Exact solution for heat transport of Newtonian uid with quadratic order thermal slip in a porous medium

In this communication, an analytical solution for the thermal transfer of Newtonian uid ow with quadratic order thermal and velocity slips is presented for the rst time. The ow of a Newtonian uid over a stretching sheet which is embedded in a porous medium is considered. Karniadakis and Beskok's quadratic order slip boundary conditions are taking into account. A closed form of analytical solution of momentum equation is used to derive the analytical solution of heat transfer equation in terms of con uent hyper-geometric function with quadratic order thermal slip boundary condition. Accuracy of present results is assured with the numerical solution obtained by Iterative Power Series method with shooting technique. The impacts of porous medium parameter, tangential momentum accommodation coe cient, energy accommodation coe cient on velocity and temperature pro les, skin friction coe cient and reduced Nusselt number are discussed. The Nusselt number increases with the higher estimations of tangential momentum and energy accommodation coe cients.


Introduction
The investigation of uid ow in the presence of slip boundary conditions has received considerable interest due its accuracy of predicting the realistic behaviour in many engineering processes. For example, the uid ow in micro pumps, micro nozzles, micro vales and hard disk experiences slip at wall. The use of no-slip condition in the above cases does not predict the actual physical situation. The consideration of velocity slip and temperature jump in this type of ow regime is very important to determine the velocity and temperature, respectively. The investigation of heat transfer in a uid ow induced by a moving surface is very important in the processes of glass blowing, continuous casting, cool oil slurries, metal spinning and plastic lms etc. A primary investigation on this type of problem was done by Sakiadis [1,2] and Crane [3]. Much attention has been given to this type of uid ow problem with various physical eects via both analytical and numerical techniques [4][5][6][7][8][9][10][11][12].
Karniadakis and Beskok [13] proposed a quadratic order slip boundary condition. Xiou et al. [14] studied the gas ow in microtube with quadratic order slip conditions. Hamdan et al. [15] modelled the micro gas ow with quadratic order slip conditions. Fang et al. [16] considered the Wu's [17] quadratic order velocity slip condition in the problem of viscous uid ow over a shrinking sheet. Fang et al. [16] derived closed form analytical solutions for momentum equation. Nandeppanavar et al. [18] studied the quadratic order velocity slip with heat transfer in a viscous uid. A closed form analytical solution was presented for the momentum equation and the energy equation was solved numerically in [18]. Turkyilmazoglu [19] derived the analytical solution for MHD viscous uid ow using Laguerre polynomials with quadratic order velocity slip. The quadratic order velocity slip eects on nanouid ow over a stretching/shrinking sheet were investigated both numerically and analytically in the articles [20,21]. In the above articles, a closed form solution for momentum equation was presented and the energy equation was solved using conuent hyper-geometric function. Numerous numerical investigations on the uid ow with quadratic order velocity slip can be found in the literature [22][23][24][25][26][27][28][29][30][31][32]. In all the above studies [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35], the eects of quadratic order temperature boundary conditions were omitted and Wu's quadratic order velocity boundary condition was considered. Arikoglu et al. [33] used dierential transform method (DTM) to analyse the Karniadakis and Beskok's quadratic order velocity slip and thermal jump impacts in a rotating disk problem. Recently, Ganesh et al. [34] have carried out a numerical investigation on the Newtonian uid over a vertical stretching sheet with quadratic order velocity and thermal slip boundary conditions of Karniadakis and Beskok.
Having all the above literature in mind, we focused in this communication on the problem of Newtonian uid ow over a stretching sheet immersed in a porous medium with suction. we aimed to derive an analytical solution in terms of conuent hyper-geometric function for energy equation in the presence of the following quadratic order temperature and velocity slip boundary conditions [13]: It is worth mentioning herein that -to achieve this target-, we rstly derived a closed form analytical solution of momentum equation following the footsteps in [16,[18][19][20][21]].

Mathematical formulation
We consider the steady, 2D laminar ow of a Newtonian uid over a stretching sheet in a Darcian porous medium with suction eects. The sheet stretching velocity u w = dax with constant`a' and stretching parameter`d' are assumed. The x-axis runs along the stretching surface with velocity`u'. The y-axis runs perpendicular to the sheet with velocity`v'. The temperature at the stretching surface takes the constant value T w , while the ambient value, attained as y tends to innity, takes the constant value T ∞ . It is assumed that the uid experiences both second order velocity and thermal slips. The governing equations of this problem can be expressed as: where µ, ρ, K, k and C p are representing the viscosity, density, permeability of the porous medium, thermal conductivity, and specic heat capacity, respectively. The quadratic order velocity and temperature boundary conditions of Eqs. (1)-(3) are considered as where is the quadratic order velocity slip the quadratic-order thermal slip factor. In addition, λ is the mean free path and β is specic heat ratio.

Solution of Momentum equation
The method of solution is based on the similarity transformations; given by: Plugging the similarity transformations (5) into Equations (2) and (4); one obtains where a v f is quadratic order velocity slip parameter and s is suction/injection parameter. An analytical solution of Eq. (6) subject to boundary conditions (7), is obtained following similar procedure given in [16,[18][19][20][21]; hence, we obtain .
Substituting Eq. (8) in Eq. (6) gives the following algebraic equation which has four distinct roots expressed as where A physically meaningful solution is obtained using the IVth of Eq. (9). Notice that the non-dimensional form of local skin friction coecient is obtained as follows:

solution of Energy Equation
Plugging similarity transformations (5) into both Eq. (3) and the temperature boundary conditions in (4), we obtain θ + P rgθ = P rg θ, where γ t = β t1 a v f is rst order thermal slip parameter, δ t = β t2 a v f is quadratic order slip parameter and Pr is the Prandtl number. Substituting ξ = −B α e −αη and Eqs. (8) into equations (10) & (11); we can derive the solution for the energy equation with quadratic order thermal slip boundary condition, in terms of conuent hyper-geometric function which is given by where Here, M is the conuent hyper-geometric function dened in [20,21] M The non-dimensional form of reduced Nusselt number is derived as

Results and Discussion
The derived analytical solutions are veried with numerical solutions of governing equations by Iterative Power Series method with shooting method [35]. A comparison with Turkyilmazoglu [19] has been included in Table 1 for −g (0) which gives condent on the accuracy of present results.  Table 1 Comparison results of −g (0) Table 2 Numerical values of local skin friction coecient and reduced Nusselt number with d=1, s=1, P r = 0.71 and β =0.5. The results are discussed through graphical representations with following xed values of parameters: d=1, s=1, P r = 0.71, β =0.5 and k 1 = 0.5. The inuences of porous medium parameter (k 1 ), tangential momentum accommodation coecient (σ v ) and energy accommodation coecient ((σ t ) on velocity prole, temperature prole, skin friction and reduced Nusselt number are discussed through analytical solutions. Both σ v and σ t are varied from 0.2 to 0.8. The rst and quadratic order velocity and thermal slip parameters are calculated using σ v and σ t . The numerical values of −g (0) are calculated by both numerical and analytical methods and presented in Table. 2.
The behaviour of velocity prole with σ v and k 1 is shown in Fig.1. For higher estimations of σ v (0.4, 0.6, 0.8), an increasing behaviour in velocity prole has been noted. The tangential momentum accommodation coecient has a signicant eect on the velocity prole for a certain rage of η. This is due the fact that, σ v is inversely proportional of the rst and quadratic order velocity slip parameters. Increase in σ v leads to decrease the velocity slip parameters. Hence the velocity prole increases. The velocity prole reduces with porous medium parameter due to Darcian resistant force in the ow region.
Features of σ v , σ t and k 1 on the temperature prole portrayed in Fig 2. For higher estimations of σ v and σ t (0.4, 0.6, 0.8), an augmentation in the temperature prole has been observed. This is because, an increase in σ v and σ t causes the rst and quadratic order velocity and thermal slip parameter to decrease. A notable eect has been seen via k 1 . The temperature prole is decreased with k 1 near the wall. Thermal boundary layer thickness is enhanced with k 1 .
The variation of skin friction coecient and reduced Nusselt number with σ v , σ t and k 1 is shown in Figs. Fig. 2 . Impacts of tangential momentum accommodation coecient (σ v ), energy accommodation coecient (σ t ) and porous medium parameter (k 1 ) on temperature prole with d=1, P r= 0.71, β=0.5 and s=1.
3 and 4 respectively. A comparison between analytical and numerical solution of present problem is also highlighted. It is observed from Fig. 3, the magnitude of g (0), increases with k 1 and reduced with σ v . The skin friction is lower in the slip ow case compared to no-slip case. The magnitude of −θ (0) decreases with k 1 and increases with σ v and σ t .

Conclusion
An analytical solution for the heat transport of Newtonian uid ow over a stretching sheet with quadratatic velocity and thermal slips conditions is obtained. The obtained solutions are veried with the numerical solutions through Iterative Power Series method with shooting method. Important ndings of present analytical study are listed below: • The velocity prole increases with tangential momentum accommodation coecient and decreases with porous medium parameter.
• The thermal boundary layer thickness enhances with tangential momentum accommodation coecient, energy accommodation coecient and porous medium parameter.
• The magnitude of skin friction increases via porous medium parameter and decreases via tangential momentum and energy accommodation coecients. Fig. 3. Impacts of tangential momentum accommodation coecient (σ v ) and porous medium parameter (k 1 ) on local skin friction coecient with d=1 and s=1.
• Higher tangential momentum and energy accommodation coecients increase the Nusselt number.