Exact Solution of Second Grade Fluid in a Rotating Frame through Porous Media Using Hodograph Transformation Method

Sayantan Sil^1*, Manoj Kumar²

¹Department of Physics, S.S.L.N.T. Mahila College, Dhanbad, India.
²Post Graduate Department of Physics, Ranchi College, Ranchi, India.

DOI: 10.4236/jamp.2015.311172                

Abstract

In this paper exact solution for a homogenous incompressible, second grade fluid in a rotating frame through porous media has been provided using hodograph-Legendre transformation method. Results are summarised in the form of theorems. Two examples have been taken and streamline patterns are shown for the solutions.

Keywords

Non-Newtonian Fluid, Rotating Frame, Hodograph Transformation, Porous Media, Exact Solution

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Jana, R.N. and Dutta, N. (1977) Couette Flow and Heat Transfer in a Rotating System. Acta Mechanica, 26, 301-306. http://dx.doi.org/10.1007/BF01177152
[2]	Vidyanidhu, V. and Nigam, S.D. (1967) Secondary Flow in a Rotating Channel. Journal of Mathematical and Physical Sciences, 1, 85-100.
[3]	Soundalgekar, V.M. and Pop, I. (1973) On Hydromagnetic Flow in a Rotating Fluid Past an Infinite Porous Wall. Journal of Applied Mathematics and Mechanics, ZAMM, 53, 718-719. http://dx.doi.org/10.1002/zamm.19730531012
[4]	Gupta, A.S. (1972) Ekman Layer on a Porous Plate. Physics of Fluids, 15, 930-931. http://dx.doi.org/10.1063/1.1694002
[5]	Bagewadi, C.S. and Siddabasappa (1993) Study of Variably Inclined Rotating MHD Flows in Magnetograph Plane. Bulletin of the Calcutta Mathematical Society, 85, 93-106.
[6]	Bagewadi, C.S. and Siddabasappa (1993) The Plane Rotating Viscous MHD Flows. Bulletin of the Calcutta Mathematical Society, 85, 513-520.
[7]	Singh, S.N., Singh, H.P. and Rambabu (1984) Hodograph Transformations in Steady Plane Rotating Hydromagnetic Flow. Astrophysics and Space Science, 106, 231-243. http://dx.doi.org/10.1007/BF00650351
[8]	Singh, K.K. and Singh, D.P. (1993) Steady Plane MHD Flows through Porous Media with Constant Speed along Each Stream Line. Bulletin of the Calcutta Mathematical Society, 85, 255-262.
[9]	Thakur, C. and Kumar, M. (2008) Plane Rotating Viscous MHD Flows through Porous Media. Pure and Applied Mathematika Sciences, LXVII, 113-124.
[10]	Singh, H.P. and Tripathi, D.D. (1988) A Class of Exact Solutions in Plane Rotating MHD Flows. Indian Journal of Pure and Applied Mathematics, 19, 677-687.
[11]	Singh, K.D. (2013) Rotating Oscillatory MHD Poisseuille Flow: An Exact Solution. Kragujevac Journal of Science, 35, 15-25.
[12]	Imran, M.A., Imran, M. and Fetecau, C. (2014) MHD OSCILLATING Flows of a Rotating Second Grade Fluid in Porous Medium. Communication in Numerical Analysis, 2014, Article ID: cna-00196. www.ispacs.com/journals/cna/2014/cna-00196/article.pdf
[13]	Rashid, A.M. (2014) Effects of Radiation and Variable Viscosity on Unsteady MHD Flow of a Rotating Fluid from Stretching Surface in Porous Media. Journal of Egyptian Mathematical Society, 2, 134-142. http://dx.doi.org/10.1016/j.joems.2013.05.008
[14]	Sil, S. and Kumar, M. (2014) A Class of Solution of Orthogonal Plane MHD Flow through Porous Media in a Rotating Frame. Global Journal of Science Frontier Research: A Physics and Space Science, 14, 17-26. https://globaljournals.org/GJSFR_Volume14/E-Journal_GJSFR_%28A%29_Vol_14_Issue_7.pdf
[15]	Chandna, O.P. and Nguyen, P.V. (1989) Hodograph Method in Non-Newtonian MHD Transverse Fluid Flows. Journal of Engineering Mathematics, 23, 119-139. http://dx.doi.org/10.1007/BF00128864
[16]	Chandna, O.P. and. Garg, M.R (1979) On Steady Plane Magnetohydrodynamic Flows with Orthogonal Magnetic and Velocity Field. International Journal of Engineering Science, 17, 251-257. http://dx.doi.org/10.1016/0020-7225(79)90088-0
[17]	Martin, M.H. (1971) The Flow of Viscous Fluid. Archive for Rational Mechanics and Analysis, 41, 266-286. http://dx.doi.org/10.1007/BF00250530
[18]	Ames, W.F. (1965) Non-Linear Partial Differential Equations in Engineering. Academic Press, New York.
[19]	Ghaffari, A.G. (1950) The Hodograph Method in Gas Dynamics. Tehran Taban Press, Tehran.
[20]	Cherry, T.M. (1961) Trans-Sonic Nozzle Flows Found by the Hodograph Method. In: Langer, R.E., Ed., Partial Differential Equations and Continuum Mechanics, University of Wisconsin Press, Madison, 216-232.
[21]	Chandna, O.P., Barron, R.M. and Smith, A.C. (1982) Rotational Plane Steady Flows of a Viscous Fluid. SIAM Journal on Applied Mathematics, 42, 1323-1336. http://dx.doi.org/10.1137/0142092
[22]	Siddiqui, A.M., Kaloni, P.N. and Chandna, O.P. (1985) Hodograph Transformation Methods in Non-Newtonian Fluids. Journal of Engineering Mathematics, 19, 203-216. http://dx.doi.org/10.1007/BF00042534
[23]	Moro, L., Siddiqui, A.M. and Kaloni, P.N. (1990) Steady Flows of a Third-Grade Fluid by Transformation Methods. ZAMM, 70, 189-198. http://dx.doi.org/10.1002/zamm.19900700309
[24]	Swaminathan, M.K., Chandna, O.P. and Sridhar, K. (1983) Hodograph Study of Transverse MHD Flows. Canadian Journal of Physics, 61, 1323-1336. http://dx.doi.org/10.1139/p83-134
[25]	Barron, R.M. and Chandna, O.P. (1981) Hodograph Transformation and Solutions in Constantly Inclined Plane Flows. Journal of Engineering Mathematics, 15, 211-220. http://dx.doi.org/10.1007/BF00042781
[26]	Chandna, O.P., Barron, R.M. and Chew, K.T. (1982) Hodograph Transformations and Solutions in Variably Inclined MHD Plane Flows. Journal of Engineering Mathematics, 16, 223-243. http://dx.doi.org/10.1007/BF00042718
[27]	Singh, H.P. and Mishra, R.B. (1987) Legendre Transformation in Steady Plane MHD Flows of a Viscous Fluid. Indian Journal of Pure and Applied Mathematics, 18, 100-109.
[28]	Thakur, C. and Mishra, R.B. (1988) On Steady Plane Rotating Hydromagnetic Flows. Astrophysics and Space Science, 146, 89-97. http://dx.doi.org/10.1007/BF00656985
[29]	Singh, S.N. and Tripathi, D.D. (1987) Hodograph Transformations in Steady Plane Rotating MHD Flows. Applied Scientific Research, 43, 347-353. http://dx.doi.org/10.1007/BF00540568
[30]	Siddiqui, A.M., Hayat, T., Siddiqui, J. and Asghar, S. (2008) Exact Solutions of Time-Dependent Navier-Stokes Equations by Hodograph-Legendre Transformation Method. Tamsui Oxford Journal of Mathematical Sciences, 24, 257-268.
[31]	Mishra, P. and Mishra, R.B. (2010) Hodograph Transformations in Unsteady MHD Transverse Flows. Applied Mathematical Sciences, 56, 2781-2795.
[32]	Sil, S., Kumar, M. and Thakur, C. (2012) Solutions of Non-Newtonian MHD Transverse Fluid Flows through Porous Media. Proceedings of the 57th Congress of ISTAM, Defence Institute of Advanced Technology, Pune, 17-20 December 2012, 13-21.
[33]	Kumar, M. (2014) Solution of Non-Newtonian Fluid Flows through Porous Media by Hodograph Transformation Method. Bulletin of Calcutta Mathematical Society, 106, 239-250.
[34]	Ram, G. and Mishra, R.S. (1977) Unsteady MHD Flow of Fluid through Porous Medium in a Circular Pipe. Indian Journal of Pure and Applied Mathematics, 8, 637-647.
[35]	Thakur, C. and Singh, B. (2000) Study of Variably Inclined MHD Flows through Porous Media in Magnetograph Plane. Bulletin of Calcutta Mathematical Society, 92, 39-50.
[36]	Thakur, C., Kumar, M. and Mahan, M.K. (2006) Exact Solution of Steady MHD Orthogonal Plane Fluid Flows through Porous Media. Bulletin of Calcutta Mathematical Society, 98, 583-596.
[37]	Bhatt, B. and Shirley, A. (2008) Plane Viscous Flows in Porous Medium. Matematicas: Ensenanza Universitaria, XVI, 51-62.