Effects of slip on nonlinear convection in nanofluid flow on stretching surfaces
- Sachin Shaw^{1},
- Peri K Kameswaran^{1, 2} and
- Precious Sibanda^{1}Email author
Received: 20 October 2014
Accepted: 7 December 2015
Published: 4 January 2016
Abstract
We investigate the effects of momentum, thermal, and solute slip boundary conditions on nanofluid boundary layer flow along a permeable surface. The conventional no-slip boundary conditions at the surface are replaced by slip boundary conditions. At moderate to high temperatures, the temperature-concentration dependence relation is nonlinear and the Soret effect is significant. The governing partial differential equations are solved numerically. The influence of significant parameters on the fluid properties as well as on the skin friction, local Nusselt number, local Sherwood number, and the local nanoparticle Sherwood number are determined. We show, among other results, that the existence and uniqueness of the solutions depends on the slip parameters, and that the region of existence of the dual solution increases with the slip parameters.
Keywords
1 Introduction
Boundary layer flow over a stretching surface is important as it occurs in several engineering process, for example, materials manufactured by extrusion. During the manufacturing process, a stretching sheet interacts with the ambient fluid both thermally and mechanically. Several studies on the dynamics of boundary layer flow over a stretching surface have appeared in the literature (Crane [1]; Dutta et al. [2]). Recently an innovative technique used for improving heat transfer is to add ultra fine solid particles to a base fluid, Choi [3]. Recent literature shows a significant rise in applications of nanofluids such as in microchannels (Ebrahimi et al. [4], enzyme biosensors (Li et al. [5]), drug delivery (Shaw and Murthy [6]) biomimetic microsystems (Huh et al. [7]) etc. An impressive review of nanofluid research is given by Das et al. [8]. Kameswaran et al. [9] studied homogeneous-heterogeneous reactions in a nanofluid flow over a porous stretching sheet. Mabood et al. [10] studied the MHD boundary layer flow of nanofluids over a nonlinear stretching sheet.
A significant number of studies have applied the no-slip boundary conditions at the wall. However, the no-slip assumption is not applicable when fluid flows in micro and nano channels and must be replaced by slip boundary conditions (Aziz [11]). Nield and Kuznetsov [12] presented an analytic solution for convection flow in channel or circular ducts saturated with a rarefied gas in a slip-flow regime. The slip condition applies to corner flows and in the extrusion of polynomial melts from a capillary tube (Thompson and Troian [13]; Nguyen and Wereley [14]). Karniadakis et al. [15] showed that hydrodynamic and thermal slip occur simultaneously. The difference between the fluid velocity at the wall and the velocity of the wall itself is directly proportional to the shear stress. The proportionality constant is called the slip length (Maxwell [16]; Hak [17]).
Beavers and Joseph [18] investigated fluid flow over a permeable wall with a slip boundary condition. The effects of a second order velocity-slip and temperature-jump on basic gaseous fluctuating micro-flows were analyzed by Hamdan et al. [19]. The effects of partial slip on steady boundary layer stagnation-point flow of an incompressible fluid and heat transfer from a shrinking sheet was investigated by Bhattacharyya et al. [20]. This was extended in Bhattacharyya et al. [21] to unsteady stagnation-point flow of a Newtonian fluid and heat transfer from a stretching sheet with partial slip conditions. Niu et al. [22] investigated slip flow and heat transfer in a non-Newtonian nanofluid in a microtube. Khan et al. [23] analyzed the effects of hydrodynamic and thermal slip boundary conditions on double-diffusive free convective flow of a nanofluid along a semi-infinite flat solid vertical plate. Ibrahim and Shankar [24] studied the effects of velocity, thermal and solutal slip condition on the MHD boundary layer flow of a nanofluid past a permeable stretching sheet. Mabood et al. [25] studied the MHD slip flow over a radiation stretching sheet by using the optimal homotopy asymptotic method.
Some thermal systems such as those encountered in reactor safety, combustion and solar collectors operate at moderate to very high temperatures. In such cases, the temperature-concentration dependence relation is nonlinear and the Soret effect is of immense important. Partha [26] studied natural convection in a non-Darcy porous medium with a nonlinear temperature-concentration-dependent density relation.
In this paper, we analyze the effects of momentum, temperature and solute slip on stagnation-point flow over a permeable stretching or shrinking sheet. We transform the governing partial differential equations into similarity equations which are then solved numerically. The effects of physical parameters on the flow, heat, mass and nanoparticle concentration are determined and presented graphically. In the present paper, we mainly focus on the effect of slip parameters on the governing system along with the nonlinear thermal convection. To the best of authors knowledge such study has not been reported earlier in the literature.
2 Mathematical formulation
3 Results and discussion
Comparison of the reduced Nusselt number \(\pmb{-\theta'(0)}\) and Sherwood number \(\pmb{-\phi'(0)}\) with Noghrehabadi et al. [ 27 ] for \(\pmb{Nc=1}\) , \(\pmb{Nr=3}\) , \(\pmb{\delta_{1}=1}\) , \(\pmb{Le=Pr=10}\) . (a) \(\pmb{\delta_{3}=\delta_{4}=\lambda=\gamma= Ln=Sr = K_{p}=0}\) and (b) \(\pmb{\delta_{3}=\delta_{4}=\lambda=0}\) , \(\pmb{Ln=1}\) , \(\pmb{Sr=0.5}\) , \(\pmb{K_{p}=0.5}\)
\(\boldsymbol {\delta_{2}}\) | Nb | Nt | Noghrehabadi et al. [ 27 ] | Present (a) | Present (b) | |||
---|---|---|---|---|---|---|---|---|
Nur | Shr | Nur | Shr | Nur | Shr | |||
0.0 | 0.1 | 0.1 | 0.952377 | 2.129394 | 0.95237683 | 2.12939377 | 1.00058819 | 2.66004285 |
0.3 | 0.520079 | 2.528638 | 0.52007905 | 2.52863816 | 0.54419785 | 3.46549450 | ||
0.5 | 0.321054 | 3.035142 | 0.32105433 | 3.03514247 | 0.31586598 | 4.38704894 | ||
0.2 | 0.1 | 0.505581 | 2.381871 | 0.50558141 | 2.38187064 | 0.54400672 | 2.85986019 | |
0.3 | 0.273096 | 2.655459 | 0.27309580 | 2.65545946 | 0.28254157 | 3.33932740 | ||
0.5 | 0.168077 | 2.888339 | 0.16807658 | 2.88833918 | 0.16127263 | 3.75314398 | ||
0.3 | 0.1 | 0.252156 | 2.410019 | 0.25215609 | 2.41001880 | 0.27177120 | 2.86844013 | |
0.3 | 0.135514 | 2.608819 | 0.13551419 | 2.60881871 | 0.13604735 | 3.19990440 | ||
0.5 | 0.083298 | 2.751875 | 0.08329860 | 2.75187540 | 0.07566950 | 3.45202615 | ||
1.0 | 0.1 | 0.1 | 2.751875 | 1.607430 | 0.71892800 | 1.60743180 | 1.14988755 | 3.00376688 |
0.3 | 0.392596 | 1.908809 | 0.39259606 | 1.90881247 | 0.63804840 | 3.78822390 | ||
0.5 | 0.242357 | 2.291156 | 0.24235674 | 2.29116127 | 0.37823785 | 4.77292460 | ||
0.2 | 0.1 | 0.381652 | 1.798019 | 0.38165211 | 1.79802095 | 0.63400147 | 3.28337188 | |
0.3 | 0.206154 | 2.004545 | 0.20615392 | 2.00454704 | 0.33706394 | 3.78349666 | ||
0.5 | 0.126877 | 2.180339 | 0.12687726 | 2.18034274 | 0.19696830 | 4.23378793 | ||
10.0 | 0.1 | 0.1 | 0.412468 | 0.922099 | 0.41247939 | 0.92225151 | 1.20480363 | 3.13488626 |
0.3 | 0.225245 | 1.094883 | 0.22524897 | 1.09516618 | 0.67291837 | 3.91228460 | ||
0.5 | 0.139050 | 1.314098 | 0.13905033 | 1.31453566 | 0.40123365 | 4.91834227 | ||
0.2 | 0.1 | 0.218955 | 1.031454 | 0.21896995 | 1.03160059 | 0.66701497 | 3.44165247 | |
0.3 | 0.118275 | 1.149881 | 0.11827923 | 1.15009326 | 0.35709376 | 3.94942212 | ||
0.5 | 1.204555 | 1.250672 | 0.07279486 | 1.25095500 | 0.21002703 | 4.41174227 | ||
0.3 | 0.1 | 0.109199 | 1.043650 | 0.10921014 | 1.04379192 | 0.33770638 | 3.47832715 | |
0.3 | 0.058689 | 1.129708 | 0.05869191 | 1.12989321 | 0.17488396 | 3.84292518 | ||
0.5 | 0.036078 | 1.191622 | 0.03607706 | 1.19185182 | 0.10098316 | 4.12849783 |
A representative set of graphical results for the velocity, temperature, solute concentration and nanoparticle volume fraction as well as the skin friction, local Nusselt number, local Sherwood number and local nanoparticle Sherwood number is presented and discussed for different parametric values. We note that solutions of equations (8) and (11) exist for all values of \(\alpha_{1}>0\), while in the case of a shrinking surface (\(\alpha_{1}<0\)), the equations have a solution only in the range \(\alpha_{1}>{\alpha_{1}}_{\mathrm{crit}}\), where \({\alpha_{1}}_{\mathrm{crit}}\) is a critical value of \(\alpha_{1}\). This critical value depends on other parameter values. There are no solutions real when \(\alpha_{1}<{\alpha_{1}}_{\mathrm{crit}}\). Dual solutions of the boundary layer equations appear in the range \({\alpha_{1}}_{\mathrm{crit}}<\alpha_{1}\). As noted by Merkin [28], Postelnicu and Pop [29], the first solution is stable and physically realizable, while the second solution is unstable. In this study our primary focus is on the dual solutions and the effects of the slip coefficients and nonlinear volumetric thermal and solute constants.
Value of critical velocity ratio \(\pmb{\alpha_{1}}\) for different momentum, thermal, and solute slip constants for \(\pmb{Kp=\lambda_{1}=\lambda_{2}=Nt=Nb=Nc=Nr=0.1}\) , \(\pmb{Pr=0.5}\) , \(\pmb{Le=Ln=1}\) , \(\pmb{Sr=0.5}\)
\(\boldsymbol {(\alpha_{1})}_{\mathbf{crit}}\) | \(\boldsymbol {\alpha_{2}}\) | \(\boldsymbol {\alpha_{3}}\) | \(\boldsymbol {\alpha_{4}}\) | dual solution range |
---|---|---|---|---|
−1.902 | 0.5 | 0.1 | 0.1 | −1.902, −1.18 |
−1.899 | 0.5 | 0.5 | 0.1 | −1.899, −1.18 |
−1.898 | 0.5 | 0.5 | 0.5 | −1.898, −1.18 |
−2.622 | 1 | 0.1 | 0.1 | −2.622, −1.18 |
−3.407 | 1.5 | 0.1 | 0.1 | −3.407, −1.18 |
The local Nusselt number, local Sherwood number, and the local nanoparticle density are nonlinear increasing functions of the momentum slip parameter for the first solution while they are a decreasing function for the case of the second solution. It is interesting to note that the generation of vorticity for the shrinking velocity is reduced by an increase in the momentum slip at the surface (when \(\alpha_{1}>1\)). The momentum slip parameter enhances the velocity at the surface which forces the solute and particle to move away from the surface. As a result, the local Nusselt number, local Sherwood number, and local nanoparticle Sherwood number increase nonlinearly with the momentum slip parameter but decrease with increases in the solute slip coefficient. It evident that the local Nusselt number decreases with increase of the partial thermal slip coefficient and this finding is similar to earlier results by Zheng et al. [31]. The nature of the second solution for the local Nusselt number, local Sherwood number, and local nanoparticle Sherwood number is quite similar and this mainly depends on the momentum slip coefficient rather than the other slip coefficient.
4 Conclusions
The effect of momentum, thermal, and solute slip on nonlinear convection boundary layer flow from a stretching and shrinking sheet has been investigated numerically. Analysis of stagnation-point slip flow from a shrinking sheet has shown that existence and uniqueness of the solution depends on the slip parameters, mainly the momentum slip and the velocity ratio parameter \(\alpha_{1}\). Dual solutions were obtained when the velocity ratio was less than a certain critical value. The region of existence of the dual solution increases with the slip parameters. The nonlinear temperature and concentration coefficients reduce the thermal and solute boundary layer thicknesses. The thermal slip coefficient reduces the momentum and thermal boundary layer thickness. The local Nusselt number, local Sherwood number, and local density of the nanoparticles increase nonlinearly with the convection coefficient.
Declarations
Acknowledgements
The authors are grateful to the University of KwaZulu-Natal for financial support.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Crane, LJ: Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645-647 (1970) View ArticleGoogle Scholar
- Dutta, BK, Roy, P, Gupta, AS: Temperature field in the flow over a stretching sheet with uniform heat flux. Int. Commun. Heat Mass Transf. 12, 89-94 (1985) View ArticleGoogle Scholar
- Choi, SUS: Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer, DA, Wang, HP (eds.) Developments and Applications of Non-Newtonian Flows, vol. 99. ASME, New York (1995) Google Scholar
- Ebrahimi, S, Sabbaghzadeh, J, Lajevardi, M, Hadi, I: Cooling performance of a microchannel heat sink with nanofluids containing cylindrical nanoparticles (carbon nanotubes). Heat Mass Transf. 46, 549-553 (2010) View ArticleGoogle Scholar
- Li, H, Liu, S, Dai, Z, Bao, J, Yang, Z: Applications of nanomaterials in electrochemical enzyme biosensors. Sensors 9, 8547-8561 (2009) View ArticleGoogle Scholar
- Shaw, S, Murthy, PVSN: The effect of shape factor on the magnetic targeting in the permeable microvessel with two-phase Casson fluid model. J. Nanotechnol. Eng. Med. 2, 041003 (2011) View ArticleGoogle Scholar
- Huh, D, Matthews, BD, Mammoto, A, Montoya-Zaval, M, Hsin, NY, Ingber, DE: Reconstituting organ-level lung functions on a chip. Science 328, 1662-1668 (2010) View ArticleGoogle Scholar
- Das, SK, Choi, SUS, Yu, W, Pradeep, T: Nanofluids: Science and Technology. Wiley, Hoboken (2007) View ArticleGoogle Scholar
- Kameswaran, PK, Shaw, S, Sibanda, P, Murthy, PVSN: Homogeneous-heterogeneous reactions in a nanofluid flow due to a porous stretching sheet. Int. J. Heat Mass Transf. 57, 465-472 (2013) View ArticleGoogle Scholar
- Mabood, F, Khan, WA, Ismail, AIM: MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: a numerical study. J. Magn. Magn. Mater. 374, 569-576 (2015) View ArticleGoogle Scholar
- Aziz, A: Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Commun. Nonlinear Sci. Numer. Simul. 15, 573-580 (2010) MathSciNetView ArticleGoogle Scholar
- Nield, DA, Kuznetsov, AV: Forced convection with slip-flow in a channel or duct occupied by a hyper-porous medium saturated by a rarefied gas. Transp. Porous Media 64, 161-170 (2006) MathSciNetView ArticleGoogle Scholar
- Thompson, PA, Troian, SM: A general boundary condition for liquid flow at solid surfaces. Nature 389, 360-362 (1997) View ArticleGoogle Scholar
- Nguyen, NT, Wereley, ST: Fundamentals and Applications of Microfluidics. Artech House, London (2009) Google Scholar
- Karniadakis, G, Beskok, A, Aluru, N: Microflows and Nanoflows. Fundamentals and Simulation. Springer, New York (2005) MATHGoogle Scholar
- Maxwell, JC: On stresses in rarefied gases arising from inequalities of temperature. Philos. Trans. R. Soc. Lond. 170, 231-256 (1879) MATHView ArticleGoogle Scholar
- Hak, GM: Flow physics. In: Gad-el-Hak, M (ed.) The MEMS Handbook. CRC Press, Boca Raton (2002) Google Scholar
- Beavers, GS, Joseph, DD: Boundary condition at a naturally permeable wall. J. Fluid Mech. 30, 197-207 (1967) View ArticleGoogle Scholar
- Hamdan, MA, Al-Nimr, MA, Hammoudeh, VA: Effect of second order velocity-slip/temperature-jump on basic gaseous fluctuating micro-flows. J. Fluids Eng. 132, 074503 (2010) View ArticleGoogle Scholar
- Bhattacharyya, K, Mukhopadhyay, S, Layek, GC: Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet. Int. J. Heat Mass Transf. 54, 308-313 (2011) MATHView ArticleGoogle Scholar
- Bhattacharyya, K, Mukhopadhyay, S, Layek, GC: Slip effects on an unsteady boundary layer stagnation-point flow and heat transfer towards a stretching sheet. Chin. Phys. Lett. 28, 094702 (2011) View ArticleGoogle Scholar
- Niu, J, Fu, C, Tan, W: Slip-flow and heat transfer of a non-Newtonian nanofluid in a microtube. PLoS ONE 7, e37274 (2012) View ArticleGoogle Scholar
- Khan, WA, Uddin, MJ, Md Ismail, AI: Hydrodynamic and thermal slip effect on double-diffusive free convective boundary layer flow of a nanofluid past a flat vertical plate in the moving free stream. PLoS ONE 8, e54024 (2013) View ArticleGoogle Scholar
- Ibrahim, W, Shankar, B: MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions. Comput. Fluids 75, 1-10 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Mabood, F, Khan, WA, Uddin, MJ, Md Ismail, AI: Optimal homotopy asymptotic method for MHD slips flow over a radiating stretching sheet with heat transfer. Far East J. Appl. Math. 90, 21-40 (2015) View ArticleGoogle Scholar
- Partha, MK: Nonlinear convection in a non-Darcy porous medium. Appl. Math. Mech. 31, 565-574 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Noghrehabadi, A, Pourrajab, R, Ghalambaz, M: Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature. Int. J. Therm. Sci. 54, 253-261 (2012) View ArticleGoogle Scholar
- Merkin, JH: On dual solutions occurring in mixed convection in a porous media. J. Eng. Math. 20, 171-179 (1985) MathSciNetView ArticleGoogle Scholar
- Postelnicu, A, Pop, I: Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge. Appl. Math. Comput. 217, 4359-4368 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Wang, CY: Stagnation flow towards a shrinking sheet. Int. J. Non-Linear Mech. 43, 377-382 (2008) View ArticleGoogle Scholar
- Zheng, L, Zhang, C, Zhang, X, Zhang, J: Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium. J. Franklin Inst. 350, 990-1007 (2013) MATHMathSciNetView ArticleGoogle Scholar