Flow and Heat Transfer of Cu-Water Nanofluid between a Stretching Sheet and a Porous Surface in a Rotating System

The aim of the present paper is to study the flow of nanofluid and heat transfer characteristics between two horizontal plates in a rotating system. The lower plate is a stretching sheet and the upper one is a solid porous plate. Copper Cu as nanoparticle and water as its base fluid have been considered. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions using similarity transformation, which is then solved analytically using the homotopy analysis method HAM . Comparison between HAM and numerical solutions results showed an excellent agreement. The results for the flow and heat transfer characteristics are obtained for various values of the nanoparticle volume fraction, suction/injection parameter, rotation parameter, and Reynolds number. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable of causing change in the flow pattern. It is found that for both suction and injection, the heat transfer rate at the surface increases with increasing the nanoparticle volume fraction, Reynolds number, and injection/suction parameter and it decreases with power of rotation parameter.


Introduction
The fluid dynamics due to a stretching sheet are important from theoretical as well as practical point of view because of their various applications to polymer technology and metallurgy.During many mechanical forming processes, such as extrusion, melt-spinning, cooling of a large metallic plate in a bath, manufacture of plastic and rubber sheets, glass blowing, continuous casting, and spinning of fibers, the extruded material issues through a Tiwari and Das 23 is used.Copper Cu as nanoparticle and water as its base fluid have been considered.The reduced ordinary differential equations are solved analytically using the homotopy analysis method HAM .The effects of the parameters governing the problem are studied and discussed.

Governing Equations
Consider the steady flow of a nanofluid between two horizontal parallel plates when the fluid and the plates rotate together around the axis, which is normal to the plates with a constant angular velocity of Ω.
A Cartesian coordinate system x, y, z is considered as follows: the x-axis is along the plate, the y-axis is perpendicular to it, and the z-axis is normal to the x-y plane see Figure 1 .The origin is located at the lower plate, and the plates are located at y 0 and y h.The lower plate is being stretched by two equal opposite forces so that the position of the point 0, 0, 0 remains unchanged.The upper plate is subjected to a constant wall suction with velocity of v 0 <0 or a constant wall injection with velocity of v 0 >0 , respectively.The lower and upper plates are maintained at constant hot T h and cold T 0 temperature, respectively.
The fluid is a water-based nanofluid containing Cu copper .The nanofluid is a twocomponent mixture with the following assumptions: i incompressible, ii no-chemical reaction, iii negligible viscous dissipation, iv negligible radiative heat transfer, v nano-solid-particles and the base fluid are in thermal equilibrium and no slip occurs between them.
The thermophysical properties of the nanofluid are given in Table 1 25 .
where u, v, and w denote the fluid velocity components along the x, y, and z directions, p * is the modified fluid pressure, and the physical meanings of the other quantities are mentioned in the Nomenclature.The absence of ∂p * /∂z in 2.4 implies that there is a net cross-flow along the z-axis.The corresponding boundary conditions of 2.1 -2.4 are u ax, v 0, w 0 at y 0, u 0, v v 0 , w 0 at y h.

2.5
The effective density ρ nf , the effective dynamic viscosity μ nf , the effective heat capacity ρC p nf , and the effective thermal conductivity k nf of the nanofluid are defined as 26 where φ is the solid volume fraction of the nanoparticles.The nondimensional variables are introduced as follows: where the prime denotes differentiation with respect to η. Substituting 2.7 into 2.2 -2.4 , we obtain The dimensionless quantities in these equations are the following: A 1 is the nanofluid parameter, R is the Reynolds number, and Kr is the rotation parameter, and they are defined as Eliminating the pressure gradient terms from 2.8 , these equations can be reduced to where A is constant.Differentiation of 2.11 with respect to η gives 12 Therefore, the governing momentum equations for this problem are given in the dimensionless form by and are subjected to the boundary conditions where λ v 0 / ah is the dimensionless suction/injection parameter.The physical quantity of interest in this problem is the skin friction coefficient C f along the stretching wall, which is defined as where τ w is the shear stress or skin friction along the stretching wall, which is given by Using 2.7 , 2.15 , and 2.16 , we get where

Heat Transfer Analysis
The energy equation of the present problem with viscous dissipation neglected is given by where α nf is the thermal diffusivity of the nanofluids and is defined as We look for a solution of 2.18 of the following form: where T 0 and T h are temperatures at the lower and upper plates, respectively.Substituting the similarity variables 2.7 and 2.20 into 2.18 , we obtain the following ordinary differential equation: subject to the boundary conditions θ 0 1, θ 1 0.

2.22
Here, A 2 and A 3 are dimensionless constants given by and Pr μ f C p /k f is the Prandtl number.
The Nusselt number at the lower plate is defined as where q w is the heat flux from the lower plate and is given by Using 2.24 , 2.25 , and 2.26 , it can be obtained

The HAM Solution of the Problem
According to some previous works like 27 , we choose the initial approximate solutions of f η , g η , and θ η as follows: and the auxiliary linear operators are

3.10
For p 0 and p 1, we, respectively, have

3.11
As p increases from 0 to 1, f η; p , g η; p , and θ η; p vary, respectively, from f 0 η , g 0 η , and θ 0 η to f η , g η , and θ η .By Taylor's theorem and using 3.11 , f η and θ η can be expanded in a power series of p as follows:

3.12
In which 1 , 2 , and 3 are chosen in such a way that these series are convergent at p 1. Convergence of the series 3.12 depends on the auxiliary parameters 1 , 2 , and 3 .
Assume that 1 and 2 are selected such that the series 3.12 is convergent at p 1, then due to 3.12 we have θ m η .

3.13
Differentiating the zeroth-order deformation 3.4 , 3.6 , and 3.8 m times with respect to p and then dividing them by m! and finally setting p 0, we have the following mth-order deformation problem:

3.14
We use MAPLE software to obtain the solution of these equations.We assume 1 2 3 , for instance, when φ 0.1, Kr 0.5, R 0.5, λ 0.5, and Pr 6.2 Cu-water .First, deformations of the coupled solutions are presented as follows:

3.15
The solutions f 2 η , g 2 η and θ 2 η were too long to be mentioned here, therefore, they are shown graphically.

Convergence of the HAM Solution
As pointed out by Liao 28 , the convergence and the rate of approximation for the HAM solution strongly depend on the values of auxiliary parameter .This region of can be found by plotting f 0 , g 0 , and θ 0 for -curve and choosing , where f 0 , g 0 , and θ 0 are constant.It is worthwhile to be mentioned that for different values of flow parameters φ, Kr, R, λ a new h-curve should be plotted as using a unique -curve for all cases may lead to a considerable error.Therefore, in this study, we have obtained admissible values of for all cases but only depicted the -curves of f 0 , g 0 , and θ 0 for one case in Figure 2 for brevity.

Results and Discussions
The governing equations and their boundary conditions are transformed to ordinary differential equations that are solved analytically using the homotopy analysis method HAM and the results compared with numerical method fourth-order Runge-Kutta 29 .The results obtained by the homotopy analysis method were well matched with the results carried out by the numerical solution obtained by the four-order Runge-kutta method as shown in Figure 3.In order to test the accuracy of the present results, we have compared the results for the temperature profiles θ η with those reported by Mehmood and Ali 24 when φ 0 regular or Newtonian fluid and different values of the Prandtl number.After this validity, results are given for the velocity, temperature distribution, wall shear stress, and Nusselt number for different nondimensional numbers.
Figure 4 shows the effect of nanoparticle volume fraction φ on a velocity profile and b temperature distribution when Kr 0.5, R 1, λ 0.5, and Pr 6. of suction is to bring the fluid closer to the surface and, therefore, to reduce the thermal boundary layer thickness, while for injection opposite trend is observed.As suction/injection parameter λ increases, the magnetic skin friction coefficient decreases and Nusselt number increases Figures 5 c and 5 d .The sensitivity of thermal boundary layer thickness to volume fraction of nanoparticles is related to the increased thermal conductivity of the nanofluid.In fact, higher values of thermal conductivity are accompanied by higher values of thermal diffusivity.The high value of thermal diffusivity causes a drop in the temperature gradients and accordingly increases the boundary thickness as demonstrated in Figure 4 b .This increase in thermal boundary layer thickness reduces the Nusselt number; however, according to 2.26 , the Nusselt number is a multiplication of temperature gradient and the thermal conductivity ratio conductivity of the of the nanofluid to the conductivity of the base fluid .Since the reduction in temperature gradient due to the presence of nanoparticles is much smaller than thermal conductivity ratio, an enhancement in Nusselt takes place by increasing the volume fraction of nanoparticles as it can be seen in Figures 5 c and 5 d .Also Figure 5 c indicates that increasing nanoparticle volume fraction leads to decrease in magnitude of the skin friction coefficient.
Figure 6 displays the effects of Reynolds number R on a velocity profile, b temperature distribution, c skin friction coefficient, and d Nusselt number when Kr 0.5, λ 0.5, φ 0.1, and Pr 6.2.It is worth to mention that the Reynolds number indicates the relative significance of the inertia effect compared to the viscous effect.Thus, both velocity and temperature profiles decrease as Re increase and in turn increasing Reynolds number leads to increase in the magnitude of the skin friction coefficient and Nusselt number Figure 6 .
Figure 7 shows the effects of rotation parameter Kr on a velocity profile, b temperature distribution c skin friction coefficient, and d Nusselt number when R 1, λ 0.5, φ 0.1, and Pr 6.2.Increasing rotation parameter leads to Coriolis force increase that causes both velocity and temperature profiles to increase.Also increasing rotation parameter leads to decreasing the magnitude of the skin friction coefficient and Nusselt number.

Conclusions
In the present paper the three-dimensional nanofluid flow between two horizontal parallel plates in which plates rotate together is considered.The problem is solved analytically using the homotopy analysis method HAM .The results compared with numerical method fourth-order Runge-Kutta results.Effects of nanoparticle volume fraction, suction/injection parameter, Reynolds number, and rotation parameter on the flow and heat transfer  Prandtl number q w : Heat flux at the lower plate R: Reynolds number u, v, w: Velocity components along x, y, and z axes, respectively u w x : Velocity of the stretching surface v 0 : Suction/injection velocity.

Figure 1 :
Figure 1: Schematic theme of the problem geometry.

Figure 2 :
Figure 2:The curve of a f 0 , b g 0 , and c θ 0 a for different orders of approximation when φ 0.1, Kr 0.5, R 0.5, λ 0.5, and Pr 6.2, b for different values of φ when Kr 0.5, R 0.5, λ 0., 5 and Pr 6.2 at the 12th order of approximation and for different values of λ when Kr 0.5, R 0.5, and Pr 6.2 at the 12th order of approximation.

Figure 5 :
Figure 5: Effect of suction/injection parameter λ on a velocity profile, b temperature distribution, c skin friction coefficient, and d Nusselt number when Kr 0.5, R 1, φ 0.1, and Pr 6.2.

Figure 6 :
Figure 6: Effect of Reynolds number R on a velocity profile, b temperature distribution, c skin friction coefficient, and d Nusselt number when Kr 0.5, λ 0.5, φ 0.1, and Pr 6.2.

Figure 7 :
Figure 7: Effect of rotation parameter Kr on a velocity profile, b temperature distribution, c skin friction coefficient, and d Nusselt number when R 1, λ 0.5, φ 0.1, and Pr 6.2.

Table 1 :
Thermophysical properties of water and nanoparticle 25 .