Generalized second-order slip for unsteady convective flow of a nanofluid: a utilization of Buongiorno’s two-component nonhomogeneous equilibrium model

Abstract The unsteady convective boundary layer flow of a nanofluid along a permeable shrinking/stretching plate under suction and second-order slip effects has been developed. Buongiorno’s two-component nonhomogeneous equilibrium model is implemented to take the effects of Brownian motion and thermophoresis into consideration. It can be emphasized that, our two-phase nanofluid model along with slip concentration at the wall shows better physical aspects relative to taking the constant volume concentration at the wall. The similarity transformation method (STM), allows us to reducing nonlinear governing PDEs to nonlinear dimensionless ODEs, before being solved numerically by employing the Keller-box method (KBM). The graphical results portray the effects of model parameters on boundary layer behavior. Moreover, results validation has been demonstrated as the skin friction and the reduced Nusselt number. We understand shrinking plate case is a key factor affecting non-uniqueness of the solutions and the range of the shrinking parameter for which the solution exists, increases with the first order slip parameter, the absolute value of the second order slip parameter as well as the transpiration rate parameter. Besides, the second-order slip at the interface decreases the rate of heat transfer in a nanofluid. Finally, the analysis for no-slip and first-order slip boundary conditions can also be retrieved as special cases of the present model.


Introduction
Today, nano uids play a bold role in modern industry and that important role comes from their great capability in thermal conductivity and capacity and also other physical characteristics. Having a low heat transfer in a uid would cause limited heat transfer and can lead to limited heat transfer e ciency. Producing a solution that comprises of suspending solid metal components will enhance the thermal conductivity characteristics of the base uid. This approach is one of the state-of-the-art procedure that has been exploited to increase the heat transfer coe cient. Due to the high thermal conductivity of metal particles, adding them to a uid would increase the thermal conductivity and also heat transfer of the resultant mixture uid [1][2][3][4][5]. This phenomenon attracts a lot of attention such as of Oztop and Abu-Nada [6], Buongiorno [7], Tiwari and Das [8], Dinarvand et al. [9], Nield and Kuznetsov [10], Grosan and Pop [11], Tamim et al. [12], Dinarvand et al. [13], Sheremet et al. [14] and Maïga et al. [15] to have deeper examination of the matter. What have been proved and still demand more research was that the presence of nano metal particle in a uid will signi cantly increase the thermal conductivity and will ultimately improve the heat transfer feature on the uid.
In Ref. [7], Buongiorno made a thorough research on convective transfer of nano uids and concludes that, so far, there has not been a reasonable rationalization for the extraordinary increase in the thermal conductivity and viscosity in the resultant nano uids. Buongiorno's main concentration was on additional improvement which occurs in connective situations. He discussed that many researchers try to justify that abnormal increase in the heat transfer. Some scientists consider the suspended metal particles as a reason for that increase, but Buongiorno disputes this reason and argues that this suspension is minutia to justify the observed enhancement. Another group contemplates turbulence as a potential explanation, however Buongiorno asserts that there would be no change in turbulence as nanoparticles appear in the nano uids, so turbulence could not justify the enhancement in the heat transfer. Another untenable justi cation for heat transfer enhancement is considering the rotation of particles. Buongiorno has worked out to calculate this e ect and proved that the rotation of the particles is too small to pro er a satisfactory explanation for the observed heat transfer improvement. By considering the suspension of particles, particle rotation, and turbulence as diminutive e ects, which cannot rational the diminutive heat transfer improvement, Buongiorno constructed a novel formulation based on the mechanics of the nanoparticle/baseuid relative velocity. He came up with a relative speed and named it the slip velocity, and suggested that the velocity of nanoparticles can be seen as the summation of velocity on the base uid and the slip velocity [7]. Buongiorno then regarded seven di erent slip components, including: inertia, Brownian di usion, thermophoresis, diffusiophoresis, Magnus e ect, uid drainage, and gravity settling. He investigated each on these seven velocities separately and deduced that when there is no turbulence, Brownian di usion and the thermophoresis are the most dominant e ects. He developed the conservation equations with regard to these two results. After proposing the model by Buongiorno [7], recently several researchers, including Nield and Kuznetsov [10], Bachok et al. [16], Pop and Khan [17], Kuznetsov and Nield [18], Dinarvand et al. [19] and others [20][21][22][23], developed their work based on Buongiorno's model.
In no-slip ow, the velocity at the wall of solid container is zero, so that uid components and the wall have equal velocity. However, there are some circumstances in which this assumption is not applicable, particularly in the case of nanoparticles. Hence, di erent researchers try to address slip boundary conditions for special problems [24][25][26]. The subject of slip is much practical in medical engineering such as making arti cial arteries and heart balloons also in the processes of pipe and wire productions, as well as extrusion of polymeric materials. Rosca and Pop [27] examined second order slip suggested by Wu [28], to scrutinize the varying surface temperature. In Ref. [29], Rosca and Pop inquire another condition involving second order slip condition to examine surface heat ux. They found the high in uence of second order slip on the properties of ow and heat transfer.
Considering these facts, the authors of this paper studied the e ects of second-order slip on unsteady convective boundary layer ow of a nano uid on a permeable shrinking/stretching sheet with suction in surface and to include the e ects of Brownian motion and thermophoresis for the nano uids, the two-component nonhomogeneous equilibrium model of Buongiorno is used. The similarity solution, which relies on nine independent dimensionless components, was also deployed. In order to solve the derived equations numerically, the Keller-box method was used. In this paper, the results and discourses are mainly concentrated on: (1) the multiple solutions [30][31][32], (2) the boundary layers behavior and (3) the skin friction and heat transfer.

Problem formulation and model development . Obtaining the mathematical formulation
Assuming the unsteady two-dimensional ow of a nano uid driven by a stretching/shrinking at sheet ( Figure 1). The velocity of plate is assumed as λuw(x), and λ is shrinking parameter that λ > and λ < indicative stretching sheet and shrinking sheet, respectively. Regarding x and y axes, we consider x-axis along the sheet and the y-axis is perpendicular to the surface of the sheet, it's positive from the surface of sheet toward the nano uid ow (Fig. 1). Here, the sheet to be permeable with v * w (t) as the identical mass ux at wall, where v * w (t) < and v * w (t) > represent suction and injection, respectively. It is also assumed that the plate has a heat source with unchanging heat ux (qw). Referring to Bonjourno's model, the mathematical relations are as follows [7] ∇.V = , In Relations (1)-(4), V = (u, v) denotes velocity, T is temperature and Cis the nanoparticle concentration. Moreover, density of base uid is shown with ρ f , µ and k are the viscosity and e ective thermal conductivity of the nano uid, respectively. D B represent the Brownian di usion coe cient and D T is thermophoretic di usion coecient, and the (ρc) f and (ρc)p are the heat capacity of the base uid and e ective heat capacity of the solid particle, respectively. A detailed explanation of equations (3) and (4) can be nd on [7] and [10]. According to scale analysis, Eqs.(1)-(4) can be written as: ∂C ∂t The uid thermal di usivity α, and nano uid heat capacit yγ are written as: De nition of the stream function ψ(x, y) as: Eqs. (5)-(8) exposed to the boundary and initial conditions Regarding Wu [28], velocity of slip u slip (x) at the plate surface is driven as: where In Eqs. (15)-(16) kn represent the Knudsen number, l = min( /kn, ), σ and δ indicates the coe cient of momentum accommodation ( ≤ σ ≤ ) and the molecular mean free path, respectively. l would be ≤ l ≤ for all values of Knudsen number. As δ is always positive it leads N to be negative. Eq. (15) is been used by other researcher such as [30,34], and lately by Rosca and Pop [24,29].
It is worth mentioning that we assume the ux of nanoparticles at the surface equal to zero and it is because of the e ect of e ect of thermophoresis. Hence, we have denotes that due to thermophoresis the nanoparticles ux is zero. In order to obtain similarity solutions, the temperature at the wall Tw(x, t), and the nanoparticles volume fraction at the wall Cw(x, t) are assigned in the following form . (18) .

Similarity transformation
We need a similarity transformation to solve Eqs. (11)- (13) subject to boundary conditions (14), for this purpose considering uw(x, t) = ax( − ct) − in which a and c are positive constants. Besides, it can be de ned a dimensionless normal distance η given by We used the equations (20) to deploy similarity solution and solve Eqs. (11)- (13).
Considering the streamline Eq. (10), it can be obtained as In which the derivation are with respect to η. We dened v * w (t) as (22) in order to be able to solve Eqs.
wherein V w describes the rate of transpiration so that V w > , V w < and V w = are relate to suction, injection and an impermeable surface, respectively. Eqs. (11)-(13) simpli ed to nonlinear ODE by using the similarity transformations (19) and (20) and are rewritten as follows and the transformed boundary conditions as follows the rst order slip (m) and second order slip (n) given by (27) in Eqs. (23)-(25) A is unsteadiness parameter, Pr is the Prandtl number, Le is the Lewis number, NbandNt represent the Brownian motion and the thermophoresis parameter, respectively and de ned as .

Quantities of engineering interest
Among many quantities in the formulation of the problem, three of them including skin friction coe cient C f , the Nusselt number Nu and the Sherwood number Sh are noteworthy in empirical cases. These parameters are dened as follow where τw is the wall shear stress, qw and q m is heat ux and nanoparticle mass ux of the wall, which as Based on similarity equations of (20), and considering Rex = ax /υ( − ct) which is local Reynold number, C f , Nu and Sh would be as

Stability analysis
To evaluate the physical realization of the rst and second solution, stability analysis in been carried out [35]. Considering the procedure in [36], variableτthat is associated with the initial value problem is introduced. Therefore, the new variables are replacing (32) into (5)- (8), we would have under these boundary conditions As speci ed by Weidman [36], stability of the steady ow solution f (η) = f (η), θ(η) = θ (η) and ϕ(η) = ϕ (η) for Eqs. (5)- (8) is tested using the following Which γ is an unknown eigenvalue parameter and F(η, τ), G(η, τ) and H(η, τ) are small compared to f (η), θ (η) and ϕ (η) respectively. Replacing (37) into Eqs. (33)-(35), we would have the following equations which are transformed into linear equations Subsequently, the pertaining boundary conditions would be as With regard to Weidman et al. [36], we assign τ = , F = F (η), G = G (η) and H = H (η) in order to achieve the linear eigenvalue problem (equations (42)- (44)) Pr Le with these boundary conditions As claimed by Haris [31], the range of possible eigenvalues can be determined by relaxing a boundary condition on F (η), G (η) or H (η). For instance, we can select to relax the condition that G (η) → when η → ∞. Thus, we are able to solve the Eqs.

Numerical procedure and validation
We have to solve a complex boundary value problem that has been represented in Eqs. (23)- (26) with nine governing parameters such as unsteadiness parameter (A), shrinking parameter (λ), mass suction velocity (Vw), thermophoresis parameter (Nt), Brownian motion parameter (Nb), rstorder slip parameter (m), second-order slip parameter (n), Lewis number (Le) and Prandtl number (Pr). Our numerical procedure is the Keller-box method (see Refs. [38][39][40]). Further, after converting Eqs. (23)-(25), to a system of rstorder ODEs, we can attempt to numerically solve them using nite di erence method with central di erence approximation. Then, the linearization process is performed with help of Newton's method. Finally, the resultant algebraic system is solved considering the boundary conditions by the block-tridiagonal elimination method. In this investigation, the pertinent mesh sizes (∆η) were chosen to 0.001 along with a relative tolerance of 0.00001, so that it has four decimal places accuracy. On the other hand, the far eld boundary condition (η∞) changes between 0.4 and 9 to successfully satisfying them. It is worth mentioning that, Pantokratoras [41] has alarmed that some graphical published results for the velocity and temperature proles obtained by previous dear investigators are wrong due to do not approaching their dependent variables to the correct values at the edge of the boundary layer, perfectly.
In order to numerical validation of the problem, Table  1 compares the similarity value of the skin friction coecient (f ( ) ) with previously published reports like Rosca and Pop [24] and Fang et al. [30], when A = and λ = − . Table 1 proves that our numerical attitude is in perfect agreement with other previous publications. Therefore, it can be deduced that our numerical results are reliable and accurate.

Results and discussion
The unsteady convective boundary layer ow of a nano uid with the utilization of Buongiorno's twocomponent nonhomogeneous equilibrium model has been studied numerically using the KBM. The in uences of the second-order slip on heat and uid ows are a major target for the present study. In this section, the results for justi ed values of parameters are obtained and discussed. The observations have been separated in three sections as: (1) the multiple solutions, (2) the boundary layers behavior: ow, thermal and concentrations elds, (3) the skin friction and heat transfer.  Table 1: The influence of the rst-order slip parameter (m), second-order slip parameter (n) and transpiration rate parameter ( Vw) on f ( ) for shrinking plate, and comparison with results of Rosca and Pop [24] and Fang et al. [30] for steady flow case(A = ), when λ = − .       sible to obtain dual solutions for the similarity equations (23)- (25). It is seen that the solution exists up to a critical value of λ(say λ c), with two solution branches for λ > λ c, a saddle-node bifurcation at λ = λ c and no solutions for λ < λ c. To determine the stability of the dual solutions, we solve the eigenvalue problem (42)-(45) and nd the smallest eigenvalue γ. Positive value of γ results in an initial decay and the ow is stable while negative value implies the growth of disturbance and the ow is unstable. The smallest eigenvalues γ for various values of λ, when Nt = Nb = . , Vw = , A = . , n = , m = . , Le = and Pr = . are presented in Table 2. The results indicate that γ is positive for the upper branch ( rst) solution and negative for the lower branch (second) solution. So, the upper branch solution is stable, while the lower branch is not. From Figures 2, 3, 6    .
. Boundary layers behavior with focus on shrinking plate case  show the velocity, temperature and concentration pro les, for di erent values of unsteadiness parameter (A) and transpiration rate parameter ( Vw). What can be noted from these gures is that the pro les for both rst and second solutions is satisfactory for the far eld boundary conditions asymptotically, and concur the results of numerical solution of Section 4.1. It is also manifest that the pro les of the second solution have a much higher boundary layer thickness which implies that the top diagrams (black sold lines) are the stable solution as opposed to the bottom diagrams (red dash lines) solution. Figures 8 and 9 indicate that as the values of the unsteadiness parameter (A) and transpiration rate parameter ( Vw). increase, the velocity of boundary layer in either solution would increase as well, which suggests that the hydrodynamic boundary layer thickness would be thinner when the amplitude of unsteadiness parameter increases. However, it is worth mentioning that the matter is not acceptable for the unsteadiness parameter e ect near the wall, when there is a reversed ow near the wall for the second solutions. As a result, when the transpiration rate parameter ( Vw) increases, the thickness of the hydrodynamic boundary layer decreases. This phenomenon is in accordance to that of a viscous uid. Hence, regardless of the types of uids, viscous uid or nano uid, suction always would lead to stabilization of the boundary layer grow. Moreover, in this research, the suction case (V w > ) has been applied, since in the boundary layer de nition, it is basically assumed that the boundary layer thickness is supposed to be practically very thin.
On the other hand, as the unsteadiness parameter (A) and transpiration rate parameter ( Vw) raise, the temperature pro les of the both solutions is compressed which is shown in Figures 10 and 11 respectively. Therefore, when unsteadiness parameter (A) and transpiration rate parameter ( Vw) enhance, the thickness of the thermal boundary layer for the both rst and second solutions decreases Figure 11 depicts the second solution of the temperature when there is heat generation inside the boundary layer, which is impossible while the viscous dissipation e ects has not been considered in the present model. This matter also can be other reason to decline the second solution of problem. Figures 12 and 13 illustrate the e ect of the unsteadiness parameter (A) and transpiration rate parameter ( Vw) on the nanoparticles concentration pro le. What we noticed here was that, these pro les increase near the surface of the shrinking sheet and climax to its maximum before falling to its ambient value zero. This e ect might come from thermophoresis e ect on the concentration boundary conditionNbϕ ( ) + Ntθ ( ) = . As the transpiration rate parameter ( Vw) intensi es, nanoparticles concentration pro les would decrease which are pertaining to the lower and upper branch solutions.

Figures 14 and 15 depict the skin friction
Re / x C f for di erent values of the rst-order slip parameter (m), the secondorder slip parameter (n) and transpiration rate parameter ( Vw) versus the unsteadiness parameter (A) between 0 to 1.5. Figures 14 and 15 demonstrate that skin friction enhance with increasing the unsteadiness parameter (A). Moreover, from Figure 14, the skin friction coe cient reduces slightly with the rst order slip parameter (m) and increase more strongly with the absolute value of the second order slip parameter (|n|). Besides, the Figure 15 depicts that the suction increases values of the skin friction coe cient that is a predictable topic in the problem conditions. In fact, the boundary layer thickness decreases with transpiration rate parameter ( Vw), where this matter enhances velocity gradient on the wall. Consequently, one can predict a higher skin ction with the increase in transpiration rate parameter ( Vw). Figures 16 and 17 are made to reveal the rami cations of the rst-order slip parameter (m), the second-order slip parameter (n), the unsteadiness parameter (A) and thermophoresis parameter ( Nt) on the reduced Nusselt numberNur(the rate of heat transfer at the surface). An increasing manner of the reduced Nusselt number Nur with the unsteadiness parameter (A) can be concluded from Figures 16 and 17. On the other hand, Figure 16 demonstrates that the declined Nusselt numberNurlessens with both the rst-order slip parameter (m) and the absolute value of the second-order slip parameter (|n|). The e ects of the thermophoresis parameter ( Nt) on the reduced Nusselt number Nur for the di erent unsteadiness parameter (A) is depicted in Figure 17. Obviously, it is observed that the reduced Nusselt number Nur decrease as the thermophoresis parameter ( Nt) decreases, which was already reported by Dinarvand et al. [5,9]. Consequently, the greatest heat transfer rate is observed for the situation in which the thermophoresis parameter ( Nt) is very small.

Conclusions
In this article, the e ects of second-order slip on unsteady convective boundary layer ow of a Buongiorno's nano uid (in which nanoparticles' Brownian motion and thermophoresis e ects have been considered) along a permeable shrinking/stretching plate in the presence of suction has been investigated. The similarity solution is em-ployed to reduce the governing system of partial di erential equations to nonlinear ordinary di erential equations with the aim of solving them numerically by the Keller-box method (KBM). The study has been focused on: (1) the multiple solutions, (2) the boundary layers behavior and (3) the skin friction and heat transfer.
The main outcomes resulting from this research are as follows. (1) The shrinking parameterλdetermines how many solutions the problem would have. Physical parameters would change the range of λ in which the solution exist and multiple solutions can be achieved by solving similarity equations. (2) The application of rst and second order slips at the wall cause the critical suction parameter to decrease. (3) For shrinking plate, as unsteadiness e ects increase, the velocity component enhances, whereas the concentration of particles and the temperature pro les decline. (4) The in uence of the unsteadiness on the nanoparticles concentration pro les turns out to be more outstanding as opposed to the velocity and temperature pro les. (5) It is necessary to consider the second order slip in modeling a nano uid because the second order slip would amplify the rate of shear stress at the wall, and also reduces the heat transfer rate in a nano uid. (6) The thermophoresis and the Brownian motion e ects were found to be key factors in the growth of heat transfer. The highest values are obtained when thermophoresis is very small and approaches to zero.