Sains Malaysiana 47(11)(2018): 2907–2916

http://dx.doi.org/10.17576/jsm-2018-4711-34

 

MHD Stagnation-Point Flow over a Stretching/Shrinking Sheet in a Micropolar Fluid with a Slip Boundary

(Aliran Titik Genangan MHD terhadap Lembaran Meregang/Mengecut dalam Bendalir Mikrokutub dengan Gelinciran Sempadan)

 

SITI KHUZAIMAH SOID¹, ANUAR ISHAK^2* & IOAN POP³

 

¹Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 UiTM Shah Alam, Selangor Darul Ehsan, Malaysia

 

²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

 

³Department of Mathematics, Babeș-Bolyai University, 400084 Cluj-Napoca, Romania

 

Diserahkan: 15 Januari 2018/Diterima: 20 Julai 2018

 

ABSTRACT

The problem of stagnation point flow over a stretching/shrinking sheet immersed in a micropolar fluid is analyzed numerically. The governing partial differential equations are transformed into a system of ordinary (similarity) differential equation and are then solved numerically using the boundary value problem solver (bvp4c) in Matlab software. The effects of various parameters on the velocity and the angular velocity as well as the skin friction coefficient and the couple stress are shown in tables and graphs. The noticeable results are found that the micropolar and the slip parameters decrease the skin friction coefficient and the couple stress in the existence of magnetic field. Dual solutions appear for certain range of the shrinking strength. A stability analysis is performed to determine which one of the solutions is stable. Practical applications include polymer extrusion, where one deals with stretching of plastic sheets and in metallurgy that involves the cooling of continuous strips.

 

Keywords: Micropolar fluid; stability analysis; stagnation point flow; stretching/shrinking

 

ABSTRAK

Masalah aliran titik genangan di atas lembaran meregang/mengecut yang direndam dalam bendalir mikrokutub dianalisis secara berangka. Persamaan menakluk pembezaan separa dijelmakan menjadi sistem persamaan pembezaan biasa (keserupaan) dan kemudian diselesaikan secara berangka menggunakan penyelesai masalah nilai sempadan (bvp4c) dalam perisian Matlab. Kesan pelbagai parameter terhadap profil halaju dan halaju sudut serta pekali geseran kulit dan tegasan kupel ditunjukkan dalam jadual dan graf. Hasil yang ketara didapati bahawa parameter mikrokutub dan gelinciran menurunkan pekali geseran kulit dan tegasan kupel dengan kehadiran medan magnet. Penyelesaian dual diperoleh untuk kekuatan tertentu regangan. Analisis kestabilan dilakukan untuk menentukan penyelesaian mana yang stabil antara dua penyelesaian yang diperoleh ini. Aplikasi praktikal termasuk penyemperitan polimer dengan peregangan lembaran plastik dan metalurgi yang melibatkan penyejukan jalur berterusan.

 

Kata kunci: Aliran titik genangan; analisis kestabilan; bendalir mikrokutub; meregang/mengecut

RUJUKAN

Abbas, N., Saleem, S., Nadeem, S., Alderremy, A.A. & Khan, A.U. 2018. On stagnation point flow of a micro polar nanofluid past a circular cylinder with velocity and thermal slip. Result in Physics 9: 1224-1232.

Ahmadi, G. 1976. Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate. International Journal of Engineering Science 14: 639-646.

Aman, F., Ishak, A. & Pop, I. 2013a. MHD stagnation point flow of a micropolar fluid toward a vertical plate with a convective surface boundary condition. Bulletin of the Malaysian Mathematical Sciences Society 36: 865-879.

Aman, F., Ishak, A. & Pop, I. 2013b. Magnetohydrodynamic stagnation-point flow towards a stretching/shrinking sheet with slip effects. International Communications in Heat and Mass Transfer 47: 68-72.

Andersson, H.I. 1995. An exact solution of the Navier-Stokes equations for magnetohydrodynamic flow. Acta Mechanica 113: 241-244.

Awaludin, I.S., Weidman, P.D. & Ishak, A. 2016. Stability analysis of stagnation-point flow over a stretching/shrinking sheet. AIP Advances 6: 045308.

Basir, M.F.M., Uddin, M.J. & Ismail, A.I.M. 2017. Unsteady magnetoconvective flow of bionanofluid with zero mass flux boundary condition. Sains Malaysiana 46(2): 327-333.

Borrelli, A., Giantesio, G. & Patria, M.C. 2015. MHD orthogonal stagnation-point flow of a micropolar fluid with the magnetic field parallel to the velocity at infinity. Applied Mathematics and Computation 264: 44-60.

Chakrabarti, A. & Gupta, A.S. 1979. Hydromagnetic flow and heat transfer over a stretching sheet. Quarterly of Applied Mathematics 37: 73-78.

Chiam, T.C. 1995. Hydromagnetic flow over a surface stretching with a power-law velocity. International Journal of Engineering Science 33: 429-435.

Das, K. 2012. Slip effects on MHD mixed convection stagnation point flow of a micropolar fluid towards a shrinking vertical sheet. Computers & Mathematics with Applications 63: 255-267.

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

Fisher, E.G. 1976. Extrusion of Plastics. New York: Wiley.

Hafidzuddin, E.H., Nazar, R., Arifin, N.M. & Pop, I. 2015. Numerical solutions of boundary layer flow over an exponentially stretching/shrinking sheet with generalized slip velocity. International Journal of Mathematical, Computation, Physical, Electrical and Computer Engineering 9(4): 240-245.

Hiemenz, K. 1911. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers Polytech Journal 326: 321-324.

Imran, M.A., Riaz, M.B., Shah, N.A. & Zafar, A.A. 2018. Boundary layer flow of MHD generalized Maxwell fluid over an exponentially accelerated infinite vertical surface with slip and Newtonian heating at the boundary. Results in Physics 8: 1061-1067.

Ishak, A. 2011. MHD boundary layer flow due to an exponentially stretching sheet with radiation effect. Sains Malaysiana 40(4): 391-395.

Ishak, A., Lok, Y.Y. & Pop, I. 2010. Stagnation-point flow over a shrinking sheet in a micropolar fluid. Chemical Engineering Communications 197: 1417-1427.

Ishak, A., Nazar, R. & Pop, I. 2009. Dual solutions in mixed convection boundary layer flow of micropolar fluids. Communications in Nonlinear Science and Numerical Simulation 14: 1324-1333.

Ishak, A., Nazar, R. & Pop, I. 2008. MHD boundary-layer flow due to a moving extensible surface. Journal of Engineering Mathematics 62: 23-33.

Jena, S.K. & Mathur, M.N. 1981. Similarity solutions for laminar free convection flow of a thermomicropolar fluid past a non-isothermal vertical flat plate. International Journal of Engineering Science 19: 1431-1439.

Peddieson, J.J. 1972. An application of the micropolar fluid model to the calculation of a turbulent shear flow. International Journal of Engineering Science 10: 23-32.

Lok, Y.Y., Amin, N., Campean, D. & Pop, I. 2005. Steady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface. International Journal of Numerical Methods for Heat & Fluid Flow 15(7): 654-670.

Malvandi, A., Hedayati, F. & Ganji, D.D. 2014. Slip effects on unsteady stagnation point flow of a nanofluid over a stretching sheet. Powder Technology 253: 377-384.

Mat Yasin, M.H., Ishak, A. & Pop, I. 2017. Boundary layer flow and heat transfer past a permeable shrinking surface embedded in a porous medium with a second-order slip: A stability analysis. Applied Thermal Engineering 115: 1407-1411.

Meeten, G.H. 2004. Squeeze flow of soft solids between rough surfaces. Rheology Acta 43: 6-16.

Merkin, J.H. 1985. On dual solutions occurring in mixed convection in a porous medium. Journal of Engineering Mathematics 20: 171-179.

Mooney, M. 1931. Explicit formulas for slip and fluidity. Journal of Rheology (1929-1932) 2(2) : 210-222.

Nair, K.A. & Sameen, A. 2015. Experimental study of slip flow at the fluid-porous interface in a boundary layer flow. Procedia IUTAM 15: 293-299.

Najib, N., Bachok, N., Arifin, N.M. & Ishak, A. 2014. Stagnation point flow and mass transfer with chemical reaction past a stretching/shrinking cylinder. Scientific Report 4(4178): 1-7.

Nazar, R., Amin, N., Filip, D. & Pop, I. 2004. Stagnation point flow of a micropolar fluid towards a stretching sheet. International Journal of Non-Linear Mechanics 39: 1227- 1235.

Papautsky, I., Brazzle, J., Ameel, T. & Frazier, A.B. 1999. Laminar fluid behavior in microchannels using micropolar fluid theory. Sensors and Actuators A: Physical 73(1-2) : 101-108.

Rahmati, A.R., Akbari, O.A., Marzban, A., Toghraie, D., Karimi, R. & Pourfattah, F. 2018. Simultaneous investigations the effects of non-Newtonian nanofluid flow in different volume fractions of solid nanoparticles with slip and no-slip boundary conditions. Thermal Science and Engineering Progress 5: 263-277.

Roşca, N.C. & Pop, I. 2013. Mixed convection stagnation point flow past a vertical flat plate with a second order slip: Heat flux case. International Journal of Heat and Mass Transfer 65: 102-109.

Sajid, M., Mahmood, K. & Abbas, Z. 2012. Axisymmetric stagnation-point flow with a general slip boundary condition over a lubricated surface. Chinese Physics Letters 29(2): 0247021-0247024.

Shercliff, J.A. 1965. A Textbook of Magnetohydrodynamics. Oxford: Pergamon Press.

Shu, J.J. & Lee, J.S. 2008. Fundamental solutions for micropolar fluids. Journal of Engineering Mathematics 61(1) : 69-79.

Tretheway, D.C. & Meinhart, C.D. 2002. Apparent fluid slip at hydrophobic microchannel walls. Physics of Fluids 14(3) : L9-L12.

Uddin, M.J., Alginahi, Y., Beg, O.A. & Kabir, M.N. 2016. Numerical solutions for gyrotactic bioconvection in nanofluid-saturated porous media with Stefan blowing and multiple slip effects. Computers and Mathematics with Applications 72(10): 2562-2581.

Wang C.Y. 2008. Stagnation flow towards a shrinking sheet. International Journal of Non-Linear Mechanics 43: 377-382.

Wang, C.Y. 2003. Stagnation flows with slip: Exact solutions of the Navier-Stokes equations. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 54(1): 184-189.

Weidman, P. 2016. Axisymmetric rotational stagnation point flow impinging on a radially stretching sheet. International Journal of Non-Linear Mechanics 82: 1-5.

Yacob, N.A., Ishak, A. & Pop, I. 2011. Melting heat transfer in boundary layer stagnation-point flow towards a stretching/ shrinking sheet in a micropolar fluid. Computers and Fluids 47(1): 16-21.

Yoshimura, A. & Prud’homme, R.K. 1988. Wall slip corrections for couette and parallel disk viscometers. Journal of Rheology 32: 53-67.

Zaimi, K., Ishak, A. & Pop, I. 2014. Boundary layer flow and heat transfer over a nonlinearly permeable stretching/shrinking sheet in a nanofluid. Scientific Reports 4(4404): 1-8.

 

*Pengarang untuk surat-menyurat; email: anuar_mi@ukm.edu.my