## Main Article Content

## Abstract

This investigation is an approach to setup an analytical solution of steady plane allied MHD fluid flow having infinite electrical conductivity in a rotating frame through porous media by Martin’s method. The governing non-linear

equations of the fluid flow are transformed into a new form called Martin’s form by employing differential geometry where the curvilinear co-ordinates (Φ, Ψ) in the plane of flow shows that, the co-ordinate lines Ψ are the streamlines of flow and the co-ordinate lines Φ are arbitrary constants. Exact solution is obtained and velocity,

vorticity, current density magnetic field and pressure distribution are found out. Also, the diagrams have been plotted to sketch the streamline patterns and to study variation of pressure function with angular velocity.

## Keywords

## Article Details

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*How to Cite*

*Journal of the Indonesian Mathematical Society*,

*30*(1), 40–62. https://doi.org/10.22342/jims.30.1.1356.40-62

* * References

- Krishna, M. V., and Chamkha, A.J., Hall and ion slip effects on MHD rotating boundary layer flow of nanofluid past an infinite vertical plate embedded in a porous medium, Results in Physics, 15 (2019), 1026-1052.
- Krishna, M. V., Ahamad, N. A., and Chamkha, A. J., Hall and ion slip effects on unsteady MHD free convective rotating flow through a saturated porous medium over an exponential accelerated plate, Alexandria Engineering Journal, 59(2) (2020), 565-577.
- Prabhakar, R. B., Hall effect on MHD transient ow past an impulsively started infinite horizontal porous plate in a rotating system, International Journal of Applied Mechanics and Engineering, 23(2) (2018).
- Krishna, M. V., Ahmad, N. A., and Chamkha, A. J., Hall and ion slip impacts on unsteady MHD convective rotating flow of heat generating/absorbing second grade fluid, Alexandria Engineering Journal, 60(1) (2020), 845-858.
- Krishna, M. V., and Chamkha, A. J., Hall and ion slip effects on MHD rotating flow of elastic viscous fluid through porous medium, International Communications in Heat and Mass Transfer, 113 (2020), 104494.
- Krishna, M. V., Ahamad, N. A., and Chamkha, A. J., Hall and ion slip effects on unsteady MHD free convective rotating flow through a saturated porous medium over an exponential accelerated plate, Alexandria Engineering Journal, 59(2) (2020), 565-577.
- Bagewadi, C. S., and Siddabasappa, The plane rotating viscous MHD flows, Bulletin of Calcutta Mathematical Society, 85 (1993), 513.
- Thakur, C., and Mishra, R. B., On steady plane rotating hydromagnetic flows, Astrophysics and Space Science, 146(1) (1988), 89-97.
- Singh, S. N., and Tripathi, D. D., Hodograph transformations in steady plane rotating MHD flows, Applied Scientific research, 43 (1993), 347-353.
- Singh, S., Singh, H., and Mishra, R. B., Hodograph transformations in steady plane rotating hydromagnetic flow, Astrophysics and Space Science, 106 (1984), 231-243.
- Ram, G., and Mishra, R.S., Unsteady flow through magnetohydrodynamic porous media, Ind. Jour. Pure and appl., 8(6) (1977), 637-647
- Nguyen, P. V., and Chandna, O. P., Non-Newtonian MHD orthogonal steady plane fluid flows, Int. J. Eng. Sci., 30 (1992), 433-453.
- Rashid, A. M., 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(1) (2014), 134-142.
- Sil, S., and Kumar, M., 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(7) (2014), 17-26.
- Sil, S., and Kumar M., Exact solution of second grade fluid in a rotating frame through porous media using hodograph transformation method, J. of Appl. Math. and Phys., 3 (2015), 1443-1453.
- Singh, K. K., and Singh, D. P., Steady plane MHD flows through porous media with constant speed along each stream line, Bull. Cal. Math. Soc., 85 (1993), 255-262.
- Kumar, M., Solution of non-Newtonian fluid flows through porous media by hodograph transformation method, Bull. Cal. Math. Soc., 106(4) (2014), 239-250.
- Thakur, C., and Singh, B., Study of variably inclined MHD flows through porous media in magnetograph plan, Bull. Cal. Math. Soc., 92 (2000) 39-50.
- Bhatt, B., and Shirley, A., Plane viscous flows in porous medium, Matematicas: Ensenanza Universitaria, 16 (1) (2008), 51-62.
- M. H. Martin, The flow of a viscous fluid., Arch. Rat. Mech. Anal., 41 (1971), 266-286.
- Bagewadi, C.S., and Siddabasappa, The plane rotating viscous MHD flows., Bull. Cal. Math. Soc., 85 (1993), 513-520.
- K.V. Govindaraju, The flow of viscous fluid, Arch. Rat mech. Anal., 45 (1972), 66-80.
- Nath V.I., and Chandna, O.P, On plane viscous magnetohydrodynamic flows., Quart. Appl. Math., 31 (1973), 351-362.
- Chandna, O.P., Barron, R.M., and Garg, M.R., Plane compressible MHD flows., Quart. Appl. Math., 36 (1979), 411-422.
- Chandna, O.P., and Labropulu, F., Exact solutions of steady plane flows using Von Mises coordinates, J. Math. Anal. Appl., 185 (1994), 36-64.
- Naeem, R.K., and Nadeem S.A., Study of steady plane flows of an incompressible fluid of variable viscosity using Martins methods, J. App. Mech. and Engr., 1(3) (1996), 397-434.
- Naeem, R.K., and Ali, S.A., A class of exact solutions to equations governing the steady plane flows of an incompressible fluid of variable viscosity via von Mises variables, Int. J. App. Mech. and Engr., 6(2) (2001), 395-436.
- Chandna, O.P., and Labropulu, F., Exact Solutions of steady plane MHD aligned flows using von coordinates, Internat. J. Math. and Math Sci., 20(1) (1997), 165-186.
- Ali, S.A., Ara A., and Khan N.A., Martins method applied to steady plane flow of a second grade fluid, Int. J. App. Math, Mech., 3(3) (2007), 71-81.
- Thakur, C., Kumar, M., and Mahan, M.K., Martins method applied to constantly inclined viscous MHD flows through porous media., Bull. Cal. Math. Soc., 100 (2) (2008), 101-114.
- Kumar, M., Thakur C., Singh, T.P., and Mahan M.K., An exact solution of steady plane aligned MHD flow using Martins method, Proceedings of UGC sponsored National Seminar on Recent trends in engineering frontiers of Physical Sciencesa, B.I.T. Sindri, Dhanbad, India, (2009), 189-195.
- Naeem, R.K., Mansoor, A., Khan W.A., and Aurangjaib, Exact solutions of steady plane flows of an incompressible fluid of variable viscosity using (, ψ) or (η, ψ)-coordinates, Quart. Appl. Math., 34 (1976), 287-299.
- Naeem, R.K., Mansoor, A., Khan, W.A., and Aurangjaib, Exact Solutions of Plane Flows of an Incompressible Fluid of Variable Viscosity in the Presence of Unknown External Force Using (, ψ) or (η, ψ)-coordinates, Thai Journal of Mathematics, 7(3), (2009), 259-284.
- Bagewadi, C.S., and Siddabasappa, Study of variably inclined rotating MHD flows in magnetograph plane, Bull. Cal. Math. Soc., 85 (1993), 93-106.
- R.K., Naeem,, On exact solutions for Navier-Stokes equations for viscous incompressible fluids, A major paper of Master of Science at the University of Windsor. Windsor. Ontario. Canada. 1984
- Sil, S., and Kumar, M., An Exact solution of steady state plane rotating aligned MHD flows using Martins method in magnetograph plane, Journal of mathematical sciences, 3 (2016), 83-89.
- Sil, S., Prajapati, M., and Kumar, M., A class of exact solution of equations governing aligned plane rotating magnetohydrodynamic flows by Martins method, Bharat Ganit Parishad Ganita, 70(1) (2020), 41-52.
- Eegunjobi, A. S., and Makinde, O. D., Inherent irreversibility in a variable viscosity Hartmann flow through a rotating permeable channel with Hall effects, Defect and Diffusion Forum, 377, (2017), 180-188.
- Mabood F., Khan W. A., and Makinde, O. D., Hydromagnetic flow of a variable viscosity nanofluid in a rotating permeable channel with hall effects, Journal of Engineering Thermophysics, 26 (4) (2017), 553-566.
- Das S., Jana R. N., and Makinde, O. D., Transient hydromagnetic reactive Couette flow and heat transfer in a rotating frame of reference, Alexandria Engineering Journal, 55(1) (2016), 635-644.
- Das S., Jana R. N., and Makinde O. D., Magnetohydrodynamic free convective flow of nanofluid past an oscillating porous flat plate in a rotating system with thermal radiation and Hall effects, Journal of Mechanics, 32(2) (2016), 197-210.
- Das, S., Jana, R. N., and Makinde, O. D., Numerical study of unsteady MHD Couette flow and heat transfer of nanofluids in a rotating system with convective cooling, International Journal of Numerical Methods for Heat and Fluid Flow, 26(5) (2016), 1567-1579.
- Eegunjobi, A. S., and Makinde, O. D., Entropy analysis of variable viscosity Hartmann flow through a rotating channel with Hall effects, Applied Mathematics and Information Science, 10(4) (2016), 1415-1423.

#### References

Krishna, M. V., and Chamkha, A.J., Hall and ion slip effects on MHD rotating boundary layer flow of nanofluid past an infinite vertical plate embedded in a porous medium, Results in Physics, 15 (2019), 1026-1052.

Krishna, M. V., Ahamad, N. A., and Chamkha, A. J., Hall and ion slip effects on unsteady MHD free convective rotating flow through a saturated porous medium over an exponential accelerated plate, Alexandria Engineering Journal, 59(2) (2020), 565-577.

Prabhakar, R. B., Hall effect on MHD transient ow past an impulsively started infinite horizontal porous plate in a rotating system, International Journal of Applied Mechanics and Engineering, 23(2) (2018).

Krishna, M. V., Ahmad, N. A., and Chamkha, A. J., Hall and ion slip impacts on unsteady MHD convective rotating flow of heat generating/absorbing second grade fluid, Alexandria Engineering Journal, 60(1) (2020), 845-858.

Krishna, M. V., and Chamkha, A. J., Hall and ion slip effects on MHD rotating flow of elastic viscous fluid through porous medium, International Communications in Heat and Mass Transfer, 113 (2020), 104494.

Bagewadi, C. S., and Siddabasappa, The plane rotating viscous MHD flows, Bulletin of Calcutta Mathematical Society, 85 (1993), 513.

Thakur, C., and Mishra, R. B., On steady plane rotating hydromagnetic flows, Astrophysics and Space Science, 146(1) (1988), 89-97.

Singh, S. N., and Tripathi, D. D., Hodograph transformations in steady plane rotating MHD flows, Applied Scientific research, 43 (1993), 347-353.

Singh, S., Singh, H., and Mishra, R. B., Hodograph transformations in steady plane rotating hydromagnetic flow, Astrophysics and Space Science, 106 (1984), 231-243.

Ram, G., and Mishra, R.S., Unsteady flow through magnetohydrodynamic porous media, Ind. Jour. Pure and appl., 8(6) (1977), 637-647

Nguyen, P. V., and Chandna, O. P., Non-Newtonian MHD orthogonal steady plane fluid flows, Int. J. Eng. Sci., 30 (1992), 433-453.

Rashid, A. M., 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(1) (2014), 134-142.

Sil, S., and Kumar, M., 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(7) (2014), 17-26.

Sil, S., and Kumar M., Exact solution of second grade fluid in a rotating frame through porous media using hodograph transformation method, J. of Appl. Math. and Phys., 3 (2015), 1443-1453.

Singh, K. K., and Singh, D. P., Steady plane MHD flows through porous media with constant speed along each stream line, Bull. Cal. Math. Soc., 85 (1993), 255-262.

Kumar, M., Solution of non-Newtonian fluid flows through porous media by hodograph transformation method, Bull. Cal. Math. Soc., 106(4) (2014), 239-250.

Thakur, C., and Singh, B., Study of variably inclined MHD flows through porous media in magnetograph plan, Bull. Cal. Math. Soc., 92 (2000) 39-50.

Bhatt, B., and Shirley, A., Plane viscous flows in porous medium, Matematicas: Ensenanza Universitaria, 16 (1) (2008), 51-62.

M. H. Martin, The flow of a viscous fluid., Arch. Rat. Mech. Anal., 41 (1971), 266-286.

Bagewadi, C.S., and Siddabasappa, The plane rotating viscous MHD flows., Bull. Cal. Math. Soc., 85 (1993), 513-520.

K.V. Govindaraju, The flow of viscous fluid, Arch. Rat mech. Anal., 45 (1972), 66-80.

Nath V.I., and Chandna, O.P, On plane viscous magnetohydrodynamic flows., Quart. Appl. Math., 31 (1973), 351-362.

Chandna, O.P., Barron, R.M., and Garg, M.R., Plane compressible MHD flows., Quart. Appl. Math., 36 (1979), 411-422.

Chandna, O.P., and Labropulu, F., Exact solutions of steady plane flows using Von Mises coordinates, J. Math. Anal. Appl., 185 (1994), 36-64.

Naeem, R.K., and Nadeem S.A., Study of steady plane flows of an incompressible fluid of variable viscosity using Martins methods, J. App. Mech. and Engr., 1(3) (1996), 397-434.

Naeem, R.K., and Ali, S.A., A class of exact solutions to equations governing the steady plane flows of an incompressible fluid of variable viscosity via von Mises variables, Int. J. App. Mech. and Engr., 6(2) (2001), 395-436.

Chandna, O.P., and Labropulu, F., Exact Solutions of steady plane MHD aligned flows using von coordinates, Internat. J. Math. and Math Sci., 20(1) (1997), 165-186.

Ali, S.A., Ara A., and Khan N.A., Martins method applied to steady plane flow of a second grade fluid, Int. J. App. Math, Mech., 3(3) (2007), 71-81.

Thakur, C., Kumar, M., and Mahan, M.K., Martins method applied to constantly inclined viscous MHD flows through porous media., Bull. Cal. Math. Soc., 100 (2) (2008), 101-114.

Kumar, M., Thakur C., Singh, T.P., and Mahan M.K., An exact solution of steady plane aligned MHD flow using Martins method, Proceedings of UGC sponsored National Seminar on Recent trends in engineering frontiers of Physical Sciencesa, B.I.T. Sindri, Dhanbad, India, (2009), 189-195.

Naeem, R.K., Mansoor, A., Khan W.A., and Aurangjaib, Exact solutions of steady plane flows of an incompressible fluid of variable viscosity using (, ψ) or (η, ψ)-coordinates, Quart. Appl. Math., 34 (1976), 287-299.

Naeem, R.K., Mansoor, A., Khan, W.A., and Aurangjaib, Exact Solutions of Plane Flows of an Incompressible Fluid of Variable Viscosity in the Presence of Unknown External Force Using (, ψ) or (η, ψ)-coordinates, Thai Journal of Mathematics, 7(3), (2009), 259-284.

Bagewadi, C.S., and Siddabasappa, Study of variably inclined rotating MHD flows in magnetograph plane, Bull. Cal. Math. Soc., 85 (1993), 93-106.

R.K., Naeem,, On exact solutions for Navier-Stokes equations for viscous incompressible fluids, A major paper of Master of Science at the University of Windsor. Windsor. Ontario. Canada. 1984

Sil, S., and Kumar, M., An Exact solution of steady state plane rotating aligned MHD flows using Martins method in magnetograph plane, Journal of mathematical sciences, 3 (2016), 83-89.

Sil, S., Prajapati, M., and Kumar, M., A class of exact solution of equations governing aligned plane rotating magnetohydrodynamic flows by Martins method, Bharat Ganit Parishad Ganita, 70(1) (2020), 41-52.

Eegunjobi, A. S., and Makinde, O. D., Inherent irreversibility in a variable viscosity Hartmann flow through a rotating permeable channel with Hall effects, Defect and Diffusion Forum, 377, (2017), 180-188.

Mabood F., Khan W. A., and Makinde, O. D., Hydromagnetic flow of a variable viscosity nanofluid in a rotating permeable channel with hall effects, Journal of Engineering Thermophysics, 26 (4) (2017), 553-566.

Das S., Jana R. N., and Makinde, O. D., Transient hydromagnetic reactive Couette flow and heat transfer in a rotating frame of reference, Alexandria Engineering Journal, 55(1) (2016), 635-644.

Das S., Jana R. N., and Makinde O. D., Magnetohydrodynamic free convective flow of nanofluid past an oscillating porous flat plate in a rotating system with thermal radiation and Hall effects, Journal of Mechanics, 32(2) (2016), 197-210.

Das, S., Jana, R. N., and Makinde, O. D., Numerical study of unsteady MHD Couette flow and heat transfer of nanofluids in a rotating system with convective cooling, International Journal of Numerical Methods for Heat and Fluid Flow, 26(5) (2016), 1567-1579.

Eegunjobi, A. S., and Makinde, O. D., Entropy analysis of variable viscosity Hartmann flow through a rotating channel with Hall effects, Applied Mathematics and Information Science, 10(4) (2016), 1415-1423.