## THERMAL SCIENCE

International Scientific Journal

### MHD MIXED CONVECTION SLIP FLOW NEAR A STAGNATION-POINT ON A NON-LINEARLY VERTICAL STRETCHING SHEET IN THE PRESENCE OF VISCOUS DISSIPATION

**ABSTRACT**

In this study, MHD mixed convection stagnation-point flow toward a non-linearly stretching vertical sheet in the presence of thermal radiation and viscous dissipation is numerically analyzed. The partial momentum and heat transfer equation are transformed into a set of ordinary differential equations by employing suitable similarity transformations. Using the Runge-Kutta Fehlberg fourth-fifth order method, numerical calculations to the desired level of accuracy are obtained for different values of dimensionless parameters. The results are presented graphically and in tabular form. The results for special cases are also compared to those obtained by other investigators and excellent agreements were observed. The effect of injection on the MHD mixed slip flow near a stagnation-point on a non-linearly vertical stretching sheet is to enhance the velocity field which results from the suppression of the skin friction on the wall surface. The heat transfer rate at the surface increases with increasing values of the non-linearity parameter. The velocity and thermal boundary-layer thicknesses are found to be decreasing with increasing values of the non-linearity parameter.

**KEYWORDS**

PAPER SUBMITTED: 2015-10-25

PAPER REVISED: 2015-12-10

PAPER ACCEPTED: 2015-12-21

PUBLISHED ONLINE: 2016-01-01

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**Issue 6**, PAGES [2731 - 2745]

- Hiemenz, K. K., Die Grenzschicht an einem in den gleichformingen Flussigkeitsstrom eingetauchten graden Kreiszylinder, Dinglers Polytechnic J., 326 (1911), pp. 321-324
- Sualia, M., et al., Unsteady stagnation point flow and heat transfer over a stretching/shrinking sheet with prescribed surface heat flux, App. Math. Comp. Int., Vol. 1 (2012), pp. 111
- Shateyi, S., Makinde, O.D., Hydromagnetic stagnation-point flow towards a radially stretching convectively heated disk, Math. Prob. Eng., Volume 2013, 8 pages, dx.doi.org/10.1155/2013/616947
- Hayat, T et al., MHD flow and heat transfer over permeable stretching sheet with slip conditions, Int. J. Numer. Meth. Fluids, 66 (2011), 8, pp. 963975
- Ali, F.M., et al., Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field. Appl. Math. Mech., 35 (2014), 2, pp. 155-166
- Pal,D., Mondal, H., Effects of temperature-dependent viscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretching sheet, J. Egyptian Math. Soc., 22 (2014), pp. 123-133
- Aman, F., et al., Magnetohydrodynamic stagnation-point flow towards a stretching/shrinking sheet with slip effects, Int. Commun. Heat . Mass Transf., 47 (2014), pp. 6872
- Sharma, R., et al., Stability analysis of magnetohydrodynamic stagnation point flow toward a stretching/shrinking sheet, Computers & Fluids 102 (2014), pp. 9498
- Ishak, A., et al., MHD stagnation point flow towards a stretching sheet, Physica A, 388 (2009), pp. 3377-3383
- Hsiao, K.L., Heat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), pp. 18031812
- Hsiao, K.L., MHD mixed convection for viscoelastic fluid past a porous wedge, Int. J. Non-Linear Mech., 46 (2011), pp. 1-8
- Hsiao, K.L., Nanofluid Flow with Multimedia Physical Features for Conjugate Mixed Convection and Radiation, Computers & Fluids, 104 (2014), pp. 1-8
- Vajravelu, K., Hdjinicolaou, A., Heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and internal heat generation, Int. Comm. Heat Mass Transf., 20 (1993), pp. 417-430
- Shen, M., et al., MHD mixed convection slip flow near a stagnation point on a nonlinearly vertical stretching sheet, Boundary Value Prob., DOI 10.1186/s13661-015-0340-6: (2015), pp.78
- Dessie,H., Kishan, K., MHD effects on heat transfer over stretching sheet embedded in porous medium with variable viscosity, viscous dissipation and heat source/sink, Ain Shains Eng. J. 5 (2014), pp. 967-977
- Jat, R. N., Chaudhary, S., Magnetohydrodynamic boundary layer flow near the stagnation point of a stretching sheet, IL NUOVO CIMENTO 123 (2008), pp. 555-566
- Abel, M.S., et al., MHD flow, and heat transfer with effects of buoyancy, viscous and joules dissipation over a nonlinear vertical stretching porous sheet with partial slip, Engineering, 3 (2011), pp. 285-291
- Ali, M., Mixed Convection boundary layer flows induced by a permeable continuous surface stretched with prescribed skin friction, World Academy of Science, Eng. Tech., 7 (2013), pp. 633-637
- Hossain, M.A., et al., The effect of radiation on free convection from a porous vertical plate, Int. J.Heat Mass Transf., 42 (1999), 1, pp. 181191
- Shateyi, S., Motsa, S. S., Thermal radiation effects on heat and mass transfer over an unsteady stretching surface, Math. Prob. Eng., Vol. 2009 (2009), 13 pages doi:10.1155/2009/965603
- Shateyi, S., Marewo, G. T., Numerical analysis of unsteady MHD flow near a stagnation-point of a two dimensional porous body with heat and mass transfer thermal radiation and chemical reaction, Boundary Value Prob., 2014 (2014), pp. 218
- Shateyi, S., Thermal radiation and buoyancy effects on heat and mass transfer over a semi-infinite stretching surface with suction and blowing, J. Appl. Math., Vol. 2008 (2008), 12 pages doi:10.1155/2008/414830
- Machireddy, G. R., Influence of thermal radiation, viscous dissipation and Hall current on MHD convection flow over a stretched vertical flat plate, Ain Shams Eng. J., 5 (2014), pp. 169175
- Ferdows, M., Md. Shakhaoath Khan., Md. Mahmud Alam and Shuyu Sun, MHD mixed convective boundary layer flow of a nanofluid through a porous medium due to an exponentially stretching sheet, Math. Prob. Eng., Vol. 3, (2012), 2551-2555
- Ferdows, M., Md. Shakhaoath Khan., O. Anwar Bég and M.M. Alam , Numerical study of transient magnetohydrodynamic radiative free convection nanofluid flow from a stretching permeable surface, J. Process Mech. Eng., (2013), 1-16
- Beg, O. A., Md. Shakhaoath Khan, Ifsana Karim, M.M. Alam, M. Ferdows., Explicit numerical study of unsteady hydromagnetic mixed convective nanofluid flow from an exponential stretching sheet in porous media, Appl. Nanosci., (2013), 1-15
- Wahiduzzaman, W., Md. Shakhaoath Khan, Ifsana Karim, P.Biswas, and M.S.Uddin., Viscous dissipation and radiation effects on MHD boundary layer flow of a nanofluid past a rotating stretching sheet, Appl. Math., Vol. 6, (2015) 547-567
- Wahiduzzaman, W., Md. Shakhaoath Khan, Ifsana Karim,., MHD convective stagnation flow of nanofluid over a shrinking surface with thermal radiation, heat generation and chemical reaction, Proce. Eng., Vol. 105, (2015), 398-405
- Shakhaoath, Md. K., I. Karim, Md. Sirajul Islam and M. Wahiduzzaman., MHD boundary layer radiative, heat generating and chemical reacting flow past a wedge moving in a nanofluid, Nano Converg., Vol. (2014), 1-13.
- Shakhaoath, Md., Md. Mahmud Alam and M. Ferdows., Effects of magnetic field on radiative flow of a nanofluid past a stretching sheet, Proce. Eng., Vol. 56, (2013) 316- 322
- Dinarvand, S., et al., Homotopy analysis method for mixed convection boundary layer flow of a nanofluid over a vertical circular cylinder, THERMAL SCIENCE, 19 (2015), 2, pp. 549-561
- Rashidi, M. M., et al., Mixed convection boundary layer flow of a micropolar fluid towards a heated shrinking sheet by homotopy analysis method, THERMAL SCIENCE, 2013, doi:10.2298/TSCI130212096R
- Andersson, H. I., Slip Flow Past a Stretching Surface, Acta Mechanica, Vol. 158, (2002),1-2, pp.121-125
- Chaudhary, S., Kumar, P., MHD Slip Flow past a Shrinking Sheet, Appl. Math., 4 (2013), pp. 574-581
- Abdel-Rahman, R. G., MHD Slip Flow of Newtonian fluid past a stretching sheet with thermal convective boundary condition, radiation, and chemical reaction, Math. Prob. Eng., Vol. 2013 (2013), 12 pages
- Sharma, R., et al., Boundary layer flow and heat transfer over a permeable exponentially shrinking sheet in the presence of thermal radiation and partial slip, J. Appl.Fluid Mech.,7 (2014), 1, pp. 125-134
- Mukhopadhyay, S., Analysis of boundary layer flow over a porous nonlinearly stretching sheet with partial slip at the boundary, Alexandria Eng. J., 52 (2013), pp. 563-569
- Wang, C. Y., Stagnation flow towards a shrinking sheet, Int. J. NonLinear Mech., 43 (2008), 5, pp. 377382
- Wong, S. W., et al., Stagnation-point flow toward a vertical, nonlinearly stretching sheet with prescribed surface heat flux. J. Appl. Math., Vol. 2013 (2013), 6 pages
- Hossain, M. A. et al., The effect of radiation on free convection flow of fluid with variable viscosity from a vertical porous plate, Int. J. Therm. Sci., 40 (2001), pp. 115-124
- Raptis, A., Flow of a micropolar fluid past a continuously moving plate by the presence of radiation, Int. J. Heat Mass Transf., 41 (1998), 18, pp. 28652866
- Faires, J. D., Burden, R. L., Numerical methods, Cengage Learning, fourth ed.,2012