Sains Malaysiana 43(8)(2014): 1249-1257

Scaling Transformation for Free Convection Flow of a Micropolar Fluid along a Moving Vertical Plate in a Porous Medium with Velocity and Thermal Slip Boundary Conditions

(Transformasi Penskalaan untuk Aliran Olakan Bebas Bendalir Mikropolar Sepanjang Plat Menegak dalam Medium Berliang dengan Syarat Sempadan Slip Halaju dan Haba)

A.A. Mutlag^1,2*, Md. Jashim Uddin^1,2 & Ahmad Izani Md. Ismail¹

¹School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

²Mathematics Department, College of Education for Pure Science, AL- Anbar University, ALAnbar Iraq

³Mathematics Department, American International University-Bangladesh, Banani Dhaka, 1213

Bangladesh

Received: 11 January 2013/Accepted: 9 December 2013

Abstract

We study and discuss the effect of thermal slip on steady free convection flow of a viscous, incompressible micropolar fluid past a vertical moving plate in a saturated porous medium. The effect of viscous dissipation is incorporated in the energy equation. The associated partial differential equations are transformed into a system of ordinary differential equations using similarity transformations generated by a group method and this system is then solved numerically. The effect of controlling parameters on the dimensionless velocity, angular velocity and temperature as well as friction factor, couple stress factor and heat transfer rate are shown graphically and discussed in detail. It is found that the dimensional velocity and angular velocity decrease whilst the temperature increases with velocity slip parameter. It is further found that thermal slip decreases the dimensional velocity and temperature but increases the dimensional angular velocity. Data from published work and our results are found to be in good agreement.

Keywords: Free convection; micropolar fluid; moving plate; porous medium; scaling group; velocity and thermal slip boundary conditions

Abstrak

Kami mengkaji dan membincangkan kesan slip haba pada aliran olakan bebas tak mampat bendalir mikro-kutub likat yang mantap melalui plat menegak yang bergerak dalam medium berliang tepu. Kesan penghambaran likat dimasuk ke dalam persamaan tenaga. Persamaan pembezaan separa bersekutu dijelmakan kepada sistem persamaan pembezaan biasa menggunakan penjelmaan keserupaan yang dijanakan menggunakan kaedah kumpulan dan sistem ini kemudiannya diselesaikan secara berangka. Kesan parameter pengawal halaju tak berdimensi, halaju sudut dan suhu serta faktor geseran, faktor regangan pasangan dan kadar pemindahan haba ditunjukkan secara grafik dan dibincangkan secara terperinci. Didapati bahawa halaju berdimensi dan halaju sudut menurun manakala suhu naik dengan slip halaju. Adalah selanjutnya didapati bahawa slip haba menurun halaju berdimensi dan suhu tetapi meningkatkan halaju sudut berdimensi. Data daripada kerja yang telah diterbitkan dan keputusan kami didapati mempunyai persamaan.

Kata kunci: Bendalir mikropolar; kumpulan penskalaan; medium berliang; perolakan bebas; plat bergerak; syarat sempadan slip termal halaju dan termal

References

Ahmad, K., Nazar, R. & Pop, I. 2012. Mixed convection in laminar film flow of a micropolar fluid. International Communications in Heat and Mass Transfer 39: 36-39.

Ali, A., Amin, N. & Pop, I. 2010. Unsteady mixed convection boundary layer from a circular cylinder in a micropolar fluid. International Journal of Chemical Engineering 2010: 417875.

Beg, O.A., Ramachandra Prasad, V., Vasu, B., Bhaskar Reddy, N., Li, Q. & Bhargava, R. 2011. Free convection heat and mass transfer from an isothermal sphere to a micropolar regime with Soret/Dufour effects. International Journal of Heat and Mass Transfer 54: 9-18.

Bejan, A. & Khair, K.R. 1985. Heat and mass transfer by natural convection in a porous medium. International Journal of Heat and Mass Transfer  28: 909-918. 

Bhattacharyya, K., Mukhopadhyay, S. & Layek, G.C. 2011. Steady boundary layer slip flow and heat transfer over a flat porous plate embedded in a porous media.  Journal of Petroleum Science and Engineering 78: 304-309. 

Butcher, J.C. 2008. Numerical Methods for Ordinary Differential Equations. England. John Wiley & Sons, Ltd.

Cheng, C. 2011. Natural convection boundary layer flow of a micropolar fluid over a vertical permeable cone with variable wall temperature. International Communications in Heat and Mass Transfer 38: 429-433.

El-Aziz, M.A. 2013. Mixed convection flow of a micropolar fluid from an unsteady stretching surface with viscous dissipation. Journal of the Egyptian Mathematical Society 21: 385-394.

Eringen, A.C. 1966. Theory of micropolar fluids. Journal of Mathematics and Mechanics 16: 1-18.

Eringen, A.C. 1972. Theory of thermomicropolar fluids. Journal of Mathematical Analysis and Applications 38: 480-496.

Hayat, T., Javed, T. & Abbas, Z. 2009. MHD flow of a micropolar fluid near a stagnation-point towards a non-linear stretching surface. Nonlinear Analysis: Real World Applications 10: 1514-1526.

Ingham, D.B. & Pop, I. 2005. Transport Phenomena in Porous Media. 3rd ed. Oxford, UK: Elsevier.

Ishak, A., Nazar, R. & Pop, I. 2006. Flow of a micropolar fluid on a continuous moving surface. Archives of Mechanics 58: 529-541.

Lukaszewicz‏, G. 1999. Micropolar Fluids: Theory and Applications. New York: Springer.

Mukhopadhyay, S. 2011. Effects of slip on unsteady mixed convective flow and heat transfer past a porous stretching surface. Nuclear Engineering and Design 241: 2660-2665.  

Mutlag, A.A., Uddin, M.J., Hamad, M.A.A. & Ismail, A.I.M. 2013. Heat transfer analysis for falkner-skan boundary layer flow past a stationary wedge with slips boundary conditions considering temperature-dependent thermal conductivity. Sains Malaysiana 42: 855-862.

Nadeem, S., Hussain, M. & Naz, M. 2010.  MHD stagnation flow of a micropolar fluid through a porous medium. Meccanica 45: 869-880.

Nield, D.A. & Bejan, A. 2006. Convection in Porous Media. 3rd ed. New York: Springer.

Rahman, M.M., Aziz, A. & Al-Lawatia, M.A. 2010. Heat transfer in micropolar fluid along an inclined permeable plate with variable fluid properties. International Journal of Thermal Sciences 49: 993-1002.

Reena & Rana, U.S. 2009. Effect of dust particles on rotating micropolar fluid heated from below saturating a porous medium. Applications and Applied Mathematics 4: 189-217.

Rosali, H., Ishak, A. & Pop, I. 2012. Micropolar fluid flow towards stretching/shrinking sheet in a porous medium with suction.  International Communications in Heat and Mass Transfer 39: 826-829.

Sajid, M., Ali, N. & Hayat, T., 2009. On exact solutions for thin film flows of a micropolar fluid.  Communications in Nonlinear Science and Numerical Simulation 14: 451-461.

Trevisan, O.V. & Bejan, A. 1990. Combined heat and mass transfer by natural convection in a porous medium. Advances in Heat Transfer 20: 315-352. 

Uddin, M.J., Hamad, M.A.A. & Ismail, A.I.M. 2012a. Investigation of heat mass transfer for combined convective slips flow: A lie group analysis.  Sains Malaysiana 41: 1145-1155.

Uddin, M.J., Khan, W.A. & Ismail, A.I.M. 2012b. MHD free convective boundary layer flow of a nanofluid past a flat vertical plate with Newtonian heating boundary condition. Plos One 7: e49499.

*Corresponding author; email: alassafi2005@yahoo.com