Heat and Mass Transfer of Upper Convected Maxwell Fluid Flow with Variable Thermo-Physical Properties over a Horizontal Melting Surface

Affiliation(s)

^{1}
Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria.

^{2}
School of Science and Technology, National Open University of Nigeria, Lagos, Nigeria.

ABSTRACT

The objective of this article is to present the dynamics of an Upper Convected Maxwell (UCM) fluid flow with heat and mass transfer over a melting surface. The influence of melting heat transfer, thermal and solutal stratification are properly accounted for by modifying the classical boundary conditions of temperature and concentration respectively. It is assumed that the ratio of inertia forces to viscous forces is high enough for boundary layer approximation to be valid. The corresponding influence of exponential space dependent internal heat source on viscosity and thermal conductivity of UCM is properly considered. The dynamic viscosity and thermal conductivity of UCM are temperature dependent. Classical temperature dependent viscosity and thermal conductivity models were modified to suit the case of both melting heat transfer and thermal stratification. The governing non-linear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformations and completed the solution numerically using the Runge-Kutta method along with shooting technique. For accurate and correct analysis of the effect of variable viscosity on fluid flow in which (*T*_{w} or *T*_{m}) < *T*_{∞} , the mathematical models of variable viscosity and thermal conductivity must be modified.

The objective of this article is to present the dynamics of an Upper Convected Maxwell (UCM) fluid flow with heat and mass transfer over a melting surface. The influence of melting heat transfer, thermal and solutal stratification are properly accounted for by modifying the classical boundary conditions of temperature and concentration respectively. It is assumed that the ratio of inertia forces to viscous forces is high enough for boundary layer approximation to be valid. The corresponding influence of exponential space dependent internal heat source on viscosity and thermal conductivity of UCM is properly considered. The dynamic viscosity and thermal conductivity of UCM are temperature dependent. Classical temperature dependent viscosity and thermal conductivity models were modified to suit the case of both melting heat transfer and thermal stratification. The governing non-linear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformations and completed the solution numerically using the Runge-Kutta method along with shooting technique. For accurate and correct analysis of the effect of variable viscosity on fluid flow in which (

Cite this paper

Adegbie, K. , Omowaye, A. , Disu, A. and Animasaun, I. (2015) Heat and Mass Transfer of Upper Convected Maxwell Fluid Flow with Variable Thermo-Physical Properties over a Horizontal Melting Surface.*Applied Mathematics*, **6**, 1362-1379. doi: 10.4236/am.2015.68129.

Adegbie, K. , Omowaye, A. , Disu, A. and Animasaun, I. (2015) Heat and Mass Transfer of Upper Convected Maxwell Fluid Flow with Variable Thermo-Physical Properties over a Horizontal Melting Surface.

References

[1] Asano, K. (2006) Mass Transfer (From Fundamentals to Modern Industrial Applications). Wiley-VCH Verlag GmbH and Co., Weinheim.

http://dx.doi.org/10.1002/3527609180

[2] Christopher, W.M. (1993) Rheology: Principles, Measurements and Applications. Wiley-VCH Publisher, Weinheim.

[3] Barnes, H.A., Hutton, J.F. and Walters, K. (1989) An Introduction to Rheology. Elsevier Science Publishing Company, New York.

[4] Steffe, J.F. (1996) Rheological Methods in Food Process Engineering. 2nd Edition, Freeman Press, East Lansing.

[5] Poole, R.J. (2012) The Deborah and Weissenberg Numbers. Rheology Bulletin, 53, 32-39.

[6] Reiner, M. (1964) The Deborah number. Physics Today, 17, 62.

http://dx.doi.org/10.1063/1.3051374

[7] Sadeghy, K., Najafi, A.H. and Saffaripour, M. (2005) Sakiadis Flow of an Upper-Convected Maxwell Fluid. International Journal of Non-Linear Mechanics, 40, 1220-1228.

http://dx.doi.org/10.1016/j.ijnonlinmec.2005.05.006

[8] Shateyi, S., Motsa, S.S. and Makukula, Z. (2015) On Spectral Relaxation Method for Entropy Generation on a MHD Flow and Heat Transfer of a Maxwell Fluid. Journal of Applied Fluid Mechanics, 8, 21-31.

[9] Abbas, Z., Sajid, M. and Hayat, T. (2006) MHD Boundary-Layer Flow of an Upper-Convected Maxwell Fluid in a Porous Channel. Theoretical and Computational Fluid Dynamics, 20, 229-238.

http://dx.doi.org/10.1007/s00162-006-0025-y

[10] Fosdick, R.L. and Rajagopal, K.R. (1979) Anomalous Features in the Model of Second Grade Fluids. Archive for Rational Mechanics and Analysis, 70, 145-152.

http://dx.doi.org/10.1007/BF00250351

[11] Hayat, T., Abbas, Z. and Sajid, M. (2006) Series Solution for the Upper-Convected Maxwell Fluid over a Porous Stretching Plate. Physics Letters A, 358, 396-403.

http://dx.doi.org/10.1016/j.physleta.2006.04.117

[12] Sadeghy, K., Hajibeygi, H. and Taghavi, S.M. (2006) Stagnation-Point Flow of Upper-Convected Maxwell Fluids. International Journal of Non-Linear Mechanics, 41, 1242-1247.

http://dx.doi.org/10.1016/j.ijnonlinmec.2006.08.005

[13] Hayat, T. and Sajid, M. (2007) Homotopy Analysis of MHD Boundary Layer Flow of an Upper-Convected Maxwell Fluid. International Journal of Engineering Science, 45, 393-401.

http://dx.doi.org/10.1016/j.ijengsci.2007.04.009

[14] Abbas, Z., Hayat, T. and Alib, N. (2008) MHD Flow and Mass Transfer of an Upper-Convected Maxwell Fluid Past a Porous Shrinking Sheet with Chemical Reaction Species. Physics Letters A, 372, 4698-4704.

http://dx.doi.org/10.1016/j.physleta.2008.05.006

[15] Sadeghy, K., Aliakbar, V. and Alizadeh-Pahlavan, A. (2009) The Influence of Thermal Radiation on MHD Flow of Maxwellian Fluids above Stretching Sheets. Communications in Nonlinear Science and Numerical Simulation, 14, 779-794.

http://dx.doi.org/10.1016/j.cnsns.2007.12.003

[16] Awais, M., Hayat, T., Qasim, M. and Hendi, A.A. (2011) Effects of Mass Transfer on the Stagnation Point Flow of an Upper-Convected Maxwell (UCM) Fluid. International Journal of Heat and Mass Transfer, 15-16, 3777-3782.

[17] Mustafa, M., Hayat, T., Shehzad, S.A. and Obaidat, S. (2012) Melting Heat Transfer in the Stagnation-Point Flow of an Upper-Convected Maxwell (UCM) Fluid Past a Stretching Sheet. International Journal for Numerical Methods in Fluids, 68, 233-243.

http://dx.doi.org/10.1002/fld.2503

[18] Motsa, S.S., Hayat, T. and Aldossary, O.M. (2012) MHD Flow of Upper-Convected Maxwell Fluid over Porous Stretching Sheet Using Successive Taylor Series Linearization Method. Applied Mathematics and Mechanics (English Edition), 33, 975-990.

http://dx.doi.org/10.1007/s10483-012-1599-x

[19] Abel, M.S., Tawade, J.V. and Nandeppanavar, M.M. (2012) MHD Flow and Heat Transfer for the Upper-Convected Maxwell Fluid over a Stretching Sheet. Meccanica, 47, 385-393.

http://dx.doi.org/10.1007/s11012-011-9448-7

[20] Pop, I., Sujatha, A., Vajravelu, K. and Prasad, K.V. (2012) MHD Flow and Heat Transfer of a UCM Fluid over a Stretching Surface with Variable Thermophysical Properties. Meccanica, 47, 1425-1439.

http://dx.doi.org/10.1007/s11012-011-9526-x

[21] Prasad, K.V., Vajravelu, K. and Sujatha, A. (2013) Influence of Internal Heat Generation/Absorption, Thermal Radiation, Magnetic Field, Variable Fluid Property and Viscous Dissipation on Heat Transfer Characteristics of a Maxwell Fluid over a Stretching Sheet. Journal of Applied Fluid Mechanics, 6, 249-256.

[22] Hayat, T., Mushtaq, A., Mustafa, M. and Alsaedi, A. (2014) Effects of Thermal Radiation on the Stagnation-Point Flow of Upper-Convected Maxwell Fluid over a Stretching Sheet. Journal of Aerospace Engineering, 27, Article ID: 04014015.

[23] Crepeau, J.C. and Clarksean, R. (1997) Similarity Solutions of Natural Convection with Internal Heat Generation. Transactions of ASME—Journal of Heat Transfer, 119, 184-185.

http://dx.doi.org/10.1115/1.2824086

[24] Salem, A.M. and El-Aziz, M.A. (2007) MHD-Mixed Convection and Mass Transfer from a Vertical Stretching Sheet with Diffusion of Chemically Reactive Species and Space- or Temperature-Dependent Heat Source. Canadian Journal of Physics, 85, 359-373.

http://dx.doi.org/10.1139/P07-048

[25] 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

[26] Animasaun, I.L., Adebile, E.A. and Fagbade, A.I. (2015) Casson Fluid Flow with Variable Thermo-Physical Property along Exponentially Stretching Sheet with Suction and Exponentially Decaying Internal Heat Generation Using the Homotopy Analysis Method. Journal of the Nigerian Mathematical Society, 35. (in press)

http://dx.doi.org/10.1016/j.jnnms.2015.02.001

[27] Yin-Chao, Y. and Tien, C. (1963) Laminar Heat Transfer over a Melting Plate, the Modified Leveque Problem. Journal of Geophysical Research, 68, 3673-3678.

http://dx.doi.org/10.1029/JZ068i012p03673

[28] Tien, C. and Yen, Y. (1965) The Effect of Melting on Forced Convection Heat Transfer. Journal of Applied Meterology, 4, 523-527.

http://dx.doi.org/10.1175/1520-0450(1965)004<0523:TEOMOF>2.0.CO;2

[29] Epstein, M. (1975) The Effect of Melting on Heat Transfer to Submerged Bodies. Letters in Heat and Mass Transfer, 2, 97-104.

[30] Pop, I., Bachok, N. and Ishak, A. (2010) Melting Heat Transfer in Boundary Layer Stagnation-Point Flow towards a Stretching/Shrinking Sheet. Physics Letter A, 374, 4075-4079.

http://dx.doi.org/10.1016/j.physleta.2010.08.032

[31] Ishak, A., Nazar, R., Bachok, N. and Pop, I. (2010) Melting Heat Transfer in Steady Laminar Flow over a Moving Surface. Heat Mass Transfer, 46, 463-468.

http://dx.doi.org/10.1007/s00231-010-0592-8

[32] Hayat, T., Hussain, M., Awais, M. and Obaidat, S. (2013) Melting Heat Transfer in a Boundary Layer Flow of a Second Grade Fluid under Soret and Dufour Effects. International Journal of Numerical Methods for Heat and Fluid Flow, 23, 1155-1168.

http://dx.doi.org/10.1108/HFF-09-2011-0182

[33] Fukusako, S. and Yamada, M. (1999) Melting Heat Transfer inside Ducts and Over External Bodies. Experimental Thermal and Fluid science, 19, 93-117.

http://dx.doi.org/10.1016/S0894-1777(99)00011-4

[34] Batchelor, G.K. (1987) An Introduction to Fluid Dynamics. Cambridge University Press, London.

[35] Animasaun, I.L. (2015) Effects of Thermophoresis, Variable Viscosity and Thermal Conductivity on Free Convective Heat and Mass Transfer of Non-Darcian MHD Dissipative Casson Fluid Flow with Suction and nth Order of Chemical Reaction. Journal of the Nigerian Mathematical Society, 34, 11-31.

http://dx.doi.org/10.1016/j.jnnms.2014.10.008

[36] Meyers, T.G., Charpin, J.P.F. and Tshela, M.S. (2006) The Flow of a Variable Viscosity Fluid between Parallel Plates with Shear Heating. Applied Mathematic Modeling, 30, 799-815.

http://dx.doi.org/10.1016/j.apm.2005.05.013

[37] Mukhopadhyay, S. (2013) Effects of Thermal Radiation and Variable Fluid Viscosity on Stagnation Point Flow Past a Porous Stretching Sheet. Meccanica, 48, 1717-1730.

http://dx.doi.org/10.1007/s11012-013-9704-0

[38] Animasaun, I.L. (2015) Dynamics of Unsteady MHD Convective Flow with Thermophoresis of Particles and Variable Thermo-Physical Properties Past a Vertical Surface Moving through Binary Mixture. Open Journal of Fluid Dynamics, 5, 106-120.

http://dx.doi.org/10.4236/ojfd.2015.52013

[39] Hayat, T., Shehzad, S.A., Al-Sulami, H.H. and Asghar, S. (2013) Influence of Thermal Stratification on the Radiative Flow of Maxwell Fluid. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 35, 381-389.

http://dx.doi.org/10.1007/s40430-013-0036-8

[40] Dunn, J.E. and Rajagopal, K.R. (1995) Fluids of Differential Type: Critical Review and Thermodynamic Analysis. International Journal of Engineering Science, 33, 689-729.

http://dx.doi.org/10.1016/0020-7225(94)00078-X

[41] Schichting, H. (1964) Boundary Layer Theory. Sixth Edition, McGraw-Hill, New York.

[42] Larson, R.G. (1988) Constitutive Equations for Polymer Melts and Solutions. Butterworths, Boston.

[43] Lienhard-IV, J.H. and Lienhard-V, J.H. (2008) A Heat Transfer Textbook. 3rd Edition, Phlogiston Press, Cambridge, Massachusetts.

[44] Animasaun, I.L. (2015) Casson Fluid Flow of Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation. Journal of Heat and Mass Transfer Research, 3. (in press)

[45] Epstein, M. and Cho, D.H. (1976) Melting Heat Transfer in Steady Laminar Flow Over a Flat Plate. Journal of Heat Transfer, 98, 531-533.

http://dx.doi.org/10.1115/1.3450595

[46] Sivagnana, K.K.P., Kandasamy, R. and Saravanan, R. (2009) Lie Group Analysis for the Effect of Viscosity and Thermophoresis Particle Deposition on Free Convective Heat and Mass Transfer in the Presence of Suction/Injection. Theoretical and Applied Mechanics, 36, 275-298.

http://dx.doi.org/10.2298/TAM0904275S

[47] Vimala, C. and Loganthan, P. (2015) MHD Flow of Nanofluids over an Exponentially Stretching Sheet Embedded in a Stratified Medium with Suction and Radiation Effects. Journal of Applied Fluid Mechanics, 8, 85-93.

[48] Na, T.Y. (1979) Computational Methods in Engineering Boundary Value Problems. Academic Press, New York.

[49] Shampine, L.F., Reichelt, M.W. and Kierzenka, J. (2010) Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c.

http://cn.mathworks.com/help/matlab/ref/bvp4c.html

[1] Asano, K. (2006) Mass Transfer (From Fundamentals to Modern Industrial Applications). Wiley-VCH Verlag GmbH and Co., Weinheim.

http://dx.doi.org/10.1002/3527609180

[2] Christopher, W.M. (1993) Rheology: Principles, Measurements and Applications. Wiley-VCH Publisher, Weinheim.

[3] Barnes, H.A., Hutton, J.F. and Walters, K. (1989) An Introduction to Rheology. Elsevier Science Publishing Company, New York.

[4] Steffe, J.F. (1996) Rheological Methods in Food Process Engineering. 2nd Edition, Freeman Press, East Lansing.

[5] Poole, R.J. (2012) The Deborah and Weissenberg Numbers. Rheology Bulletin, 53, 32-39.

[6] Reiner, M. (1964) The Deborah number. Physics Today, 17, 62.

http://dx.doi.org/10.1063/1.3051374

[7] Sadeghy, K., Najafi, A.H. and Saffaripour, M. (2005) Sakiadis Flow of an Upper-Convected Maxwell Fluid. International Journal of Non-Linear Mechanics, 40, 1220-1228.

http://dx.doi.org/10.1016/j.ijnonlinmec.2005.05.006

[8] Shateyi, S., Motsa, S.S. and Makukula, Z. (2015) On Spectral Relaxation Method for Entropy Generation on a MHD Flow and Heat Transfer of a Maxwell Fluid. Journal of Applied Fluid Mechanics, 8, 21-31.

[9] Abbas, Z., Sajid, M. and Hayat, T. (2006) MHD Boundary-Layer Flow of an Upper-Convected Maxwell Fluid in a Porous Channel. Theoretical and Computational Fluid Dynamics, 20, 229-238.

http://dx.doi.org/10.1007/s00162-006-0025-y

[10] Fosdick, R.L. and Rajagopal, K.R. (1979) Anomalous Features in the Model of Second Grade Fluids. Archive for Rational Mechanics and Analysis, 70, 145-152.

http://dx.doi.org/10.1007/BF00250351

[11] Hayat, T., Abbas, Z. and Sajid, M. (2006) Series Solution for the Upper-Convected Maxwell Fluid over a Porous Stretching Plate. Physics Letters A, 358, 396-403.

http://dx.doi.org/10.1016/j.physleta.2006.04.117

[12] Sadeghy, K., Hajibeygi, H. and Taghavi, S.M. (2006) Stagnation-Point Flow of Upper-Convected Maxwell Fluids. International Journal of Non-Linear Mechanics, 41, 1242-1247.

http://dx.doi.org/10.1016/j.ijnonlinmec.2006.08.005

[13] Hayat, T. and Sajid, M. (2007) Homotopy Analysis of MHD Boundary Layer Flow of an Upper-Convected Maxwell Fluid. International Journal of Engineering Science, 45, 393-401.

http://dx.doi.org/10.1016/j.ijengsci.2007.04.009

[14] Abbas, Z., Hayat, T. and Alib, N. (2008) MHD Flow and Mass Transfer of an Upper-Convected Maxwell Fluid Past a Porous Shrinking Sheet with Chemical Reaction Species. Physics Letters A, 372, 4698-4704.

http://dx.doi.org/10.1016/j.physleta.2008.05.006

[15] Sadeghy, K., Aliakbar, V. and Alizadeh-Pahlavan, A. (2009) The Influence of Thermal Radiation on MHD Flow of Maxwellian Fluids above Stretching Sheets. Communications in Nonlinear Science and Numerical Simulation, 14, 779-794.

http://dx.doi.org/10.1016/j.cnsns.2007.12.003

[16] Awais, M., Hayat, T., Qasim, M. and Hendi, A.A. (2011) Effects of Mass Transfer on the Stagnation Point Flow of an Upper-Convected Maxwell (UCM) Fluid. International Journal of Heat and Mass Transfer, 15-16, 3777-3782.

[17] Mustafa, M., Hayat, T., Shehzad, S.A. and Obaidat, S. (2012) Melting Heat Transfer in the Stagnation-Point Flow of an Upper-Convected Maxwell (UCM) Fluid Past a Stretching Sheet. International Journal for Numerical Methods in Fluids, 68, 233-243.

http://dx.doi.org/10.1002/fld.2503

[18] Motsa, S.S., Hayat, T. and Aldossary, O.M. (2012) MHD Flow of Upper-Convected Maxwell Fluid over Porous Stretching Sheet Using Successive Taylor Series Linearization Method. Applied Mathematics and Mechanics (English Edition), 33, 975-990.

http://dx.doi.org/10.1007/s10483-012-1599-x

[19] Abel, M.S., Tawade, J.V. and Nandeppanavar, M.M. (2012) MHD Flow and Heat Transfer for the Upper-Convected Maxwell Fluid over a Stretching Sheet. Meccanica, 47, 385-393.

http://dx.doi.org/10.1007/s11012-011-9448-7

[20] Pop, I., Sujatha, A., Vajravelu, K. and Prasad, K.V. (2012) MHD Flow and Heat Transfer of a UCM Fluid over a Stretching Surface with Variable Thermophysical Properties. Meccanica, 47, 1425-1439.

http://dx.doi.org/10.1007/s11012-011-9526-x

[21] Prasad, K.V., Vajravelu, K. and Sujatha, A. (2013) Influence of Internal Heat Generation/Absorption, Thermal Radiation, Magnetic Field, Variable Fluid Property and Viscous Dissipation on Heat Transfer Characteristics of a Maxwell Fluid over a Stretching Sheet. Journal of Applied Fluid Mechanics, 6, 249-256.

[22] Hayat, T., Mushtaq, A., Mustafa, M. and Alsaedi, A. (2014) Effects of Thermal Radiation on the Stagnation-Point Flow of Upper-Convected Maxwell Fluid over a Stretching Sheet. Journal of Aerospace Engineering, 27, Article ID: 04014015.

[23] Crepeau, J.C. and Clarksean, R. (1997) Similarity Solutions of Natural Convection with Internal Heat Generation. Transactions of ASME—Journal of Heat Transfer, 119, 184-185.

http://dx.doi.org/10.1115/1.2824086

[24] Salem, A.M. and El-Aziz, M.A. (2007) MHD-Mixed Convection and Mass Transfer from a Vertical Stretching Sheet with Diffusion of Chemically Reactive Species and Space- or Temperature-Dependent Heat Source. Canadian Journal of Physics, 85, 359-373.

http://dx.doi.org/10.1139/P07-048

[25] 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

[26] Animasaun, I.L., Adebile, E.A. and Fagbade, A.I. (2015) Casson Fluid Flow with Variable Thermo-Physical Property along Exponentially Stretching Sheet with Suction and Exponentially Decaying Internal Heat Generation Using the Homotopy Analysis Method. Journal of the Nigerian Mathematical Society, 35. (in press)

http://dx.doi.org/10.1016/j.jnnms.2015.02.001

[27] Yin-Chao, Y. and Tien, C. (1963) Laminar Heat Transfer over a Melting Plate, the Modified Leveque Problem. Journal of Geophysical Research, 68, 3673-3678.

http://dx.doi.org/10.1029/JZ068i012p03673

[28] Tien, C. and Yen, Y. (1965) The Effect of Melting on Forced Convection Heat Transfer. Journal of Applied Meterology, 4, 523-527.

http://dx.doi.org/10.1175/1520-0450(1965)004<0523:TEOMOF>2.0.CO;2

[29] Epstein, M. (1975) The Effect of Melting on Heat Transfer to Submerged Bodies. Letters in Heat and Mass Transfer, 2, 97-104.

[30] Pop, I., Bachok, N. and Ishak, A. (2010) Melting Heat Transfer in Boundary Layer Stagnation-Point Flow towards a Stretching/Shrinking Sheet. Physics Letter A, 374, 4075-4079.

http://dx.doi.org/10.1016/j.physleta.2010.08.032

[31] Ishak, A., Nazar, R., Bachok, N. and Pop, I. (2010) Melting Heat Transfer in Steady Laminar Flow over a Moving Surface. Heat Mass Transfer, 46, 463-468.

http://dx.doi.org/10.1007/s00231-010-0592-8

[32] Hayat, T., Hussain, M., Awais, M. and Obaidat, S. (2013) Melting Heat Transfer in a Boundary Layer Flow of a Second Grade Fluid under Soret and Dufour Effects. International Journal of Numerical Methods for Heat and Fluid Flow, 23, 1155-1168.

http://dx.doi.org/10.1108/HFF-09-2011-0182

[33] Fukusako, S. and Yamada, M. (1999) Melting Heat Transfer inside Ducts and Over External Bodies. Experimental Thermal and Fluid science, 19, 93-117.

http://dx.doi.org/10.1016/S0894-1777(99)00011-4

[34] Batchelor, G.K. (1987) An Introduction to Fluid Dynamics. Cambridge University Press, London.

[35] Animasaun, I.L. (2015) Effects of Thermophoresis, Variable Viscosity and Thermal Conductivity on Free Convective Heat and Mass Transfer of Non-Darcian MHD Dissipative Casson Fluid Flow with Suction and nth Order of Chemical Reaction. Journal of the Nigerian Mathematical Society, 34, 11-31.

http://dx.doi.org/10.1016/j.jnnms.2014.10.008

[36] Meyers, T.G., Charpin, J.P.F. and Tshela, M.S. (2006) The Flow of a Variable Viscosity Fluid between Parallel Plates with Shear Heating. Applied Mathematic Modeling, 30, 799-815.

http://dx.doi.org/10.1016/j.apm.2005.05.013

[37] Mukhopadhyay, S. (2013) Effects of Thermal Radiation and Variable Fluid Viscosity on Stagnation Point Flow Past a Porous Stretching Sheet. Meccanica, 48, 1717-1730.

http://dx.doi.org/10.1007/s11012-013-9704-0

[38] Animasaun, I.L. (2015) Dynamics of Unsteady MHD Convective Flow with Thermophoresis of Particles and Variable Thermo-Physical Properties Past a Vertical Surface Moving through Binary Mixture. Open Journal of Fluid Dynamics, 5, 106-120.

http://dx.doi.org/10.4236/ojfd.2015.52013

[39] Hayat, T., Shehzad, S.A., Al-Sulami, H.H. and Asghar, S. (2013) Influence of Thermal Stratification on the Radiative Flow of Maxwell Fluid. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 35, 381-389.

http://dx.doi.org/10.1007/s40430-013-0036-8

[40] Dunn, J.E. and Rajagopal, K.R. (1995) Fluids of Differential Type: Critical Review and Thermodynamic Analysis. International Journal of Engineering Science, 33, 689-729.

http://dx.doi.org/10.1016/0020-7225(94)00078-X

[41] Schichting, H. (1964) Boundary Layer Theory. Sixth Edition, McGraw-Hill, New York.

[42] Larson, R.G. (1988) Constitutive Equations for Polymer Melts and Solutions. Butterworths, Boston.

[43] Lienhard-IV, J.H. and Lienhard-V, J.H. (2008) A Heat Transfer Textbook. 3rd Edition, Phlogiston Press, Cambridge, Massachusetts.

[44] Animasaun, I.L. (2015) Casson Fluid Flow of Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation. Journal of Heat and Mass Transfer Research, 3. (in press)

[45] Epstein, M. and Cho, D.H. (1976) Melting Heat Transfer in Steady Laminar Flow Over a Flat Plate. Journal of Heat Transfer, 98, 531-533.

http://dx.doi.org/10.1115/1.3450595

[46] Sivagnana, K.K.P., Kandasamy, R. and Saravanan, R. (2009) Lie Group Analysis for the Effect of Viscosity and Thermophoresis Particle Deposition on Free Convective Heat and Mass Transfer in the Presence of Suction/Injection. Theoretical and Applied Mechanics, 36, 275-298.

http://dx.doi.org/10.2298/TAM0904275S

[47] Vimala, C. and Loganthan, P. (2015) MHD Flow of Nanofluids over an Exponentially Stretching Sheet Embedded in a Stratified Medium with Suction and Radiation Effects. Journal of Applied Fluid Mechanics, 8, 85-93.

[48] Na, T.Y. (1979) Computational Methods in Engineering Boundary Value Problems. Academic Press, New York.

[49] Shampine, L.F., Reichelt, M.W. and Kierzenka, J. (2010) Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c.

http://cn.mathworks.com/help/matlab/ref/bvp4c.html