Stokes First Problem for an Unsteady MHD Third-Grade Fluid in a Non-Porous Half Space with Hall Currents
ABSTRACT
The well-known problem of unidirectional plane flow of a fluid in a non-porous half-space due to the impulsive motion of the rigid plane wall it rests upon is discussed in the context of an unsteady MHD third-grade fluid in presence of Hall currents. The governing non-linear partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformations. The complex analytical solution is found by using the homotopy analysis method (HAM). The existing literature on the topic shows that it is the first study regarding the effects of Hall current on flow of an unsteady MHD third-grade fluid over an impulsively moving plane wall. The convergence of the obtained complex series solutions is carefully analyzed. The effects of dimensionless parameters on the velocity are illustrated through plots and the effects of the pertinent parameters on the local skin friction coefficient at the surface of the wall are presented numerically in tabular form.
The well-known problem of unidirectional plane flow of a fluid in a non-porous half-space due to the impulsive motion of the rigid plane wall it rests upon is discussed in the context of an unsteady MHD third-grade fluid in presence of Hall currents. The governing non-linear partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformations. The complex analytical solution is found by using the homotopy analysis method (HAM). The existing literature on the topic shows that it is the first study regarding the effects of Hall current on flow of an unsteady MHD third-grade fluid over an impulsively moving plane wall. The convergence of the obtained complex series solutions is carefully analyzed. The effects of dimensionless parameters on the velocity are illustrated through plots and the effects of the pertinent parameters on the local skin friction coefficient at the surface of the wall are presented numerically in tabular form.
Cite this paper
Zaman, H. , Sohail, A. and , U. (2014) Stokes First Problem for an Unsteady MHD Third-Grade Fluid in a Non-Porous Half Space with Hall Currents. Open Journal of Applied Sciences, 4, 85-95. doi: 10.4236/ojapps.2014.43010.
Zaman, H. , Sohail, A. and , U. (2014) Stokes First Problem for an Unsteady MHD Third-Grade Fluid in a Non-Porous Half Space with Hall Currents. Open Journal of Applied Sciences, 4, 85-95. doi: 10.4236/ojapps.2014.43010.
References
[1] Stokes, G.G. (1851) On the Effect of the Internal Friction of Fluids on the Motion of Pendulums. Transactions of the Cambridge Philosophical Society, Part II, 9, 8-106. [2] Tanner, R.I. (1962) Note on the Rayleigh Problem for a Visco-Elastic Fluid. Zeitschrift für Angewandte Mathematik und Physik, 13, 573-580.
http://dx.doi.org/10.1007/BF01595580 [3] Soundalgekar, V.M. (1974) Stokes Problem for Elastico-Viscous Fluid. Rheologica Acta, 13, 177-179.
http://dx.doi.org/10.1007/BF01520872 [4] Teipel, I. (1981) The Impulsive Motion of a Flat Plate in a Viscoelastic Fluid. Acta Mechanica, 39, 277-279.
http://dx.doi.org/10.1007/BF01170349 [5] Puri, P. (1984) Impulsive Motion of a Flat Plate in a Rivlin-Ericksen Fluid. Rheologica Acta, 23, 451-453.
http://dx.doi.org/10.1007/BF01329198 [6] Erdogan, M.E. (1995) Plane Surface Suddenly Set in Motion in a Non-Newtonian Fluid. Acta Mechanica, 108, 179- 187.
http://dx.doi.org/10.1007/BF01177337 [7] Zeng, Y. and Weinbaum, S. (1995) Stokes Problems for Moving Half-Planes. Journal of Fluid Mechanics, 287, 59-74.
http://dx.doi.org/10.1017/S0022112095000851 [8] Tan, W.C. and Xu, M.Y. (2002) The Impulsive Motion of Flat Plate in a Generalized Second Grade Fluid. Mechanics Research Communications, 29, 3-9.
http://dx.doi.org/10.1016/S0093-6413(02)00223-9 [9] Tan, W.C. and Xu, M.Y. (2002) Plane Surface Suddenly Set in Motion in a Viscoelastic Fluid with Fractional Maxwell Model. Acta Mechanica Sinica, 18, 342-349.
http://dx.doi.org/10.1007/BF02487786 [10] Erdogan, M.E. (2002) On the Unsteady Unidirectional Flows Generated by Impulsive Motion of a Boundary or Sudden Application of a Pressure Gradient. International Journal of Non-Linear Mechanics, 37, 1091-1106.
http://dx.doi.org/10.1016/S0020-7462(01)00035-X [11] Fetecau, C. and Fetecau, C. (2003) The First Problem of Stokes for an Oldroyd-B Fluid. International Journal of Non-Linear Mechanics, 38, 1539-1544.
http://dx.doi.org/10.1016/S0020-7462(02)00117-8 [12] Tan, W.C. and Masuoka, T. (2005) Stokes’ First Problem for an Oldroyd-B Fluid in a Porous Half Space. Physics of Fluids, 17, 023101.
http://dx.doi.org/10.1063/1.1850409 [13] Tan, W.C. and Masuoka, T. (2005) Stokes’ First Problem for a Second Grade Fluid in a Porous Half-Space with Heated Boundary. International Journal of Non-Linear Mechanics, 40, 515-522.
http://dx.doi.org/10.1016/j.ijnonlinmec.2004.07.016 [14] Zierep, J. and Fetecau, C. (2007) Energetic Balance for the Rayleigh-Stokes Problem of a Second Grade Fluid. International Journal of Engineering Science, 45, 155-162.
http://dx.doi.org/10.1016/j.ijengsci.2006.09.001 [15] Zierep, J., Bohning, R. and Fetecau, C. (2007) Rayleigh-Stokes Problem for Non-Newtonian Medium with Memory. Zeitschrift für Angewandte Mathematik und Mechanik, 87, 462-467.
http://dx.doi.org/10.1002/zamm.200710328 [16] Vieru, D., Nazar, M., Fetecau, C. and Fetecau, C. (2008) New Exact Solutions Corresponding to the First Problem of Stokes for Oldroyd-B Fluids. Computers & Mathematics with Applications, 55, 1644-1652.
http://dx.doi.org/10.1016/j.camwa.2007.04.040 [17] Shahzad, F., Hayat, T. and Ayub, M. (2008) Stokes’ First Problem for the Rotating Flow of a Third Grade Fluid. Nonlinear Analysis: Real World Applications, 9, 1794-1799.
http://dx.doi.org/10.1016/j.nonrwa.2007.05.008 [18] Hayat, T., Shahzad, F., Ayub, M. and Asghar, S. (2008) Stokes’ First Problem for a Third Grade Fluid in a Porous Half Space. Communications in Nonlinear Science and Numerical Simulation, 13, 1801-1807.
http://dx.doi.org/10.1016/j.cnsns.2007.04.015 [19] Fakhari, K., Zainal, A.A. and Kara, A.H. (2011) A Note on the Interplay between Symmetries, Reduction and Conservation Laws of Stokes’ First Problem for Third-Grade Rotating Fluids. Pramana Journal of Physics, 77, 439-445.
http://dx.doi.org/10.1007/s12043-011-0164-6 [20] Sajid, M., Ali, N., Javed T. and Abbas, Z. (2010) Stokes’ First Problem for a MHD Third Grade Fluid in a Porous Half Space. Journal of Porous Media, 1, 279-284. [21] Sato, H. (1961) The Hall Effects in the Viscous Flow of Ionized Gas between Parallel Plates under Transverse Magnetic Field. Journal of the Physical Society of Japan, 16, 1427-1433.
http://dx.doi.org/10.1143/JPSJ.16.1427 [22] Cramer, K. and Pai, S. (1973) Magnetofluid Dynamics for Engineers and Applied Physicists. McGraw-Hill, New York. [23] Ayub, M., Zaman, H. and Ahmad, M. (2010) Series Solution of Hydromagnetic Flow and Heat Transfer with Hall Effect in a Second Grade Fluid over a Stretching Sheet. Central European Journal of Physics, 8, 135-149.
http://dx.doi.org/10.2478/s11534-009-0110-0 [24] Ahmad, M., Zaman, H. and Rehman, N. (2010) Effects of Hall Current on Unsteady MHD Flows of a Second Grade Fluid. Central European Journal of Physics, 8, 422-431.
http://dx.doi.org/10.2478/s11534-009-0083-z [25] Zaman, H. (2013) Hall Effects on the Unsteady Incompressible MHD Fluid Flow with Slip Conditions and Porous Walls. Applied Mathematics and Physics, 1, 31-38. [26] Hayat, T., Zaman, H. and Ayub, M. (2010) Analytic Solution of Hydromagnetic Flow with Hall Effect over a Surface Stretching with a Power Law Velocity. Numerical Methods for Partial Differential Equations, 27, 937-959. [27] Hayat, T., Naz, R. and Asghar, S. (2004) Hall Effects on Unsteady Duct Flow of a Non-Newtonian Fluid in a Porous Medium. Applied Mathematics and Computation, 157, 103-114.
http://dx.doi.org/10.1002/num.20562 [28] Hayat, T. and Nawaz, M. (2011) Hall and Ion-Slip Effects on Three-Dimensional Flow of a Second Grade Fluid. International Journal for Numerical Methods in Fluids, 66, 183-193.
http://dx.doi.org/10.1016/j.amc.2003.08.069 [29] Pop I. and Soundalgekar, V.M. (1974) Effects of Hall Current on Hydromagnetic Flow near a Porous Plate. Acta Mechanica, 20, 315-318.
http://dx.doi.org/10.1007/BF01175933 [30] Gupta, A.S. (1975) Hydromagnetic Flow Past a Porous Flate Plate with Hall Effects. Acta Mechanica, 22, 281-287.
http://dx.doi.org/10.1007/BF01170681 [31] Debnath, L., Ray, S.C. and Chatterjee, A.K. (1979) Effects of Hall Current on Unsteady Hydromagnetic Flow Past a Porous Plate in a Rotating Fluid System. Zeitschrift für Angewandte Mathematik und Mechanik, 59, 469-471.
http://dx.doi.org/10.1002/zamm.19790590910 [32] Katagiri, M. (1969) The Effect of Hall Currents on the Magnetohydrodynamic Boundary Layer Flow Past a Semi-Infinite Flate Plate. Journal of the Physical Society of Japan, 27, 1051-1059.
http://dx.doi.org/10.1143/JPSJ.27.1051 [33] Abo-Eldahab, E.M. and Elbarbary, M.E. (2001) Hall Current Effect on Magnetohydrodynamic Free Convection Flow Past a Semi-Infinite Vertical Plate with Mass Transfer. International Journal of Engineering Science, 39, 1641-1652.
http://dx.doi.org/10.1016/S0020-7225(01)00020-9 [34] Abo-Eldahab, E.M. and Salem, A.M. (2004) Hall Effects on MHD Free Convection Flow of a Non-Newtonian Power Law Fluid at a Stretching Surface. International Communications in Heat and Mass Transfer, 31, 343-354.
http://dx.doi.org/10.1016/j.icheatmasstransfer.2004.02.005 [35] Attia, H.A. (2006) Hall Effects on the Flow of a Dusty Bingham Fluid in a Circular Pipe. Turkish Journal of Engineering and Environmental Sciences, 30, 14-21. [36] Attia, H.A. (1998) Hall Current Effects on the Velocity and Temperature Fields of an Unsteady Hartman Flow. Canadian Journal of Physics, 76, 739-746. [37] Liao, S.J. (2003) Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman and Hall/CRC Press, Florida.
http://dx.doi.org/10.1201/9780203491164 [38] Liao, S.J. (1992) The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problem. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai. [39] Liao, S.J. (2012) Homotopy Analysis Method in Nonlinear Differential Equations. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-25132-0 [40] Liao, S.J. (2013) Advances in the Homotopy Analysis Method. World Scientific Publishing Company, Singapore. [41] Liao, S.J. (2009) Notes on the Homotopy Analysis Method: Some Definitions and Theorems. Communications in Non-linear Science and Numerical Simulation, 14, 983-997. [42] Liao, S.J. (2006) An Analytic Solution of Unsteady Boundary-Layer Flows Caused by an Impulsively Stretching Plate. Communications in Nonlinear Science and Numerical Simulation, 11, 326-339. [43] Ayub, M., Zaman, H., Sajid, M. and Hayat, T. (2008) Analytical Solution of Stagnation-Point Flow of a Viscoelastic Fluid towards a Stretching Surface. Communications in Nonlinear Science and Numerical Simulation, 13, 1822-1835.
http://dx.doi.org/10.1016/j.cnsns.2007.04.021 [44] Zaman, H. and Ayub, M. (2010) Series Solution of Unsteady Free Convection Flow with Mass Transfer along an Accelerated Vertical Porous Plate with Suction. Central European Journal of Physics, 8, 931-939.
http://dx.doi.org/10.2478/s11534-010-0007-y [45] Zaman, H., Hayat, T., Ayub, M. and Gorla, R.S.R. (2011) Series Solution for Heat Transfer from a Continuous Surface in a Parallel Free Stream of Viscoelastic Fluid. Numerical Methods for Partial Differential Equations, 27, 1511-1524.
http://dx.doi.org/10.1002/num.20593 [46] Rivlin, R.S. and Ericksen, J.L. (1955) Stress Deformation Relations for Isotropic Materials. Journal of Rational Mechanics and Analysis, 4, 323-425. [47] Sutton, G.W. and Sherman, A. (1965) Engineering Magnetohydrodynamics. McGraw-Hill, New York.
[1] Stokes, G.G. (1851) On the Effect of the Internal Friction of Fluids on the Motion of Pendulums. Transactions of the Cambridge Philosophical Society, Part II, 9, 8-106. [2] Tanner, R.I. (1962) Note on the Rayleigh Problem for a Visco-Elastic Fluid. Zeitschrift für Angewandte Mathematik und Physik, 13, 573-580.
http://dx.doi.org/10.1007/BF01595580 [3] Soundalgekar, V.M. (1974) Stokes Problem for Elastico-Viscous Fluid. Rheologica Acta, 13, 177-179.
http://dx.doi.org/10.1007/BF01520872 [4] Teipel, I. (1981) The Impulsive Motion of a Flat Plate in a Viscoelastic Fluid. Acta Mechanica, 39, 277-279.
http://dx.doi.org/10.1007/BF01170349 [5] Puri, P. (1984) Impulsive Motion of a Flat Plate in a Rivlin-Ericksen Fluid. Rheologica Acta, 23, 451-453.
http://dx.doi.org/10.1007/BF01329198 [6] Erdogan, M.E. (1995) Plane Surface Suddenly Set in Motion in a Non-Newtonian Fluid. Acta Mechanica, 108, 179- 187.
http://dx.doi.org/10.1007/BF01177337 [7] Zeng, Y. and Weinbaum, S. (1995) Stokes Problems for Moving Half-Planes. Journal of Fluid Mechanics, 287, 59-74.
http://dx.doi.org/10.1017/S0022112095000851 [8] Tan, W.C. and Xu, M.Y. (2002) The Impulsive Motion of Flat Plate in a Generalized Second Grade Fluid. Mechanics Research Communications, 29, 3-9.
http://dx.doi.org/10.1016/S0093-6413(02)00223-9 [9] Tan, W.C. and Xu, M.Y. (2002) Plane Surface Suddenly Set in Motion in a Viscoelastic Fluid with Fractional Maxwell Model. Acta Mechanica Sinica, 18, 342-349.
http://dx.doi.org/10.1007/BF02487786 [10] Erdogan, M.E. (2002) On the Unsteady Unidirectional Flows Generated by Impulsive Motion of a Boundary or Sudden Application of a Pressure Gradient. International Journal of Non-Linear Mechanics, 37, 1091-1106.
http://dx.doi.org/10.1016/S0020-7462(01)00035-X [11] Fetecau, C. and Fetecau, C. (2003) The First Problem of Stokes for an Oldroyd-B Fluid. International Journal of Non-Linear Mechanics, 38, 1539-1544.
http://dx.doi.org/10.1016/S0020-7462(02)00117-8 [12] Tan, W.C. and Masuoka, T. (2005) Stokes’ First Problem for an Oldroyd-B Fluid in a Porous Half Space. Physics of Fluids, 17, 023101.
http://dx.doi.org/10.1063/1.1850409 [13] Tan, W.C. and Masuoka, T. (2005) Stokes’ First Problem for a Second Grade Fluid in a Porous Half-Space with Heated Boundary. International Journal of Non-Linear Mechanics, 40, 515-522.
http://dx.doi.org/10.1016/j.ijnonlinmec.2004.07.016 [14] Zierep, J. and Fetecau, C. (2007) Energetic Balance for the Rayleigh-Stokes Problem of a Second Grade Fluid. International Journal of Engineering Science, 45, 155-162.
http://dx.doi.org/10.1016/j.ijengsci.2006.09.001 [15] Zierep, J., Bohning, R. and Fetecau, C. (2007) Rayleigh-Stokes Problem for Non-Newtonian Medium with Memory. Zeitschrift für Angewandte Mathematik und Mechanik, 87, 462-467.
http://dx.doi.org/10.1002/zamm.200710328 [16] Vieru, D., Nazar, M., Fetecau, C. and Fetecau, C. (2008) New Exact Solutions Corresponding to the First Problem of Stokes for Oldroyd-B Fluids. Computers & Mathematics with Applications, 55, 1644-1652.
http://dx.doi.org/10.1016/j.camwa.2007.04.040 [17] Shahzad, F., Hayat, T. and Ayub, M. (2008) Stokes’ First Problem for the Rotating Flow of a Third Grade Fluid. Nonlinear Analysis: Real World Applications, 9, 1794-1799.
http://dx.doi.org/10.1016/j.nonrwa.2007.05.008 [18] Hayat, T., Shahzad, F., Ayub, M. and Asghar, S. (2008) Stokes’ First Problem for a Third Grade Fluid in a Porous Half Space. Communications in Nonlinear Science and Numerical Simulation, 13, 1801-1807.
http://dx.doi.org/10.1016/j.cnsns.2007.04.015 [19] Fakhari, K., Zainal, A.A. and Kara, A.H. (2011) A Note on the Interplay between Symmetries, Reduction and Conservation Laws of Stokes’ First Problem for Third-Grade Rotating Fluids. Pramana Journal of Physics, 77, 439-445.
http://dx.doi.org/10.1007/s12043-011-0164-6 [20] Sajid, M., Ali, N., Javed T. and Abbas, Z. (2010) Stokes’ First Problem for a MHD Third Grade Fluid in a Porous Half Space. Journal of Porous Media, 1, 279-284. [21] Sato, H. (1961) The Hall Effects in the Viscous Flow of Ionized Gas between Parallel Plates under Transverse Magnetic Field. Journal of the Physical Society of Japan, 16, 1427-1433.
http://dx.doi.org/10.1143/JPSJ.16.1427 [22] Cramer, K. and Pai, S. (1973) Magnetofluid Dynamics for Engineers and Applied Physicists. McGraw-Hill, New York. [23] Ayub, M., Zaman, H. and Ahmad, M. (2010) Series Solution of Hydromagnetic Flow and Heat Transfer with Hall Effect in a Second Grade Fluid over a Stretching Sheet. Central European Journal of Physics, 8, 135-149.
http://dx.doi.org/10.2478/s11534-009-0110-0 [24] Ahmad, M., Zaman, H. and Rehman, N. (2010) Effects of Hall Current on Unsteady MHD Flows of a Second Grade Fluid. Central European Journal of Physics, 8, 422-431.
http://dx.doi.org/10.2478/s11534-009-0083-z [25] Zaman, H. (2013) Hall Effects on the Unsteady Incompressible MHD Fluid Flow with Slip Conditions and Porous Walls. Applied Mathematics and Physics, 1, 31-38. [26] Hayat, T., Zaman, H. and Ayub, M. (2010) Analytic Solution of Hydromagnetic Flow with Hall Effect over a Surface Stretching with a Power Law Velocity. Numerical Methods for Partial Differential Equations, 27, 937-959. [27] Hayat, T., Naz, R. and Asghar, S. (2004) Hall Effects on Unsteady Duct Flow of a Non-Newtonian Fluid in a Porous Medium. Applied Mathematics and Computation, 157, 103-114.
http://dx.doi.org/10.1002/num.20562 [28] Hayat, T. and Nawaz, M. (2011) Hall and Ion-Slip Effects on Three-Dimensional Flow of a Second Grade Fluid. International Journal for Numerical Methods in Fluids, 66, 183-193.
http://dx.doi.org/10.1016/j.amc.2003.08.069 [29] Pop I. and Soundalgekar, V.M. (1974) Effects of Hall Current on Hydromagnetic Flow near a Porous Plate. Acta Mechanica, 20, 315-318.
http://dx.doi.org/10.1007/BF01175933 [30] Gupta, A.S. (1975) Hydromagnetic Flow Past a Porous Flate Plate with Hall Effects. Acta Mechanica, 22, 281-287.
http://dx.doi.org/10.1007/BF01170681 [31] Debnath, L., Ray, S.C. and Chatterjee, A.K. (1979) Effects of Hall Current on Unsteady Hydromagnetic Flow Past a Porous Plate in a Rotating Fluid System. Zeitschrift für Angewandte Mathematik und Mechanik, 59, 469-471.
http://dx.doi.org/10.1002/zamm.19790590910 [32] Katagiri, M. (1969) The Effect of Hall Currents on the Magnetohydrodynamic Boundary Layer Flow Past a Semi-Infinite Flate Plate. Journal of the Physical Society of Japan, 27, 1051-1059.
http://dx.doi.org/10.1143/JPSJ.27.1051 [33] Abo-Eldahab, E.M. and Elbarbary, M.E. (2001) Hall Current Effect on Magnetohydrodynamic Free Convection Flow Past a Semi-Infinite Vertical Plate with Mass Transfer. International Journal of Engineering Science, 39, 1641-1652.
http://dx.doi.org/10.1016/S0020-7225(01)00020-9 [34] Abo-Eldahab, E.M. and Salem, A.M. (2004) Hall Effects on MHD Free Convection Flow of a Non-Newtonian Power Law Fluid at a Stretching Surface. International Communications in Heat and Mass Transfer, 31, 343-354.
http://dx.doi.org/10.1016/j.icheatmasstransfer.2004.02.005 [35] Attia, H.A. (2006) Hall Effects on the Flow of a Dusty Bingham Fluid in a Circular Pipe. Turkish Journal of Engineering and Environmental Sciences, 30, 14-21. [36] Attia, H.A. (1998) Hall Current Effects on the Velocity and Temperature Fields of an Unsteady Hartman Flow. Canadian Journal of Physics, 76, 739-746. [37] Liao, S.J. (2003) Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman and Hall/CRC Press, Florida.
http://dx.doi.org/10.1201/9780203491164 [38] Liao, S.J. (1992) The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problem. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai. [39] Liao, S.J. (2012) Homotopy Analysis Method in Nonlinear Differential Equations. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-25132-0 [40] Liao, S.J. (2013) Advances in the Homotopy Analysis Method. World Scientific Publishing Company, Singapore. [41] Liao, S.J. (2009) Notes on the Homotopy Analysis Method: Some Definitions and Theorems. Communications in Non-linear Science and Numerical Simulation, 14, 983-997. [42] Liao, S.J. (2006) An Analytic Solution of Unsteady Boundary-Layer Flows Caused by an Impulsively Stretching Plate. Communications in Nonlinear Science and Numerical Simulation, 11, 326-339. [43] Ayub, M., Zaman, H., Sajid, M. and Hayat, T. (2008) Analytical Solution of Stagnation-Point Flow of a Viscoelastic Fluid towards a Stretching Surface. Communications in Nonlinear Science and Numerical Simulation, 13, 1822-1835.
http://dx.doi.org/10.1016/j.cnsns.2007.04.021 [44] Zaman, H. and Ayub, M. (2010) Series Solution of Unsteady Free Convection Flow with Mass Transfer along an Accelerated Vertical Porous Plate with Suction. Central European Journal of Physics, 8, 931-939.
http://dx.doi.org/10.2478/s11534-010-0007-y [45] Zaman, H., Hayat, T., Ayub, M. and Gorla, R.S.R. (2011) Series Solution for Heat Transfer from a Continuous Surface in a Parallel Free Stream of Viscoelastic Fluid. Numerical Methods for Partial Differential Equations, 27, 1511-1524.
http://dx.doi.org/10.1002/num.20593 [46] Rivlin, R.S. and Ericksen, J.L. (1955) Stress Deformation Relations for Isotropic Materials. Journal of Rational Mechanics and Analysis, 4, 323-425. [47] Sutton, G.W. and Sherman, A. (1965) Engineering Magnetohydrodynamics. McGraw-Hill, New York.