Analysis of Casson Fluid Flow over a Vertical Porous Surface with Chemical Reaction in the Presence of Magnetic Field
Affiliation(s)
1 Department of Mathematics, University for Development Studies, Navrongo, Ghana. 2 Department of Physics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana.
1 Department of Mathematics, University for Development Studies, Navrongo, Ghana. 2 Department of Physics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana.
ABSTRACT
Casson fluid flow over a vertical porous surface with chemical reaction in the presence of magnetic field has been studied. A similarity analysis was used to transform the system of partial differential equations describing the problem into ordinary differential equations. The reduced system was solved using the Newton Raphson shooting method alongside the Forth-order Runge-Kutta algorithm. The results are presented graphically and in tabular form for various controlling parameters.
Casson fluid flow over a vertical porous surface with chemical reaction in the presence of magnetic field has been studied. A similarity analysis was used to transform the system of partial differential equations describing the problem into ordinary differential equations. The reduced system was solved using the Newton Raphson shooting method alongside the Forth-order Runge-Kutta algorithm. The results are presented graphically and in tabular form for various controlling parameters.
Cite this paper
Arthur, E. , Seini, I. and Bortteir, L. (2015) Analysis of Casson Fluid Flow over a Vertical Porous Surface with Chemical Reaction in the Presence of Magnetic Field. Journal of Applied Mathematics and Physics, 3, 713-723. doi: 10.4236/jamp.2015.36085.
Arthur, E. , Seini, I. and Bortteir, L. (2015) Analysis of Casson Fluid Flow over a Vertical Porous Surface with Chemical Reaction in the Presence of Magnetic Field. Journal of Applied Mathematics and Physics, 3, 713-723. doi: 10.4236/jamp.2015.36085.
References
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http://dx.doi.org/10.1115/1.3423405 [6] Rajagopal, K.R. (1980) Viscometric Flows of Third Grade Fluids. Mechanics Research Communications, 7, 21-25. http://dx.doi.org/10.1016/0093-6413(80)90020-8 [7] Rajagopal, K.R. and Gupta A.S. (1981) On a Class of Exact Solutions to the Equations of Motion of a Second Grade Fluid. International Journal of Engineering Science, 19, 1009-1014. http://dx.doi.org/10.1016/0020-7225(81)90135-X [8] Dorier, C. and Tichy, J. (1992) Behaviour of a Bingham-Like Viscous Fluid in Lubrication Flows. Journal of Non- Newtonian Fluid Mechanics, 45, 291-310. http://dx.doi.org/10.1016/0377-0257(92)80065-6 [9] Zhou, X.-F. and Gao, L. (2007) Effect of Multipolar interaction on the Effective Thermal Conductivity of Nanofluids. Chinese Physics B, 16, 2028-2032. http://dx.doi.org/10.1088/1009-1963/16/7/037 [10] Cui, Z.W., Liu, J.X., Yao, G.J. and Wang, K.X. (2010) Borehole Guided Waves in a Non-Newtonian (Maxwell) Fluid-Saturated Porous Medium. Chinese Physics B, 19, 084301. http://dx.doi.org/10.1088/1674-1056/19/8/084301 [11] Dash, R.K., Mehta, K.N. and Jayaraman, G. (1996) Casson Fluid Flow in a Pipe Filled with a Homogeneous Porous Medium. International Journal of Engineering Science, 34, 1145-1156. [12] Casson, N. (1959) A Flow Equation for Pigment-Oil Suspensions of the Printing Ink Type. In: Mill, C.C., Ed., Rheology of Disperse Systems, Pergamon Press, Oxford, 84-104. [13] Walwander, W.P., Chen, T.Y. and Cala, D.F. (1975) An Approximate Casson Fluid Model for Tube Flow of Blood. Biorheology, 12, 111-119. [14] Vinogradov, G.V. and Malkin, A.Y. (1979) Rheology of Polymers. Mir Publisher, Moscow. [15] Blair, G.W.S. (1959) An Equation for the Flow of Blood, Plasma and Serum through Glass Capillaries. Nature, 183, 613-614. http://dx.doi.org/10.1038/183613a0 [16] Charm, S. and Kurland, G. (1965) Viscometry of Human Blood for Shear Rates of 0-100,000 sec-1. Nature, 206, 617- 618. http://dx.doi.org/10.1038/206617a0 [17] Fredrickson, A.G. (1964) Principles and Applications of Rheology. Prentice-Hall, Englewood Cliffs. [18] Hayat, T., Sajid, M. and Pop, I. (2008) Three-Dimensional Flow over a Stretching Surface in a Viscoelastic Fluid. Nonlinear Analysis: Real World Applications, 9, 1811-1822. http://dx.doi.org/10.1016/j.nonrwa.2007.05.010 [19] Bhattacharyya, K. (2013) Boundary Layer Stagnation-Point Flow of Casson Fluid and Heat Transfer towards a Shrinking/Stretching Sheet. Frontiers in Heat and Mass Transfer (FHMT), 4, Article ID: 023003. http://dx.doi.org/10.5098/hmt.v4.2.3003 [20] Arthur, E.M. and Seini, Y.I. (2014) MHD Thermal Stagnation Point Flow towards a Stretching Porous Surface. Mathematical Theory and Modeling, 4, 163-169. [21] Crane, L.J. (1970) Flow past a Stretching Plate. Zeitschrift für angewandte Mathematik und Physik, 21, 645-647. http://dx.doi.org/10.1007/BF01587695 [22] Rajagopal, K.R., Na, T.Y. and Gupta, A.S. (1984) Flow of Viscoelastic Fluid over a Stretching Sheet. Rheologica Acta, 23, 213-215. http://dx.doi.org/10.1007/BF01332078 [23] Siddappa, B. and Abel, M.S. (1985) Non-Newtonian Flow past a Stretching Plate. Zeitschrift für angewandte Mathematik und Physik, 36, 890-892. http://dx.doi.org/10.1007/BF00944900 [24] Sankara, K.K. and Watson, L.T. (1985) Micropolar Flow Past a Stretching Sheet. Zeitschrift für angewandte Mathematik und Physik, 36, 845-853. http://dx.doi.org/10.1007/BF00944898 [25] Troy, W.C., Overman, E.A., Ermentrout, H.G.B. and Keerner, J.P. (1987) Uniqueness of the Flow of a Second Order Fluid past a Stretching Sheet. The Quarterly of Applied Mathematics, 44, 753-755. [26] Andersson, H.I. and Dandapat, B.S. (1991) Flow of a Power-Law Fluid over a Stretching Sheet. Stability Applied Analysis of Continuous Media, 1, 339-347. [27] Hayat, T., Shehzad, S.A., Alsaedi, A. and Alhothuali, M.S. (2012) Mixed Convection Stagnation Point Flow of Casson Fluid with Convective Boundary Conditions. Chinese Physics Letters, 29, Article ID: 114704. http://dx.doi.org/10.1088/0256-307X/29/11/114704 [28] Bhattacharyya, K., Hayat, T. and Alsaedi, A. (2013) Analytic Solution for Magnetohydrodynamic Boundary Layer Flow of Casson Fluid over a Stretching/Shrinking Sheet with Wall Mass Transfer. Chinese Physics B, 22, Article ID: 024702. http://dx.doi.org/10.1088/1674-1056/22/2/024702 [29] Mahapatra, T.R. and Nandy, S.K. (2013) Stability of Dual Solutions in Stagnation-Point Flow and Heat Transfer over a Porous Shrinking Sheet with Thermal Radiation. Meccanica, 48, 23-32. http://dx.doi.org/10.1007/s11012-012-9579-5 [30] Ishak, A., Lok, Y.Y. and Pop, I. (2010) Stagnation-Point Flow over a Shrinking Sheet in a Micropolar Fluid. Chemical Engineering Communications, 197, 1417-1427. http://dx.doi.org/10.1080/00986441003626169 [31] Yacob, N.A., Ishak, A. and Pop, I. (2011) Melting Heat Transfer in Boundary Layer Stagnation-Point Flow towards a Stretching/Shrinking Sheet in a Micropolar Fluid. Computers & Fluids, 47, 16-21. http://dx.doi.org/10.1016/j.compfluid.2011.01.040 [32] Eldabe, N.T.M. and Salwa, M.G.E. (1995) Heat Transfer of MHD Non-Newtonian Casson Fluid Flow between Two Rotating Cylinders. Journal of the Physical Society of Japan, 64, 41. [33] Andersson, H.I., Hansen, O.R. and Holmedal, B. (1994) Diffusion of a Chemically Reactive Species from a Stretching Sheet. International Journal of Heat and Mass Transfer, 37, 659-664. http://dx.doi.org/10.1016/0017-9310(94)90137-6 [34] Salem, A.M. and El-Aziz, M.A. (2008) Effect of Hall Currents and Chemical Reaction on Hydromagnetic Flow of a Stretching Vertical Surface with Internal Heat Generation/Absorption. Applied Mathematical Modelling, 32, 1236- 1254. http://dx.doi.org/10.1016/j.apm.2007.03.008 [35] Shehzad, S.A., Hayat, T., Qasim, M. and Asghar, S. (2013) Effects of Mass Transfer on MHD Flow of Casson Fluid with Chemical Reaction and Suction. Brazilian Journal of Chemical Engineering, 30, 187-195. http://dx.doi.org/10.1590/S0104-66322013000100020
[1] Mustafa, M., Hayat, T., Pop, I. and Aziz, A. (2011) Unsteady Boundary Layer Flow of a Casson Fluid Due to an Impulsively Started Moving Flat Plate. Heat Transfer-Asian Research, 40, 563-576. http://dx.doi.org/10.1002/htj.20358 [2] Cortell, R. (2008) Analysing Flow and Heat Transfer of a Viscoelastic Fluid over a Semi-Infinite Horizontal Moving Flat Plate. International Journal of Non-Linear Mechanics, 43, 772-778. http://dx.doi.org/10.1016/J.Ijnonlinmec.2008.04.006 [3] Fox, V.G., Erickson, L.E. and Fan, L.T. (1969) The Laminar Boundary Layer on a Moving Continuous Flat Sheet Immersed in a Non-Newtonian Fluid. AIChE Journal, 15, 327-333. http://dx.doi.org/10.1002/aic.690150307 [4] Wilkinson, W. (1970) The Drainage of a Maxwell Liquid Down a Vertical Plate. Chemical Engineering Journal, 1, 255-257. http://dx.doi.org/10.1016/0300-9467(70)80008-9 [5] Djukic, D.S. (1974) Hiemenz Magnetic Flow of Power-Law Fluids. Journal of Applied Mechanics, 41, 822-823.
http://dx.doi.org/10.1115/1.3423405 [6] Rajagopal, K.R. (1980) Viscometric Flows of Third Grade Fluids. Mechanics Research Communications, 7, 21-25. http://dx.doi.org/10.1016/0093-6413(80)90020-8 [7] Rajagopal, K.R. and Gupta A.S. (1981) On a Class of Exact Solutions to the Equations of Motion of a Second Grade Fluid. International Journal of Engineering Science, 19, 1009-1014. http://dx.doi.org/10.1016/0020-7225(81)90135-X [8] Dorier, C. and Tichy, J. (1992) Behaviour of a Bingham-Like Viscous Fluid in Lubrication Flows. Journal of Non- Newtonian Fluid Mechanics, 45, 291-310. http://dx.doi.org/10.1016/0377-0257(92)80065-6 [9] Zhou, X.-F. and Gao, L. (2007) Effect of Multipolar interaction on the Effective Thermal Conductivity of Nanofluids. Chinese Physics B, 16, 2028-2032. http://dx.doi.org/10.1088/1009-1963/16/7/037 [10] Cui, Z.W., Liu, J.X., Yao, G.J. and Wang, K.X. (2010) Borehole Guided Waves in a Non-Newtonian (Maxwell) Fluid-Saturated Porous Medium. Chinese Physics B, 19, 084301. http://dx.doi.org/10.1088/1674-1056/19/8/084301 [11] Dash, R.K., Mehta, K.N. and Jayaraman, G. (1996) Casson Fluid Flow in a Pipe Filled with a Homogeneous Porous Medium. International Journal of Engineering Science, 34, 1145-1156. [12] Casson, N. (1959) A Flow Equation for Pigment-Oil Suspensions of the Printing Ink Type. In: Mill, C.C., Ed., Rheology of Disperse Systems, Pergamon Press, Oxford, 84-104. [13] Walwander, W.P., Chen, T.Y. and Cala, D.F. (1975) An Approximate Casson Fluid Model for Tube Flow of Blood. Biorheology, 12, 111-119. [14] Vinogradov, G.V. and Malkin, A.Y. (1979) Rheology of Polymers. Mir Publisher, Moscow. [15] Blair, G.W.S. (1959) An Equation for the Flow of Blood, Plasma and Serum through Glass Capillaries. Nature, 183, 613-614. http://dx.doi.org/10.1038/183613a0 [16] Charm, S. and Kurland, G. (1965) Viscometry of Human Blood for Shear Rates of 0-100,000 sec-1. Nature, 206, 617- 618. http://dx.doi.org/10.1038/206617a0 [17] Fredrickson, A.G. (1964) Principles and Applications of Rheology. Prentice-Hall, Englewood Cliffs. [18] Hayat, T., Sajid, M. and Pop, I. (2008) Three-Dimensional Flow over a Stretching Surface in a Viscoelastic Fluid. Nonlinear Analysis: Real World Applications, 9, 1811-1822. http://dx.doi.org/10.1016/j.nonrwa.2007.05.010 [19] Bhattacharyya, K. (2013) Boundary Layer Stagnation-Point Flow of Casson Fluid and Heat Transfer towards a Shrinking/Stretching Sheet. Frontiers in Heat and Mass Transfer (FHMT), 4, Article ID: 023003. http://dx.doi.org/10.5098/hmt.v4.2.3003 [20] Arthur, E.M. and Seini, Y.I. (2014) MHD Thermal Stagnation Point Flow towards a Stretching Porous Surface. Mathematical Theory and Modeling, 4, 163-169. [21] Crane, L.J. (1970) Flow past a Stretching Plate. Zeitschrift für angewandte Mathematik und Physik, 21, 645-647. http://dx.doi.org/10.1007/BF01587695 [22] Rajagopal, K.R., Na, T.Y. and Gupta, A.S. (1984) Flow of Viscoelastic Fluid over a Stretching Sheet. Rheologica Acta, 23, 213-215. http://dx.doi.org/10.1007/BF01332078 [23] Siddappa, B. and Abel, M.S. (1985) Non-Newtonian Flow past a Stretching Plate. Zeitschrift für angewandte Mathematik und Physik, 36, 890-892. http://dx.doi.org/10.1007/BF00944900 [24] Sankara, K.K. and Watson, L.T. (1985) Micropolar Flow Past a Stretching Sheet. Zeitschrift für angewandte Mathematik und Physik, 36, 845-853. http://dx.doi.org/10.1007/BF00944898 [25] Troy, W.C., Overman, E.A., Ermentrout, H.G.B. and Keerner, J.P. (1987) Uniqueness of the Flow of a Second Order Fluid past a Stretching Sheet. The Quarterly of Applied Mathematics, 44, 753-755. [26] Andersson, H.I. and Dandapat, B.S. (1991) Flow of a Power-Law Fluid over a Stretching Sheet. Stability Applied Analysis of Continuous Media, 1, 339-347. [27] Hayat, T., Shehzad, S.A., Alsaedi, A. and Alhothuali, M.S. (2012) Mixed Convection Stagnation Point Flow of Casson Fluid with Convective Boundary Conditions. Chinese Physics Letters, 29, Article ID: 114704. http://dx.doi.org/10.1088/0256-307X/29/11/114704 [28] Bhattacharyya, K., Hayat, T. and Alsaedi, A. (2013) Analytic Solution for Magnetohydrodynamic Boundary Layer Flow of Casson Fluid over a Stretching/Shrinking Sheet with Wall Mass Transfer. Chinese Physics B, 22, Article ID: 024702. http://dx.doi.org/10.1088/1674-1056/22/2/024702 [29] Mahapatra, T.R. and Nandy, S.K. (2013) Stability of Dual Solutions in Stagnation-Point Flow and Heat Transfer over a Porous Shrinking Sheet with Thermal Radiation. Meccanica, 48, 23-32. http://dx.doi.org/10.1007/s11012-012-9579-5 [30] Ishak, A., Lok, Y.Y. and Pop, I. (2010) Stagnation-Point Flow over a Shrinking Sheet in a Micropolar Fluid. Chemical Engineering Communications, 197, 1417-1427. http://dx.doi.org/10.1080/00986441003626169 [31] Yacob, N.A., Ishak, A. and Pop, I. (2011) Melting Heat Transfer in Boundary Layer Stagnation-Point Flow towards a Stretching/Shrinking Sheet in a Micropolar Fluid. Computers & Fluids, 47, 16-21. http://dx.doi.org/10.1016/j.compfluid.2011.01.040 [32] Eldabe, N.T.M. and Salwa, M.G.E. (1995) Heat Transfer of MHD Non-Newtonian Casson Fluid Flow between Two Rotating Cylinders. Journal of the Physical Society of Japan, 64, 41. [33] Andersson, H.I., Hansen, O.R. and Holmedal, B. (1994) Diffusion of a Chemically Reactive Species from a Stretching Sheet. International Journal of Heat and Mass Transfer, 37, 659-664. http://dx.doi.org/10.1016/0017-9310(94)90137-6 [34] Salem, A.M. and El-Aziz, M.A. (2008) Effect of Hall Currents and Chemical Reaction on Hydromagnetic Flow of a Stretching Vertical Surface with Internal Heat Generation/Absorption. Applied Mathematical Modelling, 32, 1236- 1254. http://dx.doi.org/10.1016/j.apm.2007.03.008 [35] Shehzad, S.A., Hayat, T., Qasim, M. and Asghar, S. (2013) Effects of Mass Transfer on MHD Flow of Casson Fluid with Chemical Reaction and Suction. Brazilian Journal of Chemical Engineering, 30, 187-195. http://dx.doi.org/10.1590/S0104-66322013000100020