## THERMAL SCIENCE

International Scientific Journal

### ANALYSIS OF HEAT TRANSFER OF HYDROMAGNETIC FLOW OVER A CURVED GENERALIZED STRETCHING OR SHRINKING SURFACE WITH CONVECTIVE BOUNDARY CONDITION

**ABSTRACT**

An investigation is carried out to discuss the heat transfer mechanism to an electrically conducting viscous fluid on a curved stretching/shrinking surface incorporated with convective boundary condition. The impact of uniform magnetic field is also considered. The mathematical formulation for the transport of heat and flow phenomena is developed by utilizing a curvilinear coordinates system. The obtained sets of partial differential equations are reconstructed into coupled nonlinear differential equations by incorporating similarity transformations. The numerical solution is attained by employing the shooting method. The obtained solutions are then used to discuss the impacts of various emerging parameters on the temperature and heat transfer across the surface. Dual nature of the solutions are obtained for definite range of convective, suction, magnetic, Prandtl and stretching or shrinking parameters. Comparison of the obtained results with the existing results for a flat sheet is found in acceptable agreement. It is noticed that with an increment in convective parameter increases the temperature of the fluid, while an increase in suction and magnetic parameters decreases the temperature of the fluid for both the solutions.

**KEYWORDS**

PAPER SUBMITTED: 2017-08-18

PAPER REVISED: 1970-01-01

PAPER ACCEPTED: 2018-07-23

PUBLISHED ONLINE: 2018-09-22

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**6**, PAGES [3775 - 3783]

- Karwe, M. V. and Jaluria, Y., Numerical solution of thermal transport associated with a continuous moving flat sheet in materials processing, J. Heat Transf. 113 (1991), pp. 612-619.
- Incropera, F. P., Fundamental of heat and mass transfer, 6th ed. John Wiley and Son, New York. (2007).
- Crane, L. J., Flow past a stretching plate, Zeits. Ange. Mathe. Phys. (ZAMP). 21 (1970), pp. 645-647.
- Miklavcic, M. and Wang, C. Y., Viscous flow due to a shrinking sheet, Quar. Appl. Mathe. 64 (2006), pp. 283-290.
- Bhattacharya, K., Effect of heat source/sink on MHD flow and heat transfer over a shrinking sheet with mass suction, Chem. Eng. Res. Bul. 15 (2011), pp. 12-17.
- Bejan, A., Convective Heat Transfer, 3rd ed. John Wiley and Son, New York. pp. 51-54 (2004).
- Aziz, A., A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition, Commun. Nonlinear Sci. Numer. Simul.14 (2009), pp. 1064-1068.
- Ishak, A., Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition, Appl. Math. Compu. 217 (2010), pp. 837-842.
- Sheikholeslami, M., Darzi, M. and Sadoughi, M., Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid; an experimental procedure, Int. J. Heat Mass Transf. 122 (2018), pp. 643-650.
- Sheikholeslami, M. and Shehzad, S. A., Simulation of water based nanofluid convective flow inside a porous enclosure via non-equilibrium model, Int. J. Heat Mass Transf. 120 (2018), pp. 1200-1212.
- Sheikholeslami, M. and Seyednezhad, M., Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM, Int. J. Heat Mass Transf. 120 (2018), pp. 772-781.
- Sheikholeslami, M., Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Molec. Liqud. 249 (2018), pp. 1212-1221.
- Sheikholeslami, M., Shamlooei, M. and Moradi, R., 3 4 Fe O - Ethylene glycol nanofluid forced convection inside a porous enclosure in existence of Coulomb force, J. Molec. Liqud. 249 (2018), pp. 429-437.
- 18] Abbas, Z., M, Rafiq and Naveed, M., Analysis of Eyring-Powell liquid flow in curved channel with Cattaneo-Christov heat flux model, Braz. J. Mech. Sci. Eng. (2018).
- Sheikholeslami, M. and Rokni, H. B., Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation, Int. J. Heat Mass Transf. 118 (2018), pp. 823-831.
- Sheikholeslami, M. and Sadoughi, M. K., Simulation of CuO - water nanofluid heat transfer enhancement in presence of melting surface, Int. J. Heat Mass Transf. 116 (2018), pp. 909-919.
- Makinde, O. A. and Aziz, A., Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, Int. J. Therm. Sci. 50 (2011), pp. 1326-1332.
- Naveed, M., Abbas, Z. and Sajid, M., Thermophoresis and Brownian effects on the Blasius flow of a nanofluid due to curved surface with thermal radiation, Euro. Phy. J. Plus (2016) 131:214.
- Naveed, M., Abbas, Z. and Sajid, M.,Nonlinear radiative heat transfer in Blasius and Sakiadis flows over a curved surface, Int. J. ThermoPhysics (2017) 38:14.
- Naveed, M., Abbas, Z. and Sajid, M., Effect of Homogeneous-heterogeneous reactions and magnetohydrodynamics on nanofluid for the Blasius flow with thermal radiation, J. Molec. Liq. 233 (2017), pp. 115-121.
- Jafar, K., Nazar, R., Ishak, A., and Pop, I., MHD flow and heat transfer over stretching/shrinking sheet with external magnetic field, viscous dissipation and Joule effects. Canad, J. Chem. Eng., 999 (2011), pp. 1-11.
- Kameswaran, P. K., Narayana, M., Sibanda, P., and Murthy, P. V. S. N., Hydromagnetic nanofluid due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects, Int. J. Heat Mass Trans. 55 (2012), pp. 7587-7595.
- Rosca, A., MHD boundary layer flow over a permeable shrinking surface, Acta Uni. Apul. 36 (2013), pp. 31-38.
- Mishra, S. R., and Jena, S., Numerical solution of boundary layer MHD flow with viscous dissipaton, Sci. World J. 756498 (2014).
- Sheikholeslami, M. and Rokni, H. B., CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of non-equilibrium model, J. Molec. Liqud. 254 (2018), pp. 446-462.
- Sheikholeslami, M. and Rokni, H. B., Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Physics of Fluids, 30 (1) (2018).
- Sheikholeslami, M., Numerical investigation for CuO-H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method, , J. Molec. Liqud. 249 (2018), pp. 739-746.
- Sheikholeslami, M., CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion, J. Molec. Liqud. 249 (2018), pp. 921-929.
- Sheikholeslami, M. and Shehzad, S. A., Numerical analysis of 3 4 2 Fe O −H O nanofluid flow in permeable media under the effect of external magnetic source, Int. J. Heat Mass Transf. 118 (2018), pp. 182-189.
- Sheikholeslami, M. and Rokni, H. B., Simulation of nanofluid heat transfer in presence of magnetic field: A review, Int. J. Heat Mass Transf. 115 (2017), pp. 1203-1233.
- Sheikholeslami, M., Influence of magnetic field on nanofluid free convection in an open porous cavity by means of Lattice Boltzmann method, J. Molec. Liqud. 234 (2017), pp. 364-374.
- Sajid, M., Ali, N., Javed, T., and Abbas, Z., Stretching a curved surface in a viscous fluid, Chin. Phy. Lett. 27 (2010), 024703 .
- Abbas, Z., Naveed, M., and Sajid, M., Heat transfer analysis for stretching flow over curved surface with magnetic field, J. Eng. Therm. 22 (2013), pp.337-345.
- Naveed, M., Abbas, Z., and Sajid, M., MHD flow of a micropolar fluid due to a curved stretching sheet with thermal radaition, J. Appl. Fluid Mech. 9 (2013), pp. 131-138.
- Abbas, Z., Naveed, M., and Sajid, M., Hydromagnetic slip flow of nanofluid over a curved stretching surface with heat generation and thermal radiation, J. Mole. Liq. 215 (2016), pp. 756-762.
- Khan, W. A., and Pop, I., Boundary layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Trans. 53 (2010), pp. 2477-2483.
- Wang, C. Y., Free convection on a vertical stretching surface, J. Appl. Math. Mech. (ZAMM). 69 (1989), pp. 418-420.