Exact Analytical Solution for Suction and Injection Flow with Thermal Enhancement of Five Nanofluids over an Isothermal Stretching Sheet with Effect of the Slip Model: A Comparative Study

and Applied Analysis 3 (ii) base fluid (i.e., water) and the nanoparticles are in a thermal equilibrium; (iii) fluid outside the boundary layer is quiescent and stretching sheet velocity is linear; (iv) velocity of the sheet is U w (x) = cx, where c > 0 is the stretching constant and x is the coordinate measured along the stretching surface. The thermophysical properties of the base fluid and nanoparticles are given in Table 1. 2.2. Basic Equations. Under the above assumptions, the governing boundary-layer equations of the considered nanofluid (continuity, momentum, and energy) can be written, respectively, in the dimensional form as


Introduction
Because of its numerous applications, the problem of flow and heat transfer in boundary-layer over a stretching surface has attracted many researchers. Examples of these applications are metallurgical processes, such as drawing of continuous filaments through quiescent fluids, annealing and tinning of copper wires, glass blowing, manufacturing of plastic and rubber sheets, crystal growing, and continuous cooling and fiber spinning [1]. Further, there are wide range of applications in many engineering processes, such as polymer extrusion, wire drawing, continuous casting, manufacturing of foods and paper, glass fiber production, stretching of plastic films, and many others. In particular, during the manufacture of these sheets, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. Hence, the final product with the desired characteristics strictly depends upon the stretching rate, the rate of cooling in the process, and the process of stretching [2]. Therefore, the choice of a proper cooling/heating liquid is essential as it has a direct impact on the rate of heat transfer.
The pioneer studies of stretching sheets were done by Sakiadis [3,4] for a moving, inextensible sheet and later extended by Crane [5] to a fluid flow over a linearly stretched sheet. Following the classical boundary-layer theory, many properties were later investigated using the no-slip condition on the wall. However, when the fluid is particulate such as emulsions, suspensions, foams, and polymer solutions, the no-slip condition is inadequate [6]. In such cases, investigations show that the no-slip condition is no longer valid, specially at the micro-and nanoscale, and instead, a certain degree of tangential slip must be allowed [7]. In particular, the fluid flow behaviour deviates significantly from the traditional no-slip flow. Therefore, in the recent years, the interest has been given to the study of this type of flow and some useful results have been recently introduced by many authors, as mentioned in the next paragraphs.
Convective heat transfer in nanofluids is a topic of major contemporary interest both in applied sciences and engineering, where a very good review was presented by Wang and Mujumdar [8,9] and Saidur et al. [10]. Choi [11] may be the first author to introduce the word "nanofluid" that represents the fluid in which nanoscale particles (diameter < 50 nm) are suspended in the base fluid. With the rapid advances in nanotechnology, many inexpensive combinations of liquid/particles are now available. The base fluids used are usually water, ethylene glycol, toluene, and oil. Recent research on nanofluids showed that nanoparticles changed the fluid characteristics because thermal conductivity of these particles was higher than convectional fluids. Nanoparticles are of great scientific interest as they are effectively a bridge between bulk materials and atomic or molecular structures. The common nanoparticles that have been used are Aluminum, Copper, Silver, and Titanium or their oxides. Experimental studies by Eastman et al. [12] and Xuan and Li [13] showed that even with the small volumetric fraction of nanoparticles (usually <5%), the thermal conductivity of the base liquid can be enhanced by 10-20%. The enhanced thermal conductivity of nanofluids together with the thermal conductivity of the base liquid and turbulence induced by their motion contributes to a remarkable improvement in the convective heat transfer coefficient. Further, Majumder et al. [14] showed experimentally that nanofluidic flow usually exhibits partial slip against the solid surface, which can be characterized by the so-called slip length (around 3.4-68 micrometers for different liquids). Therefore, the no-slip condition is no longer valid for fluid flows at the micro-and nanoscale.
In addition to the above discussion about the slip model, Noghrehabadi et al. [15] discussed the effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet at constant wall temperature to extend the work done by Khan and Pop [16]. Nandeppanavar et al. [17] have tabulated the literature of the first order slip; consequently Fang et al. [18] only considered the effect of the second order slip on the flow on a shrinking sheet. Hence, the paper by Nandeppanavar et al. [17] may be the first work to investigate the analysis of second order slip flow and heat transfer over a stretching sheet. Recently, Turkyilmazoglu [19] has analytically studied the heat and mass transfer of magnetohydrodynamic second order slip flow. He has mentioned that there exists a unique solution for any combination of the considered parameters if the stretching sheet is considered. Very recently, Roşca and Pop [20] investigated the steady flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip being investigated using a second order slip flow model. This very important study showed clearly that the second order slip flow model is necessary to predict the flow characteristics accurately.
To show the enhancement of using nanofluids in comparison with pure base fluid, Yacob et al. [21] compared numerically the thermal enhancement of two types of nanofluids, namely, Ag-water and Cu-water, over an impermeable stretching sheet. In addition, with the effect of magnetic field, Hamad [22] studied boundary layer and heat transfer of nanofluids over impermeable isothermal stretching sheet for the metallic and metallic oxide nanoparticles. Further, Noghrehabadi et al. [23] examined theoretically the flow and heat transfer of two types of nanofluids, namely, Silver water and Silicon Dioxide water. They solved the governing equations by applying a combination of a symbolic power series and Padé approximation method. Very recently, Vajravelu et al. [24] studied the effect of variable viscosity on the flow and heat transfer of viscous Ag-water and Cu-water nanofluids. They indicated that nanoparticle volume fraction is to increase the heat transfer and hence enhance the thermal boundary-layer thickness.
The aim of this work is to introduce a direct and effective approach to analytically obtain the exact solution for the flow over an isothermal stretching sheet with effect of no-slip, first order slip parameter, and second order slip parameter. In addition, it is to examine the effect of the existence of the most five common nanoparticles, namely, Silver, Copper, Alumina, Titania, and Silicon Dioxide, in a base of water. Further, we discuss the interested physical parameters, that is, the velocity, temperature, reduced skin friction coefficient, and reduced Nusselt number. The structure of the paper is as follows. Description of the problem, basic equations, and similarity solution are presented in Section 2. In Section 3, second, third, and fourth degree algebraic equations, including the investigated parameters, are governed on deducing the exact solution of the flow with no, first order, and second order slips, respectively. In addition, exact analytical solution of the temperature equation, represented in a simple gamma function, is proposed in the same section. This research is to be considered as an extension to the work done by Hamad [22] and Noghrehabadi et al. [23], besides the comparison with Wang [25] and Reddy Gorla and Sidawi [26] in the special cases.

Governing System of Equations
2.1. Description of the Problem. Consider a two-dimensional incompressible, laminar, and steady boundary-layer flow past an isothermal stretching sheet coinciding with the plane = 0, with the flow being confined to > 0. This sheet is in a water-based nanofluids, which can contain different volume fractions of nanofluids, such as Silver (Ag), Silicon Dioxide (SiO 2 ), Copper (Cu), Alumina (Al 2 O 3 ), and Titania (TiO 2 ). In addition, we assume that the [15,17] (i) sheet surface has temperature at the wall and at ambient fluid ∞ , where > ∞ ; Abstract and Applied Analysis 3 (ii) base fluid (i.e., water) and the nanoparticles are in a thermal equilibrium; (iii) fluid outside the boundary layer is quiescent and stretching sheet velocity is linear; (iv) velocity of the sheet is ( ) = , where > 0 is the stretching constant and is the coordinate measured along the stretching surface.
The thermophysical properties of the base fluid and nanoparticles are given in Table 1.

Basic Equations.
Under the above assumptions, the governing boundary-layer equations of the considered nanofluid (continuity, momentum, and energy) can be written, respectively, in the dimensional form as where ( , ) denotes the Cartesian coordinates along the sheet and normal to it, and V are the velocity components of the nanofluid in the -and -directions, respectively, is the pressure of the nanofluid, is the temperature of the nanofluid, is the specific heat at constant pressure, is the effective density, is the effective dynamic viscosity, ( ) is the heat capacitance, and is the thermal conductivity, where (∼) denotes the nanofluid and is defined as follows [28,29]: where is the solid volume fraction, is the dynamic viscosity, and are the densities, ( ) and ( ) are the heat capacitances, and and are the thermal conductivities, where (∼) and (∼) denote the basic fluid and solid fractions, respectively. The appropriate boundary conditions of (1)-(3) are as follows: where slip is the slip velocity introduced in the next section.

Slip Model.
In the present work, we consider Wu's slip model [30] (valid for arbitrary Knudsen number, ) which is given by where = min[1/ , 1] and 0 ≤ ≤ 1 and are the momentum accommodation and molecular mean free path, respectively. Based on the definition of , it is noticed that for any given value of , we have 0 ≤ ≤ 1. Therefore, the molecular mean free path is always positive. Therefore, we know that < 0, and hence the second term in the right hand side of (6) is a positive number.

Similarity Solution.
The dimensionless variables can be introduced as follows [22,31]: where is the similarity variable, ( ) is the dimensionless stream function, and ( ) is the dimensionless temperature. Further, is the stream function which is defined in the usual way as = / and V = − / to identically satisfy (1). From (7), we therefore obtain where the prime denotes differentiation with respect to . Hence, the mass transfer velocity at the wall becomes Now on substituting (7) and (8) into (2) and (3), we obtain the following nonlinear ordinary differential equations: where = V( ) / is the Prandtl number. The boundary conditions (5) then turn into where (= −V /√ V) is the wall mass transfer parameter, which refers to the suction and injection when > 0 and < 0, respectively, and 0 < 1 (= √ /V) and 0 > 2 (= /V) are the first order slip and second order slip parameters, respectively.
where and are the skin friction, or the shear stress, and heat flux from the surface, respectively, which are given by Therefore, on using (7) and (8) in (12) and (13), the reduced skin friction coefficient and reduced Nusselt number [22,31] are where Re = 2 /V is the local Reynolds number based on the stretching velocity.

Exact Solution
Equations (2) and (3) can be rewritten as where which is exactly solved, subject to the boundary conditions (11) in the next sections.

Exact Solution of the Flow: ( ).
Following the spirit analysis as introduced by Wang [32] and, Aly and Ebaid [33,34], the exact solution of equation can be deduced as follows: which satisfies (15) and the first condition in (11). Further, from the second condition in (11), the parameter satisfies algebraic equations for three models discussed in the next three subsections.
When there is no slip between the base fluid and nanoparticles, that is, 1 = 2 = 0, then satisfies the following second degree algebraic equation: This expressions is more easier than those given in [17][18][19]23].

Effect of the First Slip
Only: 2 =0. If the first order slip is only to be considered, then has to achieve the following third degree algebraic equation: By solving (20) and taking into account Descartes' rule of signs and from the fact that 1 > 0 and > 0, there is only one positive root; see Van Gorder et al. [35].

Effect of the Second Slip
In this case, satisfies the following fourth degree algebraic equation: Following the analysis in [19], the corresponding four roots of (21) are given by Abstract and Applied Analysis 5 with the following dummy variables:

Important Note.
Solutions exist for all values of (suction and injection), 1 ≥ 0 (first slip), and 2 ≤ 0 (second slip). It should be noted that any number of decimal places can be therefore obtained in a direct way on applying any software package, like Mathematica, Maple, or Matlab. Mathematica 6 has been used in the current analysis, where the positive root of (19), (20), and (21) gives the physically meaningful and therefore this is only the root to be considered. In addition, Turkyilmazoglu [19] has recently proved that there exists a unique solution for any combination of the considered parameters if the stretching sheet is considered (which was also spotted in [17]). Therefore, on obtaining the roots, skin friction coefficient of the physical significance is easily given by In an indirect and difficult analysis, this was the same result obtained by Fang et al. [18] and Nandeppanavar et al. [17], in the special case when = 1 (i.e., when = 0).

Exact Solution of the Heat Transfer: ( ).
Substituting (18) into (16), we obtain then by the integration of this equation, we get Further, on integrating the last equation again, we obtain where (0) = 1, and On supposing that then (28) becomes On substituting (30) into (27), taking into account the definition of Γ function, we obtain Applying the condition (∞) = 0, we get Hence with the help of Γ properties, (31) is given in the final exact form as where Γ here is the generalized incomplete gamma function, and , , and are well defined in (17) and (19), respectively. It should be noted here that (0), which is the importance term as mentioned in Section 2, can be easily formulated from (32) and also by differentiating (33), as

Results and Discussion
In this paper, the flow and heat equations of nanofluids over an isothermal stretching sheet with effect of the slip model were analytically solved. Exact solutions were obtained for stream function, in the presence of first order and second order slips, and temperature, in a direct and very effective way using gamma function. In addition, five metallic and nonmetallic nanoparticles have been considered in this analysis, namely, Silver (Ag), Copper (Cu), Alumina (Al 2 O 3 ), Titania (TiO 2 ), and Silicon Dioxide (SiO 2 ). Comparison with the published results via four tables was presented considering the no-slip model. The two types of slip model are then considered; these cases are to be discussed in the next sections with the effect of the various physical parameters, where the Prandtl number of the base fluid (water) is kept at 6.2.

Case 1: When
In the case of 1 = 2 = 0, that is, no-slip model effects, (24) and (32) with (19) have been programmed. The results of − (0) and − (0) for variation of , different nanoparticles, and Ag-water nanoparticles are compared with Hamad [22], Wang [25],   Reddy Gorla and Sidawi [26], and Noghrehabadi et al. [23] in Tables 2, 3, and 4, respectively. These tables indicate an excellent agreement between the present exact solutions and the previous results but in a simple and direct analysis rather than the long analytical presentation in [22] and even the difficult combination of a symbolic power series and Padé approximation method in [23].

Case 2:
When 1 ̸ = 0 and 2 =0. Figures 1 and 2 show the effect of the volume friction of Cu-water and Al 2 O 3 -water nanoparticles, respectively, on velocity distribution ( ) at different values of when = 6.2 and 1 = 1. These figures indicate that the increase in nanoparticle volume friction decreases the velocity magnitude in the case of Cu-water nanofluid; however it increases the velocity magnitude in  the case of Al 2 O 3 -water nanofluid. The difference between their behaviour is because of the difference between density ratios of the proposed nanoparticles to the density of water, which affects the momentum equation. Further, in both cases of Cu-water and Al 2 O 3 -water nanofluids, the increase in decreases the magnitude of ( ) and hydrodynamic boundary layer. It should be mentioned here that the other nanoparticles have been also examined and it was found that Ag-water nanofluid behaves like Cu-water nanofluid, while TiO 2 -water and SiO 2 -water nanofluids behave as Al 2 O 3water nanofluid. However, the velocity distribution ( ) of these nanoparticles is as follows: | Ag < | Cu < | TiO 2 < | Al 2 O 3 ≅ | SiO 2 . This is presented in Figure 3, which shows a comparison of the variation of the velocity profiles for all  the studied nanoparticles at specific values of the physical parameters.
The effect of the volume friction of Cu-water nanoparticles on temperature distribution ( ) at different values of when = 6.2 and 1 = 1 is plotted in Figure 4. This figure indicates that the increase in increases ( ) and hence the thermal boundary-layer thickness. This result is compatible with those obtained very recently by Vajravelu et al. [24] and also agrees with the physical behaviour as addition of increases the thermal conductivity of the pure fluid and this results in increasing the thermal diffusion in the boundary layer. Further, increase in decreases ( ) as well as the thermal boundary-layer thickness. All of the other investigated nanoparticles behave like Cu-water nanofluid with | Ag > | Cu > | Al 2 O 3 ≅ | SiO 2 > | TiO 2 , as illustrated  in Figure 5, which presents a comparison of the variation of temperature distribution for all the studied nanoparticles at specific values of the physical parameters. Figures 6 and 7 present the effect of the first slip 1 on the velocity and temperature distributions at different values of for Cu-water nanoparticles when = 6.2, and = 0.1, respectively. Figure 6 shows that the increase in 1 significantly decreases the velocity near = 0 and then slightly increases it as → ∞ . However, the increase in 1 increases significantly the temperature and thermal boundary-layer thickness in the injection case, that is, when < 0, with a little increasing in the suction case when > 0, and no effect when ≫ 1. From these figures, one can also notice that increase in decreases ( ) and ( ), as well as the hydrodynamic and thermal boundary-layer thickness, respectively. This means that, although the increase in the first slip slows down the velocity, it increases the temperature with impressive effect in the injection case.
The most important parameters of hydrodynamic and thermal boundary layer, namely, reduced skin friction coefficient and reduced Nusselt number, have been indicated in Figures 8-11. In particular, Figures 8 and 9 present the effect of the first slip 1 and , respectively, for reduced skin friction coefficient as a function of 0 ≤ ≤ 0.2 for Cuwater nanofluid, as a representative for the behaviour of all investigated nanoparticles, when = 6.2 and = 0.5, while Figures 10 and 11 show also the effect of 1 and on reduced Nusselt number at the same specific values. Figures 8  and 9 show that increase in 1 and decreases and increases, respectively, the local skin friction. With significant effects, the same result is also observed in Figures 10 and 11 for the reduced Nusselt number.
Comparison of the variation of reduced skin friction coefficient and reduced Nusselt number for the studied nanoparticles, at selected values of the physical parameters, is plotted in Figures 12 and 13. These figures show that the reduced skin friction coefficient (RSFC) of these nanoparticles is as follows: RSFC| Ag > RSFC| Cu > RSFC| TiO 2 > RSFC| Al 2 O 3 ≅ RSFC| SiO 2 . However, vise versa behaviour is noticed in Figure 13. The difference between these types of nanoparticles is because the difference between their densities. Observing Table 1 and (15)- (17), as the nanoparticle becomes heavy, this results in increase and decrease in reduced skin friction coefficient and reduced Nusselt number, respectively.

Case 3: When 1 ̸
= 0, and 2 ̸ = 0. In the presence of second order slip, as shown in Figures 14, 15, 16, and 17, behaviour of the velocity and temperature is similar to those in 1, 3, 4, and 5, respectively. However, for Ag-water and Cu-water nanoparticles, a decreasing difference is noticed on comparing Figures 15 and 3. This means that the second slip affects significantly the heaviest nanoparticles. 19 compare the variation of reduced skin friction coefficient and reduced Nusselt number, respectively, for the studied nanoparticles, at selected values of the physical parameters. These figures show that the RSFC of these nanoparticles is as follows: RSFC| Ag < RSFC| Cu < RSFC| TiO 2 < RSFC| Al 2 O 3 ≅ RSFC| SiO 2 . This is vise versa behaviour comparing with Figure 12 with significant change in the Ag-water and Cuwater nanoparticles, as stated before. Although the same behaviour is observed in Figures 13 and 19, the nanoparticles with highest density are more affected in the presence of the second slip. The results of this section demonstrate clearly that the second order slip flow model is necessary to predict

Conclusion
The governing equations for nanofluids flow over an isothermal stretching sheet with the effect of the slip model were examined in this paper. In a direct and very effective manner, we analytically obtained the exact solutions for the flow and temperature equations. Further, five nanoparticles were considered and compared in the present analysis. Therefore, the most interesting physical parameters were discussed in the presence of no slip, first order slip, and second order slip, parameters.
With no-slip effect, the present exact solutions are in a very good agreement with the results presented in [22,23,25,26]. Some of the interesting results of applying the slip model are as follows: (1) increase in the nanoparticle volume friction decreases the velocity of Cu/Ag-water nanoparticles, increases it for Al 2 O 3 /TiO 2 /SiO 2 -water nanoparticles, and increases the temperature, and hence the thermal boundary-layer thickness, for the whole five investigated nanoparticles; (2) increase in the wall mass decreases the velocity and temperature, as well as the thermal and hydrodynamic boundary-layer thickness, and increases the local skin friction; (3) increase in the slips slows down the velocity, increases the temperature with an impressive effect in the injection case, and decreases the local skin friction and the reduced Nusselt number, with significate effects; (4) as the nanoparticle becomes heavier, this results in increase and decrease in the reduced skin friction coefficient and reduced Nusselt number, respectively; (5) the second order slip parameter affects considerably the flow characteristics, specially for the heaviest nanoparticles.
The final note is for practical and industrial applications; Silver is the suitable nanoparticle if slowing down the velocity and increasing the temperature are needed; on the other hand, Silicon Dioxide is the appropriate nanoparticle if vise versa behaviour is to be considered.