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An optimal study for three-dimensional flow of Maxwell nanofluid subject to rotating frame
An optimal study for three-dimensional flow of Maxwell nanofluid subject to rotating frame T. Hayat, Taseer Muhammad, M. Mustafa, A. Alsaedi PII: DOI: Reference:
S0167-7322(16)33232-9 doi: 10.1016/j.molliq.2017.01.005 MOLLIQ 6797
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
19 October 2016 26 December 2016 2 January 2017
Please cite this article as: T. Hayat, Taseer Muhammad, M. Mustafa, A. Alsaedi, An optimal study for three-dimensional flow of Maxwell nanofluid subject to rotating frame, Journal of Molecular Liquids (2017), doi: 10.1016/j.molliq.2017.01.005
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An optimal study for three-dimensional flow of Maxwell nanofluid subject to rotating frame b
Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan
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a
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T. Hayata,b , Taseer Muhammada∗ , M. Mustafac and A. Alsaedib
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of
c
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Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia School of Natural Sciences (SNS), National University of Sciences and Technology
∗
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(NUST), Islamabad 44000, Pakistan Corresponding author E-mail: taseer [email protected] (Taseer Muhammad)
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Abstract: Here we are concerned with optimal homotopy solutions for flow of Maxwell nanofluid in rotating frame. Flow is induced by uniform stretching of the boundary surface in one direction. Buongiorno model is adopted which features the novel aspects of Brownian diffusion
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and thermophoresis. Boundary layer approximations are invoked to simplify the governing system of partial differential equations. Appropriate relations are introduced to nondimensionalize the rele-
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vant boundary layer expressions. Newly suggested condition associated with zero nanoparticles mass flux at the boundary is imposed. Uniformly valid convergent solution expressions are developed by
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means of optimal homotopy analysis technique (OHAM). Plots have been portrayed in order to explain the role of embedded flow parameters on the solutions. Heat transfer rate at the surface has been computed and analyzed. Our findings show that the temperature and concentration fields are smaller for Newtonian fluid when compared with the upper-convected Maxwell (UCM). Moreover Brownian diffusion has mild influence of heat flux at the boundary. Viscoelastic effect has tendency to reduce heat transfer rate from the stretching boundary.
Keywords: Maxwell fluid; Rotating frame; Nanoparticles; Nonlinear analysis; OHAM.
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Nomenclature velocity components
x, y, z
µ
dynamic viscosity
ρf
ν
kinematic viscosity
λ1
T
temperature
C
T∞
ambient fluid temperature
C∞
ambient fluid concentration
αm
thermal diffusivity
k
thermal conductivity
(ρc)p
effective heat capacity of nanoparticles
(ρc)f
heat capacity of fluid
DB
Brownian diffusion coefficient
DT
thermophoretic diffusion coefficient
uw
surface velocity
Tw
surface temperature
a
positive constant
Ω
angular velocity
ζ
similarity variable
f ′, g
dimensionless velocities
θ
dimensionless temperature
φ
dimensionless concentration
β
Deborah number
λ
rotation parameter
Sc
Schmidt number
Pr
Prandtl number
Nb
Brownian motion parameter
Nt
thermophoresis parameter
N ux
local Nusselt number
Rex
local Reynolds number
density of base fluid relaxation time concentration
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1
coordinate axes
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u, v, w
Introduction
Nanotechnology is an attractive area of research now a days owing to its rich applications in the industrial and technological processes. The current researchers are engaged to investigate the mechanisms through the nanomaterials. A solid-liquid mixture of tiny size nanoparticles and base liquid is known as nanofluid. The colloids of base liquid and nanoparticles have essential physical attributes which have potential role in coating related applications, metal working processes, ceramics, magnetic drug targeting, paintings, many heat transfer applications such as nuclear reactor cooling, air-conditioning, transportation etc. Nanofluids are pronounced as super coolants in light of the fact that their effective thermal conductivity is much larger in comparison to the traditional liquids. The reduction of the system size and improvement in thermal transport can be accomplished with an aid of nanoparticles. Choi [1] experimentally explored the mechanism of nanoparticles and concluded that the insertion
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of nanoparticles into ordinary base liquids is highly useful technique to enhance the cooling capability of conventional liquids. Buongiorno [2] developed a two-component model for in-
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vestigating thermal energy transport in nanofluids. The model is based on two important
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slip mechanisms namely Brownian motion and thermophoresis. Recently, a large number of investigations regarding the flow analysis of nanofluids have been presented (see [3 − 30] and
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several studies therein).
The investigations on rotating flows near stretchable or inextensible boundaries have
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been growing considerably since the last decade due to their occurrence in engineering and geophysical processes. Such flows commonly involve in rotor-stator system, food processing, rotating machinery, disk cleaners, gas turbine design and several others. Available data
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confirms that flows in rotating frame by a stretchable surface have been rarely attempted. Wang [31] provided analytical solutions for rotating fluid flow past a stretchable surface by
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employing perturbation technique. Then Takhar et al. [32] examined the magnetic field characteristics in rotating flow bounded by a stretchable surface. Nazar et al. [33] reported the time-dependent rotating flow past an impulsively deforming surface utilizing Keller-box
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numerical procedure. Locally similar solutions for rotating fluid flow by an exponentially deforming surface have been addressed by Javed et al. [34] . Zaimi et al. [35] prepared the self-
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similar solutions for rotating viscoelastic fluid flow due to an impermeable stretchable surface. Rosali et al. [36] numerically addressed the rotating flow past an exponentially shrinking porous surface. Mustafa [37] examined the rotating flow of viscoelastic liquid induced by a stretchable surface through thermal relaxation effects. Hayat et al. [38] employed nonFourier heat flux theory for three dimensional (3D) rotating flow of Jeffrey material. Mustafa et al. [39] examined the nonlinear radiation characteristics in rotating nanofluid flow past a stretchable surface. Recently, Shafique et al. [40] studied the heat and mass transfer aspects in rotating flow of Maxwell material with binary chemical reaction and activation energy. Inspired by the above mentioned literature, our objectives of present attempt are four folds. Firstly to model and analyze the rotating flow of Maxwell nanofluid adjacent to a deforming surface. Maxwell fluid model although predicts the relaxation time features but it is not capable of describing shear thinning/shear thickening, retardation time etc [41, 42] . Secondly to consider the rotating frame rather than the fixed frame. Thirdly to employ the Buongiorno’s model for nanofluid transport phenomena. Fourth to derive convergent series solutions for the velocities, temperature and concentration by a powerful optimal homotopy
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analysis method (OHAM) [43 − 50]. The contributions of various pertinent parameters are studied and discussed. Further the heat transfer rate at the surface is analyzed through
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Formulation
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2
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numerical values.
We explore the rotating flow of upper-convected Maxwell (UCM) nanofluid over a linearly stretchable surface. Nanofluid model describes the attributes of Brownian motion and ther-
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mophoresis. We choose the Cartesian coordinate system such that the surface is aligned with the xy−plane and fluid is considered in the space z ≥ 0. The surface deforms linearly in the
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x−direction with rate a. Further the fluid is subjected to uniform rotation about z−axis with constant angular velocity Ω. The associated equations governing the Maxwell nanofluid flow in rotating frame are [40, 50] :
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∂u ∂v ∂w + + = 0, ∂x ∂y ∂z 2 2 2 u2 ∂∂xu2 + v 2 ∂∂yu2 + w2 ∂∂zu2 ∂u ∂u ∂u ∂ 2u ∂2u ∂2u ∂2u u +v +w −2Ωv = ν 2 −λ1 +2uv ∂x∂y + 2vw ∂y∂z + 2uw ∂x∂z ∂x ∂y ∂z ∂z ∂v ∂v ∂u ∂u −2Ω u ∂x + v ∂y + w ∂v − u + 2Ω v ∂z ∂x ∂y ∂2v 2 ∂2v 2 ∂2v u2 ∂x 2 + v ∂y 2 + w ∂z 2 ∂v ∂v ∂ 2v ∂v ∂2v ∂2v ∂2v u +v +w +2Ωu = ν 2 −λ1 +2uv ∂x∂y + 2vw ∂y∂z + 2uw ∂x∂z ∂x ∂y ∂z ∂z ∂u ∂u ∂v ∂v +2Ω u ∂u + v + w − u + 2Ω v ∂x ∂y ∂z ∂x ∂y ! 2 ∂ 2T ∂T ∂C (ρc)p ∂T ∂T ∂T DT ∂T DB , u +v +w = αm 2 + + ∂x ∂y ∂z ∂z (ρc)f ∂z ∂z T∞ ∂z 2 ∂ C DT ∂ 2 T ∂C ∂C ∂C u + . +v +w = DB ∂x ∂y ∂z ∂z 2 T∞ ∂z 2
(1)
, (2)
, (3) (4) (5)
We have the following prescribed conditions: u = uw (x) = ax, v = 0, w = 0, T = Tw , DB
∂C DT ∂T + = 0 at z = 0, ∂z T∞ ∂z
u → 0, v → 0, T → T∞ , C → C∞ as z → ∞.
(6) (7)
Here u, v and w represent the components of velocity in x−, y− and z−directions respectively, λ1 stands for fluid relaxation time, ν = µ/ρf for kinematic viscosity, ρf for density of base liquid, µ for dynamic viscosity, T for temperature, αm = k/(ρc)f for thermal diffusivity, k for
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thermal conductivity, (ρc)p for effective heat capacity of nanoparticles, (ρc)f for heat capacity of fluid, C for concentration, DB for Brownian diffusion coefficient, DT for thermophoretic
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diffusion coefficient, Tw for wall temperature, T∞ for ambient fluid temperature, C∞ for
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ambient fluid concentration and a for positive constant. Considering [40, 50] : (8)
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u = axf ′ (ζ), v = axg(ζ), w = − (aν)1/2 f (ζ) , " 1/2 ∞ ∞ , φ(ζ) = C−C , ζ = νa z. θ(ζ) = TTw−T −T∞ C∞
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Equation (1) is identically verified while Eqs. (2) − (7) take to the following forms " f ′′′ + f f ′′ − f ′2 + 2λ (g − βf g ′ ) + β 2f f ′ f ′′ − f 2 f ′′′ = 0,
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" " " g ′′ + f g ′ − f ′ g − 2λ f ′ + β f ′2 − f f ′′ + g 2 + β 2f f ′ g ′ − f 2 g ′′ = 0, " θ′′ + Pr f θ′ + N bθ′ φ′ + N tθ′2 = 0,
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φ′′ + Scf φ′ +
N t ′′ θ = 0, Nb
(9) (10) (11) (12) (13)
f ′ (∞) → 0, g(∞) → 0, θ(∞) → 0, φ(∞) → 0.
(14)
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f (0) = g(0) = 0, f ′ (0) = 1, θ(0) = 1, N bφ′ (0) + N tθ′ (0) = 0,
In the above expressions λ stands for rotation parameter, β for Deborah number, Pr for
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Prandtl number, N b for Brownian motion parameter, N t for thermophoresis parameter and Sc for Schmidt number. These dimensionless variables can be specified by using the definitions given below:
Nb =
λ=
Ω , a
(ρc)p DB C∞ , (ρc)f ν
β = λ1 a, Pr = ανm , (ρc)p DT (Tw −T∞ ) Nt = , (ρc)f νT∞
Sc =
ν . DB
The local Nusselt number N ux is defined by
(15)
x ∂T Re N ux = − = −θ′ (0) , x (Tw − T∞ ) ∂z z=0
−1/2
(16)
where Rex = uw x/ν depicts the local Reynolds numbers. It is also observed that nondimensional mass flux described by Sherwood number Shx is now identically zero.
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Series solutions by OHAM
Our objective here is to develop the convergent series solutions of Eqs. (9) to (12) with conditions (13) and (14) through the optimal homotopy analysis technique (OHAM). Suitable
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initial approximations and the auxiliary linear operators for f (ζ) , g(ζ), θ(ζ) and φ(ζ) are as follows:
(17)
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f0 (ζ) = 1 − exp (−ζ) , g0 (ζ) = 0,
dζ
φ
2
dζ
2
(18)
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θ
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Nt θ0 (ζ) = exp (−ζ) , φ0 (ζ) = − N exp (−ζ) , b 3 2 df , Lg = ddζ g2 − g, Lf = ddζf3 − dζ 2 2 L = d θ − θ, L = d φ − φ.
The above linear operators satisfy the following properties:
g
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Lf [A∗1 + A∗2 exp (ζ) + A∗3 exp (−ζ)] = 0, ∗ ∗ L [A exp (ζ) + A exp (−ζ)] = 0, 4
5
Lθ [A∗6 exp (ζ) + A∗7 exp (−ζ)] = 0,
(19)
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Lφ [A∗8 exp (ζ) + A∗9 exp (−ζ)] = 0,
in which A∗j (j = 1 − 9) are arbitrary constants. The zeroth and mth-order deformation
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problems can be easily formulated in view of the above linear operators. The deformation
Optimal convergence control parameters
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4
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problems have been solved by BVPh2.0 of the software Mathematica.
Let us denote the non-zero auxiliary parameters for functions f, g, θ and φ by ~f , ~g , ~θ and ~φ respectively. These parameters are useful to regulate the convergence and rate of homotopy solutions. To obtain the optimal values of ~f , ~g , ~θ and ~φ , we have employed the idea of minimizing the squared residual of the governing equations defined by (see Liao [43] for details): εfm =
1 k+1
k X j=0
Nf
k X 1 Ng εgm = k + 1 j=0 εθm =
εφm =
1 k+1 1 k+1
k X
m X
k X
m X
j=0
j=0
Nθ
Nφ
fˆ(ζ),
i=0
i=0
m X
fˆ(ζ),
i=0
m X
fˆ(ζ),
fˆ(ζ),
i=0
m X
gˆ(ζ)
i=0
gˆ(ζ),
i=0
m X
gˆ(ζ)
i=0
i=0
m X
m X
m X
!
!
ˆθ(ζ),
i=0
gˆ(ζ),
m X i=0
ζ=jδζ
ζ=jδζ
m X
2
,
(20)
2
,
(21)
ˆ φ(ζ),
i=0
ˆθ(ζ),
m X i=0
ˆ φ(ζ),
!
ζ=jδζ
!
ζ=jδζ
2
,
(22)
2
.
(23)
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Here Nf , Ng , Nθ and Nφ denote the non-linear operators corresponding to Eqs. (9) − (12)
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respectively. Following Liao [43] (24)
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εtm = εfm + εgm + εθm + εφm ,
where εtm stands for total squared residual error, k = 20 and δζ = 0.5. The total average
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squared residual error is minimized by employing Mathematica package BVPh2.0. A case has been considered where β = λ = 0.2, N t = 0.1, N b = 0.3 and Pr = Sc = 1.0. The optimal
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values of convergence-control parameters at 2nd order of approximations are ~f = −1.50853, ~g = −1.17662, ~θ = −1.15433 and ~φ = −1.02305 and the total averaged squared residual error is εtm = 2.21×10−4 . Table 1 shows the individual average squared residual at the optimal
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values of convergence control parameters when m = 2. It is observed that the averaged squared residual error reduces with higher order approximations.
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Table 1. Individual averaged squared residual errors considering optimal values of auxiliary parameters.
εgm
εθm
εφm
2
1.47 × 10−5
8.14 × 10−6
1.04 × 10−4
9.41 × 10−5
6
3.01 × 10−8
5.41 × 10−8
2.34 × 10−6
6.87 × 10−6
10
4.52 × 10−10
1.77 × 10−9
1.65 × 10−7
1.14 × 10−6
16
1.72 × 10−11
1.61 × 10−11
4.83 × 10−9
8.01 × 10−8
20
6.60 × 10−13
1.99 × 10−12
5.72 × 10−10
1.42 × 10−8
26
7.19 × 10−14
7.96 × 10−14
3.08 × 10−11
1.10 × 10−9
30
5.50 × 10−15
1.62 × 10−14
4.93 × 10−12
2.03 × 10−10
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εfm
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m
5
Discussion
This portion explores the influences of various pertinent flow parameters like Deborah number β, rotation parameter λ, Prandtl number Pr, Schmidt number Sc, Brownian motion parameter N b and thermophoresis parameter N t on the nondimensional velocities f ′ (ζ) and g (ζ) , temperature θ (ζ) and concentration φ (ζ) fields. Fig. 1 illustrates that how Deborah number β effect the velocity field f ′ (ζ). From this Fig. we can say that an increment in the values of Deborah number β shows decreasing trend in velocity field f ′ (ζ) and momentum layer thickness. It is seen that as the Deborah number β enhances, the fluid relaxation time
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enlarges which yields a lower velocity field f ′ (ζ) and reduces momentum layer thickness. Fig. 2 presents the change in the velocity field f ′ (ζ) for varying values of rotation parameter λ.
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From this Fig. it has been examined that an enhancement in the value of rotation parameter
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λ shows decreasing trend for velocity field f ′ (ζ) and momentum layer thickness. Larger values of rotation parameter λ correspond to higher rotation to stretching rates ratio. In other
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words, the rotation rate becomes higher in comparison to the stretching rate when λ enlarges. Therefore the larger rotation effect leads to decrease velocity field f ′ (ζ) and hydrodynamic
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layer thickness. Fig. 3 is sketched to examine that how velocity field g(ζ) is affected as the Deborah number β is varied. Since the sheet is stretched only in the x−direction and thus flow in y−direction is only anticipated due to the rotating frame. Velocity field g(ζ) which is
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related with y−component of velocity decreases in magnitude away from the stretching surface. However it shows opposite effects in the region of the stretching surface. Moreover the
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negative value of g(ζ) depicts that flow is only in the negative y−direction due to rotational effects. Fig. 4 shows the deviation in function g(ζ) with the variation in rotation parameter λ. It is visualized that rotation parameter has an important role in accelerating flow along
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the y−direction. An increment in rotation parameter λ leads to an oscillatory pattern in the velocity field g(ζ) (a fact that is also observed by Zaimi et al. [35]). Fig. 5 shows the curves
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of temperature field θ (ζ) for varying Deborah number β. From this Fig. it has been observed that by increasing Deborah number β, penetration depth of temperature becomes thicker. Fig. 6 plots the temperature field θ (ζ) for distinct values of rotation parameter λ. Larger values of rotation parameter λ constitutes a higher temperature field and thicker thermal layer thickness. Fig. 7 illustrates that how the temperature field θ (ζ) gets effected with the variation in Prandtl number Pr . It is noted that by enhancing Prandtl number Pr, the temperature field θ (ζ) decreases and thermal layer thickness reduces. Physically, as Prandtl number Pr has inverse relationship with thermal diffusivity, therefore, an increment in Pr leads to weaker thermal diffusion and hence thinner penetration depth of θ (ζ) . Thicker thermal boundary layer attributed to larger Prandtl number is accompanied with higher slope of temperature near the wall. Fig. 8 is plotted to depict the impact of thermophoreis parameter N t on temperature field θ (ζ) . Increasing values of thermophoresis parameter N t lead to a higher temperature field θ (ζ) and thicker thermal boundary layer. The reason behind this outcome is that an enhancement in N t yields a stronger thermophoretic force on nanoparticles in the direction opposite to the imposed temperature gradient. This shifts nanoparticles
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towards the cold ambient fluid due to which thermal layer thickness increases. Fig. 9 plots the concentration field φ (ζ) for wide range of Deborah number β. Bigger values of Deborah
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number β constitute a larger concentration field and more concentration layer thickness. Fig.
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10 depicts the change in concentration field φ (ζ) for varying rotation parameter λ. From this Fig. it has been seen that by increasing rotation parameter λ, an enhancement in concentra-
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tion field φ (ζ) and its related concentration layer thickness is noticed. Fig. 11 demonstrates that how the variation in Schmidt number Sc affects the concentration field φ (ζ) . It has been
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noted that the increasing values of Schmidt number lead to reduce the concentration field φ(ζ). Physically Schmidt number is based on Brownian diffusivity. Increase in Schmidt number Sc yields weaker Brownian diffusivity. Such weaker Brownian diffusivity corresponds
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to lower concentration field φ(ζ) and thinner concentration layer thickness. Fig. 12 plots the concentration field φ (ζ) for varying values of thermophoreis parameter N t. It has been
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clearly noticed that the higher thermophoresis parameter N t leads to a larger concentration field φ(ζ) and its associated layer thickness. Fig. 13 shows that the bigger Brownian motion parameter N b leads to a reduction in the concentration field φ(ζ) and its related concentra-
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tion layer thickness. Table 2 is computed to validate the present results with the previous published results in a limiting sense. From this Table, we examined that the present series
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solutions have good agreement with the numerical solutions of Megahed [41] in a limiting sense. Table 3 provides numerical computations of local Nusselt number (−θ′ (0)) for several values of β, λ, N t, N b, Pr and Sc. Here we found that the local Nusselt number has higher values for larger Prandtl Pr and Schmidt Sc numbers while opposite trend is noticed for Deborah number β and rotation parameter λ. Moreover the local Nusselt number remains constant when the Brownian motion parameter N b is varied.
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Table 2. Comparative values of −f ′′ (0) for various values of β when λ = 0. OHAM
Megahed [41]
0.0
1.0000
0.999978
0.2
1.0519
1.051945
0.4
1.1019
1.101848
0.6
1.1501
1.150160
0.8
1.1967
1.196690
1.2
1.2853
1.285253
1.6
1.3686
2.0
1.4476
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SC
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β
1.368641
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1.447616
Table 3. Numerical data for local Nusselt number (−θ′ (0)) for various values of β, λ, N t, N b, Pr and Sc.
0.0
Nt
Nb
Pr
Sc
−θ′ (0)
0.2
0.1
0.3
1.0
1.0
0.5583
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0.2
λ
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β
0.5390
0.5
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0.2
0.2
0.2
0.2
0.2
0.0
0.5115 0.1
0.3
1.0
1.0
0.5580
0.2
0.5390
0.4
0.5025
0.2
0.2
0.2
0.2
0.0
0.3
1.0
1.0
0.5467
0.3
0.5235
0.5
0.5082
0.1
0.1
0.1
0.3
1.0
1.0
0.5390
0.7
0.5390
1.0
0.5390
0.3
0.3
0.5
1.0
0.3205
1.0
0.5390
1.5
0.7153
1.0
0.5
0.5329
1.0
0.5390
1.5
0.5394
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Conclusions
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Boundary layer flow of upper-convected Maxwell (UCM) nanofluid past a linearly deforming
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surface in rotating frame is investigated. The key points of presented analysis are listed below:
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• Velocities in the x− and y−directions reduce while the temperature θ (ζ) and concentration φ (ζ) enhance when the Deborah number β enlarges.
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• Increasing values of rotation parameter λ lead to smaller velocity components while opposite trend is observed for temperature and concentration profiles.
and thermal layer thickness.
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• An increment in Prandtl number Pr leads to a decreasing trend in temperature θ (ζ)
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• Both concentration field φ (ζ) and its associated concentration layer thickness reduce by increasing Schmidt number Sc.
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• Temperature θ (ζ) and concentration φ (ζ) show opposite behavior when Brownian motion parameter N b is varied.
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• By increasing the thermophoresis parameter N t, an enhancement is observed in both temperature and concentration distributions. • Heat transfer rate (−θ′ (0)) reduces with an increase in thermophoresis parameters while it is independent of the Brownian motion parameter N b. • The present results reduces to the Newtonian fluid case when Deborah number β = 0.
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variable magnetic field, Appl. Math. Comp., 298 (2017) 272-282. [29] M. Sheikholeslami, T. Hayat and A. Alsaedi, Numerical study for external magnetic source influence on water based nanofluid convective heat transfer, Int. J. Heat Mass Transfer, 106 (2017) 745-755. [30] M. Sheikholeslami and H.B. Rokni, Nanofluid two phase model analysis in existence of induced magnetic field, Int. J. Heat Mass Transfer, 107 (2017) 288-299. [31] C.Y. Wang, Stretching a surface in a rotating fluid, Z. angew. Math. Phys., 39 (1988) 177-185. [32] H.S. Takhar, A.J. Chamkha and G. Nath, Flow and heat transfer on a stretching surface in a rotating fluid with a magnetic field, Int. J. Thermal Sci., 42 (2003) 23-31. [33] R. Nazar, N. Amin and I. Pop, Unsteady boundary layer flow due to a stretching surface in a rotating fluid, Mech. Res. Commun., 31 (2004) 121-128.
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sheet with suction, J. Mol. Liquids, 211 (2015) 965-969.
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upper-convected Maxwell fluid, AIP Adv., 5 (2015) 047109. [38] T. Hayat, S. Qayyum, M. Imtiaz and A. Alsaedi, Three-dimensional rotating flow of
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Jeffrey fluid for Cattaneo-Christov heat flux model, AIP Adv., 6 (2016) 025012. [39] M. Mustafa, A. Mushtaq, T. Hayat and A. Alsaedi, Rotating flow of magnetite-water
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nanofluid on entropy generation, Int. J. Heat Mass Transfer, 81 (2015) 449-456.
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Heat Fluid Flow, 26 (2016) 2355-2369.
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Jeffrey nanofluid between two parallel disks, Appl. Sci., 6 (2016) 346. [48] T. Hayat, Z. Hussain, T. Muhammad and A. Alsaedi, Effects of homogeneous and hetero-
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geneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness, J. Mol. Liq., 221 (2016) 1121-1127. [49] T. Hayat, M.I. Khan, M. Waqas, T. Yasmeen and A. Alsaedi, Viscous dissipation effect
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heat generation/absorption, Int. J. Thermal Sci., 111 (2017) 274-288.
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Highlights
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• Three-dimensional rotating flow of Maxwell nanofluid is modeled.
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• Flow is induced by a linear stretching surface.
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• Nanofluid model consists of Brownian motion and thermophoresis. • Computations and analysis are made through optimal homotopy analysis method
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(OHAM).
S0167-7322(16)33232-9 doi: 10.1016/j.molliq.2017.01.005 MOLLIQ 6797
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
19 October 2016 26 December 2016 2 January 2017
Please cite this article as: T. Hayat, Taseer Muhammad, M. Mustafa, A. Alsaedi, An optimal study for three-dimensional flow of Maxwell nanofluid subject to rotating frame, Journal of Molecular Liquids (2017), doi: 10.1016/j.molliq.2017.01.005
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An optimal study for three-dimensional flow of Maxwell nanofluid subject to rotating frame b
Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan
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a
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T. Hayata,b , Taseer Muhammada∗ , M. Mustafac and A. Alsaedib
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of
c
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Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia School of Natural Sciences (SNS), National University of Sciences and Technology
∗
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(NUST), Islamabad 44000, Pakistan Corresponding author E-mail: taseer [email protected] (Taseer Muhammad)
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Abstract: Here we are concerned with optimal homotopy solutions for flow of Maxwell nanofluid in rotating frame. Flow is induced by uniform stretching of the boundary surface in one direction. Buongiorno model is adopted which features the novel aspects of Brownian diffusion
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and thermophoresis. Boundary layer approximations are invoked to simplify the governing system of partial differential equations. Appropriate relations are introduced to nondimensionalize the rele-
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vant boundary layer expressions. Newly suggested condition associated with zero nanoparticles mass flux at the boundary is imposed. Uniformly valid convergent solution expressions are developed by
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means of optimal homotopy analysis technique (OHAM). Plots have been portrayed in order to explain the role of embedded flow parameters on the solutions. Heat transfer rate at the surface has been computed and analyzed. Our findings show that the temperature and concentration fields are smaller for Newtonian fluid when compared with the upper-convected Maxwell (UCM). Moreover Brownian diffusion has mild influence of heat flux at the boundary. Viscoelastic effect has tendency to reduce heat transfer rate from the stretching boundary.
Keywords: Maxwell fluid; Rotating frame; Nanoparticles; Nonlinear analysis; OHAM.
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Nomenclature velocity components
x, y, z
µ
dynamic viscosity
ρf
ν
kinematic viscosity
λ1
T
temperature
C
T∞
ambient fluid temperature
C∞
ambient fluid concentration
αm
thermal diffusivity
k
thermal conductivity
(ρc)p
effective heat capacity of nanoparticles
(ρc)f
heat capacity of fluid
DB
Brownian diffusion coefficient
DT
thermophoretic diffusion coefficient
uw
surface velocity
Tw
surface temperature
a
positive constant
Ω
angular velocity
ζ
similarity variable
f ′, g
dimensionless velocities
θ
dimensionless temperature
φ
dimensionless concentration
β
Deborah number
λ
rotation parameter
Sc
Schmidt number
Pr
Prandtl number
Nb
Brownian motion parameter
Nt
thermophoresis parameter
N ux
local Nusselt number
Rex
local Reynolds number
density of base fluid relaxation time concentration
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1
coordinate axes
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u, v, w
Introduction
Nanotechnology is an attractive area of research now a days owing to its rich applications in the industrial and technological processes. The current researchers are engaged to investigate the mechanisms through the nanomaterials. A solid-liquid mixture of tiny size nanoparticles and base liquid is known as nanofluid. The colloids of base liquid and nanoparticles have essential physical attributes which have potential role in coating related applications, metal working processes, ceramics, magnetic drug targeting, paintings, many heat transfer applications such as nuclear reactor cooling, air-conditioning, transportation etc. Nanofluids are pronounced as super coolants in light of the fact that their effective thermal conductivity is much larger in comparison to the traditional liquids. The reduction of the system size and improvement in thermal transport can be accomplished with an aid of nanoparticles. Choi [1] experimentally explored the mechanism of nanoparticles and concluded that the insertion
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of nanoparticles into ordinary base liquids is highly useful technique to enhance the cooling capability of conventional liquids. Buongiorno [2] developed a two-component model for in-
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vestigating thermal energy transport in nanofluids. The model is based on two important
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slip mechanisms namely Brownian motion and thermophoresis. Recently, a large number of investigations regarding the flow analysis of nanofluids have been presented (see [3 − 30] and
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several studies therein).
The investigations on rotating flows near stretchable or inextensible boundaries have
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been growing considerably since the last decade due to their occurrence in engineering and geophysical processes. Such flows commonly involve in rotor-stator system, food processing, rotating machinery, disk cleaners, gas turbine design and several others. Available data
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confirms that flows in rotating frame by a stretchable surface have been rarely attempted. Wang [31] provided analytical solutions for rotating fluid flow past a stretchable surface by
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employing perturbation technique. Then Takhar et al. [32] examined the magnetic field characteristics in rotating flow bounded by a stretchable surface. Nazar et al. [33] reported the time-dependent rotating flow past an impulsively deforming surface utilizing Keller-box
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numerical procedure. Locally similar solutions for rotating fluid flow by an exponentially deforming surface have been addressed by Javed et al. [34] . Zaimi et al. [35] prepared the self-
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similar solutions for rotating viscoelastic fluid flow due to an impermeable stretchable surface. Rosali et al. [36] numerically addressed the rotating flow past an exponentially shrinking porous surface. Mustafa [37] examined the rotating flow of viscoelastic liquid induced by a stretchable surface through thermal relaxation effects. Hayat et al. [38] employed nonFourier heat flux theory for three dimensional (3D) rotating flow of Jeffrey material. Mustafa et al. [39] examined the nonlinear radiation characteristics in rotating nanofluid flow past a stretchable surface. Recently, Shafique et al. [40] studied the heat and mass transfer aspects in rotating flow of Maxwell material with binary chemical reaction and activation energy. Inspired by the above mentioned literature, our objectives of present attempt are four folds. Firstly to model and analyze the rotating flow of Maxwell nanofluid adjacent to a deforming surface. Maxwell fluid model although predicts the relaxation time features but it is not capable of describing shear thinning/shear thickening, retardation time etc [41, 42] . Secondly to consider the rotating frame rather than the fixed frame. Thirdly to employ the Buongiorno’s model for nanofluid transport phenomena. Fourth to derive convergent series solutions for the velocities, temperature and concentration by a powerful optimal homotopy
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analysis method (OHAM) [43 − 50]. The contributions of various pertinent parameters are studied and discussed. Further the heat transfer rate at the surface is analyzed through
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Formulation
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2
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numerical values.
We explore the rotating flow of upper-convected Maxwell (UCM) nanofluid over a linearly stretchable surface. Nanofluid model describes the attributes of Brownian motion and ther-
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mophoresis. We choose the Cartesian coordinate system such that the surface is aligned with the xy−plane and fluid is considered in the space z ≥ 0. The surface deforms linearly in the
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x−direction with rate a. Further the fluid is subjected to uniform rotation about z−axis with constant angular velocity Ω. The associated equations governing the Maxwell nanofluid flow in rotating frame are [40, 50] :
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∂u ∂v ∂w + + = 0, ∂x ∂y ∂z 2 2 2 u2 ∂∂xu2 + v 2 ∂∂yu2 + w2 ∂∂zu2 ∂u ∂u ∂u ∂ 2u ∂2u ∂2u ∂2u u +v +w −2Ωv = ν 2 −λ1 +2uv ∂x∂y + 2vw ∂y∂z + 2uw ∂x∂z ∂x ∂y ∂z ∂z ∂v ∂v ∂u ∂u −2Ω u ∂x + v ∂y + w ∂v − u + 2Ω v ∂z ∂x ∂y ∂2v 2 ∂2v 2 ∂2v u2 ∂x 2 + v ∂y 2 + w ∂z 2 ∂v ∂v ∂ 2v ∂v ∂2v ∂2v ∂2v u +v +w +2Ωu = ν 2 −λ1 +2uv ∂x∂y + 2vw ∂y∂z + 2uw ∂x∂z ∂x ∂y ∂z ∂z ∂u ∂u ∂v ∂v +2Ω u ∂u + v + w − u + 2Ω v ∂x ∂y ∂z ∂x ∂y ! 2 ∂ 2T ∂T ∂C (ρc)p ∂T ∂T ∂T DT ∂T DB , u +v +w = αm 2 + + ∂x ∂y ∂z ∂z (ρc)f ∂z ∂z T∞ ∂z 2 ∂ C DT ∂ 2 T ∂C ∂C ∂C u + . +v +w = DB ∂x ∂y ∂z ∂z 2 T∞ ∂z 2
(1)
, (2)
, (3) (4) (5)
We have the following prescribed conditions: u = uw (x) = ax, v = 0, w = 0, T = Tw , DB
∂C DT ∂T + = 0 at z = 0, ∂z T∞ ∂z
u → 0, v → 0, T → T∞ , C → C∞ as z → ∞.
(6) (7)
Here u, v and w represent the components of velocity in x−, y− and z−directions respectively, λ1 stands for fluid relaxation time, ν = µ/ρf for kinematic viscosity, ρf for density of base liquid, µ for dynamic viscosity, T for temperature, αm = k/(ρc)f for thermal diffusivity, k for
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thermal conductivity, (ρc)p for effective heat capacity of nanoparticles, (ρc)f for heat capacity of fluid, C for concentration, DB for Brownian diffusion coefficient, DT for thermophoretic
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diffusion coefficient, Tw for wall temperature, T∞ for ambient fluid temperature, C∞ for
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ambient fluid concentration and a for positive constant. Considering [40, 50] : (8)
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u = axf ′ (ζ), v = axg(ζ), w = − (aν)1/2 f (ζ) , " 1/2 ∞ ∞ , φ(ζ) = C−C , ζ = νa z. θ(ζ) = TTw−T −T∞ C∞
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Equation (1) is identically verified while Eqs. (2) − (7) take to the following forms " f ′′′ + f f ′′ − f ′2 + 2λ (g − βf g ′ ) + β 2f f ′ f ′′ − f 2 f ′′′ = 0,
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" " " g ′′ + f g ′ − f ′ g − 2λ f ′ + β f ′2 − f f ′′ + g 2 + β 2f f ′ g ′ − f 2 g ′′ = 0, " θ′′ + Pr f θ′ + N bθ′ φ′ + N tθ′2 = 0,
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φ′′ + Scf φ′ +
N t ′′ θ = 0, Nb
(9) (10) (11) (12) (13)
f ′ (∞) → 0, g(∞) → 0, θ(∞) → 0, φ(∞) → 0.
(14)
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f (0) = g(0) = 0, f ′ (0) = 1, θ(0) = 1, N bφ′ (0) + N tθ′ (0) = 0,
In the above expressions λ stands for rotation parameter, β for Deborah number, Pr for
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Prandtl number, N b for Brownian motion parameter, N t for thermophoresis parameter and Sc for Schmidt number. These dimensionless variables can be specified by using the definitions given below:
Nb =
λ=
Ω , a
(ρc)p DB C∞ , (ρc)f ν
β = λ1 a, Pr = ανm , (ρc)p DT (Tw −T∞ ) Nt = , (ρc)f νT∞
Sc =
ν . DB
The local Nusselt number N ux is defined by
(15)
x ∂T Re N ux = − = −θ′ (0) , x (Tw − T∞ ) ∂z z=0
−1/2
(16)
where Rex = uw x/ν depicts the local Reynolds numbers. It is also observed that nondimensional mass flux described by Sherwood number Shx is now identically zero.
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Series solutions by OHAM
Our objective here is to develop the convergent series solutions of Eqs. (9) to (12) with conditions (13) and (14) through the optimal homotopy analysis technique (OHAM). Suitable
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initial approximations and the auxiliary linear operators for f (ζ) , g(ζ), θ(ζ) and φ(ζ) are as follows:
(17)
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f0 (ζ) = 1 − exp (−ζ) , g0 (ζ) = 0,
dζ
φ
2
dζ
2
(18)
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θ
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Nt θ0 (ζ) = exp (−ζ) , φ0 (ζ) = − N exp (−ζ) , b 3 2 df , Lg = ddζ g2 − g, Lf = ddζf3 − dζ 2 2 L = d θ − θ, L = d φ − φ.
The above linear operators satisfy the following properties:
g
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Lf [A∗1 + A∗2 exp (ζ) + A∗3 exp (−ζ)] = 0, ∗ ∗ L [A exp (ζ) + A exp (−ζ)] = 0, 4
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Lθ [A∗6 exp (ζ) + A∗7 exp (−ζ)] = 0,
(19)
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Lφ [A∗8 exp (ζ) + A∗9 exp (−ζ)] = 0,
in which A∗j (j = 1 − 9) are arbitrary constants. The zeroth and mth-order deformation
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problems can be easily formulated in view of the above linear operators. The deformation
Optimal convergence control parameters
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4
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problems have been solved by BVPh2.0 of the software Mathematica.
Let us denote the non-zero auxiliary parameters for functions f, g, θ and φ by ~f , ~g , ~θ and ~φ respectively. These parameters are useful to regulate the convergence and rate of homotopy solutions. To obtain the optimal values of ~f , ~g , ~θ and ~φ , we have employed the idea of minimizing the squared residual of the governing equations defined by (see Liao [43] for details): εfm =
1 k+1
k X j=0
Nf
k X 1 Ng εgm = k + 1 j=0 εθm =
εφm =
1 k+1 1 k+1
k X
m X
k X
m X
j=0
j=0
Nθ
Nφ
fˆ(ζ),
i=0
i=0
m X
fˆ(ζ),
i=0
m X
fˆ(ζ),
fˆ(ζ),
i=0
m X
gˆ(ζ)
i=0
gˆ(ζ),
i=0
m X
gˆ(ζ)
i=0
i=0
m X
m X
m X
!
!
ˆθ(ζ),
i=0
gˆ(ζ),
m X i=0
ζ=jδζ
ζ=jδζ
m X
2
,
(20)
2
,
(21)
ˆ φ(ζ),
i=0
ˆθ(ζ),
m X i=0
ˆ φ(ζ),
!
ζ=jδζ
!
ζ=jδζ
2
,
(22)
2
.
(23)
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Here Nf , Ng , Nθ and Nφ denote the non-linear operators corresponding to Eqs. (9) − (12)
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respectively. Following Liao [43] (24)
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εtm = εfm + εgm + εθm + εφm ,
where εtm stands for total squared residual error, k = 20 and δζ = 0.5. The total average
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squared residual error is minimized by employing Mathematica package BVPh2.0. A case has been considered where β = λ = 0.2, N t = 0.1, N b = 0.3 and Pr = Sc = 1.0. The optimal
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values of convergence-control parameters at 2nd order of approximations are ~f = −1.50853, ~g = −1.17662, ~θ = −1.15433 and ~φ = −1.02305 and the total averaged squared residual error is εtm = 2.21×10−4 . Table 1 shows the individual average squared residual at the optimal
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values of convergence control parameters when m = 2. It is observed that the averaged squared residual error reduces with higher order approximations.
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Table 1. Individual averaged squared residual errors considering optimal values of auxiliary parameters.
εgm
εθm
εφm
2
1.47 × 10−5
8.14 × 10−6
1.04 × 10−4
9.41 × 10−5
6
3.01 × 10−8
5.41 × 10−8
2.34 × 10−6
6.87 × 10−6
10
4.52 × 10−10
1.77 × 10−9
1.65 × 10−7
1.14 × 10−6
16
1.72 × 10−11
1.61 × 10−11
4.83 × 10−9
8.01 × 10−8
20
6.60 × 10−13
1.99 × 10−12
5.72 × 10−10
1.42 × 10−8
26
7.19 × 10−14
7.96 × 10−14
3.08 × 10−11
1.10 × 10−9
30
5.50 × 10−15
1.62 × 10−14
4.93 × 10−12
2.03 × 10−10
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εfm
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m
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Discussion
This portion explores the influences of various pertinent flow parameters like Deborah number β, rotation parameter λ, Prandtl number Pr, Schmidt number Sc, Brownian motion parameter N b and thermophoresis parameter N t on the nondimensional velocities f ′ (ζ) and g (ζ) , temperature θ (ζ) and concentration φ (ζ) fields. Fig. 1 illustrates that how Deborah number β effect the velocity field f ′ (ζ). From this Fig. we can say that an increment in the values of Deborah number β shows decreasing trend in velocity field f ′ (ζ) and momentum layer thickness. It is seen that as the Deborah number β enhances, the fluid relaxation time
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enlarges which yields a lower velocity field f ′ (ζ) and reduces momentum layer thickness. Fig. 2 presents the change in the velocity field f ′ (ζ) for varying values of rotation parameter λ.
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From this Fig. it has been examined that an enhancement in the value of rotation parameter
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λ shows decreasing trend for velocity field f ′ (ζ) and momentum layer thickness. Larger values of rotation parameter λ correspond to higher rotation to stretching rates ratio. In other
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words, the rotation rate becomes higher in comparison to the stretching rate when λ enlarges. Therefore the larger rotation effect leads to decrease velocity field f ′ (ζ) and hydrodynamic
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layer thickness. Fig. 3 is sketched to examine that how velocity field g(ζ) is affected as the Deborah number β is varied. Since the sheet is stretched only in the x−direction and thus flow in y−direction is only anticipated due to the rotating frame. Velocity field g(ζ) which is
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related with y−component of velocity decreases in magnitude away from the stretching surface. However it shows opposite effects in the region of the stretching surface. Moreover the
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negative value of g(ζ) depicts that flow is only in the negative y−direction due to rotational effects. Fig. 4 shows the deviation in function g(ζ) with the variation in rotation parameter λ. It is visualized that rotation parameter has an important role in accelerating flow along
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the y−direction. An increment in rotation parameter λ leads to an oscillatory pattern in the velocity field g(ζ) (a fact that is also observed by Zaimi et al. [35]). Fig. 5 shows the curves
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of temperature field θ (ζ) for varying Deborah number β. From this Fig. it has been observed that by increasing Deborah number β, penetration depth of temperature becomes thicker. Fig. 6 plots the temperature field θ (ζ) for distinct values of rotation parameter λ. Larger values of rotation parameter λ constitutes a higher temperature field and thicker thermal layer thickness. Fig. 7 illustrates that how the temperature field θ (ζ) gets effected with the variation in Prandtl number Pr . It is noted that by enhancing Prandtl number Pr, the temperature field θ (ζ) decreases and thermal layer thickness reduces. Physically, as Prandtl number Pr has inverse relationship with thermal diffusivity, therefore, an increment in Pr leads to weaker thermal diffusion and hence thinner penetration depth of θ (ζ) . Thicker thermal boundary layer attributed to larger Prandtl number is accompanied with higher slope of temperature near the wall. Fig. 8 is plotted to depict the impact of thermophoreis parameter N t on temperature field θ (ζ) . Increasing values of thermophoresis parameter N t lead to a higher temperature field θ (ζ) and thicker thermal boundary layer. The reason behind this outcome is that an enhancement in N t yields a stronger thermophoretic force on nanoparticles in the direction opposite to the imposed temperature gradient. This shifts nanoparticles
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towards the cold ambient fluid due to which thermal layer thickness increases. Fig. 9 plots the concentration field φ (ζ) for wide range of Deborah number β. Bigger values of Deborah
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number β constitute a larger concentration field and more concentration layer thickness. Fig.
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10 depicts the change in concentration field φ (ζ) for varying rotation parameter λ. From this Fig. it has been seen that by increasing rotation parameter λ, an enhancement in concentra-
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tion field φ (ζ) and its related concentration layer thickness is noticed. Fig. 11 demonstrates that how the variation in Schmidt number Sc affects the concentration field φ (ζ) . It has been
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noted that the increasing values of Schmidt number lead to reduce the concentration field φ(ζ). Physically Schmidt number is based on Brownian diffusivity. Increase in Schmidt number Sc yields weaker Brownian diffusivity. Such weaker Brownian diffusivity corresponds
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to lower concentration field φ(ζ) and thinner concentration layer thickness. Fig. 12 plots the concentration field φ (ζ) for varying values of thermophoreis parameter N t. It has been
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clearly noticed that the higher thermophoresis parameter N t leads to a larger concentration field φ(ζ) and its associated layer thickness. Fig. 13 shows that the bigger Brownian motion parameter N b leads to a reduction in the concentration field φ(ζ) and its related concentra-
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tion layer thickness. Table 2 is computed to validate the present results with the previous published results in a limiting sense. From this Table, we examined that the present series
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solutions have good agreement with the numerical solutions of Megahed [41] in a limiting sense. Table 3 provides numerical computations of local Nusselt number (−θ′ (0)) for several values of β, λ, N t, N b, Pr and Sc. Here we found that the local Nusselt number has higher values for larger Prandtl Pr and Schmidt Sc numbers while opposite trend is noticed for Deborah number β and rotation parameter λ. Moreover the local Nusselt number remains constant when the Brownian motion parameter N b is varied.
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Table 2. Comparative values of −f ′′ (0) for various values of β when λ = 0. OHAM
Megahed [41]
0.0
1.0000
0.999978
0.2
1.0519
1.051945
0.4
1.1019
1.101848
0.6
1.1501
1.150160
0.8
1.1967
1.196690
1.2
1.2853
1.285253
1.6
1.3686
2.0
1.4476
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1.368641
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1.447616
Table 3. Numerical data for local Nusselt number (−θ′ (0)) for various values of β, λ, N t, N b, Pr and Sc.
0.0
Nt
Nb
Pr
Sc
−θ′ (0)
0.2
0.1
0.3
1.0
1.0
0.5583
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0.2
λ
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β
0.5390
0.5
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0.2
0.2
0.2
0.2
0.2
0.0
0.5115 0.1
0.3
1.0
1.0
0.5580
0.2
0.5390
0.4
0.5025
0.2
0.2
0.2
0.2
0.0
0.3
1.0
1.0
0.5467
0.3
0.5235
0.5
0.5082
0.1
0.1
0.1
0.3
1.0
1.0
0.5390
0.7
0.5390
1.0
0.5390
0.3
0.3
0.5
1.0
0.3205
1.0
0.5390
1.5
0.7153
1.0
0.5
0.5329
1.0
0.5390
1.5
0.5394
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Conclusions
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Boundary layer flow of upper-convected Maxwell (UCM) nanofluid past a linearly deforming
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surface in rotating frame is investigated. The key points of presented analysis are listed below:
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• Velocities in the x− and y−directions reduce while the temperature θ (ζ) and concentration φ (ζ) enhance when the Deborah number β enlarges.
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• Increasing values of rotation parameter λ lead to smaller velocity components while opposite trend is observed for temperature and concentration profiles.
and thermal layer thickness.
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• An increment in Prandtl number Pr leads to a decreasing trend in temperature θ (ζ)
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• Both concentration field φ (ζ) and its associated concentration layer thickness reduce by increasing Schmidt number Sc.
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• Temperature θ (ζ) and concentration φ (ζ) show opposite behavior when Brownian motion parameter N b is varied.
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• By increasing the thermophoresis parameter N t, an enhancement is observed in both temperature and concentration distributions. • Heat transfer rate (−θ′ (0)) reduces with an increase in thermophoresis parameters while it is independent of the Brownian motion parameter N b. • The present results reduces to the Newtonian fluid case when Deborah number β = 0.
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FIGURE 1
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FIGURE 12
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FIGURE 13
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Highlights
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• Three-dimensional rotating flow of Maxwell nanofluid is modeled.
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• Flow is induced by a linear stretching surface.
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• Nanofluid model consists of Brownian motion and thermophoresis. • Computations and analysis are made through optimal homotopy analysis method
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(OHAM).