Dual solutions of stagnation-point flow of a nanofluid over a stretching surface
© Kameswaran et al.; licensee Springer 2013
Received: 10 May 2013
Accepted: 7 August 2013
Published: 22 August 2013
The paper discusses the effects of homogeneous-heterogeneous reactions on stagnation-point flow of a nanofluid over a stretching or shrinking sheet. The model presented describes mass transfer in copper-water and silver-water nanofluids. The governing system of equations is solved numerically, and the study shows that dual solutions exist for certain suction/injection, stretching/shrinking and magnetic parameter values. Comparison of the numerical results is made with previously published results for special cases.
Keywordshomogeneous-heterogeneous reactions nanofluids volume fraction model stretching or shrinking sheet magnetic field strength
Problems involving fluid flow over stretching or shrinking surfaces can be found in many manufacturing processes such as in polymer extrusion, wire and fiber coating, foodstuff processing, etc. Crane  was the first to consider the steady two-dimensional flow of a Newtonian fluid driven by a stretching elastic flat sheet which moved in its own plane with velocity varying linearly with the distance from a fixed point. This study was subsequently extended by many authors to explore various aspects of heat transfer in a fluid surrounding a stretching sheet (Tsou et al. , Erickson et al. , Mucoglu and Chen , Grubka and Bobba , Karwe and Jaluria , Chen , Abo-Eldahab and El-Aziz , Salem and El-Aziz , Ali , Ishak et al. ).
The magnetohydrodynamic effect has important engineering applications in electrical motors. Heat transfer over a stretching or shrinking sheet subject to an external magnetic field, viscous dissipation and joule effects was studied by Jafar et al. . They observed that the flow and heat transfer characteristics for a shrinking sheet were quite different from those of a stretching sheet. Lok et al.  analyzed MHD stagnation-point flow from a shrinking sheet. They found that dual solutions existed for small values of the magnetic parameter. The stagnation-point flow over a stretching or shrinking sheet in a nanofluid was investigated by Bachok et al. . They showed that adding nanoparticles to a base fluid increased the skin friction and heat transfer coefficients. Recently, Narayana and Sibanda  investigated the laminar flow of a nanoliquid film over an unsteady stretching sheet. They noticed that the effect of an increase in the nanoparticle volume fraction was to reduce the axial velocity and free stream velocity in the case of a Cu-water nanoliquid. However, the opposite appeared to be true in the case of an -water nanofluid. Kameswaran et al.  studied the effects of viscous dissipation and a chemical reaction in hydromagnetic nanofluid flow due to a stretching or shrinking sheet. They found that the velocity profiles decreased with an increase in the nanoparticle volume fraction, while the opposite was true in the case of temperature and concentration profiles. The study further showed that liquids with nanoparticle suspensions were better suited for effective cooling of the stretching sheet due to their enhanced conductivity and thermal properties.
Most chemically reacting systems involve both homogeneous and heterogeneous reactions (combustion, catalysis and biochemical systems). The simple combustion model helps us to understand the combustion phenomenon in many complex engineering applications such as in aircraft and rocket engines. A model for isothermal homogeneous-heterogeneous reactions in the boundary layer flow of a viscous fluid past a flat plate was presented by Merkin . He modeled the homogeneous reaction by a cubic autocatalysis process and the heterogeneous reaction by a first-order process. Chaudhary and Merkin  analyzed homogeneous-heterogeneous reactions in boundary layer flow. They presented a numerical solution of the boundary layer equations near the leading edge of a flat plate. Ziabakhsh et al.  studied the diffusion of a chemically reactive species into a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos  studied isothermal chemical reactions on laminar boundary layer flow. The two-dimensional stagnation-point flow near an infinite permeable wall with a homogeneous-heterogeneous reaction was studied by Khan and Pop , while Khan and Pop  and Bachok et al.  studied the effects of homogeneous-heterogeneous reactions on fluid flow due to a stretching sheet. Recently, the effects of homogeneous-heterogeneous reactions in nanofluid flow due to a porous stretching sheet were studied by Kameswaran et al. . They found that the concentration at the surface decreased with the strength of the heterogeneous reaction. In the case of a shrinking sheet, they showed that the velocity profiles decreased with increasing nanoparticle volume fraction in the case of a Cu-water nanofluid.
This article presents a study of homogeneous-heterogeneous reactions on MHD nanofluid stagnation point flow due to a stretching or shrinking sheet. The transformed nonlinear conservation equations are solved numerically.
2 Mathematical formulation
Here, the subscripts nf, f and s represent the thermophysical properties of the nanofluid, the base fluid and nanoparticles, respectively.
where , is the dimensionless stream function and .
where is the local Reynolds number.
3 Results and discussion
Thermophysical properties of water, Cu-water and Ag-water nanofluids, Oztop and Abu-Nada 
Comparison of obtained by Bachok et al.  with the present results for particular values of ,
Tables 2 and 3 give the coefficient for different parameter values. Table 2 gives a comparison of the present results with those obtained by Jafar et al.  and Wang  when , for different values of the stretching parameter. We observe that for increasing λ, the present results are in good agreement with results in the literature.
Table 4 gives the values of for selected λ when , . We note here that with decreasing λ the first solution decreases while the second solution increases. These results are in good agreement with the results obtained by Bachok et al.  in the absence of the particular physical parameter.
Figure 2 shows the effects of both stretching and shrinking on the velocity profiles in the case of a Cu-water nanofluid. We observe that in both cases, the velocity profiles increase with the parameter λ. Further, we note that for a shrinking sheet, the velocity in the case of a Cu-water nanofluid is larger than that of a clear fluid. The opposite is, however, true for the case of a stretching sheet. The momentum boundary layer thickness decreases as λ increases and the flow has an inverted boundary layer structure when . The findings in the case of a clear fluid are similar to the results obtained by Jat and Chaudhary .
The effects of homogeneous-heterogeneous reactions in MHD nanofluid flow due to a stretching or shrinking sheet have been studied. The transformed governing nonlinear differential equations have been solved numerically. Dual solutions for the velocity and concentration distributions have been obtained for some values of the stretching/shrinking, suction/injection and magnetic parameters. The effects of physical and fluid parameters on the velocity, concentration and skin friction have been analyzed. It was observed that for both the cases of shrinking and stretching sheets, the fluid velocity increased with the magnetic parameter. The concentration at the surface decreased as the strength of heterogeneous reactions increased for both Cu-water and Ag-water nanofluids. The boundary layer thickness for the first solution is always thinner than that for the second solution.
The authors are grateful to the University of KwaZulu-Natal for financial support.
- Crane LJ: Flow past a stretching plate. Z. Angew. Math. Phys. 1970, 21: 645-647. 10.1007/BF01587695View ArticleGoogle Scholar
- Tsou FK, Sparrow EM, Goldstein RJ: Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transf. 1967, 10: 219-235. 10.1016/0017-9310(67)90100-7View ArticleGoogle Scholar
- Erickson LE, Fan LT, Fox VG: Heat and mass transfer on a moving continuous flat plate with suction or injection. Ind. Eng. Chem. Fundam. 1966, 5: 19-25. 10.1021/i160017a004View ArticleGoogle Scholar
- Mucoglu A, Chen TS: Mixed convection on inclined surfaces. J. Heat Transf. 1979, 101: 422-426. 10.1115/1.3450992View ArticleGoogle Scholar
- Grubka LJ, Bobba KM: Heat transfer characteristics of a continuous, stretching surface with variable temperature. J. Heat Transf. 1985, 107: 248-250. 10.1115/1.3247387View ArticleGoogle Scholar
- Karwe MV, Jaluria Y: Fluid flow and mixed convection transport from a moving plate in rolling and extrusion processes. J. Heat Transf. 1988, 110: 655-661. 10.1115/1.3250542View ArticleGoogle Scholar
- Chen CH: Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transf. 1998, 33: 471-476. 10.1007/s002310050217View ArticleGoogle Scholar
- Abo-Eldahab EM, El-Aziz MA: Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. Int. J. Therm. Sci. 2004, 43: 709-719. 10.1016/j.ijthermalsci.2004.01.005View ArticleGoogle Scholar
- Salem AM, El-Aziz MA: Effect of Hall currents and chemical reaction on hydromagnetic flow of a stretching vertical surface with internal heat generation/absorption. Appl. Math. Model. 2008, 32: 1236-1254. 10.1016/j.apm.2007.03.008MathSciNetView ArticleMATHGoogle Scholar
- Ali ME: Heat transfer characteristics of a continuous stretching surface. Wärme- Stoffübertrag. 1994, 29: 227-234.View ArticleGoogle Scholar
- Ishak A, Nazar R, Pop I: Boundary layer flow and heat transfer over an unsteady stretching vertical surface. Meccanica 2009, 44: 369-375. 10.1007/s11012-008-9176-9MathSciNetView ArticleMATHGoogle Scholar
- Jafar K, Nazar R, Ishak A, Pop I: MHD flow and heat transfer over stretching/shrinking sheets with external magnetic field, viscous dissipation and joule effects. Can. J. Chem. Eng. 2012, 90: 1336-1346. 10.1002/cjce.20609View ArticleGoogle Scholar
- Lok YY, Ishak A, Pop I: MHD stagnation-point flow towards a shrinking sheet. Int. J. Numer. Methods Heat Fluid Flow 2011, 21: 61-72. 10.1108/09615531111095076View ArticleGoogle Scholar
- Bachok N, Ishak A, Pop I: Stagnation point flow over a stretching/shrinking sheet in a nanofluid. Nanoscale Res. Lett. 2011., 6: Article ID 623Google Scholar
- Narayana M, Sibanda P: Laminar flow of a nanoliquid film over an unsteady stretching sheet. Int. J. Heat Mass Transf. 2012, 55: 7552-7560. 10.1016/j.ijheatmasstransfer.2012.07.054View ArticleGoogle Scholar
- Kameswaran PK, Narayana M, Sibanda P, Murthy PVSN: Hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects. Int. J. Heat Mass Transf. 2012, 55: 7587-7595. 10.1016/j.ijheatmasstransfer.2012.07.065View ArticleGoogle Scholar
- Merkin JH: A model for isothermal homogeneous-heterogeneous reactions in boundary layer flow. Math. Comput. Model. 1996, 24: 125-136. 10.1016/0895-7177(96)00145-8MathSciNetView ArticleMATHGoogle Scholar
- Chaudhary MA, Merkin JH: A simple isothermal model for homogeneous-heterogeneous reactions in boundary layer flow. I. Equal diffusivities. Fluid Dyn. Res. 1995, 16: 311-333. 10.1016/0169-5983(95)00015-6MathSciNetView ArticleMATHGoogle Scholar
- Ziabakhsh Z, Domairry G, Bararnia H, Babazadeh H: Analytical solution of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. J. Taiwan Inst. Chem. Eng. 2010, 41: 22-28. 10.1016/j.jtice.2009.04.011View ArticleGoogle Scholar
- Chambre PL, Acrivos A: On chemical surface reactions in laminar boundary layer flows. J. Appl. Phys. 1956, 27: 1322-1328. 10.1063/1.1722258View ArticleGoogle Scholar
- Khan WA, Pop I: Flow near the two-dimensional stagnation-point on an infinite permeable wall with a homogeneous-heterogeneous reaction. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 3435-3443. 10.1016/j.cnsns.2009.12.022View ArticleGoogle Scholar
- Khan WA, Pop I: Effects of homogeneous-heterogeneous reactions on the viscoelastic fluid toward a stretching sheet. J. Heat Transf. 2012., 134: Article ID 064506Google Scholar
- Bachok N, Ishak A, Pop I: On the stagnation-point flow towards a stretching sheet with homogeneous-heterogeneous reactions effects. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 4296-4302. 10.1016/j.cnsns.2011.01.008View ArticleMATHGoogle 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. 2013, 57: 465-472. 10.1016/j.ijheatmasstransfer.2012.10.047View ArticleGoogle Scholar
- Tiwari RK, Das MK: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 2007, 50: 2002-2018. 10.1016/j.ijheatmasstransfer.2006.09.034View ArticleMATHGoogle Scholar
- Brinkman HC: The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 1952, 20: 571-581. 10.1063/1.1700493View ArticleGoogle Scholar
- Oztop HF, Abu-Nada E: Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int. J. Heat Fluid Flow 2008, 29: 1326-1336. 10.1016/j.ijheatfluidflow.2008.04.009View ArticleGoogle Scholar
- Wang CY: Stagnation flow towards a shrinking sheet. Int. J. Non-Linear Mech. 2008, 43: 377-382. 10.1016/j.ijnonlinmec.2007.12.021View ArticleGoogle Scholar
- Jat RN, Chaudhary S: MHD flow and heat transfer over a stretching sheet. Appl. Math. Sci. 2009, 3: 1285-1294.MathSciNetGoogle Scholar
- Bhattacharyya K: Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet. Int. Commun. Heat Mass Transf. 2011, 38: 917-922. 10.1016/j.icheatmasstransfer.2011.04.020View ArticleGoogle Scholar