Microsystem Technologies https://doi.org/10.1007/s00542-018-3717-5
(0123456789().,-volV)(0123456789().,-volV)
TECHNICAL PAPER
Two dimensional boundary layer flow with heat and mass transfer of magneto hydrodynamic non-Newtonian nanofluid through porous medium over a semi-infinite moving plate N. T. Eldabe1 • M. E. Gabr2 • S. A. Zaher2 Received: 20 October 2017 / Accepted: 5 January 2018 Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract Steady two-dimensional motion of an incompressible non-Newtonian Nanofluid through porous medium over a semiinfinite moving plate is studied . The system is stressed by an external uniform magnetic field. The heat and mass transfer are considered with the flow of non-Newtonian fluid which obeys the Eyring–Powell model. The problem is mathematically formulated and solved numerically using the ParametricNDSolve-Mathematica package. The effects of the physical parameters of the problem such as, permeability, chemical reaction as well as the fluid material parameters Hartmann number, Eckert number and Reynolds number are discussed and illustrated in figures with some important applications. The results show that the fluid velocity increases with the increases of porosity parameter as well as the material parameter, but it decreases with the increase of the magnetic field. Increasing the magnetic field decreases porosity, temperature and concentration. The effects of some physical quantities on fluid velocity distribution can be neglected.
1 Introduction In the early twentieth century, Prandtl published the first paper on a new concept known as the boundary layer just a year after the first flight by the Wright brothers. Since then, as a result of its important applications in industry, studying the behavior of fluids around objects still attracts the attention of many researchers and engineers. The need to improve fluid properties becomes a great challenge. One of the most effective methods to improve fluid properties is adding nanoparticles to the base fluid. Adding nano/micro or larger sized particle materials to the base fluid improves fluid properties (Vleggaar 1977). Adding small amount of nanoparticles to convention heat transfer liquids increases its
& M. E. Gabr
[email protected] N. T. Eldabe
[email protected] S. A. Zaher
[email protected] 1
Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt
2
Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
thermal conductivity up to approximately two times (Choi 1995; Choi et al. 2001). Kuzentsov and Nield (2010) studied the influence of nanoparticles on natural convection boundary-layer flow past a vertical plate. Weidman et al. (2006) solved the problem of self-similar boundary layer flow over a moving semi-infinite flat plate. Bachok et al. (2010) studied the motion of a nanofluid past a moving semiinfinite flat plate in a uniform free stream. Studying the influence of a magnetic field on nanofluid flow is important in many engineering applications. The interest of researchers in stretching flows with boundary layer approximation substantially increased in view of its significant applications in industry such as polymer industry, wire drawing, fiber and paper production. Eldabe et al. (2012) discussed the viscous dissipation effect on free convection heat and mass transfer of magneto hydrodynamic (MHD) non-Newtonian fluid flow through porous medium. The discussion of the behavior of nanofluids around objects still attracts the attention of scientists and engineers via different aspects (Cimpean and Pop 2012; Farooq et al. 2014, 2015; Haq et al. 2014, 2015; Khan et al. 2014; Niu et al. 2012; Nadeem et al. 2014; Rashidi et al. 2013; Turkyilmazoglu 2013; Turkyilmazoglu 2014; Sheikholeslami et al. 2012, 2013; Thirupathi et al. 2017; Xu et al. 2013). The Eyring–Powell model gives more advantage than other non-Newtonian fluids because it based on the kinetic
123
Microsystem Technologies
theory of gases which acts like viscous fluids at high shear rates. In the last few years, many scientists give more attention to study the Eyring–Powell fluid model (Eldabe et al. 2008; Islam et al. 2009; Powell and Eyring 1944; Patel and Timol 2009). Maity et al. (2016) studied the flow of a thin nanoliquid film over an unsteady stretching sheet. The problem of unsteady flow of thin nanoliquid film over a stretching sheet in the presence of thermal radiation is studied by Maity (2016). The results obtained in this study show that the film thinning rate increases with the increase of the nanolayer thickness. But when the sheet is heated it decreases with the increase of the nanoparticle radius. Maity (2017) studied the problem of thermocapillary flow of thin Cu-water nanoliquid film during spin coating process. The main purpose of this study is to investigate the problem of MHD Eyring–Powell nanofluid through porous medium with heat and mass transfer over a moving semiinfinite moving plate. Also, discuss the influence of the physical parameters of the considered problem such as Hartmann number, fluid material parameter, Brownian motion parameter, thermophoretic parameter and Eckert number on both temperature and concentration.
2 Formulation of the problem Consider two-dimensional steady boundary-layer flow of an incompressible nanofluid over a moving semi-infinite flat plate. Choose the Cartesian coordinates system where the x-axis is taken along the continuous surface direction and the y-axis is normal to it. Let the fluid occupies the region y 0 and influenced by a uniform magnetic field Bo acting normal to the moving surface (Fig. 1). Let, U be the velocity of the uniform free stream and Uw = kU, be the flat plate velocity, where k is the plate velocity parameter. Also assume that the moving surface temperature T and the nanoparticles fraction C take the
constant values Tw and Cw, respectively. While in the ambient fluid their values are T1 and C1 respectively. The stress tensor in Eyring–Powell model can be written as (Eldabe et al. 2003; Hayat et al. 2012): oui 1 1 oui sij ¼ l þ sinh1 ; ð1Þ c oxj oxj b where, 1 oui 1 oui 1 1 oui 3 ; ’ sinh1 c oxj c oxj 6 c oxj
1 oui c ox 1; j
ð2Þ
l is the dynamic viscosity, b and c are the characteristics of the Eyring–Powell model. The governing equations can be written as follows: The continuity equation: ~ ¼ 0; rV qf
~ oV ~ rV ~ þV ot
ð3Þ !
l~ ~ ~ V þ J B: ¼ rp þ r ~ s k
ð4Þ
The heat equation with heat generation:
ðqcÞf
oT ~ þ V rT ot
¼ kr2 T
DT þ ðqcÞp DB rC rT þ rT rT T1 2 ~þJ : þ ~ s rV r
ð5Þ The concentration equation with chemical reaction: oC ~ DT 2 þ V rC ¼ DB r2 C þ r T þ ko ðC C1 Þ; ot T1
ð6Þ
~; ~ where, V J and ~ B are the velocity, current and magnetic field vectors, k, ko and r are the fluid thermal conductivity, the chemical reaction and fluid electric conductivity, qf ; ðqcÞf and ðqcÞp are fluid density, fluid heat capacity and nanoparticles effective heat capacity, DB and DT are Brownian and thermophoretic diffusion coefficients. Taking in consideration the steady-state flow of the boundary layer, the governing Eqs. (4)–(6) can be written as: ou ov þ ¼ 0; ox oy ! ou ou 1 o 2 u 1 2 l u þv rBo þ ¼ tþ u ox oy bcqf o y2 qf k 2 2 1 ou o u ; 3 2bc qf o y o y2
ð7Þ
ð8Þ Fig. 1 Physical model and coordinate system
123
Microsystem Technologies
" # oT oT o2 T oc oT DT oT 2 þv ¼ a 2 þ s1 D B þ u ox oy oy oy oy T1 oy ! l 1 ou 2 þ þ ðqcÞf cbðqcÞf oy 4 r 1 ou þ B2o u2 3 ; ðqcÞf 6c bðqcÞf oy oC oC o2 C D T o2 T þv ¼ DB 2 þ u þ ko ðC C1 Þ; ox oy oy T1 oy2
ð1 þ eÞf 000 ðgÞ þ f ðgÞf 00 ðgÞ ðM þ KÞf 0 ðgÞ 2 edf 00 ðgÞf 000 ðgÞ ¼ 0; ð9Þ
u00 ðgÞ þ
where u and v are the velocity components along the x and y axis respectively, a ¼ k=ðqcÞf is the thermal diffusivity of the fluid, t is the kinematic viscosity coefficient and s1 ¼ ðqcÞp =ðqcÞf : The boundary conditions of the problem are: ð11Þ
ow Using the stream function w where u ¼ ow oy ; v ¼ ox ; Eqs. (8)–(10) can be written as: ! ow o2 w ow o2 w 1 o3 w 1 2 l ow ¼ tþ rBo þ 2 oy oxoy ox oy cbqf oy3 qf k oy 2 2 3 1 o w ow 3 ; 2c bqf oy2 oy3
ð12Þ 2 #
" ow oT ow oT o2 T oC oT DT oT ¼ a 2 þ s1 D B þ oy ox ox oy oy oy oy T1 oy ! 2 1 o2 w þ lþ cbðqcÞf oy2 2 2 4 r 1 ow 2 ow þ B 3 ; ðqcÞf o oy 6c bðqcÞf oy2 ð13Þ ow oC ow oC o2 C D T o2 T ¼ DB 2 þ þ ko ðC C1 Þ: oy ox ox oy oy T1 oy2 ð14Þ With the help of the similarity transformations: pffiffiffiffiffiffiffiffiffiffiffi T T1 ; w ¼ 2Umxf ðgÞ; hðgÞ ¼ Twrffiffiffiffiffiffiffiffiffi T1 C C1 U ; ; g¼y uðgÞ ¼ 2mx Cw C1
1 00 h ðgÞ þ f ðgÞh0 ðgÞ þ Nb h0 ðgÞ0 uðgÞ þ Nt h02 ðgÞ pr þ Ec ð1 þ eÞf 002 ðgÞ
ð17Þ
þ MEc f 02 ðgÞ efdEc f 004 ðgÞ ¼ 0;
ð10Þ
v ¼ 0; u ¼ Uw ¼ kU; T ¼ Tw ; C ¼ Cw at y ¼ 0; and u ! U; T ! T1 ; C ! C1 at y ! 1:
ð16Þ
Nt 00 h ðgÞ þ Le f u0 ðgÞ þ 2cRex Le uðgÞ ¼ 0; Nb
ð18Þ
subject to the boundary conditions: f ð0Þ ¼ 0;
f 0 ð0Þ ¼ k;
0
and f ðgÞ ! 1;
hð0Þ ¼ 1;
hðgÞ ! 0;
uð0Þ ¼ 1;
uðgÞ ! 0;
as g ! 0;
as g ! 1; ð19Þ
where M ¼
2xrB2o Uqf
;
K¼
2xl ; Uqf k
2
Ec ¼
U ; ðCp Þf ðTw T1 Þ
m Ux mk m Le ¼ ; c ¼ 2 ; Pr ¼ ; ; Rex ¼ DB m a U ðqCÞp DB ðCw C1 Þ ðqCÞp Dt ðCw C1 Þ ; Nt ¼ Nb ¼ ðqcÞf m ðqcÞf T1 e¼
1 ; lbc
d¼
U2 ; 2c2
f¼
Ux ; 6t ð20Þ
where M; K; Ec ; Le ; Rex ; c; Pr ; Nb and Nt are Hertmann number, permeability parameter, Eckert number, Lewis number, Reynolds number, chemical reaction parameter, Prandtl number, Brownian motion parameter and thermophoresis parameter. e; d and f are the fluid material parameters. Moreover the physical quantities cf (skin-friction coefficient), Nux (local Nussetl number) and Shx (Local Sherwood number) are defined as: sw xqw xqm ; Shx ¼ ; cf ¼ ; Nux ¼ qU 2 kðTw T1 Þ DB ðCw C1 Þ ð21Þ where, 1 ou 1 1 ou 3 ; sw ¼ l þ bc oy 6b c oy oT : qm ¼ DB oy y¼0
qw ¼ j
oT ; oy y¼0
ð15Þ Also, with the help of the similarity transformations (15) we can write:
equations (12)–(14) can be reduced to the following nondimensional nonlinear ordinary differential equations:
123
Microsystem Technologies
(a)
f'
1.0
2.0
0.5
0.5
1
0.8
1.5
1.5
0.6
1.0
0.4
0.5
0.2 0.0
0.2
0.4
0.6
0.8
1
1.5
1.0
1.2
1.4
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.0 0.5 1
0.5
1.5
0.0
0.5
1.0 0.0
(b)
0.2
0.4
0.6
0.8
1.2
1.4
f'
1.0
0.8
M
0.5
M
1.5
M
2
2.0 1.5
0.6
1.0
0.4
0.5
0.2 0.0
1.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.0
1.4
0.2
0.4
0.6
1.0 0.5 0.0
M
0.5
M
1.5
M
2
0.5 1.0 1.5 0.0
123
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.8
1.0
M
0.5
M
1.5
M
2
1.2
1.4
Microsystem Technologies b Fig. 2 a Represents the velocity distribution, temperature and con-
centration against the fluid material parameter e for different values e ¼ 0:6; 1; 1:6 as: Pr ¼ Le ¼ 1; ¼ 0:5; Nb ¼ 0:5; Ec ¼ Rex ¼ 2; k ¼ 0:2; M ¼ k ¼ c ¼ d ¼ f ¼ Nt ¼ 0:5: b Represents the velocity distribution, temperature and concentration against M for different values of M = 0.6, 1, 1.6 as: Pr ¼ Le ¼ 1; ¼ 0:5; Nb ¼ 0:5; Ec ¼ Rex ¼ 2; k ¼ 0:2; e ¼ k ¼ c ¼ d ¼ f ¼ Nt ¼ 0:5: c Represents the velocity distribution, temperature and concentration against k for different values of k = 0.6, 1, 1.6, 2, 2.6 as: Pr ¼ Le ¼ 1; ¼ 0:5; Nb ¼ 0:5; Ec ¼ Rex ¼ 2; k ¼ 0:2; e ¼ M ¼ c ¼ d ¼ f ¼ Nt ¼ 0:5
pffiffiffiffiffiffiffiffiffi 2Rex Cf ¼ ð1 þ eÞf 00 ð0Þ edff 003 ð0Þ; rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 2 2 h0 ð0Þ ¼ Nux ; u0 ð0Þ ¼ Shx : Rex Rex
ð22Þ
where, Cf ; sw ; qw and qm are skin-friction coefficient, shear stress, plate heat flux and plate mass flux respectively. It should be noted when studying the motion of a Newtonian fluid (e ¼ 0) with no external magnetic field (M ¼ 0), no chemical reaction (c ¼ 0), no porosity (K ¼ 0 ) and no viscous dissipation (Ec ¼ 0 ) we obtain the work of Bachok et al. (2010). In addition if no nanoparticles are considered we obtain the work of Weidman et al. (2006)
(c)
f'
1.0
k k
0.8
0.6
3 Results and discussion In this problem we studied the motion of non-Newtonian Nanofluid with heat and mass transfer over a semi-infinite flat plate, this system is stressed by uniform external magnetic field. The Ohmic and Viscous dissipation as well as the chemical reaction are taken in consideration. The problem is modulated mathematically by a system of nonliner partial differential equations (PDE). Suitable similarity transformations are used to transform this system into a system of non-liner ordinary differential equations (ODEs). The Mathematica program ParametricNDSolve package is used to solve the obtained (ODEs) numerically with respect to appropriate boundary conditions. The distributions of velocity, temperature and concentration as well as skin fraction and rate of the heat and mass transfer are obtained. The effects of the problem physical parameters Pr , Rex , Nt , Nb , Le , e, d, f, Ec , M, k and c are discussed. The results are illustrated through a set of Figs. 2, 3, 4 and 5. It should be noted that when e ¼ 0; Ec ¼ 0; M ¼ 0; K ¼ 0 and c ¼ 0 in Eqs. (16)–(18) we obtain the same results of Bachok et al. (2010) (Fig. 6) which confirms the accuracy of the used method and the obtained results of the considered problem.
1.5
0.5 1
k
1.5
k
2
k
2.5
1.0
k
0.5
k
1
k
1.5
k
2
k
2.5
0.5 0.4 0.0 0.2 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
3.0 2.5 2.0 1.5
k
0.5
k
1
k
1.5
k
2
k
2.5
1.0 0.5 0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
b Fig. 2 continued
123
Microsystem Technologies
1.0
(a)
2.0
0.5 0.0
Ec
1
Ec
2
Ec
3
1.5
Ec
1
Ec
2
Ec
3
1.0
0.5 0.5
1.0 1.5 0.0
1.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
(b)
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.0 1
0.8
1
2
0.6
0.5
0.8
1.5
0.6
1.5
0.4
0.4
0.2
0.2 0.0 0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.2 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig. 3 a Represents the concentration and temperature against Ec for different values Ec = 1, 2, 3 as: Pr ¼ Le ¼ 1; k ¼ 0:2; e ¼ Rex ¼ b Represents the 2; M ¼ k ¼ c ¼ d ¼ f ¼ Nt ¼ Nb ¼ 0:5:
concentration and temperature against c for different values c = 0.6, 1, 1.5, 2 as: Pr ¼ Le ¼ 1; k ¼ 0:2; e ¼ Rex ¼ Ec ¼ 2; M ¼ k ¼ d ¼ f ¼ Nt ¼ Nb ¼ 0:5
Figure 2a: shows the relation between the velocity f 0 ðgÞ, the temperature hðgÞ and the concentration uðgÞ with respect to the fluid material parameter e. It is observed that the increasing of e both velocity and temperature are increases while decreases fluid concentration. Figure 2b, c shows the effects of magnetic field M and permeability parameter k respectively on the velocity f 0 ðgÞ, the temperature hðgÞ and the concentration uðgÞ. It is observed that, as a result of the induced Lorentz forces created by the transverse magnetic field M, the increases in M decrease the velocity and fluid concentration but increases temperature. Moreover increasing the fluid permeability parameter k increases velocity profile while decreases concentration. The porous media creates huge resistance to fluid flow and consequently minor changes occur in momentum boundary layer. This also explains the trivial changes in temperature profile as shown in the Fig. 2c.
Figure 3a, b shows the influence of Eckert number Ec and the chemical reaction parameter c on both temperature hðgÞ and concentration uðgÞ. It is noticed that the increase in Eckert number Ec decreases concentration but increases temperature. Also increasing the chemical reaction c slightly increases fluid temperature and concentration. Thus, more heat is generated in the boundary layer region due to the viscous dissipation and the transfer of chemical energy into thermal energy. Figure 4a, b presents the influence of Prandtl number Pr and Brownian motion parameter Nb on both temperature hðgÞ and concentration uðgÞ. The increases of Prandtl number increases temperature but decreases concentration. Also Brownian motion parameter Nb increases fluid temperature but decreases fluid concentration. These phenomena presented that enhanced thermal conductivity of a nanofluid is mainly due to Brownian motion which producing micromixing.
123
Microsystem Technologies
2.0
(a)
1.5
pr
1
pr
1.5
pr
2
2.4 2.2 2.0
pr
1
pr
1.5
pr
2
1.8
1.0
1.6 1.4
0.5
1.2 0.0 0.0
5
0.2
0.4
0.6
0.8
1.0
1.2
1.4
(b)
1.0 0.0
0.4
0.6
0.8
1.0
1.2
5
4
Nb
0.5
Nb
1
Nb
1.5
4
3
3
2
2
1
1
0 0.0
0.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.0
0.2
0.4
0.6
0.8
1.0
Nb
2
Nb
4
Nb
6
1.2
1.4
Fig. 4 a Represents the concentration and temperature against pr for different values pr = 1, 1.5, 2 as: Le ¼ 1; k ¼ 0:2; e ¼ Rex ¼ Ec ¼ b Represents the 2; M ¼ k ¼ d ¼ c ¼ f ¼ Nt ¼ Nb ¼ 0:5.
concentration and temperature against Nb for different values Nb = 0.6, 1, 1.6 and 2, 4, 8 as: Le ¼ 1; k ¼ 0:2; e ¼ Rex ¼ Ec ¼ 2; M ¼ k ¼ d ¼ c ¼ f ¼ Nt ¼ pr ¼ 0:5
Figure 5a, b illustrate the relation between the temperature hðgÞ and the concentration uðgÞ for different values of thermophoresis parameter Nt and the Lewis number Le. The nanoparticle concentration and the temperature are decreases as a result of the increases of thermophoretic parameter while the increase in Lewis number values decrease the concentration and temperature.
magnetic field through porous medium. The governing partial differential equations of the problem are transformed to a system of ordinary non linear differential equations. The Mathematica program package Parametric NDSolve is used to solve the obtained system numerically. The Ohmic heating and viscous dissipation with chemical reaction are considered. The effects of the physical parameter of the problem, say Hartmann number, permeability parameters are discussed as well as the chemical parameter, Eckert parameter, Reynolds number and the fluid material parameters e; d and f on. The main results of this study are summarized as follows:
4 Conclusions In this paper, the motion of nano non-Newtonian Eyring– Powell fluid with heat and mass transfer over a semi-infinite moving plate is studied under the effect of a uniform
123
Microsystem Technologies
2.0
(a)
1.5
Nt
0.5
Nt
1
Nt
1.5
0 2 4
1.0
6 0.5
8
0.0 0.0
2.0
0.2
0.4
0.6
0.8
1.0
1.2
10 0.0
1.4
(b)
0.2
0.4
0.6
0.8
1.0
Nt
0.5
Nt
1
Nt
1.5
1.2
1.4
1.0
1.5
Le
1
Le
1.5
Le
2
0.5 0.0
1.0
Le
1
Le
1.5
Le
2
0.5 1.0
0.5
1.5 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
2.0 0.0
1.4
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig. 5 a Represents the concentration and temperature against Nt for different values Nt = 0.6, 1, 1.6 as Le ¼ 1; k ¼ 0:2; e ¼ Rex ¼ Ec ¼ 2; M ¼ k ¼ d ¼ c ¼ f ¼ Np ¼ pr ¼ 0:5. b Represents the
concentration and temperature against Le for different values Le = 1, 1.6, 2 as: k ¼ 0:2; e ¼ Rex ¼ Ec ¼ 2; M ¼ k ¼ d ¼ c ¼ f ¼ Np ¼ pr ¼ Nt ¼ 0:5
•
•
•
The effects of the physical quantities (Echert number Ec, Prandtle number Pr, Lewise number Le, Reynolds number Rex, Chemical reaction c, Brawnian motion parameter Nb, thermophoretic parameter Nt and the fluid material parameter f, d) on velocity are very small and can be neglected. Increasing the fluid material parameters e decreases fluid concentration while increases both velocity and temperature. Increasing M decreases velocity and concentration but increases fluid temperature.
123
•
•
The magnetic field has an opposite effect for the porosity parameter. Increasing the porosity parameter k increases the velocity and decreases temperature but increases fluid concentration. Increasing Eckert number Ec decreases fluid temperature and concentration.
Microsystem Technologies
1.0
(a)
1.0 Le= 2
0.8
Le= 6
0.6
0.4
0.2
0.2
1.2
0
1
2
3
Le= 10 Le= 30
0.6
Le= 10
0.4
0.0
Le= 2
0.8
Le= 4
4
(b)
Le= 100
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
1.0
1.0
Nb= 0.5
0.8
Nb= 2.5
Nb= 0.5
0.8
Nb= 1
Nb= 1 Nb= 1.5
0.6
Nb= 2
0.6
0.4
0.4
0.2
0.2 0.0
0
1
2
3
4
0.0
0
1
2
3
4
Fig. 6 Validation with Bachok et al. (2010). a The nanoparticle fraction profiles uðgÞ and the temperature profile hðgÞ for various values of Le at Pr = 2, Nb = 0.5, Nt = 0.5 and k = 0.5. b The
nanoparticle fraction profiles uðgÞ and the temperature profile hðgÞ for various values of Nb = at Pr = 2, Le = 2, Nt = 0.5 and k = 0.5
5 Applications
References
There are many applications of the considered problem in industries The most important one is in petroleum industry, where drilling fluid plays a very important part. Nanotechnology could improve the quality and the properties of drilling fluids.
Bachok N, Ishak A, Pop I (2010) Boundary-layer flow of nanofluids over a moving surface in a flowing fluid. Int J Therm Sci 49:1663–1668 Choi SUS (1995) Enhancing thermal conductivity of fluids with nanoparticles. In: Proceedings of the ASME international mechanical engineering congress and exposition. San Francisco USA ASE FED 231/MD, vol 66, pp 99–105 Choi SUS, Zhang ZG, Lockwood W, Yu FE, Grulke EA (2001) Anomalously thermal conductivity enhancement in nanotube suspensions. Appl Phys Lett 79:2252–2254 Cimpean DS, Pop I (2012) Fully developed mixed convection flow of a nanofluid through an inclined channel filled with a porous medium. Int J Heat Mass Transf 55:907–914 Eldabe NTM, Hassan AA, Mona AA (2003) Effect of couple stresses on the MHD of a non-Newtonian unsteady flow between two
•
Many researches work are done in rheology science where synthetic polymers and their solutions in different solvents are necessary in industrial application.
Acknowledgements The authors would like to express their sincere thanks to the reviewers for their valuable comments and suggestions.
123
Microsystem Technologies parallel porous plates. Zeitschrift fu¨r Naturforschung A 58(4):204–210 Eldabe NT, El-Naby MAA, Ashraf F, Amaney S (2008) Numerical study of the flow of magnetohydrodynamic non-Newtonian fluid obeying the Eyring–Powell model through a non-Darcy porous medium with coupled heat and mass transfer. J Porous Media 70:691–700. https://doi.org/10.1615/JPorMedia.v11.i7 Eldabe NTM, Sallam NS, Abou-Zeid MY (2012) Numerical study of viscous dissipation effect on free convection heat and mass transfer of MHD non-Newtonian fluid flow through a porous medium. J the Egypt Math Soc 20:139–151 Farooq U, Hayat T, Alsaedi A, Liao S (2014) Heat and mass transfer of two-layer flows of third-grade nanofluids in a vertical channel. Appl Math Comput 242:528–540 Farooq U, Zhao YL, Hayat T, Alsaedi A, Liao SJ (2015) Application of the HAM-based Mathematica package BVPh 2.0 on MHD Falkner–Skan flow of nano-fluid. Comput Fluid 111:69–75 Haq RU, Khan ZH, Khan WA (2014) Thermophysical effects of carbon nanotubes on MHD flow over a stretching surface. Phys E 63:215–222 Haq RU, Nadeem S, Khan ZH, Noor NFM (2015) Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubes. Phys B 457(15):40–47 Hayat T, Iqbal Z, Qasim M, Obaidat S (2012) Steady flow of an Eyring Powell fluid over a moving surface with convective boundary conditions. Int J Heat Mass Transf 55:1817–1822 Islam S, Shah A, Zhou CY, Ali I (2009) Homotopy perturbation analysis of slider bearing with Powell–Eyring fluid. Z Angew Math Phys ZAMP 60:1178–1193 Khan JA, Mustafa M, Hayat T, Farooq MA, Alsaedi A, Liao SJ (2014) On model for three- dimensional flow of nanofluid. An application to solar energy. J Mol Liq 194:41–47 Kuzentsov AV, Nield DA (2010) Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci 49:243–247 Maity S (2016) Unsteady flow of thin nanoliquid film over a stretching sheet in the presence of thermal radiation. Eur Phys J Plus 131(2):49 (1–15) Maity S (2017) Thermocapillary flow of thin Cu-water nanoliquid film during spin coating process. Int Nano Lett 7(1):9–23
123
Maity S, Ghatani Y, Dandpat BS (2016) Thermocapillary flow of a thin nanoliquid film over an unsteady stretching sheet. J Heat Transf (ASME). https://doi.org/10.1115/1.4032146 Nadeem S, Haq RU, Khan ZH (2014) Heat transfer analysis of waterbased nanofluid over an exponentially stretching sheet. Alexandria Eng J 53:219–224 Niu J, Fu C, Tan W (2012) Slip-flow and heat transfer of a nonNewtonian nanofluid in a microtube. PLoS One 7(5):e37274. https://doi.org/10.1371/journal.pone.0037274 Patel M, Timol MG (2009) Numerical treatment of Powell–Eyring fluid flow using method of asymptotic boundary conditions. Appl Numer Math 59:2584–2592 Powell RE, Eyring H (1944) Mechanism for relaxation theory of viscosity. Nature 154:427–428 Rashidi MM, Abelman S, Mehr NF (2013) Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid. Int J Heat Mass Transf 62:515–525 Sheikholeslami M, Gorji-Bandpay M, Ganji D (2012) Magnetic field effects on natural convection around a horizontal circular cylinder inside a square enclosure filled with nanofluid. Int Commun Heat Mass Transf 39:978–986 Sheikholeslami M, Hatami M, Ganji DD (2013) Analytical investigation of MHD nanofluid flow in a semi-porous channel. Powder Technol 246:327–336 Thirupathi T, Be´g OA, Kadir A (2017) Numerical study of heat source/sink effects on dissipative magnetic nanofluid flow from a non-linear inclined stretching/shrinking sheet. J Mol Liq 232(2017):159–173 Turkyilmazoglu M (2013) Unsteady convection flow of some nanofluids past a moving vertical flat plate with heat transfer. J Heat Transf. https://doi.org/10.1115/1.4025730 Turkyilmazoglu M (2014) Nanofluid flow and heat transfer due to a rotating disk. Comput Fluids 94:139–146 Vleggaar J (1977) Laminar boundary layer behavior on continuous accelerating. Surf Chem Eng Sci 32:1517–1525 Weidman TPD, Kubitschek DG, Davis AMJ (2006) The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int J Eng Sci 44:730–737 Xu H, Pop I, You XC (2013) Flow and heat transfer in a nano-liquid film over an unsteady stretching surface. Int J Heat Mass Transf 60:646–652
