Unsteady Stagnation-Point Flow and Heat Transfer Over a Permeable Exponential Stretching/Shrinking Sheet in Nanoﬂuid with Slip Velocity Effect: A Stability Analysis

: A model of unsteady stagnation-point ﬂow and heat transfer over a permeable exponential stretching/shrinking sheet with the presence of velocity slip is considered in this paper. The nanoﬂuid model proposed by Tiwari and Das is applied where water with Prandtl number 6.2 has been chosen as the base ﬂuid, while three different nanoparticles are taken into consideration, namely Copper, Alumina, and Titania. The ordinary differential equations are solved using boundary value problem with fourth order accuracy (bvp4c) program in Matlab to ﬁnd the numerical solutions of the skin friction and heat transfer coefﬁcients for different parameters such as stretching/shrinking, velocity slip, nanoparticle volume fraction, suction/injection, and also different nanoparticles, for which the obtained results (dual solutions) are presented graphically. The velocity and temperature proﬁles are presented to show that the far ﬁeld boundary conditions are asymptotically fulﬁlled, and validate the ﬁndings of dual solutions as displayed in the variations of the skin friction and heat transfer coefﬁcients. The last part is to perform the stability analysis to determine a stable and physically-realizable solution. This paper was presented to study the characteristics and behaviour of unsteady stagnation-point flow and heat transfer over a permeable exponential stretching/shrinking sheet in nanofluids with slip effect, and also to find a solution which is stable and physically realizable for some parameters, namely, velocity slip, stretching/shrinking, suction, injection, and with different types of nanoparticles. The system of partial differential equations was transformed to ordinary differential equations using the similarity variables. The values of the skin friction coefficient and


Introduction
The stagnation-point flow is a flow that explains the behavior of the fluid motion near the stagnation region. This type of flow happens when the flow hits the solid surface and the fluid velocity at the stagnation-point equals zero. Some applications has been reported where the stagnation-point flow is applied, for instance in dentistry by Yang et al. [1], and air purification as presented by Montecchio et al. [2]. Recently, the study on the flow over an exponential stretching/shrinking sheet has received more attention than for ordinary stretching/shrinking sheets due to its wider applications, as mentioned by Pavithra et al. [3]. Further, Zaib et al. [4] studied the unsteady boundary layer flow and heat transfer passes through an exponential shrinking sheet in a Copper water nanofluid with the presence of a suction effect. The effects of joule heating and thermal radiation on the Magnetohydrodynamics (MHD) boundary layer flow over an exponential stretching sheet in a porous medium immersed in nanofluid was investigated by Rao et al. [5]. Furthermore, Aleng et al. [6] applied where L is the slip length (see Bhattacharyya et al. [8]). The reference temperature, ambient temperature, and temperature distribution near the surface are  Based on the above assumptions, the governing equations can be written as follows (see Zaib et al. [4] and Bachok et al. [7]): Based on the above assumptions, the governing equations can be written as follows (see Zaib et al. [4] and Bachok et al. [7]): ∂u ∂x ∂T ∂t subject to boundary conditions where u and v are velocity component in x and y axes respectively, t denotes time v w = v 0 e x/2l /(1 − ct) 1/2 is the variable suction velocity with v 0 > 0 being a constant, T is the nanofluid temperature, µ n f , α n f , k n f and ρc p n f represent the viscosity, thermal diffusivity, thermal conductivity and heat capacity of the nanofluid, respectively. Equation (5) is given by Oztop et al. [42]: ρc p n f = ρc p f 1 − ϕ + ϕ ρc p s where ϕ denotes the volume of the nanoparticle, µ f is the viscosity of the fluid, ρC p n f , ρC p s and ρC p f is the heat capacity of the nanofluid, nanoparticle and fluid k s and k f are the thermal conductivities of the nanoparticle and fluid, respectively. The densities of the fluid and nanoparticle are denoted as ρ f and ρ s , respectively.
The physical quantities of interest are the local skin friction coefficient C f and local Nusselt number Nu x , which are given by Equation (14) (see Rao et al. [5]): where τ w is the shear stress and q w is the heat flux at the surface which is represented by Equation (15).
where Re x = U ∞ l/v f denotes the local Reynolds number.

Stability Analysis
The results display the presence of dual solutions for the tested parameter. Due to the importance of identifying the physically-realizable solution, a stability analysis is performed. Following Merkin [30], a new dimensionless variable, τ, is introduced, which is related to an initial value problem. The new dimensionless variables can be represented in Equation (18): Equation (18) is then substituted into (2) and (3) to obtain 1 Pr along with the boundary conditions: The stability of solutions f (η) = f 0 (η) and θ(η) = θ 0 (η) that satisfy the boundary value problem Equations (11)-(13) is tested by introducing (see Merkin [30] and Weidman et al. [31]): where γ is an unknown eigenvalue parameter, F(η, τ) and G(η, τ) are small corresponding to f 0 (η) and θ 0 (η), respectively. The linearized problem can be obtained by substituting Equation (22) into (19) and (20), as below: 1 Pr subject to boundary conditions The stability of the steady flow f 0 (η) and θ 0 (η) can be tested by setting τ = 0, for which F(η) = F 0 (η) and G(η) = G 0 (η) are obtained, indicating the initial growth or decay as in Equation (22). Hence, the linearized problem (23) and (25) can be written as: 1 Pr along with boundary conditions The possible range of the smallest eigenvalues can be found by relaxing a boundary condition of F 0 (η) or G 0 (η) from condition η → ∞ as proposed by Harris et al. [32]. In this paper, it was decided that F 0 (η) → 0 should be relaxed, and the new boundary condition will be replaced by F 0 (0) = 1.

Numerical Methodology
The dual solutions of the ordinary Equations (11) and (12), along with the boundary conditions (13), are found using the bvp4c program in the Matlab software (Software, MathWorks, MA, USA). The program is a finite-different code which applies the three-stage Labatto IIIa formula and its algorithm, depending on an iteration structure, in solving the system of equations. The collocation of formulas and polynomials gives a C1-continuous solution with fourth-order accuracy. In this program, the missing values of the skin friction coefficient f (0) and heat transfer coefficient −θ (0) are found by setting some values of initial guesses for the different tested parameters, and asymptotically fulfilled the far-field boundary conditions. The finding process is repeated until no more solutions can be found either beyond a point, namely, the critical point ε c , or for a certain domain if the critical point is hard to find, or the point does not actually exist. Some points near the critical values for the first and second solutions are then tested using the stability analysis to find the smallest eigenvalues, γ, to determine the stability of the solutions. The smallest eigenvalues can be found by solving the linearized problem (26)-(28) using the same program for both the upper and lower solutions. Positive eigenvalue indicate that the solution is stable, while negative values shows an unstable solution.

Results and Discussion
The effects of the nanoparticle volume fraction ϕ, velocity slip parameter σ, unsteadiness parameter A, stretching/shrinking parameter ε, suction/injection parameter s, and different types of nanoparticles (Copper, Alumina, and Titania) are studied, and have been displayed in Figures 1-10. As suggested in the Tiwari and Das model, water with a Prandtl number of 6.2 (Pr = 6.2) was chosen as the base fluid, and the value of the nanoparticle volume fraction was between 0 and 0.
where the absence of nanoparticles in the base fluid can be known when ϕ = 0. The thermo physical properties of water, Copper, Alumina, And Titania are tabulated in Table 1, for which the value of each substance is taken from Oztop and Abu-Nadda [42]. Validation has been made from previous work by Bachok et al. [7], whereby the steady problem (A = 0) over a permeable surface (s = 0) with no-slip condition (σ = 0) was considered, as shown in Table 2; we observed excellent agreement. Table 1. Thermo physical properties of water, Copper, Alumina, and Titania (see Oztop and Abu-Nadda [42]).  Table 2. Values of f (0) and −θ (0) for different values of ε and ϕ for Cu-water. The effects of the nanoparticle volume fraction ϕ on the skin friction coefficient f (0) and heat transfer coefficient −θ (0) with stretching/shrinking parameter ε in Copper-water nanofluid are depicted in Figure 2. Both figures show that dual solutions can be obtained when ε c ≤ ε, while no solution can be found when ε < ε c , i.e., where the critical point ε c represents the end point of the existed solution. The range of solutions was shown to widen when the amount of Copper nanoparticles in the water increases. Apart from that, the skin friction coefficient f (0) for both solutions in the shrinking surface was found to be higher than in the stretching surface, while the heat transfer coefficient shows the opposite trend. Moreover, increasing the volume fraction of Copper nanoparticles in water increases the skin friction coefficient for the first solution, but decreases the second solution. Meanwhile, for the heat transfer coefficient, increasing ϕ shows a decreasing trend for both solutions.

Bachok et al. [7] Present Results
are depicted in Figure 2. Both figures show that dual solutions can be obtained when c ε ε ≤ , while no solution can be found when c ε ε < , i.e., where the critical point c ε represents the end point of the existed solution. The range of solutions was shown to widen when the amount of Copper nanoparticles in the water increases. Apart from that, the skin friction coefficient "(0) f for both solutions in the shrinking surface was found to be higher than in the stretching surface, while the heat transfer coefficient shows the opposite trend. Moreover, increasing the volume fraction of Copper nanoparticles in water increases the skin friction coefficient for the first solution, but decreases the second solution. Meanwhile, for the heat transfer coefficient, increasing ϕ shows a decreasing trend for both solutions.
showed an uptrend for the first solution and downward one for the second, as the velocity slip σ increases. In addition, the skin friction coefficient for the shrinking surface was found higher than "(0) f for the stretching surface. It can be observed that the heat transfer for the shrinking surface is slightly decreased compared to that of the stretching surface. Hence, the friction between the nanofluid and the surface is larger when the surface is shrunk compared to the stretched surface. Meanwhile, based on Figure 3b, the heat transfer rate at the surface was found to be faster for a stretching surface than a shrinking surface; thus, the cooling process becomes better when the surface is in stretched state.  Based on the figures, dual solutions exist when ε c ≤ ε, while no solution could be reported for ε < ε c . It can also be observed that as the slip effect becomes higher, the range of solutions is consequently expanded, also decreasing the skin friction coefficient f (0) for the first solution, but increasing it for the second solution within the range ε < 1. The heat transfer coefficient −θ (0) showed an uptrend for the first solution and downward one for the second, as the velocity slip σ increases. In addition, the skin friction coefficient for the shrinking surface was found higher than f (0) for the stretching surface. It can be observed that the heat transfer coefficient −θ (0) for the shrinking surface is slightly decreased compared to that of the stretching surface. Hence, the friction between the nanofluid and the surface is larger when the surface is shrunk compared to the stretched surface. Meanwhile, based on Figure 3b, the heat transfer rate at the surface was found to be faster for a stretching surface than a shrinking surface; thus, the cooling process becomes better when the surface is in stretched state.
found higher than "(0) f for the stretching surface. It can be observed that the heat transfer coefficient for the shrinking surface is slightly decreased compared to that of the stretching surface. Hence, the friction between the nanofluid and the surface is larger when the surface is shrunk compared to the stretched surface. Meanwhile, based on Figure 3b, the heat transfer rate at the surface was found to be faster for a stretching surface than a shrinking surface; thus, the cooling process becomes better when the surface is in stretched state. According to these figures, the unsteadiness parameter A only expanded the range of solutions slightly compared to the effects of the nanoparticle volume fraction ϕ and the velocity slip parameter σ . Increasing the unsteadiness parameter A increases the skin friction coefficient for the first solution, and decreases if for the second. In addition, as A increases, the heat transfer rate at the surface for the first solution is also increased, but the second solution shows the opposite behavior. The skin friction coefficient "(0) f was shown to decrease when the value of ε increases, which indicates that the skin friction over a shrinking surface is higher than that over a stretching surface. In the meantime, the heat transfer coefficient '(0) θ − increases slightly when ε increases, which shows that the heat transfer rate for the shrinking surface is higher than that of the stretching surface. Variations of the skin friction coefficient f (0) and heat transfer coefficient −θ (0) due to the effect of unsteadiness of parameter A with stretching/shrinking surfaces, are shown in Figure 4. In the range of ε c ≤ ε, dual solutions can be found, while no solution may be obtained when ε c ≤ ε. According to these figures, the unsteadiness parameter A only expanded the range of solutions slightly compared to the effects of the nanoparticle volume fraction ϕ and the velocity slip parameter σ. Increasing the unsteadiness parameter A increases the skin friction coefficient for the first solution, and decreases if for the second. In addition, as A increases, the heat transfer rate at the surface for the first solution is also increased, but the second solution shows the opposite behavior. The skin friction coefficient f (0) was shown to decrease when the value of ε increases, which indicates that the skin friction over a shrinking surface is higher than that over a stretching surface. In the meantime, the heat transfer coefficient −θ (0) increases slightly when ε increases, which shows that the heat transfer rate for the shrinking surface is higher than that of the stretching surface.
at the surface for the first solution is also increased, but the second solution shows the opposite behavior. The skin friction coefficient "(0) f was shown to decrease when the value of ε increases, which indicates that the skin friction over a shrinking surface is higher than that over a stretching surface. In the meantime, the heat transfer coefficient '(0) θ − increases slightly when ε increases, which shows that the heat transfer rate for the shrinking surface is higher than that of the stretching surface.   Figure 5a shows the decreasing skin friction coefficient for the first and second solutions when the value of ε is increased. This indicates that the resulting skin friction coefficient for the shrinking case is higher than that of the skin friction coefficient, which is caused in the stretching case by either a permeable or an impermeable surface. Figure 5b displays the opposite trend, where the heat transfer coefficient increases as ε increases for both solutions. This indicates that the heat transfer rate over shrinking surface is higher than that over a stretching surface for each s parameter. In addition, it is clearly seen in Figure 5 that the solution for the second solution was easier to find with a higher value of s . Furthermore, as the effect of the suction parameter     Figure 5a shows the decreasing skin friction coefficient for the first and second solutions when the value of ε is increased. This indicates that the resulting skin friction coefficient for the shrinking case is higher than that of the skin friction coefficient, which is caused in the stretching case by either a permeable or an impermeable surface. Figure 5b displays the opposite trend, where the heat transfer coefficient increases as ε increases for both solutions. This indicates that the heat transfer rate over shrinking surface is higher than that over a stretching surface for each s parameter. In addition, it is clearly seen in Figure 5 that the solution for the second solution was easier to find with a higher value of s. Furthermore, as the effect of the suction parameter (s > 0) becomes stronger, or the effect of injection parameter (s < 0) becomes weaker, the skin friction coefficient increases for the first and second solutions (s < 1.5), and the heat transfer coefficient increases for both solutions. trend, where the heat transfer coefficient increases as ε increases for both solutions. This indicates that the heat transfer rate over shrinking surface is higher than that over a stretching surface for each s parameter. In addition, it is clearly seen in Figure 5 that the solution for the second solution was easier to find with a higher value of s . Furthermore, as the effect of the suction parameter ( 0) s > becomes stronger, or the effect of injection parameter ( 0) s < becomes weaker, the skin friction coefficient increases for the first and second solutions ( 1.5) s < , and the heat transfer coefficient increases for both solutions. for the shrinking surface is much higher than that over the stretching surface, which indicates that the shear stress between the surface and fluid is larger for the flow over a shrinking sheet than a stretching sheet. Meanwhile, the local Nusselt number for the stretching surface is obviously higher than that for the shrinking surface. This denotes that the heat transfer rate at the surface is greater when the sheet is stretched. The increasing of nanoparticle volume fraction from 0 to 0.2 in water affects both the local skin friction coefficient as well as the local Nusselt number, where it increases the local skin friction for each nanoparticle over both stretching and shrinking surfaces. However, only the Copper-water nanofluid showed an increase in transferring the heat at the surface; the two other types of studied nanoparticles, Alumina and Titania, caused a decrease in the value for the local Nusselt number. These figures also show that different types of nanoparticles have a significant affect on shear stress and the heat transfer rate at the surface. The copper-water nanofluid was found to have the highest local skin friction coefficient and local Nusselt number for each value of the nanoparticle volume fraction over both surfaces. This means that the Copper-water nanofluid causes larger shear stress at the surface, and also has the ability to transfer heat at the surface more effectively than Titania and Alumina, since Copper has the highest thermal conductivity, as can be seen in Table 1. 3.  (16) and (17) are illustrated in Figure 6 with different nanoparticle types, namely, Copper, Alumina, and Titania, for various nanoparticle volume fractions in the base fluid (0 ≤ ϕ ≤ 0.2) over both stretching (ε > 0.5) and shrinking (ε < 0.5) surfaces. The local skin friction for the shrinking surface is much higher than that over the stretching surface, which indicates that the shear stress between the surface and fluid is larger for the flow over a shrinking sheet than a stretching sheet. Meanwhile, the local Nusselt number for the stretching surface is obviously higher than that for the shrinking surface. This denotes that the heat transfer rate at the surface is greater when the sheet is stretched. The increasing of nanoparticle volume fraction from 0 to 0.2 in water affects both the local skin friction coefficient as well as the local Nusselt number, where it increases the local skin friction for each nanoparticle over both stretching and shrinking surfaces. However, only the Copper-water nanofluid showed an increase in transferring the heat at the surface; the two other types of studied nanoparticles, Alumina and Titania, caused a decrease in the value for the local Nusselt number. These figures also show that different types of nanoparticles have a significant affect on shear stress and the heat transfer rate at the surface. The Copper-water nanofluid was found to have the highest local skin friction coefficient and local Nusselt number for each value of the nanoparticle volume fraction over both surfaces. This means that the Copper-water nanofluid causes larger shear stress at the surface, and also has the ability to transfer heat at the surface more effectively than Titania and Alumina, since Copper has the highest thermal conductivity, as can be seen in Table 1.
nanoparticles, Alumina and Titania, caused a decrease in the value for the local Nusselt number. These figures also show that different types of nanoparticles have a significant affect on shear stress and the heat transfer rate at the surface. The copper-water nanofluid was found to have the highest local skin friction coefficient and local Nusselt number for each value of the nanoparticle volume fraction over both surfaces. This means that the Copper-water nanofluid causes larger shear stress at the surface, and also has the ability to transfer heat at the surface more effectively than Titania and Alumina, since Copper has the highest thermal conductivity, as can be seen in Table 1.  (13) are asymptotically fulfilled. Each profile illustrates that the velocity and thermal boundary layer thicknesses for the first solution are smaller than for the second solution for every tested parameter, implying that the second solution is unstable, as mentioned by Bachok et al. [7].  Figure 8a,b. The velocity for the stretching case is seen to be higher than other cases for both solutions. In contrast, the temperature for the stretching surface is lower than that for shrinking and static surfaces for the first and second solutions. Physically, the stretching surface increases the velocity of the fluid, decreases the energy capability, and postpones the heat from spread up. The effects of different values of s on the velocity and temperature profiles are also displayed in Figures 9a-10b for both stretching and shrinking sheets. According to Figures 9a and 10a, the velocity profile for the suction case 1 s = is higher than that for the injection case 1 s = − , as well as over the impermeable surface 0 s = for the first solution, showing the opposite behaviour for the second solution. Figures 9b and 10b display the decreasing temperature profile as the value of s rises for both solutions, which indicates that the heat transfer rate at the surface becomes higher, and leads to a reduction in the temperature for the mass injection case compared to the mass suction case. These profiles also are depicted to show that the far field boundary conditions from Equation (13) are asymptotically fulfilled. Each profile illustrates that the velocity and thermal boundary layer thicknesses for the first solution are smaller than for the second solution for every tested parameter, implying that the second solution is unstable, as mentioned by Bachok et al. [7]. Figure 7a shows that by increasing σ, the fluid velocity increases for the first solution but decreases for the second. Meanwhile, Figure 7b presents the opposite behaviour, i.e., where the thermal boundary layer thickness is decreasing for the first solution and increasing for the second as the slip parameter becomes more significant. Therefore, the presence of velocity slip between the fluid and the surface accelerates the flow of the fluid, which decreases the deficiency between the velocity in the ambient flow and in the boundary layer. In addition, the consideration of velocity slip in real life applications is one means of reducing the skin friction coefficient at the surface, and can extend the lifetime of the material because it delays the corrosion process. The velocity and temperature profiles for a stretching surface (ε = 0.5), shrinking surface (ε = −0.5) and static surface (ε = 0) are shown in Figure 8a,b. The velocity for the stretching case is seen to be higher than other cases for both solutions. In contrast, the temperature for the stretching surface is lower than that for shrinking and static surfaces for the first and second solutions. Physically, the stretching surface increases the velocity of the fluid, decreases the energy capability, and postpones the heat from spread up. The effects of different values of s on the velocity and temperature profiles are also displayed in Figures 9 and 10 for both stretching and shrinking sheets. According to Figures 9a and 10a, the velocity profile for the suction case s = 1 is higher than that for the injection case s = −1, as well as over the impermeable surface s = 0 for the first solution, showing the opposite behaviour for the second solution. Figures 9b and 10b display the decreasing temperature profile as the value of s rises for both solutions, which indicates that the heat transfer rate at the surface becomes higher, and leads to a reduction in the temperature for the mass injection case compared to the mass suction case.

ε =
Since dual solutions are obtained, a stability analysis is implemented to identify the stability of the two solutions. According to the aforementioned authors, the upper solution is considered stable, and the solution from the first solution is physically realizable. However, the second solution is considered unstable, and is not physically realizable in real life applications. The stability of each solution is determined by the value of the smallest eigenvalue γ regardless of whether the value is positive or negative. A positive value of γ indicates that there is initial decay of the disturbance in the solution, meaning that the solution with a positive value is stable. On the other hand, a negative eigenvalue represents an initial growth of disturbance, causing the solution to become unstable. Table 3

Conclusions
This paper was presented to study the characteristics and behaviour of unsteady stagnation-point flow and heat transfer over a permeable exponential stretching/shrinking sheet in nanofluids with slip effect, and also to find a solution which is stable and physically realizable for some parameters, namely, velocity slip, stretching/shrinking, suction, injection, and with different types of nanoparticles. The system of partial differential equations was transformed to ordinary differential equations using the similarity variables. The values of the skin friction coefficient and Since dual solutions are obtained, a stability analysis is implemented to identify the stability of the two solutions. According to the aforementioned authors, the upper solution is considered stable, and the solution from the first solution is physically realizable. However, the second solution is considered unstable, and is not physically realizable in real life applications. The stability of each solution is determined by the value of the smallest eigenvalue γ regardless of whether the value is positive or negative. A positive value of γ indicates that there is initial decay of the disturbance in the solution, meaning that the solution with a positive value is stable. On the other hand, a negative eigenvalue represents an initial growth of disturbance, causing the solution to become unstable. Table 3 shows the smallest eigenvalues for the upper and lower solutions for different nanoparticle volume fractions, ϕ, and for some values of shrinking parameter ε < 0 where these values are close to the critical values ε c for each case. Based on the table, the smallest eigenvalues for the upper (first) solution are positive, while negative values are obtained for the smallest eigenvalues from the lower (second) solution. Therefore, the first solution is stable, and the results obtained are physically realizable, while those of the second solution are not.

Conclusions
This paper was presented to study the characteristics and behaviour of unsteady stagnation-point flow and heat transfer over a permeable exponential stretching/shrinking sheet in nanofluids with slip effect, and also to find a solution which is stable and physically realizable for some parameters, namely, velocity slip, stretching/shrinking, suction, injection, and with different types of nanoparticles. The system of partial differential equations was transformed to ordinary differential equations using the similarity variables. The values of the skin friction coefficient and heat transfer coefficient were obtained by a set of initial guesses in the bvp4c program in Matlab, and were presented graphically.