## THERMAL SCIENCE

International Scientific Journal

### FLOW AND HEAT TRANSFER OVER A STRETCHING SURFACE WITH VARIABLE THICKNESS IN A MAXWELL FLUID AND POROUS MEDIUM WITH RADIATION

**ABSTRACT**

The effect of thermal radiation on flow and heat transfer of Maxwell fluid over a stretching surface with variable thickness embedded in a porous medium is considered. The governing non-linear PDE are transformed into a non-linear ODE by using a similarity transformation. These equations were solved numerically with fourth/fifth-order Runge-Kutta method. A comparison of obtained numerical results is made with the previously results in some special cases and excellent agreement is noted. The effects of elasticity, radiation parameter, porosity parameter, wall thickness parameter, and thermal conductivity parameter on the velocity and temperature profiles are presented. Moreover, the skin-friction and Nusselt number are presented.

**KEYWORDS**

PAPER SUBMITTED: 2017-02-28

PAPER REVISED: 2018-02-04

PAPER ACCEPTED: 2018-05-02

PUBLISHED ONLINE: 2018-05-13

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**Issue 5**, PAGES [3105 - 3116]

- B. C. Sakiadis, "Boundary Layer Behavior on Continuous Solid Surfaces: II Boundary Layer on a Continuous Flat Surface," American Institute of Chemical Engineers Journal, Vol. 7, No. 2, 1961, pp. 221-225. doi:10.1002/aic.690070211
- Crane, L. J. (1970). Flow past a stretching plate, Z. Angew. Math. Phys., 21(1970), pp. 645-647.
- E.M.A. Elbashbeshy, and M.A.A. Bazid, Heat transfer over a continuously moving plate embedded in non-Darcian porous medium, International Journal of Heat and Mass Transfer, 43(2000), pp. 3087-3092.
- M. Anwar Hossain, et al. , The effect of radiation on free convection flow of fluid with variable viscosity from a porous vertical plate, Int. J. Therm. Sci., 40 (2001), pp. 115-124.
- Rafael Cortell, Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing, Fluid Dynamics Research, 37(2005), pp. 231-245.
- Z. M. Yusof, et al. , Radiation Effect on Unsteady MHD Flow over a Stretching Surface, International Journal of Computer, Electrical, Automation, Control and Information Engineering, 6(2012),12.
- Elsayed M.A. Elbashbeshy, et al. , Effects of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over an exponentially stretching surface with suction in the presence of internal heat generation/absorption, Journal of the Egyptian Mathematical Society, 20(2012), pp. 215-222.
- I. C. Mandal, Swati M., Heat transfer analysis for fluid flow over an exponentially stretching porous sheet with surface heat flux in porous medium, Ain Shams Engineering Journal, 4(2013), pp. 103-110.
- E. M. A. Elbashbeshy, et al. , Unsteady Flow of Micropolar Maxwell Fluid over Strtetching Surface in The Presence of Magnetic Field, International Journal of Electronic Emgineering and Computer Science , 2(2017), 4, pp. 28-34.
- Kai-Long Hsiao, Combined Electrical MHD Heat Transfer Thermal Extrusion System Using Maxwell Fluid with Radiative and Viscous Dissipation Effects, Applied Thermal Engineering, DOI: 10.1016/j.applthermaleng.2016.08.208, (2016)
- Kai-Long Hsiao, To Promote Radiation Electrical MHD Activation Energy Thermal Extrusion Manufacturing System Efficiency by Using Carreau-Nanofluid with Parameters Control Method, Energy, 130 (2017), pp. 486-499
- S. A. Shehzad, et al. , Radiative Maxwell Fluid Flow with Variable Thermal Conductivity due to a Stretching Surface in a Porous Medium, J. Aerosp. Eng., 27(2014).
- Shimaa E. Waheed, Flow and heat transfer in a Maxwell liquid film over an unsteady stretching sheet in a porous medium with radiation, Waheed SpringerPlus, 5(2016):1061.
- N. F. M. Noor, et al. , Heat flux performance in a porous medium embedded Maxwell fluid flow over a vertically stretched plate due to heat absorption, J. Nonlinear Sci. Appl., 9(2016), pp. 2986-3001.
- Kai-Long Hsiao, Micropolar Nanofluid Flow with MHD and Viscous Dissipation Effects Towards a Stretching Sheet with Multimedia Feature, International Journal of Heat and Mass Transfer, 112 (2017), pp. 983-990
- Kai-Long Hsiao, Stagnation Electrical MHD Nanofluid Mixed Convection with Slip Boundary on a Stretching Sheet, Applied Thermal Engineering, 98 (2016), pp. 850-861, doi:10.1016/j.applthermaleng.2015.12.138.
- Kai-Long Hsiao, Corrigendum to ‘‘Stagnation Electrical MHD Nanofluid Mixed Convection with Slip Boundary on a Stretching Sheet''
- T. Fang, et al. , Boundary layer flow over a stretching sheet with variable thickness, Applied Mathematics and Computation, 218(2012), pp. 7241-7252.
- M.M. Khader and A. M. Megahed, Numerical solution for boundary layer flow due to a nonlinearly stretching sheet with variable thickness and slip velocity, Eur. Phys. J. Plus, pp.128-100, 2013.
- M. Khader, et al. , Numerical Study for Simulation the MHD Flow and Heat-Transfer Due to a Stretching Sheet on Variable Thickness and Thermal Conductivity with Thermal Radiation, Applied Mathematics, 6(2015), pp. 2045-2056.
- A. Eid and M. Mohamed Khader, Numerical studies using finite different method for viscous dissipation and thermal radiation effects on the slip flow and heat transfer due to a stretching sheet embedded in a porous medium with variable thickness and variable thermal conductivity, NTMSCI, 4(2016), No. 1, pp. 38-50.
- E.M.A. Elbashbeshy, et al., Flow and heat transfer over a moving surface with nonlinear velocity and variable thickness in a Nanofluid in the presence of thermal radiation, Can. J. Phys. 92(2014), pp. 124-130.
- M.S. Abdel-wahed, et al., Flow and heat transfer over a moving surface with non-linear velocity and variable thickness in a nanofluids in the presence of Brownian motion, Applied Mathematics and Computation, 254(2015), pp. 49-62.
- C. Sulochana and N. Sandeep, Dual Solutions for Radiation MHD Forced Convective Flow of A Nanofluid over A slandering Stretching Sheet in Porous Medium, Journal of Naval Architecture and Marine Engineering, Journal of Naval Architecture and Marine Engineering, 12 (2015), pp. 115-124.
- T. Hayat, et al. , Impact of Cattaneo-Christov heat flux in the flow over a stretching sheet with variable thickness, AIP ADVANCES, 5(2015), 087159.
- T.Salahuddin, et al., MHD flow of Cattanneo-Christovheat flux modelfor Williamson fluid over astretching sheet with variable thickness :Using numerical approach, Journal of Magnetism and Magnetic Materials , 401(2016), pp. 991-997.
- Raptis A (1998) Flow of a micropolar fluid past a continuously moving plate by the presence of radiation. Int J Heat Mass Transf 41:2865-2866
- Raptis A (1999) Radiation and viscoelastic flow. Int Commun Heat Mass Transf 26:889-895
- M. M. Khader and A. M. Megahed, Approximate Solutions for the Flow and Heat Transfer due to a Stretching Sheet Embedded in a Porous Medium with Variable Thickness, Variable Thermal Conductivity and Thermal Radiation using Laguerre Collocation Method, Applications and Applied Mathematics: An International Journal, 10(2016), 2, pp. 817 - 834.
- E. M. A. Elbashbeshy, et al. , Effect of thermal radiation on flow, heat, and mass transfer of a Nanofluid over a stretching horizontal cylinder embedded in a porous medium with suction/injection, Journal of Porous Media, 18(2015), 3, pp. 215-229.
- F.Mabood, et al., MHD boundarylayer flow andheat transfer ofnanofluids over a nonlinear stretching sheet :a numerical study, J. Magn. Magn. Mater., 374(2015), pp. 569-576.
- C.Y.Wang, Free convection on a vertical stretching surface with suction and blowing, Appl. Math. Mech., 69(1989), pp. 418-420.