Boundary layer flow beneath a uniform free stream permeable continuous moving surface in a nanofluid

1.

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 [5-7], 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 heat-treated materials travelling between a feed roll and a wind-up roll or on conveyor-belts 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 Runge-Kutta 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 two-point 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 semi-infinite 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 [5-7] and Tivari and Das [22]):

boundary conditions of these equations are:

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 Abu-Nada [24]):

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:

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:

with the new boundary conditions as follows:

We look for a similarity solution to Eqs. (9-12) along with the boundary conditions (13) of the following form (see Weidman et al. [26]):

where  is the stream function which is defined in the usual form as and . Thus, we have:

where primes denote differentiation with respect to . In order that Eqs. (9-12) have similarity solutions; it is necessary that has the following form:

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:

and the boundary conditions (13) become:

here Pr is the Prandtl number,  is the Lewis number,  is the Bronian parameter, and  is the thermophoresis parameter, which are defined as:

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:

where , q_w, and q_m are the surface shear stress, the surface heat flux, and the surface mass flux would be:

 Using variables (8) and (14), we obtain:

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 (17-19) 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 Abu-Nada [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  (skin-friction 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

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 skin-friction coefficient  with respect to  for Ag, , and -water nanoparticles when ,and The skin-friction  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

 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

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.   

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.

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.

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