.3
1.0
1.2
0.1
0.2
0.1
0.1
1.0
1.0
1.2
0.1
0.2
0.1
0.1
1.0
0.2
0.0 0.1 0.2 0.3
1.2
0.1
0.2
0.1
0.1
1.0
0.2
0.3
1.0 1.5 2.0 1.0
0.1
0.2
0.1
0.1
1.0
0.2
0.3
1.0
1.3 1.5 1.7 1.2
0.2
0.1
0.1
1.0
0.2
0.3
1.0
1.2
0.2 0.3 0.4 0.1
0.1
0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.0 0.2 0.4 0.2
0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.2
0.1 1.0 3.0 0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.2
0.1
0.2 0.3 0.4 0.1
0.2132826 0.2119981 0.2112568 0.2109825 0.211397 0.2117267 0.2119981 0.1718841 0.1517178 0.2166362 0.2266575 0.2378075 0.3863935 0.5304262 0.6502942 0.2108292 0.2119981 0.2127863 0.2119981 0.2119351 0.2117922 0.2119981 0.2119981 0.2119981 0.2119939 0.2119908 0.2119885
0.2131824 0.2118712 0.2111134 0.2108818 0.2112853 0.2116066 0.2118712 0.1717823 0.1516284 0.2165315 0.2266048 0.2378187 0.3859495 0.5295584 0.648955 0.2105249 0.2118712 0.2127297 0.2118712 0.2118075 0.211663 0.2118712 0.2118712 0.2118712 0.2118669 0.2118638 0.2118614
2.0 3.0 4.0
Table 3 Numerical values of the local Sherwood number for different values of A; b; N R ; hw ; c; L; Nt; Nb and Sc when We ¼ 1:0; k ¼ 0:3 and Pr ¼ 3:0. j/0 ð0Þj
Parameters A
b
NR
hw
c
L
Nt
Nb
Sc
n ¼ 0:5
n ¼ 1:5
0.0 0.2 0.3 0.2
0.3
1.0
1.2
0.1
0.2
0.1
0.1
1.0
1.0
1.2
0.1
0.2
0.1
0.1
1.0
0.2
0.0 0.1 0.2 0.3
1.2
0.1
0.2
0.1
0.1
1.0
0.2
0.3
1.0 1.5 2.0 1.0
0.1
0.2
0.1
0.1
1.0
0.2
0.3
1.0
1.3 1.5 1.7 1.2
0.2
0.1
0.1
1.0
0.2
0.3
1.0
1.2
0.2 0.3 0.4 0.1
0.1
0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.0 0.2 0.4 0.2
0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.2
0.1 1.0 3.0 0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.2
0.1
0.2 0.3 0.4 0.1
0.08758551 0.08683194 0.08639908 0.08623937 0.08648082 0.08667329 0.08683194 0.08792725 0.08853906 0.08665592 0.08628157 0.08587438 0.1526004 0.2033918 0.2434015 0.08615018 0.08683194 0.08729382 0.08683194 0.8679508 2.601347 0.04341597 0.02894398 0.02170798 0.08682944 0.08682765 0.08682627
0.08752656 0.08675772 0.08631555 0.08618075 0.08641574 0.08660313 0.08675772 0.08785772 0.0884725 0.08657982 0.0862014 0.08578962 0.1523621 0.2029567 0.2427659 0.08597331 0.08675772 0.08726056 0.08675772 0.8672049 2.599083 0.04337886 0.02891924 0.02168943 0.08675519 0.08675338 0.08675198
2.0 3.0 4.0
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M. Khan et al. / Results in Physics 7 (2017) 2261–2270
those of White. This gives the confident on our numerical computations. To analyze the influences of pertinent parameters on the local Nusselt and local Sherwood numbers, numerical computations
have been executed for these parameters for both shear thickening ðn > 1Þ and shear thinning ð0 < n < 1Þ fluids. On the evidence of Tables 2 and 3, it is noted that the local Nusselt number and magnitude of local Sherwood number depreciate by increasing the val-
1
1
0.9
0.9
λ = 0.0
0.8 0.7
λ = 0.3
0.7
β = 0.0, 0.1, 0.2, 0.3
f '(η)
0.6
f '(η)
λ = 0.0
0.8
λ = 0.3
0.5 0.4 0.3
β = 0.0, 0.1, 0.2, 0.3
0.6 0.5 0.4 0.3
A = 0.2, L = 0.2, We = 1.0
A = 0.2, L = 0.2, We = 1.0
0.2
0.2
0.1
0.1
(b) n = 1.5
(a) n = 0.5
0
0
1
2
3
4
0
5
0
1
2
η
η
3
4
5
0
Fig. 2. Effects of the wedge angle parameter b on the velocity f ðgÞ profiles.
1
1
0.9 0.8
f '(η)
0.5 0.4
L = 0.0, 0.2, 0.4, 0.6
0.6 0.5 0.4 0.3
A = 0.2, β = 0.3, We = 1.0
0.2
A = 0.2, β = 0.3, We = 1.0
0.2
0.1 0
λ = 0.3
0.7
L = 0.0, 0.2, 0.4, 0.6
0.6
0.3
λ = 0.0
0.8
λ = 0.3
0.7
f '(η)
0.9
λ = 0.0
0.1
(a) n = 0.5 0
1
2
η
3
4
0
5
(b) n = 1.5 0
1
2
η
3
4
5
0
Fig. 3. Effects of the velocity slip parameter L on the velocity f ðgÞ profiles.
0.35
0.3
N R = 0.1, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
0.25
0.2
0.2
θ(η)
0.25
θw = 1.3, 1.5, 1.7, 1.9
0.15
0.1
0.05
0.05
0
1
2
3
4
η
5
6
7
8
N R = 0.1, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
θw = 1.3, 1.5, 1.7, 1.9
0.15
0.1
0
(b) n = 1.5 A = 0.2, λ = 0.2, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
A = 0.2, λ = 0.2, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
0.3
θ(η)
0.35
(a) n = 0.5
0
0
1
2
3
4
η
Fig. 4. Effects of the temperature ratio parameter hw on the temperature hðgÞ profiles.
5
6
7
8
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Nusselt number and magnitude of local Sherwood number are the growing functions of the Biot number as well as velocity slip parameter in both cases. It is also observed that the local Nusselt number depreciates by improving the values of the thermophoresis parameter but opposite behavior has been predicted in the magnitude of local Sherwood number. Additionally, the magnitude
ues of the unsteadiness parameter, radiation parameter and Schmidt number both for shear thickening and shear thinning fluids. Whereas, the local Nusselt number is an enhancing function of the temperature ratio parameter but opposite trend has been noticed in the magnitude of the local Sherwood number both for shear thickening and shear thinning fluids. Furthermore, the local
(a) n = 0.5
(b) n = 1.5
A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
θw = 1.2, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
θw = 1.2, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
0.1
θ(η)
θ(η)
0.1
0.05
0.05
N R = 1.0, 1.5, 2.0, 2.5
0
0
1
2
N R = 1.0, 1.5, 2.0, 2.5
3
η
0
4
0
1
2
3
η
4
Fig. 5. Effects of the radiation parameter N R on the temperature hðgÞ profiles.
0.4
0.4
(a) n = 0.5 A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, θw = 1.2
0.35
N R = 1.0, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
0.3
θ(η)
0.2
0.2 0.15
γ = 0.1, 0.2, 0.3, 0.4
0.1
0.1
0.05
0.05
0
1
2
3
η
0
4
0.05
0.05
0
0
γ = 0.1, 0.2, 0.3, 0.4
-0.05
-0.1
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2, β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.15
γ = 0.1, 0.2, 0.3, 0.4
0
2
3
η
4
γ = 0.1, 0.2, 0.3, 0.4
-0.1
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2, β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.15
Sc = 1.0, Nt = Nb = 0.1
Sc = 1.0, Nt = Nb = 0.1
-0.2
-0.2
(c) n = 0.5 -0.25
1
-0.05
φ(η)
θ(η)
N R = 1.0, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
0.25
0.15
φ(η)
A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, θw = 1.2
0.3
0.25
0
(b) n = 1.5
0.35
0
1
(d) n = 1.5 2
η
3
4
-0.25
0
1
2
η
Fig. 6. Effects of the Biot number c on the temperature hðgÞ and nanoparticle concentration /ðgÞ profiles.
3
4
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observed that the velocity of the fluid increases for growing the values of the velocity slip parameter in static and moving wedge for both cases of shear thickening and shear thinning fluids. It is also noticed that the velocity slip parameter has a substantial impact on the solution. For L ¼ 0, no slip condition case is achieved. The variation in the dimensionless temperature for distinct values of the temperature ratio parameter hw is depicted in Fig. 4(a) and (b) for both shear thickening and shear thinning fluids. On the evidence of these Figs., it is noticed that growing values of temperature ratio parameter relates to higher wall temperature as compare to ambient fluid. As a result fluid temperature enhances. It is also noted that the thermal boundary layer thickness enhances for growing the values of the temperature ratio parameter. Fig. 5(a) and (b) demonstrate the influences of radiation parameter on the dimensionless temperature profiles. On the behalf of these Figs., one can decide that the temperature and associated thermal boundary layer thickness depreciate for enhancing the values of the radiation parameter N R in shear thickening and shear thinning fluids. Fig. 6(a) and (b) are plotted to examine the influence of Biot number c on the temperature and nanoparticle concentration profiles. From these Figs., it is predicted that the temperature, nanoparticle concentration and their related boundary layer thicknesses increase in shear thickening and shear thinning fluids. For c ¼ 0, the surface of the wedge is totally insolated. In fact, the
of local Sherwood number is a depreciating function of the Brownian motion parameter in both cases. The influence of the wedge angle parameter b is illustrated in Fig. 2(a) and (b) where velocity profiles are shown in both cases of shear thickening and shear thinning fluids for static and moving wedge. From these Figs., it can be seen that velocity of the fluid enhances by growing the values of the wedge angle parameter for both the cases of static and moving wedge in shear thickening and shear thinning fluids. It is also clear that the velocity of the fluid is larger in the case of moving wedge when compared to static wedge. It is important to state here that k is the constant moving wedge parameter or the velocity ratio parameter with k > 0 and k < 0 relates to a moving wedge with the same and opposite directions to the free stream velocity, respectively. Note that the wedge angle parameter is the Hartree pressure gradient parameter which relates to b ¼ Xp with a total wedge angle X. On perspective of White [26], positive values of b indicates that the pressure gradient is favorable or negative then the flow will be accelerating along the surface. Negative values of b shows that the pressure gradient is adverse then the flow will be decelerating. Additionally, b ¼ 0 ðm ¼ 0Þ corresponds to boundary layer flow over a horizontal flat plate and b ¼ 1 ðm ¼ 1Þ relates to boundary layer flow near the stagnation point of a vertical flat plate. To analyze the impact of velocity slip parameter L on the velocity profiles, Fig. 3(a) and (b) are plotted. From these Figs., it can be
0.15
0.15
(a) n = 0.5
(b) n = 1.5 A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
N R = 1.0, Pr = 3.0, Sc = 1.0, θw = 1.2, Nb = 0.1
N R = 1.0, Pr = 3.0, Sc = 1.0, θ w = 1.2, Nb = 0.1 0.1
θ(η)
θ(η)
0.1
Nt = 1.0, 3.0, 5.0, 7.0
0.05
0
0
1
2
η
0
4
1
1
0
0
-1 -2
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-3
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0, Sc = 1.0, γ = Nb = 0.1
-4 -5 -6
1
η
3
2
4
3
η
4
Nt = 1.0, 3.0, 5.0, 7.0
-2
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-3
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0, Sc = 1.0, γ = Nb = 0.1
-5
2
1
-4
(c) n = 0.5 0
0
-1
Nt = 1.0, 3.0, 5.0, 7.0
φ(η)
φ(η)
3
Nt = 1.0, 3.0, 5.0, 7.0
0.05
-6
(d) n = 1.5 0
1
2
η
3
Fig. 7. Effects of the thermophoresis parameter Nt on the temperature hðgÞ and nanoparticle concentration /ðgÞ profiles.
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both shear thickening and shear thinning fluids. The higher Schmidt number corresponds to lower nanoparticles concentration and its affiliated boundary layer thickness.
internal thermal resistance of the wedge surface is very large and convective heat transfer does not take place from the wedge surface to the cold fluid far away from the wedge. The influences of the thermophoresis parameter Nt on the temperature and nanoparticle concentration are depicted through Fig. 7(a) and (b). These Figs., reveal that an increment in the thermophoresis parameter causes to enhance the temperature profiles. From these Figs., it is also observed that the nanoparticle concentration and its related boundary layer thickness enhance for growing the thermophoresis parameter. Physically, the thermophoresis force grows with the improvement of Nt which tends to move nanoparticles from hot portion to cold portion and hence enhances the magnitude of the nanoparticles concentration profile. Fig. 8(a) and (b) indicate the variation of nanoparticles concentration profiles for some values of the Brownian motion parameter Nb. It is observed that an enhancement in the Brownian motion parameter depreciates the nanoparticles concentration distribution and its related boundary layer thickness. Brownian motion occurs due to the presence of nanoparticles and resulted in the decrement of the nanoparticles concentration thickness. To analyze the impact of the Schmidt number Sc on nanoparticle concentration profiles, Fig. 9(a) and (b) are presented. It is noted that the nanoparticle concentration and the related boundary layer thickness shrink with the enhancement of the Schmidt number for
Main findings The numerical solutions for the unsteady Falkner-Skan flow of Carreau nanofluid past a static or moving wedge in the presence of nonlinear thermal radiation have been investigated. The velocity slip and convective boundary conditions were also taken into account. Additionally, a zero nanoparticle mass flux condition at the boundary was implemented. An efficient Matlab solver bvp4c was used to solve the governing problem. The effects of pertinent parameters controlling the velocity, temperature and nanoparticles concentration distributions were presented graphically. Some of the key findings of the present investigations are listed below: The local Nusselt number and magnitude of local Sherwood number were depreciated by enhancing the values of the unsteadiness parameter, radiation parameter and Schmidt number both for shear thickening and shear thinning fluids and opposite was true for Biot number and velocity slip parameter.
0.02
0.02
(a) n = 0.5
0.01 0
0
-0.01
-0.01
Nb = 0.1, 0.2, 0.3, 0.4
Nb = 0.1, 0.2, 0.3, 0.4
-0.02
-0.03
φ(η)
-0.02
φ(η)
(b) n = 1.5
0.01
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-0.03
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-0.04
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.04
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.05
Sc = 1.0, γ = Nt = 0.1
-0.05
Sc = 1.0, γ = Nt = 0.1
-0.06
-0.06
-0.07
-0.07
-0.08
0
1
2
3
η
-0.08
4
0
1
2
3
η
4
Fig. 8. Effects of the Brownian motion parameter Nb on the nanoparticles concentration /ðgÞ profiles.
0.02
0.02
(a) n = 0.5
(b) n = 1.5
0.01
0.01
0
0 -0.01
-0.01
Sc = 1.0, 2.0, 3.0, 4.0
Sc = 1.0, 2.0, 3.0, 4.0 -0.02
-0.03
φ(η)
φ(η)
-0.02
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-0.03
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-0.04
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.04
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.05
γ = Nt = Nb = 0.1
-0.05
γ = Nt = Nb = 0.1
-0.06
-0.06
-0.07
-0.07
-0.08
0
1
2
η
3
4
-0.08
0
1
2
η
Fig. 9. Effects of the Schmidt number Sc on the nanoparticle concentration /ðgÞ profiles.
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The temperature and its related thermal boundary layer thickness were depreciating function of the radiation parameter both in shear thickening and shear thinning fluids and opposite is true for the temperature ratio parameter. The temperature and nanoparticles concentration were uplifted with the increment of the Biot number and thermophoresis parameter. The nanoparticles concentration was depressed with the increment of Brownian motion parameter and Schmidt number.
Acknowledgements The authors wish to convey their genuine thanks to the reviewers for their essential suggestions and comments to progress the superiority of this manuscript. The work of Dr. Alshomrani was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. References [1] Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. ASME Pub Fed 1995;231:99–106. [2] Akoh H, Tsukasaki Y, Yatsuya S, Tasaki A. Magnetic properties of ferromagnetic ultrafine particles prepared by a vacuum evaporation on running oil substrate. J Cryst Growth 1978;45:495–500. [3] Eastman JA, Choi SUS, Li S, Thompson LJ, Lee S. Enhanced thermal conductivity through the development of nanofluids. In: Materials research society symposium-proceedings, Materials research society, Pittsburgh, PA, USA, Boston, MA, USA, vol. 457; 1997. p. 3–11. [4] Wang XQ, Mujumdar AS. Heat transfer characteristics of nanofluids: a review. Int J Therm Sci 2007;46:1–19. [5] Buongiorno J. Convective transport in nanofluids. J Heat Transfer 2006;128 (3):240–50. [6] Tiwari RK, Das MK. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transfer 2007;50:2002–18. [7] Makinde OD, Aziz A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int J Therm Sci 2011;50:1326–32. [8] Sandeep N, Kumar BR, Kumar MSJ. A comparative study of convective heat and mass transfer in non-Newtonian nanofluid flow past a permeable stretching sheet. J Mol Liq 2015;212:585–91. [9] Khan M, Azam M, Alshomrani AS. On unsteady heat and mass transfer in Carreau nanofluid flow over expanding or contracting cylinder. J Mol Liq 2017;231:474–84. [10] Umavathi JC, Sheremet MA. Influence of temperature dependent thermal conductivity of a nanofluid in a vertical rectangular duct. Int J Non Linear Mech 2016;78:17–28. [11] Khan M, Azam M. Unsteady heat and mass transfer mechanisms in MHD Carreau nanofluid flow. J Mol Liq 2017;225:554–62. [12] Blasius H. Grenzschichten in Flussigkeiten mit kleiner Reibung. Z Math Phys 1908;56:1–37.
[13] Falkner VM, Skan SW. Some approximate solutions of the boundary-layer equations. Philos Mag 1931;12:865–96. [14] Riley N, Weidman PD. Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary. SIAM J Appl Math 1989;49:1350–8. [15] Asaithambi A. A finite difference method for the Falkner-Skan equation. Appl Math Comput 1998;92:135–41. [16] Zaturska MB, Banks WHH. A new solution branch of the Falkner-Skan equation. Acta Mech 2001;152:197–201. [17] Yang HT, Chien LC. Analytic solutions of the Falkner-Skan equation when b ¼ 1 and c ¼ 0. SIAM J Appl Math 1975;29:558–96. [18] Raju CSK, Sandeep N. A comparative study on heat and mass transfer of the Blasius and Falkner-Skan flow of a bio-convective Casson fluid past a wedge. Eur Phys J Plus 2016;131:405. [19] Khan M, Azam M, Munir A. On unsteady Falkner-Skan flow of MHD Carreau nanofluid past a static/moving wedge with convective surface condition. J Mol Liq 2017;230:48–58. [20] Sheremet MA, Pop I, Rosca NC. Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid: Buongiorno’s mathematical model. J Taiwan Inst Chem Eng 2016;61:211–22. [21] Raju CSK, Sandeep N. Falkner-Skan flow of a magnetic-Carreau fluid past a wedge in the presence of cross diffusion effects. Eur Phys J Plus 2016;131:267. [22] Khan M, Azam M, Alshomrani AS. Effects of melting and heat generation/ absorption on unsteady Falkner-Skan flow of Carreau nanofluid over a wedge. Int J Heat Mass Transfer 2017;110:437–46. [23] Smith WJ. Effect of gass radiation in boundary layer on aerodynamic heat transfer. J Aerosol Sci 1952;20:579–80. [24] Viskanta R, Grosh RJ. Boundary layer in thermal radiation absorbing and emitting media. Int J Heat Mass Transfer 1962;5:795–806. [25] Cortell R. Fluid flow and radiative nonlinear heat transfer over a stretching sheet. J King Saud Uni Sci 2014;26:161–7. [26] Pantokratoras A. Natural convection along a vertical isothermal plate with linear and non-linear Rosseland thermal radiation. Int J Therm Sci 2014;84:151–7. [27] Khan WA, Alshomrani AS, Khan M. Assesment on characteristics of heterogeneous-homogeneous process in three-dimensional flow of Burgers fluid. Results Phys 2016;6:772–9. [28] Kumaran G, Sandeep N, Ali ME. Computational analysis of magnetohydrodynamic Casson and Maxwell flows over a stretching sheet with cross diffusion. Results Phys 2017;7:147–55. [29] Turkyilmazoglu M. Flow of micropolar fluid due to a porous stretching sheet and heat transfer. Int J Non Linear Mech 2016;83:59–64. [30] Kumaran G, Sandeep N. Thermophoresis and Brownian moment effects on parabolic flow of MHD Casson and Williamson fluids with cross diffusion. J Mol Liq 2017;233:262–9. [31] Khan WA, Khan M. Impact of thermophoresis particle deposition on threedimensional radiative flow of Burgers fluid. Results Phys 2016;6:829–36. [32] Ali ME, Sandeep N. Cattaneo-Christov model for radiative heat transfer of magnetohydrodynamic Casson-ferrofluid: a numerical study. Results Phys 2017;7:21–30. [33] Turkyilmazoglu M. Dual and triple solutions for MHD slip flow of nonNewtonian fluid over a shrinking surface. Comput Fluids 2012;70:53–8. [34] Carreau PJ. Rheological equations from molecular network theories. Trans Soc Rheol 1972;116:99–127. [35] Khan M, Azam M. Unsteady boundary layer flow of Carreau fluid over a permeable stretching surface. Results Phys 2016;6:1168–74. [36] White FM. Viscous fluid flow. New York, USA: McGraw-Hill; 1991.
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Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics
Unsteady slip flow of Carreau nanofluid over a wedge with nonlinear radiation and new mass flux condition M. Khan a, M. Azam a,⇑, A.S. Alshomrani b a b
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 27 April 2017 Received in revised form 2 June 2017 Accepted 23 June 2017 Available online 4 July 2017 Keywords: Unsteady wedge flow Carreau nanofluid Non-linear radiation Velocity slip and nanoparticles mass flux conditions
a b s t r a c t This article addresses a numerical investigation for the unsteady 2D slip flow of Carreau nanofluid past a static and/or moving wedge with the nonlinear radiation. A zero nanoparticle mass flux and convective boundary conditions are implemented. Further, the most recently devised model for nanofluid is adopted that incorporates the effects of Brownian motion and thermophoresis. A set of suitable transformation is demonstrated to alter the nonlinear partial differential equations into nonlinear ordinary differential equations and then tackled numerically by employing bvp4c in Matlab package. The numerical computations for the wall heat flux (Nusselt number) and wall mass flux (Sherwood number) are also performed. Effects of several controlling parameters on the velocity, temperature and nanoparticles concentration are explored and discussed in detail. Our study reveals that the temperature and the associated thermal boundary layer thickness are enhancing function of the temperature ratio parameter for both shear thickening and shear thinning fluids. Moreover, it is noticed that the velocity in case of moving wedge is higher than static wedge. Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction From last two decades, modern evaluation in engineering has pushed the advanced thermal systems to enhance their performance. This can be achieved by the possibility of using nanofluids in such systems as the characteristic feature of nanofluids in thermal conductivity enhancement. The concept of nanofluid was devised by Choi [1] in 1995. There are two main methods used to develop nanofluids which are the single-step and the two-step method (see Akoh et al. [2] and Eastman et al. [3]). Both of these techniques possess advantages and disadvantages as reported by Wang and Mujumdar [4]. Transport properties of nanofluids can be analyzed by two different models proposed by Buongiorno [5] and Tiwari and Das [6]. Here, we are adopting Buongiorno’s model to study the mechanism of heat transfer in nanofluids. Buongiorno proposed a model that ignores the limitations of dispersion and homogeneous models. He demonstrated the seven slip mechanisms that generate a parallel velocity between the base fluid and nanoparticles. These are inertia, Magnus effect, thermophoresis, Brownian diffusion, fluid drainage, gravity and diffusiophoresis. He concluded that thermophoresis and Brownian diffusion are
⇑ Corresponding author. E-mail address: [email protected] (M. Azam).
main slip mechanisms in nanofluids. On the evidence of these facts, he demonstrated two-component four-equation nonhomogeneous equilibrium model subject to mass, momentum and heat transport in nanofluids. In view of such facts, several studies on nanofluids were presented by many authors. Makinde and Aziz [7] reported the problem of boundary layer flow of nanofluid in the presence of convective surface conditions. They observed the nanoparticle concentration is enhanced with the enhancment of the Biot number. A comparative analysis for convective heat and mass transfer in nanofluid was studied by Sandeep et al. [8]. They concluded that the rate of heat and mass transfer in Oldroyd-B nanofluid is larger when compared to Maxwell and Jeffrey nanofluids. Khan et al. [9] conducted a numerical investigation on unsteady flows of Carreau nanofluid over contracting or expanding cylinder. They observed that the rate of heat transfer increases for growing values of Biot number. Umawathi and Sheremet [10] considered a problem of nanofluid in a rectangular conduct to analyze the impact of variable thermal conductivity. They noticed that the heat transfer rate is smaller for regular fluid when compared to nanofluid. Recently, Khan and Azam [11] presented the numerical study of unsteady heat and mass transfer of Carreau nanofluid in the presence of magnetic field. They noticed that the temperature and nanoparticles concentration are higher due to the presence of magnetic field. The Carreau viscosity model can characterize the rheology of several polymeric solutions such as pure poly ethylene oxide, 1%
http://dx.doi.org/10.1016/j.rinp.2017.06.038 2211-3797/Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Nomenclature x y u v t V T C
s
p I
l l0 l1 c_ C
g
W 0
f h / We NR Pr b C fx Nux Shx Re
distance along the surface distance normal to the surface velocity component in x-direction velocity component in y-direction time velocity field temperature field concentration field Cauchy stress tensor pressure identity tensor apparent viscosity zero shear rate viscosity infinite shear rate viscosity shear rate time material constant local similarity variable stream function dimensionless velocity dimensionless temperature dimensionless concentration local Weissenberg number radiation parameter Prandtl number wedge angle parameter local skin friction coefficient local Nusselt number local Sherwood number local Reynolds number
methylcellulose tylose in glycerol solution and 0:3% hydroxyethylcellulose. These polymers are intensively used in capillary electrophoresis to enhance the resolution in the partition of proteins. It predicts a specific area where the shear rate and viscosity have linear relationship. It also predicts the power law region. In the existing data, different researchers considered the problem of Falkner-Skan for several values of wedge angle parameter b using different methods like as the classical Blasius problem for b ¼ 0 [12]. The study of Falkner-Skan flow [13] of viscous fluid past a static or moving wedge has been presented by Riley and Weidman [14]. A finite difference scheme was used by Asaithambi [15] to find the numerical solution of the Falkner-Skan problem for 0:19884 6 b 6 2. Zaturska and Banks [16] presented the analytic solutions for b > 1. Yang and Chien [17] explored the analytical solutions for the wedge angle parameter b ¼ 1. Raju and Sandeep [18] conducted a comparative study for the heat and mass transfer analysis of Falkner-Skan and Blasius flow of Casson fluid over a wedge. They noted that rate of heat and mass transfer in Falkner-Skan flow is high when compared to Blasius flow. Khan et al. [19] reported a numerical investigation on unsteady flows of MHD Carreau nanofluid past a wedge. They concluded that thermal boundary layer thickness is an enhancing function of the Biot number. Sheremet et al. [20] presented a numerical investigation for the problem of unsteady natural convection of nanofluid in wavy-walled cavity using implicit finite difference scheme. They observed that an increment in the wavy contraction ratio leads to enhance in the wave amplitude. Raju and Sandeep [21] presented a work on Falkner-Skan flow of MHD Carreau fluid over a wedge with cross diffusion effects. They predicted that improving values of wedge angle parameter grows the velocity field. Khan et al. [22] presented a numerical study to analyze the impact of heat generation or absorption and melting phenomenon on
A1 n l
c q m am
Uw Tw Cw L T1 C1 DT DB k cp k
r
a; c hw Nt Nb Sc A
sw
qw qm k
first Rivlin-Ericksen tensor power law index slip length Biot number density of fluid kinematic viscosity thermal diffusivity stretching velocity surface temperature surface nanoparticle concentration slip parameter ambient fluid temperature ambient fluid concentration thermophoresis diffusion coefficient Brownian diffusion coefficient thermal conductivity specific heat capacity mean absorption coefficient Stefan–Boltzman constant constants temperature ratio parameter thermophoresis parameter Brownian motion parameter Schmidt number unsteadiness parameter wall shear stress wall heat flux wall mass flux velocity ratio parameter
Falkner-Skan flow of Carreau nanofluid past a wedge. They noted that nanoparticle concentration distribution decrease by growing values of melting parameter. On the other hand, the radiative characteristics of non-Newtonian fluids have been paid much attention because of its variable characteristics found in base fluids. Thermal radiations have key role in controlling heat transfer mechanism in polymer processing industrial area as the quality of final product depends on the factors of heat controlling. The influences of radiation in boundary layer flow were reported by many authors. Probably, Smith [23] reported the first study on the impact of radiation involving the boundary layer flow. Viskanta and Grosh [24] explored the influences of thermal radiation on heat transfer and temperature profiles in Falkner-Skan flow considering Rosseland approximation. They noticed that thermal radiation becomes an additional part in missile reentry, gas cooled nuclear reactors etc. Cortell [25] reported a numerical study for the characteristics of nonlinear thermal radiation in boundary layer flow over a stretching sheet. He observed that radiation parameter and temperature ratio parameter have opposite behavior qualitatively in temperature profiles. Pantokratoras [26] demonstrated a numerical investigation for the natural convection flow over a vertical plate subject to linear and nonlinear thermal radiation. He explored that when the wall shear stress enhances the wall heat transfer depreciates and vice versa. Khan et al. [27] analyzed the impact of homogeneous and heterogeneous reaction on the flow of Burger fluid. Kumaran et al. [28] reported a numerical study for MHD Maxwell and Casson fluid flow with cross diffusion effects. They noted that rate of heat and mass transfer was high in Maxwell fluid when compared to Casson fluid. Turkyilmazoglu [29] considered a problem of flow and heat transfer of micropolar fluid over a porous stretching sheet. He found the analytical unique solution of the considered problem. The effects of thermophoresis and Brownian
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motion on flow of Williamson and Casson fluids were analyzed numerically by Kumaran and Sandeep [30]. They examined that the performance of Casson fluid was better as compared to Williamson fluid. Khan and Khan [31] examined the impact of thermophoresis particle deposition on three dimensional flow of Burgers fluid in the presence of thermal radiation. They concluded that concentration field depreciated for improving values of thermophoretic parameter. Ali and Sandeep [32] demonstrated a numerical study to investigate the Cattaneo-Christov model for MHD Casson fluid. Turkyilmazoglu [33] also presented the multiple solutions for slip flow over a shrinking surface considering magnetic field effects. On the basis of motivation of the aforesaid investigations, the aim of present study is to present numerical solutions on the characteristics of nonlinear thermal radiation on the unsteady twodimensional Falkner-Skan flow of Carreau nanofluid past a static or moving wedge for 0 6 b 6 2. Additionally, velocity slip and convective boundary conditions are implemented at the boundary. Furthermore, zero nanoparticles mass flux condition is also implemented which leads to a more realistic physical problem. It is important to mention that Carreau viscosity model is an important class of generalized Newtonian fluid which characterizes the shear thinning as well as shear thickening nature of the fluid. This model reduces to viscous fluid when the power law index n ¼ 1 and We ¼ 0. Suitable transformations are utilized to alter the governing nonlinear boundary-layer equations to non-linear ordinary differential equations. Finally, these resulting equations are solved numerically by adopting an effective numerical approach bvp4c Matlab package. A comparison between the current results and existing study in limiting case is conducted with very good agreement. Mathematical analysis We consider the unsteady two-dimensional Falkner-Skan flow of a non-Newtonian Carreau fluid with the insertion of nanoparticles. Heat and mass transfer properties are explored through the Brownian motion and thermophoresis effects. The fluid flow is induced by a stretching wedge subject to stretching velocity
partial slip at the surface of wedge is implemented. The flux of nanoparticle volume fraction at the surface is assumed to be zero. The surface temperature T w of the wedge is also assumed to be higher than the ambient temperature ðT w > T 1 Þ. The Cauchy stress tensor s for Carreau viscosity model [34,35] can be written as n1
s ¼ pI þ lA1 ; l ¼ l0 ½1 þ ðCc_ Þ2 2 ; c_ ¼
ð1Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 trðA21 Þ; Ri Rj c_ ij c_ ji ¼ 2 2
ð2Þ
where I is the identity tensor, p the pressure, l0 the zero shear rate viscosity, C the time material parameter, c_ the shear rate, n the power law index and A1 the first Rivlin-Ericksen tensor. For the unsteady two-dimensional laminar flow, the velocity, temperature and nanoparticles concentration fields are demonstrated as
V ¼ ½uðx; y; tÞ; v ðx; y; tÞ; 0;
T ¼ Tðx; y; tÞ;
C ¼ Cðx; y; tÞ:
ð3Þ
Utilizing the aforesaid assumptions and boundary layer approximations, the governing equations of the Carreau nanofluid model can be represented as [19]
›u ›v þ ¼ 0; ›x ›y
ð4Þ
" 2 #n1 2 @u ›u ›u @U e @U e @2u 2 @u þu þv ¼ þ Ue þm 2 1þC @t ›x ›y @y @y @t @x 2 2 @ u @u þ m ð n 1 Þ C2 2 @y @y " 2 #n3 2 @u 1 þ C2 ; @y
ð5Þ
" 2 # @T @T @T @2T @C @T DT @T 1 @qr þu þv ¼ am 2 þ s DB þ ; @t @x @y @y @y @y T 1 @y qcp @y ð6Þ
m
bx U w ðx; tÞ ¼ 1ct . It is stated here that U w ðx; tÞ > 0 indicates the stretching wedge surface velocity and U w ðx; tÞ < 0 relates to a contracting wedge surface velocity. The free stream velocity of the axm considered problem is U e ðx; tÞ ¼ 1ct where a; b; c and m are positive constants. In Fig. 1, the total angle of the wedge is defined by 2m is the wedge angle parameter. The nonlinear X ¼ bp and b ¼ mþ1
thermal radiation is also taken into account. The temperature at the surface of the wedge is controlled by a convective heating analysis which is referred to heat transfer coefficient hf . The velocity
@C @C @C @ 2 C DT @ 2 T þu þv ¼ DB 2 þ : @t @x @y @y T 1 @y2
ð7Þ
The appropriate boundary conditions in the current problem are as follows
u ¼ U w þ U slip ; DB
v ¼ 0;
k
@T ¼ hf ðT w T Þ; @y
@C DT @T þ ¼ 0 at y ¼ 0; @y T 1 @y
u ! Ue ;
T ! T1;
C ! C1
Here, s ¼ ðqcÞp =ðqcÞf
ð8Þ as y ! 1:
ð9Þ
is the ratio of the effective heat capacity
of the nanoparticle to the effective heat capacity of the base fluid, am ¼ qkcp the thermal diffusivity with cp the specific heat, q the density of fluid and k the thermal conductivity, DT the thermophoresis diffusion coefficient, DB the Brownian diffusion coefficient, qr the radiative heat flux and C 1 the ambient fluid concentration. The partial slip velocity U slip for Carreau fluid model in present problem can be stated as:
U slip Fig. 1. Physical description of flow problem.
" 2 #n1 2 @u 2 @u ¼l ; 1þC @y @y
ð10Þ
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M. Khan et al. / Results in Physics 7 (2017) 2261–2270
where l is the slip length having dimension of length. Utilizing the Rosseland approximation for the radiation [23], the simplified form of radiative heat flux can be written as
qr ¼
4r @T 4 ; 3k @y
ð11Þ
where k and r indicate the mean absorption coefficient and Stefan-Boltzmann constant, respectively. It is stated that unlike classical cases the nonlinear structure of thermal radiation is considered here. Note that, the impact of thermal radiation in the linearized Rosseland approximation is simply a rescaling of the Prandtl number by a factor including the radiation parameter. Thus it is noted here that the energy equation with the addition of nonlinear thermal radiation is highly nonlinear in including the additional temperature ratio parameter. It is important to mention here that for a planer boundary layer flow over a horizontal flat surface [23], from Eq. (7), we obtain
qr ¼
16r 3 @T : T @y 3k
ð12Þ
By substituting of Eq. (12), in Eq. (6) results in the energy equation with nonlinear thermal radiation in the form.
! # 16r T 3 @T am þ 3k qcp @y " 2 # @C @T DT @T þ s DB þ : @y @y T 1 @y
@T @T @T @ þu þv ¼ @t @x @y @y
"
ð13Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 1Þue 2mxue ; Wðx; y; tÞ ¼ f ðgÞ; g¼y 2m x mþ1 T T1 C C1 hðgÞ ¼ ; /ðgÞ ¼ ; Tw T1 C1
ð14Þ
where W is the stream function, T ¼ T 1 ð1 þ ðhw 1ÞhÞ and hw ¼ TT1w > 1 the temperature ratio parameter. In perspective of Eq. (14), Eqs. (5), (7) and (13) reduce to the following ordinary equations.
n on on3 n 0 2 o 2 00 2 00 2 000 00 1 þ nWe2 ðf Þ f þ ff þ b 1 f 1 þ We2 ðf Þ n o g 00 0 A f þ f 1 ¼ 0; 2 A 2 h00 þ Pr f h0 Pr gh0 þ Nbh0 /0 þ Ntðh0 Þ 2 i 4 d h þ ð1 þ ðhw 1ÞhÞ3 h0 ¼ 0; 3NR dg A Nt 00 h ¼ 0; /00 þ Scf /0 Sc g/0 þ 2 Nb
f ð0Þ ¼0;
0
hð1Þ ! 0;
Sc ¼ hw ¼
Pr ¼
m 2c ; A¼ ; ðm þ 1Þaxm1 am
hf sDT ðT w T 1 Þ sDB C 1 ; Nb ¼ ; c¼ am T 1 am k m DB
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mx ; ð1 þ mÞue
;
Tw ; T1
NR ¼
kk 4r
T3 1
;
b¼
2m ; mþ1
b k¼ : a
ð20Þ
The physical quantities of engineering interest are the local skin friction coefficient C f , the local Nusselt number Nu and the local Sherwood number Sh. They are defined as
sw jy¼0 xqw xqm ; Nu ¼ ; ; Sh ¼ qu2e kðT w T 1 Þ DB C 1
ð21Þ
with sw ; qw and qm as the wall shear stress, wall heat flux and wall mass flux, respectively and they are given by
" 2 #n1 2 @u 2 @u ; 1þC sw ¼ l0 @y @y y¼0 @T @C þ q j ; q ¼ D : qw ¼ k B r m y¼0 @y y¼0 @y y¼0
ð22Þ
The dimensionless surface drag, heat transfer rate and mass transfer rate are thus given by
h in1 2 2 00 00 ð2 bÞ1=2 Re1=2 C f ¼ f ð0Þ 1 þ We2 ðf ð0ÞÞ ;
n o 4 ½1 þ ðhw 1Þhð0Þ3 ; ð2 bÞ1=2 Re1=2 Nu ¼ h0 ð0Þ 1 þ 3N R ð2 bÞ1=2 Re1=2 Sh ¼ /0 ð0Þ; where Re ¼
xU e
m
ð23Þ ð24Þ ð25Þ
is the local Reynolds number.
ð15Þ
Discussion of numerical results
ð16Þ
The behavior of unsteady slip flow of Carreau nanofluid in the presence of nonlinear radiation and new mass flux condition is studied numerically. The numerical computations have been performed using Matlab function bvp4c for distinct values of physical parameters namely the local Weissenberg number We, unsteadiness parameter A, power law index n, wedge angle parameter b,
ð17Þ
2
00
2
f ð0Þ ¼ k þ Lf ð0Þ 1 þ We ðf ð0ÞÞ
h0 ð0Þ ¼ cð1 hð0ÞÞ; f ð1Þ ! 1;
h
Nt ¼
Table 1 Comparison values of ð2 bÞ1=2 Re1=2 Nu for different values of Pr and b when n ¼ 1; c ¼ N R ¼ 106 ; hw ¼ 1:7 and We ¼ Nt ¼ Nb ¼ k ¼ L ¼ Sc ¼ A ¼ 0.
where prime indicates the differentiation with respect to g. The associated conditions of the present problem are the 00
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 ðm þ 1ÞU 3e ; We ¼ 2mx rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 1ÞU e L¼l ; 2mx
Cf ¼
The non-dimensional suitable variables can be demonstrated in the following way
0
the Brownian motion parameter, c the Biot number, N R the radiation parameter, k the velocity ratio parameter and Sc the Schmidt number. These parameters are stated as follows:
in1 2
Nb/0 ð0Þ þ Nth0 ð0Þ ¼ 0; /ð1Þ ! 0:
b¼0
; ð18Þ ð19Þ
In the above equations, We indicates the local Weissenberg number, Pr the Prandtl number, A the unsteadiness parameter, L the velocity slip parameter, Nt the thermophoresis parameter, Nb
b ¼ 0:3
Pr
White [36]
Present results
White [36]
Present results
0.1 0.3 0.6 0.72 1.0 2.0 6.0 10.0
0.1980 0.3037 0.3916 0.4178 0.4696 0.5972 0.8672 1.0297
0.198129 0.303718 0.391675 0.418091 0.469600 0.597234 0.867278 1.029747
0.2090 0.3278 0.4289 0.4592 0.5195 0.6690 0.9872 1.1791
0.209152 0.327829 0.428924 0.459551 0.519519 0.669045 0.987268 1.179130
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M. Khan et al. / Results in Physics 7 (2017) 2261–2270
Prandtl number Pr, radiation parameter N R , temperature ratio parameter hw , Biot number c, velocity slip parameter L, Brownian motion parameter Nb, Schmidt number Sc, velocity ratio parameter k and thermophoresis parameter Nt. For the better understanding the computed results are depicted through tables and plots and discussed in detail.
To assess the accuracy of the method which we have used, a comparison with the existing data is conducted in limiting manners. Our results of the local Nusselt number for some selected values of the Prandtl number and the wedge angle parameter are compared in limiting cases with those of White [36] (see Table 1). There is an excellent agreement between the present results and
Table 2 Numerical values of the local Nusselt number for different values of A; b; N R ; hw ; c; L; Nt; Nb and Sc when We ¼ 1:0; k ¼ 0:3 and Pr ¼ 3:0. ð2 bÞ1=2 Re1=2 Nu
Parameters A
b
NR
hw
c
L
Nt
Nb
Sc
n ¼ 0:5
n ¼ 1:5
0.0 0.2 0.3 0.2
0.3
1.0
1.2
0.1
0.2
0.1
0.1
1.0
1.0
1.2
0.1
0.2
0.1
0.1
1.0
0.2
0.0 0.1 0.2 0.3
1.2
0.1
0.2
0.1
0.1
1.0
0.2
0.3
1.0 1.5 2.0 1.0
0.1
0.2
0.1
0.1
1.0
0.2
0.3
1.0
1.3 1.5 1.7 1.2
0.2
0.1
0.1
1.0
0.2
0.3
1.0
1.2
0.2 0.3 0.4 0.1
0.1
0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.0 0.2 0.4 0.2
0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.2
0.1 1.0 3.0 0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.2
0.1
0.2 0.3 0.4 0.1
0.2132826 0.2119981 0.2112568 0.2109825 0.211397 0.2117267 0.2119981 0.1718841 0.1517178 0.2166362 0.2266575 0.2378075 0.3863935 0.5304262 0.6502942 0.2108292 0.2119981 0.2127863 0.2119981 0.2119351 0.2117922 0.2119981 0.2119981 0.2119981 0.2119939 0.2119908 0.2119885
0.2131824 0.2118712 0.2111134 0.2108818 0.2112853 0.2116066 0.2118712 0.1717823 0.1516284 0.2165315 0.2266048 0.2378187 0.3859495 0.5295584 0.648955 0.2105249 0.2118712 0.2127297 0.2118712 0.2118075 0.211663 0.2118712 0.2118712 0.2118712 0.2118669 0.2118638 0.2118614
2.0 3.0 4.0
Table 3 Numerical values of the local Sherwood number for different values of A; b; N R ; hw ; c; L; Nt; Nb and Sc when We ¼ 1:0; k ¼ 0:3 and Pr ¼ 3:0. j/0 ð0Þj
Parameters A
b
NR
hw
c
L
Nt
Nb
Sc
n ¼ 0:5
n ¼ 1:5
0.0 0.2 0.3 0.2
0.3
1.0
1.2
0.1
0.2
0.1
0.1
1.0
1.0
1.2
0.1
0.2
0.1
0.1
1.0
0.2
0.0 0.1 0.2 0.3
1.2
0.1
0.2
0.1
0.1
1.0
0.2
0.3
1.0 1.5 2.0 1.0
0.1
0.2
0.1
0.1
1.0
0.2
0.3
1.0
1.3 1.5 1.7 1.2
0.2
0.1
0.1
1.0
0.2
0.3
1.0
1.2
0.2 0.3 0.4 0.1
0.1
0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.0 0.2 0.4 0.2
0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.2
0.1 1.0 3.0 0.1
1.0
0.2
0.3
1.0
1.2
0.1
0.2
0.1
0.2 0.3 0.4 0.1
0.08758551 0.08683194 0.08639908 0.08623937 0.08648082 0.08667329 0.08683194 0.08792725 0.08853906 0.08665592 0.08628157 0.08587438 0.1526004 0.2033918 0.2434015 0.08615018 0.08683194 0.08729382 0.08683194 0.8679508 2.601347 0.04341597 0.02894398 0.02170798 0.08682944 0.08682765 0.08682627
0.08752656 0.08675772 0.08631555 0.08618075 0.08641574 0.08660313 0.08675772 0.08785772 0.0884725 0.08657982 0.0862014 0.08578962 0.1523621 0.2029567 0.2427659 0.08597331 0.08675772 0.08726056 0.08675772 0.8672049 2.599083 0.04337886 0.02891924 0.02168943 0.08675519 0.08675338 0.08675198
2.0 3.0 4.0
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M. Khan et al. / Results in Physics 7 (2017) 2261–2270
those of White. This gives the confident on our numerical computations. To analyze the influences of pertinent parameters on the local Nusselt and local Sherwood numbers, numerical computations
have been executed for these parameters for both shear thickening ðn > 1Þ and shear thinning ð0 < n < 1Þ fluids. On the evidence of Tables 2 and 3, it is noted that the local Nusselt number and magnitude of local Sherwood number depreciate by increasing the val-
1
1
0.9
0.9
λ = 0.0
0.8 0.7
λ = 0.3
0.7
β = 0.0, 0.1, 0.2, 0.3
f '(η)
0.6
f '(η)
λ = 0.0
0.8
λ = 0.3
0.5 0.4 0.3
β = 0.0, 0.1, 0.2, 0.3
0.6 0.5 0.4 0.3
A = 0.2, L = 0.2, We = 1.0
A = 0.2, L = 0.2, We = 1.0
0.2
0.2
0.1
0.1
(b) n = 1.5
(a) n = 0.5
0
0
1
2
3
4
0
5
0
1
2
η
η
3
4
5
0
Fig. 2. Effects of the wedge angle parameter b on the velocity f ðgÞ profiles.
1
1
0.9 0.8
f '(η)
0.5 0.4
L = 0.0, 0.2, 0.4, 0.6
0.6 0.5 0.4 0.3
A = 0.2, β = 0.3, We = 1.0
0.2
A = 0.2, β = 0.3, We = 1.0
0.2
0.1 0
λ = 0.3
0.7
L = 0.0, 0.2, 0.4, 0.6
0.6
0.3
λ = 0.0
0.8
λ = 0.3
0.7
f '(η)
0.9
λ = 0.0
0.1
(a) n = 0.5 0
1
2
η
3
4
0
5
(b) n = 1.5 0
1
2
η
3
4
5
0
Fig. 3. Effects of the velocity slip parameter L on the velocity f ðgÞ profiles.
0.35
0.3
N R = 0.1, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
0.25
0.2
0.2
θ(η)
0.25
θw = 1.3, 1.5, 1.7, 1.9
0.15
0.1
0.05
0.05
0
1
2
3
4
η
5
6
7
8
N R = 0.1, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
θw = 1.3, 1.5, 1.7, 1.9
0.15
0.1
0
(b) n = 1.5 A = 0.2, λ = 0.2, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
A = 0.2, λ = 0.2, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
0.3
θ(η)
0.35
(a) n = 0.5
0
0
1
2
3
4
η
Fig. 4. Effects of the temperature ratio parameter hw on the temperature hðgÞ profiles.
5
6
7
8
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M. Khan et al. / Results in Physics 7 (2017) 2261–2270
Nusselt number and magnitude of local Sherwood number are the growing functions of the Biot number as well as velocity slip parameter in both cases. It is also observed that the local Nusselt number depreciates by improving the values of the thermophoresis parameter but opposite behavior has been predicted in the magnitude of local Sherwood number. Additionally, the magnitude
ues of the unsteadiness parameter, radiation parameter and Schmidt number both for shear thickening and shear thinning fluids. Whereas, the local Nusselt number is an enhancing function of the temperature ratio parameter but opposite trend has been noticed in the magnitude of the local Sherwood number both for shear thickening and shear thinning fluids. Furthermore, the local
(a) n = 0.5
(b) n = 1.5
A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
θw = 1.2, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
θw = 1.2, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
0.1
θ(η)
θ(η)
0.1
0.05
0.05
N R = 1.0, 1.5, 2.0, 2.5
0
0
1
2
N R = 1.0, 1.5, 2.0, 2.5
3
η
0
4
0
1
2
3
η
4
Fig. 5. Effects of the radiation parameter N R on the temperature hðgÞ profiles.
0.4
0.4
(a) n = 0.5 A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, θw = 1.2
0.35
N R = 1.0, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
0.3
θ(η)
0.2
0.2 0.15
γ = 0.1, 0.2, 0.3, 0.4
0.1
0.1
0.05
0.05
0
1
2
3
η
0
4
0.05
0.05
0
0
γ = 0.1, 0.2, 0.3, 0.4
-0.05
-0.1
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2, β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.15
γ = 0.1, 0.2, 0.3, 0.4
0
2
3
η
4
γ = 0.1, 0.2, 0.3, 0.4
-0.1
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2, β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.15
Sc = 1.0, Nt = Nb = 0.1
Sc = 1.0, Nt = Nb = 0.1
-0.2
-0.2
(c) n = 0.5 -0.25
1
-0.05
φ(η)
θ(η)
N R = 1.0, Pr = 3.0, Sc = 1.0, Nt = Nb = 0.1
0.25
0.15
φ(η)
A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, θw = 1.2
0.3
0.25
0
(b) n = 1.5
0.35
0
1
(d) n = 1.5 2
η
3
4
-0.25
0
1
2
η
Fig. 6. Effects of the Biot number c on the temperature hðgÞ and nanoparticle concentration /ðgÞ profiles.
3
4
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observed that the velocity of the fluid increases for growing the values of the velocity slip parameter in static and moving wedge for both cases of shear thickening and shear thinning fluids. It is also noticed that the velocity slip parameter has a substantial impact on the solution. For L ¼ 0, no slip condition case is achieved. The variation in the dimensionless temperature for distinct values of the temperature ratio parameter hw is depicted in Fig. 4(a) and (b) for both shear thickening and shear thinning fluids. On the evidence of these Figs., it is noticed that growing values of temperature ratio parameter relates to higher wall temperature as compare to ambient fluid. As a result fluid temperature enhances. It is also noted that the thermal boundary layer thickness enhances for growing the values of the temperature ratio parameter. Fig. 5(a) and (b) demonstrate the influences of radiation parameter on the dimensionless temperature profiles. On the behalf of these Figs., one can decide that the temperature and associated thermal boundary layer thickness depreciate for enhancing the values of the radiation parameter N R in shear thickening and shear thinning fluids. Fig. 6(a) and (b) are plotted to examine the influence of Biot number c on the temperature and nanoparticle concentration profiles. From these Figs., it is predicted that the temperature, nanoparticle concentration and their related boundary layer thicknesses increase in shear thickening and shear thinning fluids. For c ¼ 0, the surface of the wedge is totally insolated. In fact, the
of local Sherwood number is a depreciating function of the Brownian motion parameter in both cases. The influence of the wedge angle parameter b is illustrated in Fig. 2(a) and (b) where velocity profiles are shown in both cases of shear thickening and shear thinning fluids for static and moving wedge. From these Figs., it can be seen that velocity of the fluid enhances by growing the values of the wedge angle parameter for both the cases of static and moving wedge in shear thickening and shear thinning fluids. It is also clear that the velocity of the fluid is larger in the case of moving wedge when compared to static wedge. It is important to state here that k is the constant moving wedge parameter or the velocity ratio parameter with k > 0 and k < 0 relates to a moving wedge with the same and opposite directions to the free stream velocity, respectively. Note that the wedge angle parameter is the Hartree pressure gradient parameter which relates to b ¼ Xp with a total wedge angle X. On perspective of White [26], positive values of b indicates that the pressure gradient is favorable or negative then the flow will be accelerating along the surface. Negative values of b shows that the pressure gradient is adverse then the flow will be decelerating. Additionally, b ¼ 0 ðm ¼ 0Þ corresponds to boundary layer flow over a horizontal flat plate and b ¼ 1 ðm ¼ 1Þ relates to boundary layer flow near the stagnation point of a vertical flat plate. To analyze the impact of velocity slip parameter L on the velocity profiles, Fig. 3(a) and (b) are plotted. From these Figs., it can be
0.15
0.15
(a) n = 0.5
(b) n = 1.5 A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
A = 0.2, λ = 0.3, We = 1.0, L = 0.2, β = 0.3, γ = 0.1
N R = 1.0, Pr = 3.0, Sc = 1.0, θw = 1.2, Nb = 0.1
N R = 1.0, Pr = 3.0, Sc = 1.0, θ w = 1.2, Nb = 0.1 0.1
θ(η)
θ(η)
0.1
Nt = 1.0, 3.0, 5.0, 7.0
0.05
0
0
1
2
η
0
4
1
1
0
0
-1 -2
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-3
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0, Sc = 1.0, γ = Nb = 0.1
-4 -5 -6
1
η
3
2
4
3
η
4
Nt = 1.0, 3.0, 5.0, 7.0
-2
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-3
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0, Sc = 1.0, γ = Nb = 0.1
-5
2
1
-4
(c) n = 0.5 0
0
-1
Nt = 1.0, 3.0, 5.0, 7.0
φ(η)
φ(η)
3
Nt = 1.0, 3.0, 5.0, 7.0
0.05
-6
(d) n = 1.5 0
1
2
η
3
Fig. 7. Effects of the thermophoresis parameter Nt on the temperature hðgÞ and nanoparticle concentration /ðgÞ profiles.
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both shear thickening and shear thinning fluids. The higher Schmidt number corresponds to lower nanoparticles concentration and its affiliated boundary layer thickness.
internal thermal resistance of the wedge surface is very large and convective heat transfer does not take place from the wedge surface to the cold fluid far away from the wedge. The influences of the thermophoresis parameter Nt on the temperature and nanoparticle concentration are depicted through Fig. 7(a) and (b). These Figs., reveal that an increment in the thermophoresis parameter causes to enhance the temperature profiles. From these Figs., it is also observed that the nanoparticle concentration and its related boundary layer thickness enhance for growing the thermophoresis parameter. Physically, the thermophoresis force grows with the improvement of Nt which tends to move nanoparticles from hot portion to cold portion and hence enhances the magnitude of the nanoparticles concentration profile. Fig. 8(a) and (b) indicate the variation of nanoparticles concentration profiles for some values of the Brownian motion parameter Nb. It is observed that an enhancement in the Brownian motion parameter depreciates the nanoparticles concentration distribution and its related boundary layer thickness. Brownian motion occurs due to the presence of nanoparticles and resulted in the decrement of the nanoparticles concentration thickness. To analyze the impact of the Schmidt number Sc on nanoparticle concentration profiles, Fig. 9(a) and (b) are presented. It is noted that the nanoparticle concentration and the related boundary layer thickness shrink with the enhancement of the Schmidt number for
Main findings The numerical solutions for the unsteady Falkner-Skan flow of Carreau nanofluid past a static or moving wedge in the presence of nonlinear thermal radiation have been investigated. The velocity slip and convective boundary conditions were also taken into account. Additionally, a zero nanoparticle mass flux condition at the boundary was implemented. An efficient Matlab solver bvp4c was used to solve the governing problem. The effects of pertinent parameters controlling the velocity, temperature and nanoparticles concentration distributions were presented graphically. Some of the key findings of the present investigations are listed below: The local Nusselt number and magnitude of local Sherwood number were depreciated by enhancing the values of the unsteadiness parameter, radiation parameter and Schmidt number both for shear thickening and shear thinning fluids and opposite was true for Biot number and velocity slip parameter.
0.02
0.02
(a) n = 0.5
0.01 0
0
-0.01
-0.01
Nb = 0.1, 0.2, 0.3, 0.4
Nb = 0.1, 0.2, 0.3, 0.4
-0.02
-0.03
φ(η)
-0.02
φ(η)
(b) n = 1.5
0.01
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-0.03
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-0.04
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.04
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.05
Sc = 1.0, γ = Nt = 0.1
-0.05
Sc = 1.0, γ = Nt = 0.1
-0.06
-0.06
-0.07
-0.07
-0.08
0
1
2
3
η
-0.08
4
0
1
2
3
η
4
Fig. 8. Effects of the Brownian motion parameter Nb on the nanoparticles concentration /ðgÞ profiles.
0.02
0.02
(a) n = 0.5
(b) n = 1.5
0.01
0.01
0
0 -0.01
-0.01
Sc = 1.0, 2.0, 3.0, 4.0
Sc = 1.0, 2.0, 3.0, 4.0 -0.02
-0.03
φ(η)
φ(η)
-0.02
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-0.03
A = 0.2, λ = 0.3, W e = 1.0, L = 0.2,
-0.04
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.04
β = 0.3, θw = 1.2, N R = 1.0, Pr = 3.0,
-0.05
γ = Nt = Nb = 0.1
-0.05
γ = Nt = Nb = 0.1
-0.06
-0.06
-0.07
-0.07
-0.08
0
1
2
η
3
4
-0.08
0
1
2
η
Fig. 9. Effects of the Schmidt number Sc on the nanoparticle concentration /ðgÞ profiles.
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The temperature and its related thermal boundary layer thickness were depreciating function of the radiation parameter both in shear thickening and shear thinning fluids and opposite is true for the temperature ratio parameter. The temperature and nanoparticles concentration were uplifted with the increment of the Biot number and thermophoresis parameter. The nanoparticles concentration was depressed with the increment of Brownian motion parameter and Schmidt number.
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