Document Type : Full Lenght Research Article
Authors
^{1} Department of Mathematics, BabesBolyai University, 400048 ClujNapoca, Romania
^{2} Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
Abstract
Keywords
1.
Boundary layer flow beneath a uniform free stream permeable continuous moving surface in a nanofluid
Ioan Pop^{1*}, Sarkhosh Seddighi^{2,3,4}, Norfifah Bachok^{2}, Fudziah Ismail^{2}
^{1}Department of Mathematics, BabesBolyai University, 400048 ClujNapoca, Romania ^{2}Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia ^{3}Department of Mathematics, Science and Research Branch, Islamic Azad University, Bushehr Branch, Bushehr, Iran ^{4}Nuclear Science Research School, Nuclear Science and Technology Research Institute (NSTRI), P.O. Box 14395836, Tehran, Iran

History: Received 5 February 2014 Received in revised form 4 April 2014 Accepted 8 April 2014
Keywords: Suction/injection Moving surface Nanofluid RungeKutta method Shooting techniques Dual solutions.

A B S T R A C T
The main purpose of this paper is to introduce a boundary layer analysis for the fluid flow and heat transfer characteristics of an incompressible nanofluid flowing over a permeable isothermal surface moving continuously. The resulting system of nonlinear ordinary differential equations is solved numerically using the fifth–order Runge–Kutta method with shooting techniques using Matlab and Maple softwares. Numerical results are obtained for the velocity, temperature, and concentration distributions, as well as the friction factor, local Nusselt number, and local Sherwood number for several values of the parameters, namely the velocity ratio parameter, suction/injection parameter, and nanofluid parameters. The obtained results are presented graphically in tabular forms and the physical aspects of the problem are discussed.
© 2014 Published by Semnan University Press. All rights reserved. 

Journal of Heat and Mass Transfer Research
Journal homepage: http://jhmtr.journals.semnan.ac.ir 

Introduction

Many industrial processes involve the transfer of heat by means of a flowing fluid in either the laminar or turbulent regime as well as flowing or stagnant boiling fluids. The processes cover a large range of temperatures and pressures. Many of these applications would benefit from a decrease in the thermal resistance of the heat transfer fluids. This situation would lead to smaller heat transfer systems with lower capital cost and improved energy efficiencies. An innovative technique, which uses a mixture of nanoparticles and a base fluid, was first introduced by Choi [1] to develop advanced heat transfer fluids with substantially higher conductivities. The resulting mixture of the base fluid and nanoparticles having unique physical and chemical properties is referred to as a nanofluid. Flow and heat in a nanofluid over a stretching/shrinking sheet have become a hot topic and have attracted the interest of many researchers recently. The interest in this field has been stimulated due to its applications in industrial processes such as in power generation, chemical processes, and heating or cooling processes. The solid nanoparticles have been suspended into the base fluid which has poor heat transfer properties in order to increase its thermal conductivity. Choi et al. [2] reported that the thermal conductivity of the base fluids increases up to approximately two times with the addition of a small amount (less than 1% by volume fraction) of nanoparticles to the base fluids. A good literature on convective flow and applications of nanofluids were done in the books by Das et al. [3], and Nield and Bejan [4], and in the review papers by Buongiorno [57], Kakaç and Pramuanjaroenkij [8], Wong and Leon [9], Saidur et al. [10], Wen et al. [11], Mahian et al. [12], and many others.
The flow induced by a moving boundary is important in the study of extrusion processes and is a subject of considerable interest in the contemporary literature (Fang et al. [13]). For example, materials which are manufactured by extrusion processes as well as heattreated materials travelling between a feed roll and a windup roll or on conveyorbelts possess the characteristics of stretching/shrinking surfaces. Polymer sheets and filaments are also manufactured by continuous extrusion from a die to a windup roller which is located a finite distance away (Sparrow and Abraham [14]). For both impermeable and permeable shrinking sheets, multiple solutions were discovered (Liao and Pop [15]). Most of these solutions are based on the boundary layer assumption and therefore do not constitute exact solutions of the Navier–Stokes equations (Wang [16]). The influence of thermal radiation on boundary layer flow over a shrinking sheet in a nanofluid has been studied by Zaimi et al. [17]. The partial differential equations are transformed to the ODE and are solved by shooting alongside with sixth order of RungeKutta integration technique. It is observed that radiation has dominant effect on the heat transfer and the mass transfer rates. In general, suction tends to increase the skin friction and heat transfer coefficients, whereas injection acts in the opposite manner. Bachok et al. [18, 19] have studied the boundary layer flow over a stretching/shrinking surface in a nanofluid. Finally, we mention the paper by Ibrahim et al. [20] on the MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet.
In this study, five different types of nanoparticles, Silver Ag, Copper Cu, Copper Oxide CuO, Titania and Alumina were considered in the boundary layer flow over a permeable continuous moving surface with suction and injection. The governing boundary layer equations have been transformed to a twopoint boundary value problem using similarity variables. These have been numerically solved using fourth order Runge–Kutta method with shooting technique. The effects of governing parameters on fluid velocity, temperature, and particle concentration have been discussed.
2. Mathematical formulation
Consider a steady flow of a nanofluid in the region past a moving semiinfinite permeable flat plate, as shown in Fig. 1, where and are the Cartesian coordinates measured along the plate and are normal to it, respectively. It is assumed that the plate moves into or out of the origin at the uniform speed , where is the constant velocity of the external (inviscid) flow and is the constant moving parameter. corresponds to the downstream movement of the plate from the origin and corresponds to the plate moving into the origin (opposing flow). It is also assumed that the mass flux velocity is , where corresponds to the suction and corresponds to the injection or withdrawal of the fluid, respectively. Further, we assume that the uniform temperature and the uniform nanofluid volume fraction at the surface of the plate are and while the uniform temperature and the uniform nanofluid volume fraction far from the surface of the plate are and , respectively (see Kuznetsov and Nield [21]). Under these assumptions, the basic nanofluid conservation equations are (Buongiorno [57] and Tivari and Das [22]):
(1) 

(2) 

(3) 

(4) 

(5) 
boundary conditions of these equations are:
(6)

here, and are the velocity components along the and axes, respectively, T is temperature of the nanofluid, is the nanoparticle volume fraction, is the pressure, is the density of the nanofluid, is the Brownian diffusion coefficient and is the thermophoretic diffusion coefficient, , where is the heat capacity of the fluid and is the effective heat capacity of the nanoparticle material, respectively. is the effective thermal diffusivity of the nanofluid, is its effective viscosity of the nanofluid, and is its effective density of the nanofluid, which are given by (Khanafer et al. [23] or Oztop and AbuNada [24]):
(7) 
where is the dynamic viscosity of the base fluid being proposed by Brinkman [25], is the thermal conductivity of the nanofluid, and are the thermal conductivities of the base fluid and of the solid particles, respectively. is the heat capacitance of the nanofluid. Strictly, expressions (7) are restricted to spherical (or near spherical) nanoparticles with other expressions being required for other shapes of nanoparticles.
We define now the following boundary layer variables:
(8) 
where is the characteristic length of the plate and is the Reynolds number. Substituting Eq. (9) for Eqs. (1) to (5) and using the boundary layer approximation in which , we obtain the following dimensionless boundary layer equations for the problem under consideration:
(9) 

(10) 

(11) 

(12) 

with the new boundary conditions as follows:
(13) 
We look for a similarity solution to Eqs. (912) along with the boundary conditions (13) of the following form (see Weidman et al. [26]):
(14)

where is the stream function which is defined in the usual form as and . Thus, we have:
(15) 
where primes denote differentiation with respect to . In order that Eqs. (912) have similarity solutions; it is necessary that has the following form:
(16) 
where is the constant mass flux parameter with for suction and for injection, respectively.
Substituting variables (14) for Eqs. (9) to (12), we obtain the following ordinary (similarity) differential equations:
(17) 

(18)



(19) 

and the boundary conditions (13) become:
(20)

here Pr is the Prandtl number, is the Lewis number, is the Bronian parameter, and is the thermophoresis parameter, which are defined as:
(21) 
The quantities of practical interest in this study are the skin friction coefficient , the local Nusselt number , and the local Sherwood number , which are defined as:
(22)

where , q_{w}, and q_{m} are the surface shear stress, the surface heat flux, and the surface mass flux would be:
(23) 
Using variables (8) and (14), we obtain:
(24) 
where is the local Reynolds number.
It is worth mentioning to this end that for (pure viscous fluid), Eq. (17) with the corresponding boundary conditions (20) for reduces to Eq. (4a) with the boundary conditions (4b,c,d) from the paper by Wedman et al. [26].
3. Results and discussion
Numerical solutions to the nonlinear ordinary differential equations (1719) with the boundary conditions (20) were obtained using the fifth–order Runge–Kutta [27] with the shooting technique. We find the missing slopes and , for some values of the governing parameters, namely, the nanoparticle volume fraction , the moving parameter , and the suction/injection parameter using the Maple and Matlab softwares. Five types of nanoparticles were considered, namely, Ag, Cu, CuO, , and . Following Oztop and AbuNada [24], the value of the Prandtl number is taken as 6.2 (for water) and the values of the volume fraction parameter is from 0 to 0.2 ( 0) in which corresponds to the pure (Newtonian) fluid. It is worth mentioning that we have used data related to thermophysical properties of the fluid and nanoparticles as listed in table 1 to compute each case of the nanofluid. The numerical results are summarized in Table 2 and Figs. 2 to 16.
Figs. (2) to (7) show the variation of (skinfriction coefficient) with respect to for Ag, Cu, CuO, , and  water nanofluids and different values of when , and . It is seen that the solution is unique when . There are two solutions (upper and lower branches) when (opposite flow), and no solution when , where is the critical value of for which the solution exists. The values of are positive when , and they become negative when
Figure 1. Physical model and coordinate system: a) flat plate moving out of the origin; b) flat plate moving into the origin 
Figure 2. Variation of the reduced skinfriction coefficient with for and different values of when Nt = 0.1, Nb=0.3, Le=1, and Pr=6.2. 
Figure 3. Variation of the reduced skinfriction coefficient with for Agwater nanofluid and different values of when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and . 
the value of exceeds 1, for all values of the suction/injection parameter . Physically, the positive value of means that the fluid exerts a drag force on the plate, and the negative value means the opposite. The zero value of when does not mean separation, but it corresponds to the equal velocity of the plate and the free stream. A comparison of the obtained values for several values of with those reported by Weidman et al. [26] is given in Table 2. It is seen that the results are in a very good agreement so that we are confident that the present results are accurate.
Fig. (8) shows the variation of the reduced skinfriction coefficient with respect to for Ag, , and water nanoparticles when ,and The skinfriction increases when the nonoparticles have the order of . As it is clear from Fig. (7), the difference between two nonoparticles and is so less compared to other particles.
Figs. (9) and (10) show the variation of with respect to for Ag water and  water nanofluids and different values of the when ,and . It is seen that the solution is unique when , while dual solutions are found to exist when ¸ i.e. when the plate and the free stream move in the opposite directions. The values of are positive for all values (positive or negative) of and for all values of the suction/injection parameter .
The variation of and with respect to for Ag and  water nanoparticles and different values of nanoparticle volume fraction ( ) when , and (impermeable plate) has been shown in Figs. (11) to (14). The values of are positive when , and they become negative when the value of exceeds 1, for both values of the parameter considered. The values of are positive for all values of and for both values of nanoparticle volume fraction . The values of and increase when the variable increases from 0 to 0.2. The variation of for different values of ( ) is very fiddling.
Figs. (2) to (14) also show that for a particular value of , the solution exists up to the certain critical value of
Figure 4. Variation of the reduced skinfriction coefficient with for Cuwater nanofluid and different values of when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and .

Figure 5. Variation of the reduced skinfriction coefficient with for CuOwater nanofluid and different values of when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and . 
Figure 6. Variation of the reduced skinfriction coefficient with for water nanofluid and different values of when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and . 
for . Beyond this value, the boundary layer approximations break down, and thus the numerical solution cannot be obtained. The boundary layer separates from the surface at . Based on our computations, the critical values of are presented in Table (2), which show that for all nanoparticles considered, the values of increase as increases. Hence, suction delays the boundary layer separation, while injection accelerates it.
Finally, Figs. (15) and (16) present the velocity and the temperature profiles for Ag, Cu, CuO, , and  water nanofluids when
, , and . It can be seen that all these profiles asymptoticaly satisfied all asymptotically the far field boundary conditions equation (19). In these figures the solid lines and the dash lines are for the upper and lower branch solutions, respectively.
These velocity and temperature profiles support the existence of dual nature of solutions presented in Figs. (2) up to (9). The velocity profiles for the upper and lower branch solutions when in Fig. (15) show that the velocity gradient at the surface is positive, which
Figure 7. Variation of the reduced skinfriction coefficient with for water nanofluid and different values of when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and . 
Figure 8. Variation of the reduced skinfriction coefficient with for water nanofluid when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and . 
produces positive value of the skin friction coefficient. The temperature gradient at the surface as shown in Figs. (14) and (15) is in agreement with the curves shown in Figs. (2) up to (9).
4. Conclusion
This paper have been theoretically the existence of dual similarity solutions in boundary layer flow over a moving surface immersed in a nanofluid with suction and injection effects have been theoretically studied. The governing boundary layer equations were solved numerically using the fifth–order Runge–Kutta method with shooting technique using the Matlab 12a software.
Figure 9. Variation of with for water nanofluid when Nt= 0.1, Nb=0.3, Le=1, Pr=6.2, and . 
Figure 10. Variation of with for water nanofluid when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and . 
Discussion were carried out for the effects of nanoparticle volume fraction , suction/injection parameter , and the moving parameter on the skin friction coefficient and the local Nusselt number . It was found that dual solutions exist when the plate and the free stream move in the opposite directions. It was also shown that introducing the suction increases the range of for which the solution exists, and in consequence delays the boundary layer separation, while it was found that the injection acts in the opposite manner.
Figure 11. Variation of the reduced skinfriction coefficient with for water nanofluid and different values of when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and . 
Figure 12. Variation of the reduced skinfriction coefficient with for water nanofluid and different values of when Nt= 0.1, Nb=0.3, Le=1, Pr= 6.2, and . 
Figure 13. Variation of with for water nanofluid and different values of when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and .

Figure 14. Variation of with for water nanofluid and different values of when Nt=0.1, Nb=0.3, Le=1, Pr=6.2, and .

Acknowledgements
The authors wish to express their thanks to the anonymous Reviewers for their valuable comments and suggestions. The financial supports received from the Ministry of Higher Education, Malaysia (Project codes: FRGS/1/2012/SG04/UPM/03/1) and the Research University Grant (RUGS) from the University Putra Malaysia are gratefully acknowledged.
nomenclature 

Specific heat capacity 

skin friction coefficient 
Table 1.Thermophysical properties of fluid and nanoparticles (Oztop and AbuNada [22])
Table 2. Comparison of the values of for various when (pure fluid)

dimensionless stream function 

thermal conductivity 

local Nusselt number 

Prandtl number 

surface heat flux 

local Reynolds number 

plate temperature 

fluid temperature 

ambient temperature 

velocity components along the and directions, respectively 

plate velocity 

free stream velocity 


Components of velocity 
Cartesian coordinates along and normal to the surface, respectively 

Greek letters 

Thermal diffusivity 

nanoparticle volume fraction 

Dynamic viscosity 

Kinematics viscosity 

Density 

Dimensionless temperature 

velocity ratio parameter 

surface shear stress 

stream function 

similarity variable 

Subscript 

s 
solid 
f 
Fluid 
nf 
nanofluid 
ambient condition 

condition at the surface of the plate 

Superscript 

differentiation with respect to 
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