Superposability in Hydrodynamic and MHD Flow
ABSTRACT
In this paper, phenomena of superposability and self superposability in hydrodynamics and magneto hydrodynamics have been discussed. One of the most important applications of superposability in hydrodynamics is the construction of exact analytic solution of the basic equation of fluid dynamics. Kapur and Bhatia have given a simple idea that if two velocity vectors have self superposable and mutually superposable motion then sum or difference of these two is self superposable and vice versa and if each of the vector is superposable on the third then their sum and difference are also superposable on the third. For superposability in magneto-hydrodynamics many mathematicians like Ram Moorthy, Ram Ballabh, Mittal, Kapur & Bhatia and Gold & Krazyblocki have defined it in various ways, especially Kapur & Bhatia generalized the well-known work on superposability by Ram Ballabh to the case of viscous incompressible electrically conducting fluids in the presence of magnetic field. We found the relationship of two basic vectors for two important curvilinear coordinate systems for their use in our work. We’ve found the equations of div, curl and grad for a unit vector in parabolic cylinder coordinates and ellipsoidal coordinates for further use.
In this paper, phenomena of superposability and self superposability in hydrodynamics and magneto hydrodynamics have been discussed. One of the most important applications of superposability in hydrodynamics is the construction of exact analytic solution of the basic equation of fluid dynamics. Kapur and Bhatia have given a simple idea that if two velocity vectors have self superposable and mutually superposable motion then sum or difference of these two is self superposable and vice versa and if each of the vector is superposable on the third then their sum and difference are also superposable on the third. For superposability in magneto-hydrodynamics many mathematicians like Ram Moorthy, Ram Ballabh, Mittal, Kapur & Bhatia and Gold & Krazyblocki have defined it in various ways, especially Kapur & Bhatia generalized the well-known work on superposability by Ram Ballabh to the case of viscous incompressible electrically conducting fluids in the presence of magnetic field. We found the relationship of two basic vectors for two important curvilinear coordinate systems for their use in our work. We’ve found the equations of div, curl and grad for a unit vector in parabolic cylinder coordinates and ellipsoidal coordinates for further use.
Cite this paper
Rastogi, S. , Kaul, B. and Rajan, S. (2015) Superposability in Hydrodynamic and MHD Flow. Open Journal of Fluid Dynamics, 5, 151-170. doi: 10.4236/ojfd.2015.52018.
Rastogi, S. , Kaul, B. and Rajan, S. (2015) Superposability in Hydrodynamic and MHD Flow. Open Journal of Fluid Dynamics, 5, 151-170. doi: 10.4236/ojfd.2015.52018.
References
[1] Ballabh, R. (1952) Two Dimensional Superposable Motions. Journal of the Indian Mathematical Society, 16, 191-197. [2] Bhatia, B.L. (1971) Continuity Condition in Parabolic Cylinder System. The Mathematics Student, 39, 167-180. [3] Gupta, S. and Rajan, S. (2007) On Some Magnetohydrodynamics Configuration in Parabolic Cylindical Ducts. Acta Ciencia Indica, XXXIIIM, 919-926. [4] Mittal, P.K., Singh, V. and Rajan, S. (1997) A Note on Vorticity of Hydromagnetic Converting Slip Flow through a Horizontal Channel. Acta Ciencia Indica, XXIIIM, 117. [5] Mittal, P.K. (1983) On Some Magnetohydrostatic Configurations in Parabolic Coordinated. Bulletin of Calcutta Mathematical Society, 75, 339-352. [6] Mittal, P.K. (1986) On Some Self-Superposable Fluid Motions in Paraboloidal Ducts. International Journal of Theoretical Physics, 34, 181-191. [7] Mittal, P.K. and Khan, M.I. (1986) On Some Self-Superposable Flows in Conical Ducts. Jhanabha, 16, 91-102. [8] Mittal, P.K., Thapaliyal, P.S. and Khan, M.I. (1987) On Some Magnetohydrodynamics Configurations in Conical Ducts. Indian Journal of Physics and Natural Science, 8B, 7-14. [9] Mittal, P.K., Chandra, S. and Salam, S.B. (1987) On Some Magnetohydrodynamics Configuration in Parabolic Ducts. Indian Journal of Physics and Natural Science, 8B, 48-53. [10] Ballabh, R. (1963) U.P. Scientific Committee Monograph 1963. [11] Truesdall, C. (1954) The Kinematics of Vorticity. Indiana University Press, Bloomington. [12] Gold, R.R. and von Krzywoblocki, M.Z. (1958) On Superposability and Self-Superposability Conditions for Hydrodynamic Equations Based on Continuum. I. Journal für die reine und angewandte Mathematik, 199, 139-164. [13] Gold, R.R. and von Krzywoblocki, M.Z. (1958) On Superposability and Self-Superposability Conditions for Hydrodynamic Equations Based on Continuum. II. Journal für die reine und angewandte Mathematik, 200, 140-169. [14] Gupta, S. and Rajan, S. (2007) Rotating Disk Flow with Heat Transfer of a Non-Newtonian Fluid in Porous Medium. Acta Ciencia Indica, XXXIIIM, 1031-1036. [15] Kapur, J.N. and Bhatia, B.L. (1965) Superposability and Self-Superposability in Fluiddynamics-II. Proceedings of the National Institute of Sciences of India, 31A, 126-151. [16] Kapur, J.N. (1959) Superposability in Magnetohydrodynamics. Applied Scientific Research, 8, 198-208. [17] Kapur, J.N. (1960) Superposability in Magnetohydrodynamics II. Applied Scientific Research, 9, 139-147. [18] Bhatnagar, P.L. (1960) Superposability and Harmonic Analysis of Flows of a Viscous Liquid in the Presence of Magnetic Field. Jubilee Commemoration Volume of the Calcutta Mathematical Society, 205-216. [19] Kapur, J.N. (1961) Superposability and Self-Superposability in Fluid Dynamics. The Mathematics Seminar, 2, 1-31. [20] Kapur, J.N. (1962) Some Properties of Force-Free Fields. The Mathematics Seminar, 2, 135-138. [21] Kapur, J.N. (1962) Characterisation of Axially-Symmetric Self-Superposable Flows in Magnetohydrodynamic. Bulletin of Calcutta Mathematical Society, 59-66. [22] Chandrasekhar, S. (1981) Hydrodynamics and Hydromagnetics Stability. Dover Publication, New York. [23] Rao, G.T. (1960) Superposability of the Equations of Magneto-Hydrodynamics. Journal of the Mathematical Society of Japan, 12, 97-103. [24] Ramamoorthy, P. (1960) Superposability of Two Axi-Symmetric Flows under Axi-Symmetric Magnetic Fields. Applied Scientific Research, 9, 153-156. [25] Mittal, P.K. (1977) A Note on Vorticity in Conducting Fluid. The Mathematics Student, 45, 84-88. [26] Mittal, P.K. (1981) A Note on Steady Laminar Magneto Hydrodynamic Flow. Bulletin of Calcutta Mathematical Society, 73, 179-184. [27] Mittal, P.K. (1978) On the Vorticity of the MHD Flow in a Rectangular Duct. The Mathematics Education, 12, 5-9. [28] Mittal, P.K., Thapaliyal, P.S. and Aarwal, G.K. (1987) On Some Self-Superposable Fluid Motions in Toroidal Ducts. Proceedings of the National Academy of Sciences, India, Section A, 57, 224-229. [29] Mittal, P.K. and Rastogi, S.C. (1987) Self-Superposable Motions in Paraboloidal Ducts. Indian Journal of Physics and Natural Science, 8B, 27-32. [30] Mittal, P.K., Negi, B.S. and Shamshi, S.R. (1986) Self-Superposable Flows in Ducts having Confocal Ellipsoidal Shape. Indian Journal of Physics and Natural Science, 7B, 1-10. [31] Rastogi, S., Kaul, B.N. and Rajan, S. (2015) Few Magnetohydrostatic Forms in Confocal Paraboloidal Ducts. International Journal of Mathematics and Computer Application Research, 5, 25-34. [32] Singh, V. and Rajan, S. (1997) On Some Hydrodynamic Flows in Ellipsoidal Ducts. Acta Ciencia Indica, XXXIIIM, 111.
[1] Ballabh, R. (1952) Two Dimensional Superposable Motions. Journal of the Indian Mathematical Society, 16, 191-197. [2] Bhatia, B.L. (1971) Continuity Condition in Parabolic Cylinder System. The Mathematics Student, 39, 167-180. [3] Gupta, S. and Rajan, S. (2007) On Some Magnetohydrodynamics Configuration in Parabolic Cylindical Ducts. Acta Ciencia Indica, XXXIIIM, 919-926. [4] Mittal, P.K., Singh, V. and Rajan, S. (1997) A Note on Vorticity of Hydromagnetic Converting Slip Flow through a Horizontal Channel. Acta Ciencia Indica, XXIIIM, 117. [5] Mittal, P.K. (1983) On Some Magnetohydrostatic Configurations in Parabolic Coordinated. Bulletin of Calcutta Mathematical Society, 75, 339-352. [6] Mittal, P.K. (1986) On Some Self-Superposable Fluid Motions in Paraboloidal Ducts. International Journal of Theoretical Physics, 34, 181-191. [7] Mittal, P.K. and Khan, M.I. (1986) On Some Self-Superposable Flows in Conical Ducts. Jhanabha, 16, 91-102. [8] Mittal, P.K., Thapaliyal, P.S. and Khan, M.I. (1987) On Some Magnetohydrodynamics Configurations in Conical Ducts. Indian Journal of Physics and Natural Science, 8B, 7-14. [9] Mittal, P.K., Chandra, S. and Salam, S.B. (1987) On Some Magnetohydrodynamics Configuration in Parabolic Ducts. Indian Journal of Physics and Natural Science, 8B, 48-53. [10] Ballabh, R. (1963) U.P. Scientific Committee Monograph 1963. [11] Truesdall, C. (1954) The Kinematics of Vorticity. Indiana University Press, Bloomington. [12] Gold, R.R. and von Krzywoblocki, M.Z. (1958) On Superposability and Self-Superposability Conditions for Hydrodynamic Equations Based on Continuum. I. Journal für die reine und angewandte Mathematik, 199, 139-164. [13] Gold, R.R. and von Krzywoblocki, M.Z. (1958) On Superposability and Self-Superposability Conditions for Hydrodynamic Equations Based on Continuum. II. Journal für die reine und angewandte Mathematik, 200, 140-169. [14] Gupta, S. and Rajan, S. (2007) Rotating Disk Flow with Heat Transfer of a Non-Newtonian Fluid in Porous Medium. Acta Ciencia Indica, XXXIIIM, 1031-1036. [15] Kapur, J.N. and Bhatia, B.L. (1965) Superposability and Self-Superposability in Fluiddynamics-II. Proceedings of the National Institute of Sciences of India, 31A, 126-151. [16] Kapur, J.N. (1959) Superposability in Magnetohydrodynamics. Applied Scientific Research, 8, 198-208. [17] Kapur, J.N. (1960) Superposability in Magnetohydrodynamics II. Applied Scientific Research, 9, 139-147. [18] Bhatnagar, P.L. (1960) Superposability and Harmonic Analysis of Flows of a Viscous Liquid in the Presence of Magnetic Field. Jubilee Commemoration Volume of the Calcutta Mathematical Society, 205-216. [19] Kapur, J.N. (1961) Superposability and Self-Superposability in Fluid Dynamics. The Mathematics Seminar, 2, 1-31. [20] Kapur, J.N. (1962) Some Properties of Force-Free Fields. The Mathematics Seminar, 2, 135-138. [21] Kapur, J.N. (1962) Characterisation of Axially-Symmetric Self-Superposable Flows in Magnetohydrodynamic. Bulletin of Calcutta Mathematical Society, 59-66. [22] Chandrasekhar, S. (1981) Hydrodynamics and Hydromagnetics Stability. Dover Publication, New York. [23] Rao, G.T. (1960) Superposability of the Equations of Magneto-Hydrodynamics. Journal of the Mathematical Society of Japan, 12, 97-103. [24] Ramamoorthy, P. (1960) Superposability of Two Axi-Symmetric Flows under Axi-Symmetric Magnetic Fields. Applied Scientific Research, 9, 153-156. [25] Mittal, P.K. (1977) A Note on Vorticity in Conducting Fluid. The Mathematics Student, 45, 84-88. [26] Mittal, P.K. (1981) A Note on Steady Laminar Magneto Hydrodynamic Flow. Bulletin of Calcutta Mathematical Society, 73, 179-184. [27] Mittal, P.K. (1978) On the Vorticity of the MHD Flow in a Rectangular Duct. The Mathematics Education, 12, 5-9. [28] Mittal, P.K., Thapaliyal, P.S. and Aarwal, G.K. (1987) On Some Self-Superposable Fluid Motions in Toroidal Ducts. Proceedings of the National Academy of Sciences, India, Section A, 57, 224-229. [29] Mittal, P.K. and Rastogi, S.C. (1987) Self-Superposable Motions in Paraboloidal Ducts. Indian Journal of Physics and Natural Science, 8B, 27-32. [30] Mittal, P.K., Negi, B.S. and Shamshi, S.R. (1986) Self-Superposable Flows in Ducts having Confocal Ellipsoidal Shape. Indian Journal of Physics and Natural Science, 7B, 1-10. [31] Rastogi, S., Kaul, B.N. and Rajan, S. (2015) Few Magnetohydrostatic Forms in Confocal Paraboloidal Ducts. International Journal of Mathematics and Computer Application Research, 5, 25-34. [32] Singh, V. and Rajan, S. (1997) On Some Hydrodynamic Flows in Ellipsoidal Ducts. Acta Ciencia Indica, XXXIIIM, 111.