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

Abstract

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.

1. 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 die. Provoked by the process of polymer extrusion in which extradite emerges from a narrow slit, Crane [1] first analyzed the two-dimensional fluid flow over a linearly stretching surface. Later, this problem has been extensively studied in various directions, for example, for non-Newtonian fluids, porous space, and magneto-hydrodynamics [2–5]. It is worth mentioning that, in recent years, interests in flow and heat transfer through porous media have grown considerably, due largely to the demands of such diverse areas such as geophysics, chemical and petroleum industries, building construction, and nuclear reactors [6, 7]. There are very few studies in which authors have considered the channel flow. Borkakoti and Bharali [8] studied the two-dimensional channel flows with heat transfer analysis of a hydromagnetic fluid where the lower plate was a stretching sheet. The flow between two rotating disks has many important technical applications such as in lubrication. Keeping this fact in mind Vajravelu and Kumar [9] studied the effects of rotation on the two-dimensional channel flows. They solved the governing equations analytically and numerically. Fluid heating and cooling are important in many industries fields such as manufacturing and transportation. Effective cooling techniques are absolutely needed for cooling any sort of high-energy device. Common heat transfer fluids such as water, ethylene glycol, and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals thermal conductivities are up to three times higher than the fluids, so it is naturally desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid but has the thermal conductivity of a metal.

Recently, due to the rising demands of modern technology, including chemical production, power station, and microelectronics, there is a need to develop new types of fluids that will be more effective in terms of heat exchange performance. Nanofluids are produced by dispersing the nanometer-scale solid particles into base liquids with low thermal conductivity such as water, ethylene glycol, oils, etc. The term “nanofluid” was first coined by Choi [10] to describe this new class of fluids. The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. [11]. Nanofluids are envisioned to describe fluids in which nanometer-sized particles (usually less than 100 nm in size) are suspended in convectional heat transfer basic fluids. Numerous methods have been taken to improve the thermal conductivity of these fluids by suspending nano/microsized particles in liquids. There have been published several numerical studies on the modeling of natural convection heat transfer in nanofluids recently such as [12–14]. Most scientific problems and phenomena are inherently in form of nonlinearity. Except a limited number of these problems, most of them do not have exact solution. Therefore, these nonlinear equations should be solved using the other methods. Liao [15, 16] proposed a new asymptotic technique for nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs), named the homotopy analysis method (HAM). Based on the homotopy in topology, the homotopy analysis method contains obvious merits over perturbation techniques: its validity does not depend on small/larger parameters. Thus, the HAM method can be applied to analyze more of the nonlinear problems in science and engineering. Another advantage of the homotopy analysis method is that it provides larger freedom to select initial approximations, auxiliary linear operators, and some other auxiliary parameters. This method does not need small parameters such as the Adomian decomposition method [17] and homotopy perturbation method [18] so it can overcome the restrictions and limitations of perturbation methods. These Analytical methods have already been successfully applied to solve some engineering problems [19–22].

The objective of the present paper is to study the nanofluid flow and heat transfer due to a stretching cylinder with uniform suction/injection. The nanofluid model proposed by 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.

2. Flow Analysis

2.1. 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 𝑣₀(<0) or a constant wall injection with velocity of𝑣₀(>0), respectively. The lower and upper plates are maintained at constant hot(𝑇_ℎ) and cold(𝑇₀) temperature, respectively.

Figure 1 

Schematic theme of the problem geometry.

The fluid is a water-based nanofluid containing Cu (copper). The nanofluid is a two-component 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].

Table 1 

Thermophysical properties of water and nanoparticle [25].

Under these assumptions and using the nanofluid model proposed by Tiwari and Das [23], the governing equations of motion in a rotating frame of reference are𝜕𝑢𝑢𝜕𝑢𝜕𝑥+𝜕𝜕𝑣𝑦+𝜕𝜕𝑤𝑧=0,(2.1)𝜕𝑥+𝜈𝜕𝜕𝑦𝑢+2Ω𝑤=−𝜌1_nf𝜕𝑝^∗𝜕𝑥+𝜐^nf𝜕²𝑢𝜕𝑥²+𝜕²𝑢𝑢𝜕𝑣𝜕𝑦²,(2.2)𝜕𝑦=−𝜌1_nf𝜕𝑝^∗𝜕𝑦+𝜐^nf𝜕²𝑣𝜕𝑥²+𝜕²𝑣𝑢𝜕𝑤𝜕𝑦²,(2.3)𝜕𝑥+𝜈𝜕𝜕𝑤𝑦−2Ω𝑤=𝜐^nf𝜕²𝑤𝜕𝑥²+𝜕²𝑤𝜕𝑦²,(2.4) 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) are𝑢𝑢==𝑎0𝑥,,𝑣𝑣==𝑣₀0,𝑤=0at𝑦=0,,𝑤=0at𝑦=ℎ.(2.5) The effective density 𝜌_nf, the effective dynamic viscosity𝜇_nf, the effective heat capacity (𝜌𝐶_𝑝)_nf,and the effective thermal conductivity 𝑘_nf of the nanofluid are defined as [26]𝜌_nf=(1−𝜙)𝜌_𝑓+𝜙𝜌_𝑠,𝜇_nf=𝜇^𝑓(1−𝜙)^2.5𝜌𝐶_𝑝,𝑘_nfnf=(1−𝜙)𝜌𝐶_𝑝_𝑓+𝜙𝜌𝐶_𝑝_𝑠,𝑘_𝑓=𝑘^𝑠+2𝑘_𝑓−2𝜙𝑘_𝑓−𝑘_𝑠𝑘_𝑠+2𝑘_𝑓+2𝜙𝑘_𝑓−𝑘_𝑠,(2.6) where 𝜙 is the solid volume fraction of the nanoparticles.

The nondimensional variables are introduced as follows:𝜂=𝑦ℎ,𝑢=𝑎𝑥𝑓^(𝜂),𝜈=−𝑎ℎ𝑓(𝜂),𝑤=𝑎𝑥𝑔(𝜂),(2.7) where the prime denotes differentiation with respect to 𝜂. Substituting (2.7) into (2.2)–(2.4), we obtain−1𝜌_nfℎ𝜕𝜕𝑝𝑥^∗=𝑎²𝑥(𝑓′)²−𝑓𝑓^−𝑓^𝑅𝐴₁(1−𝜙)^2.5+2Kr−1𝑅𝑔,𝜌_nfℎ𝜕𝜕𝑝𝜂^∗=𝑎²ℎ𝑓𝑓′+𝑓^𝑔𝑅𝐴₁(1−𝜙)^2.5,(2.8)_−𝑅𝐴₁(1−𝜙)^2.5(𝑓′𝑔−𝑓𝑔′)+2Kr𝐴₁(1−𝜙)^2.5𝑓′=0.(2.9) The dimensionless quantities in these equations are the following: 𝐴₁ is the nanofluid parameter, 𝑅 is the Reynolds number, and Kris the rotation parameter, and they are defined as𝐴₁=(1−𝜙)+𝜙𝜌^𝑠𝜌_𝑓,𝑅=𝑎ℎ²𝜈_𝑓,Kr=Ωℎ²𝜈_𝑓.(2.10) Eliminating the pressure gradient terms from (2.8), these equations can be reduced to𝑓^−𝑅𝐴₁(1−𝜙)^2.5𝑓^2−𝑓𝑓^−2Kr𝐴₁(1−𝜙)^2.5𝑔=𝐴,(2.11) where 𝐴 is constant. Differentiation of (2.11) with respect to 𝜂 gives𝑓^𝑖𝑣−𝑅𝐴₁(1−𝜙)^2.5𝑓′𝑓^−𝑓𝑓^−2Kr𝐴₁(1−𝜙)^2.5𝑔′=0(2.12) Therefore, the governing momentum equations for this problem are given in the dimensionless form by𝑓^𝑖𝑣−𝑔𝑅𝐴₁(1−𝜙)^2.5𝑓′𝑓^−𝑓𝑓^−2Kr𝐴₁(1−𝜙)^2.5𝑔′=0,_−𝑅𝐴₁(1−𝜙)^2.5(𝑓′𝑔−𝑓𝑔′)+2Kr𝐴₁(1−𝜙)^2.5𝑓′=0(2.13) and are subjected to the boundary conditions𝑓𝑓((10))==𝜆0,,𝑓𝑓^_(0)=0,𝑔(0)=0,(1)=0,𝑔(1)=0,(2.14) where 𝜆=𝑣₀/(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 as𝐶_𝑓=𝜏^𝑤𝜌_𝑓𝑢^2𝑤,(2.15) where 𝜏_𝑤 is the shear stress or skin friction along the stretching wall, which is given by𝜏_𝑤=𝜇_nf𝜕𝜕𝑦𝑢_𝑦=0.(2.16) Using (2.7), (2.15), and (2.16), we get𝐶_𝑓=1𝐴₁(1−𝜙)^2.5𝑓^(0),(2.17) where 𝐶_𝑓=(𝑅𝑥/ℎ)𝐶_𝑓.

2.2. Heat Transfer Analysis

The energy equation of the present problem with viscous dissipation neglected is given by𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝜕𝑇𝑥+𝑤𝜕𝜕𝑇𝑧=𝛼^nf𝜕²𝑇𝜕𝑥²+𝜕²𝑇𝜕𝑦²+𝜕²𝑇𝜕𝑧²,(2.18) where 𝛼_nf is the thermal diffusivity of the nanofluids and is defined as𝛼_nf=𝑘^nf𝜌𝐶_𝑝_nf.(2.19)

We look for a solution of (2.18) of the following form:𝜃(𝜂)=𝑇−𝑇⁰𝑇_ℎ−𝑇₀,(2.20) where 𝑇₀ 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: 𝜃^+Pr𝑅𝐴²𝐴₃𝑓𝜃^=0(2.21) subject to the boundary conditions𝜃(0)=1,𝜃(1)=0.(2.22) Here, 𝐴₂ and 𝐴₃are dimensionless constants given by𝐴₂=(1−𝜙)+𝜙𝜌𝐶^𝑝_𝑠𝜌𝐶_𝑝_𝑓,𝐴₃=𝑘^nf𝑘_𝑓=𝑘^𝑠+2𝑘_𝑓−2𝜙𝑘_𝑓−𝑘_𝑠𝑘_𝑠+2𝑘_𝑓+2𝜙𝑘_𝑓−𝑘_𝑠,(2.23) and Pr=𝜇_𝑓𝐶_𝑝/𝑘_𝑓 is the Prandtl number.

The Nusselt number at the lower plate is defined as Nu=−ℎ𝑞^𝑤𝑘_𝑓𝑇₀−𝑇_ℎ,(2.24) where 𝑞_𝑤is the heat flux from the lower plate and is given by 𝑞_𝑤=−𝑘_nf𝜕𝜕𝑇𝑦_𝑦=0.(2.25) Using (2.24), (2.25), and (2.26), it can be obtainedNu=−𝑘^nf𝑘_𝑓𝜃^(0),(2.26)

3. 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:𝑓₀(𝜂)=(1−2𝑔𝜆)𝜂³+(3𝜆−2)𝜂²+𝜂,₀𝜃(𝜂)=0,₀(𝜂)=1−𝜂,(3.1) and the auxiliary linear operators are𝐿𝐿₁₂(𝑓)=𝑓^𝑖𝑣,𝐿₃(𝑔)=𝑔^,(𝜃)=𝜃^.(3.2) These auxiliary linear operators satisfy𝐿₁𝐶₀+𝐿₂𝐶₁𝜂+𝐶₂𝜂²+𝐶₃𝜂³,𝐿₃𝐶₄+𝐶₅𝜂,𝐶₆+𝐶₇𝜂,(3.3) where 𝐶_𝑖(𝑖=0,1,2,3,4,5,6,7) are constants. Introducing nonzero auxiliary parameters ℏ₁, ℏ₂, and ℏ₃, we develop the zeroth-order deformation problems as follows:(1−𝑓𝑝(0);𝐿𝑝)𝑓=(𝜂0;,𝑝)𝑓−(𝑓1;₀𝑝(𝜂))==𝜆,𝑝ℏ𝑓₁𝑁_1[𝑓(𝜂;𝑝),𝑔(𝜂;𝑝),𝜃(𝜂;𝑝)],(3.4)(1−𝑝)𝐿𝑔(𝜂;𝑝)−𝑔₀(0;𝑝)=0,𝑓^(1;𝑝)=1,(3.5)(1−𝑝)𝐿𝜃(𝜂;𝑝)𝑔−(𝜃0₀;𝑝(𝜂))==0,𝑝ℏ₂𝑁𝑔₂(1[𝑓;𝑝()𝜂=;𝑝0),,𝑔(𝜂;𝑝),𝜃(𝜂;𝑝)],((33..67))𝜃(0;(𝑝𝜂))==1,𝑝ℏ₃𝑁𝜃₃(1[𝑓;𝑝()𝜂;=𝑝)0,,𝑔(𝜂;𝑝),𝜃(𝜂;𝑝)],((33..89)) where nonlinear operators 𝑁₁,𝑁₂, and 𝑁₃are defined as𝑁=₁𝜕[𝑓(𝜂;𝑝),𝑔(𝜂;𝑝),𝜃(𝜂;𝑝)₄]𝑓(𝜂;𝑝)𝜕𝜂⁴−𝑅𝐴₁(1−𝜙)^2.5𝜕𝑓𝜕(𝜂𝜂;𝑝)𝜕²𝑓(𝜂;𝑝)𝜕𝜂²𝑓(𝜂;𝑝)𝜕³𝑓(𝜂;𝑝)−2𝐾𝑟𝐴₁𝜕𝜂²(1−𝜙)^2.5𝜕𝑔(𝜂;𝑝)𝑁𝜕𝜂₂,=𝜕[𝑓(𝜂;𝑝),𝑔(𝜂;𝑝),𝜃(𝜂;𝑝)₂]𝑓(𝜂;𝑝)𝜕𝜂²−𝑅𝐴₁(1−𝜙)^2.5+2Kr𝐴₁𝜕𝑓𝜕(𝜂𝜂;𝑝)𝑔(𝜂;𝑝)−𝜕𝑔𝜕(𝜂𝜂;𝑝)𝑓(𝜂;𝑝)(1−𝜙)^2.5𝜕𝑓(𝜂;𝑝)𝑁𝜕𝜂₃,=𝜕[𝑓(𝜂;𝑝),𝑔(𝜂;𝑝),𝜃(𝜂;𝑝)₂]𝜃(𝜂;𝑝)𝜕𝜂²+Pr𝑅𝐴²𝐴₃𝜕𝜃𝜕(𝜂𝜂;𝑝)𝑓(𝜂;𝑝).(3.10) For 𝑝=0 and 𝑝=1, we, respectively, have𝑓𝑔((𝜂𝜂;;𝑝𝑝))==𝑓𝑔₀₀(𝜂),𝑓(𝜂;1)=𝑓(𝜂),𝜃(𝜂;𝑝)=𝜃₀(𝜂),𝑔(𝜂;1)=𝑔(𝜂),(𝜂),𝜃(𝜂;1)=𝜃(𝜂).(3.11) As 𝑝 increases from 0 to 1, 𝑓(𝜂;𝑝),𝑔(𝜂;𝑝), and 𝜃(𝜂;𝑝) vary, respectively, from 𝑓₀(𝜂),𝑔₀(𝜂),and 𝜃₀(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:𝑓(𝜂;𝑝)=𝑓₀(𝜂)+^∞_𝑚₌₁𝑓_𝑚𝑓_𝑚(𝜂)𝑝^𝑚,(𝜏)=1𝑚!𝜕^𝑚𝑓(𝜂;𝑝)𝜕𝑝^𝑚𝑔(𝜂;𝑝)=𝑔₀,(𝜂)+^∞_𝑚₌₁𝑔_𝑚𝑔_𝑚(𝜂)𝑝^𝑚,(𝜏)=1𝑚!𝜕^𝑚𝑔(𝜂;𝑝)𝜕𝑝^𝑚𝜃(𝜂;𝑝)=𝜃₀,(𝜂)+^∞_𝑚₌₁𝜃_𝑚𝜃_𝑚(𝜂)𝑝^𝑚,(𝜏)=1𝑚!𝜕^𝑚𝜃(𝜂;𝑝)𝜕𝑝^𝑚.(3.12) In which ℏ₁, ℏ₂, and ℏ₃ are chosen in such a way that these series are convergent at 𝑝=1. Convergence of the series (3.12) depends on the auxiliary parametersℏ₁, ℏ₂, andℏ₃.

Assume that ℏ₁ and ℏ₂ are selected such that the series (3.12) is convergent at 𝑝=1, then due to (3.12) we have𝑓(𝜂)=𝑓₀(𝜂)+^∞_𝑚₌₁𝑓_𝑚𝑔(𝜂)=𝑔₀(𝜂),(𝜂)+^∞_𝑚₌₁𝑔_𝑚𝜃(𝜂)=𝜃₀(𝜂),(𝜂)+^∞_𝑚₌₁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: 𝑓(0;𝑝)=𝐿0,₁𝑓𝑓_𝑚((1𝜂;)𝑝−)=𝜒_𝑚𝜆𝑓,𝑓_𝑚−1(_𝜂)=ℏ₁𝑅^𝑓𝑚(𝜂),𝑅(0;𝑝)=0,𝑓^(1;𝑝)=1,_𝑓𝑚(𝜂)=𝑓^{𝐼𝑉𝑚−1}−𝑅𝐴₁(1−𝜙)^2.5^𝑚−1_𝑛=0𝑓_{𝑚−1−𝑛}𝑓^𝑛−^𝑚−1_𝑛=0𝑓_{𝑚−1−𝑛}𝑓^{𝑛}−2Kr𝐴¹𝐿₂(1−𝜙)^2.5𝑔^𝑚−1,𝑅𝑔𝑔_𝑚(0(;𝜂)𝑝−)=𝜒_𝑚0𝑔,_𝑚−1(𝑔𝜂()1;=𝑝ℏ)₂=𝑅0^𝑔𝑚,_𝑔𝑚(𝜂),(𝜂)=𝑔^{𝑚−1}−𝑅𝐴₁(1−𝜙)^2.5^𝑚−1_𝑛=0𝑔_{𝑚−1−𝑛}𝑓^𝑛−^𝑚−1_𝑛=0𝑓_{𝑚−1−𝑛}𝑔^𝑛+2Kr𝐴¹𝐿₃(1−𝜙)^2.5𝑓^𝑚−1,𝑅𝑔𝜃_𝑚(0(;𝜂)𝑝−)=𝜒_𝑚0𝜃,_𝑚−1(𝑔𝜂()1;=𝑝ℏ)₃=𝑅^𝜃𝑚0,_𝜃𝑚(𝜂),(𝜂)=𝜃^{𝑚−1}+Pr𝑅𝐴²𝐴₃^𝑚−1_𝑛=0𝑓_{𝑚−1−𝑛}𝜃^𝑛.(3.14) We use MAPLE software to obtain the solution of these equations. We assume ℏ₁=ℏ₂=ℏ₃=ℏ, for instance, when 𝜙=0.1,Kr=0.5,𝑅=0.5,𝜆=0.5, andPr=6.2(Cu-water). First, deformations of the coupled solutions are presented as follows:𝑓₁(𝜂)=−0.0009583529265ℏ𝜂⁶+0.00570117560ℏ𝜂⁵+0.0160093065ℏ𝜂⁴𝑔₁−0.04543500228ℏ𝜂³+0.024634207ℏ𝜂²,𝜃(𝜂)=0.690014107ℏ𝜂²−0.2300047024ℏ𝜂³−0.46000094046ℏ𝜂,₁(𝜂)=0.08477232594ℏ𝜂⁴−0.3390893037ℏ𝜂³+0.2543169778ℏ𝜂.(3.15) The solutions 𝑓₂(𝜂),𝑔₂(𝜂)and 𝜃₂(𝜂) were too long to be mentioned here, therefore, they are shown graphically.

4. 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 parameterℏ. This region of ℏ can be found by plotting𝑓^(0), 𝑔^(0), and 𝜃′(0) for ℏ (ℏ-curve) and choosing ℏ, where𝑓^(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 ℏ-curve for all cases may lead to a considerable error. Therefore, in this study, we have obtained admissible values of ℏ for all cases but only depicted the ℏ-curves of 𝑓^(0), 𝑔^(0), and 𝜃′(0)for one case in Figure 2 for brevity.

(a)

(b)

(c)

Figure 2 

The ℏ 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.

5. 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.

(a)

(b)

(c)

Figure 3 

Comparison between numerical results and HAM solution results for (a) 𝑔(𝜂) and (b) 𝜃(𝜂) whenPr=6.2; (c) temperature profiles with Mehmood and Ali [24] for 𝜙=0 (regular fluid), Kr=0 (nonrotating fluid) when 𝜑=0,𝜆=0.5,𝑀=1,𝑅=0.5, and Kr=0.5.

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 whenKr=0.5,𝑅=1,𝜆=0.5, and Pr=6.2. Effects of suction/injection parameter (𝜆) on (a) velocity profile, (b) Temperature distribution, (c) skin friction coefficient, and (d) Nusselt number whenKr=0.5,𝑅=1,𝜙=0.1, and Pr=6.2 are shown in Figure 5. It has been found that when the volume fraction of the nanoparticle increases from 0 to 0.2, the thickness of the momentum boundary and thermal boundary layer increases (Figure 4). Figures 5(a) and 5(b) show that all boundary layer thicknesses decrease as 𝜆 increases from negative (injection) to positive (suction) values. We know that the effect 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.

(a)

(b)

Figure 4 

Effect of nanoparticle volume fraction (𝜙) on (a) velocity profile and (b) temperature distribution whenKr=0.5,𝑅=1,𝜆=0.5, and Pr=6.2.

(a)

(b)

(c)

(d)

Figure 5 

Effect of suction/injection parameter (𝜆) on (a) velocity profile, (b)temperature distribution, (c) skin friction coefficient, and (d) Nusselt number whenKr=0.5,𝑅=1,𝜙=0.1, and Pr=6.2.

Figure 6 displays the effects of Reynolds number (𝑅) on (a) velocity profile, (b) temperature distribution, (c) skin friction coefficient, and (d) Nusselt number whenKr=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 asReincrease and in turn increasing Reynolds number leads to increase in the magnitude of the skin friction coefficient and Nusselt number (Figure 6).

(a)

(b)

(c)

(d)

Figure 6 

Effect of Reynolds number (𝑅) on (a) velocity profile, (b) temperature distribution, (c) skin friction coefficient, and (d) Nusselt number whenKr=0.5,𝜆=0.5,𝜙=0.1, and Pr=6.2.

Figure 7 shows the effects of rotation parameter (Kr) on (a) velocity profile, (b) temperature distribution (c) skin friction coefficient, and (d) Nusselt number when𝑅=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.

(a)

(b)

(c)

(d)

Figure 7 

Effect of rotation parameter (Kr) on (a) velocity profile, (b) temperature distribution, (c) skin friction coefficient, and (d) Nusselt number when𝑅=1,𝜆=0.5,𝜙=0.1, and Pr=6.2.

6. 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 characteristics have been examined. Some conclusions obtained from this investigation are summarized as follows. (a)The magnitude of the skin friction coefficient increases as the rotation parameter increases, but it decreases as each of nanoparticle volume fraction, Reynolds number, and injection/suction parameter increases. (b)Nusselt number has direct relationship with nanoparticle volume fraction, Reynolds number, and injection/suction parameter, while it has reverse relationship with power of rotation parameter.

Nomenclature

𝐴1,𝐴2,𝐴3:	Dimensionless constants
𝐶𝑝:	Specific heat at constant pressure
𝐶𝑓,𝐶𝑓:	Skin friction coefficients
𝑓(𝜂),𝑔(𝜂):	Similarity functions
𝐿1,𝐿2,𝐿3:	Auxiliary linear operators
ℏ:	Nonzero auxiliary parameter
ℎ:	Distance between the plates
𝑘:	Thermal conductivity
Kr:	Rotation parameter
𝑁1,𝑁2,𝑁3:	Nonlinear operators
Nu:	Nusselt number
𝑝∗:	Modified fluid pressure
Pr:	Prandtl number
𝑞𝑤:	Heat flux at the lower plate
𝑅:	Reynolds number
𝑢,𝑣,𝑤:	Velocity components along x, y, and z axes, respectively
𝑢𝑤(𝑥):	Velocity of the stretching surface
𝑣0:	Suction/injection velocity.

Greek Symbols

α:	Thermal diffusivity
η:	Dimensionless variable
θ:	Dimensionless temperature
ρ:	Density
ϕ:	Nanoparticle volume fraction
λ:	Dimensionless suction/injection parameter
μ:	Dynamic viscosity
υ:	Kinematic viscosity
σ:	Electrical conductivity
𝜏𝑤:	Skin friction or shear stress along the stretching surface
Ω:	Constant rotation velocity.

Subscripts

∞:	Condition at infinity
nf:	Nanofluid
f:	Base fluid
s:	Nano-solid-particles.