Upper Bounds for the Complex Growth Rate of a Disturbance in Ferrothermohaline Convection

Ram, Kaka; Prakash, Jyoti; Kumari, Kultaran; Kumar, Pankaj

doi:10.2478/sgem-2022-0005

Abstract

It is proved analytically that the complex growth rate σ= σ_r+iσ_i (σ_r and σ_i are the real and imaginary parts of σ, respectively) of an arbitrary oscillatory motion of neutral or growing amplitude in ferrothermohaline convection in a ferrofluid layer for the case of free boundaries is located inside a semicircle in the right half of the σ_rσ_i-plane, whose center is at the origin and $radius = \sqrt{\frac{R_{s} [1 - M_{1}^{'} (1 - \frac{1}{M_{5}})]}{P_{r}^{'}}},$ {\rm{radius}}\, = \,\sqrt {{{{R_s}\left[{1 - M_1^{'}\left({1 - {1 \over {{M_5}}}} \right)} \right]} \over {P_r^{'}}}}, where R_s is the concentration Rayleigh number, P_r^′ is the solutal Prandtl number, M₁^′ is the ratio of magnetic flux due to concentration fluctuation to the gravitational force, and M₅ is the ratio of concentration effect on magnetic field to pyromagnetic coefficient. Further, bounds for the case of rigid boundaries are also derived separately.

References

Banerjee M.B., Katoch D.C., Dube G.S., Banerjee, K. (1981), Bounds for growth rate of perturbation in thermohaline convection, Proc. Roy. Soc. London A, 378, 301–304.
Search in Google Scholar Back to article
Finlayson B.A. (1970), Convective instability of ferromagnetic fluids, J. Fluid Mech., 40, 753–767.
Search in Google Scholar Back to article
Gupta M.D., Gupta A.S. (1979), Convective instability of a layer of a ferromagnetic fluid rotating about a vertical axis, Int. J. Eng. Sci., 17, 271–277.
Search in Google Scholar Back to article
Gupta J.R., Sood.S.K., Shandil.R.G., Banerjee M.B., Banerjee K. (1983), Bounds for the growth of a perturbation in some double-diffusive convection problems, J. Aust. Math. Soc. Ser. B, 25, 276–285.
Search in Google Scholar Back to article
Lalas D.P., Carmi S. (1971), Thermoconvective stability of ferrofluids, Phys. Fluids, 14(2), 436–437.
Search in Google Scholar Back to article
Nataraj R., Bhavya S. (2019), Effect of Exponentially Temperature-Dependent Viscosity on the Onset of Penetrative Ferro-Thermal-Convection in a Saturated Porous Layer via Internal Heating, Journal of Electromagnetic Analysis and Applications, 11, 101–116.
Search in Google Scholar Back to article
Odenbach S. (2002), Ferrofluids: Magnetically controllable fluids and their applications, Springer-Verlag, Berlin, Heidelberg.
Search in Google Scholar Back to article
Odenbach S. (2002a), Magnetoviscous effects in ferrofluids, Springer-Verlag, Berlin, Heidelberg.
Search in Google Scholar Back to article
Prakash J. (2012), On stationary convection and oscillatory motions in ferromagnetic convection in a ferrofluid layer, J. Magn. Magn. Mater. 324(8), 1523–1527.
Search in Google Scholar Back to article
Prakash J. (2013), On arresting the complex growth rates in ferromagnetic convection in a ferrofluid saturated porous layer, J. Porous Media, 16(3), 217–226.
Search in Google Scholar Back to article
Prakash J., and Gupta S. (2013), On arresting the complex growth rates in ferromagnetic convection with magnetic field dependent viscosity in a rotating ferrofluid layer, J. Magn. Magn. Mater. 345, 201–207.
Search in Google Scholar Back to article
Prakash J. (2014), On exchange of stabilities in ferromagnetic convection in a rotating ferrofluid saturated porous layer, J. Appl. Fluid Mech. 7(1), 147–154.
Search in Google Scholar Back to article
Prakash J., Vaid K., Bala R. (2014), Upper limits to the complex growth rates in triply diffusive convection, Proc. Indian Nat. Sci. Acad., 80(1), 115–122.
Search in Google Scholar Back to article
Prakash J., Bala R., Kumari K. (2017), Upper bounds for the complex growth rates in ferromagnetic convection in a rotating porous medium: Darcy-Brinkman Model, Bull. Cal. Math. Soc. 109(2), 153–170.
Search in Google Scholar Back to article
Prakash J., Kumar R., Kumari K. (2017a), Thermal convection in a ferromagnetic fluid layer with magnetic field dependent viscosity: A correction applied, Studia Geotech. et Mech. 39(3), 39–46.
Search in Google Scholar Back to article
Rahman, H., and Suslov S.A. (2015), Thermomagnetic convection in a layer of ferrofluids placed in a uniform oblique external magnetic field, J. Fluid Mech. 764, 316–348.
Search in Google Scholar Back to article
Rosensweig R.E., Zahn M., Volger T. (1978), Stabilization of fluid penetration through a porous medium using magnetisable fluids, in: Thermomechanics of magnetic fluids (Ed. B. Berkovsky), Hemisphere, Washington, DC, 195–211.
Search in Google Scholar Back to article
Rosensweig. R. E. (1985), Ferrohydrodynamics, Cambridge University Press, Cambridge.
Search in Google Scholar Back to article
Sekar. R. and Vaidyanathan G. (1993), Convective instability of a magnetized ferrofluid in a rotating porous medium, Int. J. Eng. Sci. 31, 1139–1150.
Search in Google Scholar Back to article
Sekar R., Vaidyanathan G., Ramanathan A. (1993), The ferroconvection in fluids saturating a rotating densely packed porous medium, Int. J. Eng. Sci. 13, 241–250.
Search in Google Scholar Back to article
Sekar R., Vaidyanathan G., Ramanathan A. (1996), Ferroconvection in an anisotropic porous medium, Int. J. Engng. Sci. 34(4), 399–405.
Search in Google Scholar Back to article
Sekar R., Vaidyanathan G., Ramanathan A. (2000), Effect of rotation on ferrothermohaline convection, J. Magn. Magn. Mater. 218, 266–272.
Search in Google Scholar Back to article
Sekar R., Raju K., Vasanthakumari R. (2013), A linear analytical study on Soret-driven ferrothermohaline convection in an anisotropic porous medium, J. Magn. Magn. Mater. 331, 122–128.
Search in Google Scholar Back to article
Sekar, R. and Raju K. (2015), Effect of sparse distribution pores in thermohaline convection in a micropolar ferromagnetic fluid, J. Appl. Fluid Mech., 8(4), 899–910.
Search in Google Scholar Back to article
Sekar R., and Murugan D. (2018), Stability analysis of ferrothermohaline convection in a Darcy porous medium with Soret and MFD viscosity effects, Tecnica Italiana-Ita. J. Engng. Sci. 61+1(2), 151–161.
Search in Google Scholar Back to article
Shliomis M. I. (1974), Magnetic Fluids, Sov. Phys. Uspekhi, 17, 153–169.
Search in Google Scholar Back to article
Sunil, Bharti P.K., Sharma R.C. (2004), Thermosolutal convection in ferromagnetic fluid, Arch. Mech., 56(2), 117–135.
Search in Google Scholar Back to article
Sunil, Divya, and Sharma R.C. (2004a), Effect of rotation on ferromagnetic fluid heated and soluted from below saturating a porous medium, J. Geophys. Eng., 1, 116–127.
Search in Google Scholar Back to article
Vaidyanathan G., Sekar R., Balasubramanian R. (1991), Ferroconvective instability of fluids saturating a porous medium, Int. J. Engng. Sci., 29, 1259–1267.
Search in Google Scholar Back to article
Vaidyanathan G., Sekar R., Ramanathan A. (1995), Ferro thermohaline convection in a porous medium, J. Magn. Magn. Mater. 149, 137–142.
Search in Google Scholar Back to article
Vaidyanathan G., Sekar R., Ramanathan A. (1997), Ferrothermohaline convection, J. Magn. Magn. Mater. 176, 321–330.
Search in Google Scholar Back to article

Upper Bounds for the Complex Growth Rate of a Disturbance in Ferrothermohaline Convection

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