viscosity with temperature in analysing thermal convective instability in horizontal fluid layers, but the studies were limited to ordinary viscous fluids Kassoy and Zebib (1975), Blythe and Simpkins (1981), Patil and Vaidyanathan (1981), Patil and Vaidyanathan (1982). To our knowledge, no attention has been given to convective instability problems involving FFs, despite the importance of FFs in many heat transfer applications. For example, in a rotating shaft seal involving FFs the temperature may rise above 1000C at high shaft surface speeds. A similar situation may arise in the use of FFs in loudspeakers Lebon and Cloot (1986). The onset of FTC in a horizontal FF layer with temperature dependent viscosity in exponentially is examined Shivakumara et al. (2012). In many natural phenomena, the study of penetrative FTC in a saturated porous layer Nanjundappa et al. (2011) with the internal heating source and applied Brinkman extended Darcy model in the momentum equation analyzed the internal heat generation effect on the onset of FTC in an FF saturated porous layer Nanjundappa et al. (2011), Nanjundappa et al. (2012). Savitha et al. (2021) investigated the penetrative FTC in an FF-saturated high porosity anisotropic porous layer via uniform internal heating. Thus, the purpose of the present chapter is to study a general problem of coupled thermo- gravitational and surface-tension FC in an FF layer with magnetic field dependent (MFD) viscosity. The study helps to understanding the control of FTC by MFD viscosity, which is constructive in various problems associated by heat transfer particularly in material-science processing. In the current study, the bottom surface is rigid with either constant temperature or uniform heat flux, while the upper is un-deformable free surface of surface tension forces. Besides, the Neumann-type of boundary condition is imposed on the upper surface. Several investigators have studied both types of instabilities in isolation or together in a horizontal FF layer. 2.
PROBLEM
FORMULATION Consider a layer of horizontal Boussinesq FF of
constant depth
where sT, s0 are positive constants. The Maxwell’s equations for magnetic field are implemented by were
With
and The equation of momentum with variable viscosity is
The heat equation with internal heating
Q
is
The conservation of mass equation
is The state equation is
Here The undisturbed quiescent state
Here we note that, To study the stability of the quiescent state and perturb the relevant variables in the corresponding governing equations with framework of the linear theory
Let the components of be perturbed the magnetization and magnetic field, respectively. Using these in Equation 2,Equation 6, linearizing, we obtain
From Equation 17,Equation 19 and it is considered
that Experimentally, Rosenwieg
et al. (1969) has demonstrated the
exponential variation in magneto-viscosity, As before, using
Equation 16 ,Equation 8 and applying basic state solutions, and linearizing, we
obtain
Where Finally, Equation 2,Equation 3, after using Equation 16 together with Equation 17,Equation 19, yields (after neglecting
primes) As the customary of convective instability analysis for
each variable of
Substituting into Equation 20,Equation 22, we get Thus, Equation 24,Equation 26 are the governing linearized perturbation equations and they are non- dimensional zed using the following quantities:
After using Equation 25 in Equation 22, Equation 24, we obtain (ignoring the asterisks)
Where We set Here, Substituting into Equation 26 ,Equation 28, we obtain The boundary conditions for these equations are 1. Lower boundary rigid-ferromagnetic at fixed temperature as
2. Lower boundary rigid-ferromagnetic at fixed heat flux as After linearizing the equations for balancing the surface tension gradient with shear stress at the free surfaces (Pearson 1958), we have where For most of the liquids as the temperature rises, the variation between the liquid and its vapor phase decreases. Thus, the suitable boundary conditions of surface tension at the free surfaces are Using Equation 39 and non-dimensionlizing the equations, we get
where, Upper boundary free ferromagnetic at fixed heat flux is were, Bi denoted as the Biot number 3. METHOD OF SOLUTION The GT is applied to obtain the problem of eigenvalue
is to study the linear system of Equation 32 with Equation 35 and Equation 41.
The unknown factors
Substitute in Equation 32 Equation 32, multiplying the
resulting equations respectively by
From Equation 43,Equation 45 have a non-trivial solution if
Were
Where The eigenvalue is extracted from Equation 45. A trivial function (i) For lower insulating case: (ii) For lower
conducting case: Here 4. NUMERICAL RESULTS AND DISCUSSION It may be illustrated that Equation
41 with w = 0 leads to the Marangoni
number To solve the eigenvalue problem from Equation 41 by employing the Galerkin-type of WRM. In order to confirm the
numerical technique is applied, the values
Figure 1,Figure 6 illustrates the
neutral stability curves corresponding for different
In Figure 8,Figure 11 analogous to solid
curves are corresponding to lower conducting and dotted curves corresponding to
lower insulating. The plot of The effect of MFD viscosity parameter
The
effect of increase in nonlinearity of fluid magnetization (i.e. M3 ) is shown in Figure 10 for different The locus plot of 5. CONCLUSIONS The linear stability theory is applied to study the
effect of MFD viscosity on coupled buoyancy-gravitational and surface-tension
forces on FTC in a FF layer through the strength of internal heat source on the
system under the conditions of lower insulating/conducting case. The FF layer is heated from below and its top
surface is subjected to a surface-tension force decreasing linearly with
temperature. The problem of resulting eigenvalue is obtained numerically by
utilizing the Galerkin WRT technique. It is shown that the effect of MFD
viscosity is to enhance the onset of FTC and hence MFD viscosity plays a
stabilizing role on the system. The increase in buoyancy-gravitational force,
the forces of magnetic and surface-tension effect is to destabilize the system.
Their effects are complementary in the sense that the critical ACKNOWLEDGEMENTS The authors (CEN) and (MKR) wish to thank the Management and Principal of Dr. Ambedkar Institute of Technology, and MES Pre-University College of Arts, Commerce and Science, Bangalore, respectively, for their encouragement. REFERENCES Blums, E. (2002). Heat And Mass Transfer Phenomena. https://link.springer.com/chapter/10.1007/3-540-45646-5_7 Blythe, P. A. And Simpkins, P. G. (1981). Convection In A Porous Layer For A Temperature Dependent Viscosity", International Journal Of Heat And Mass Transfer,24(3), 497-506. https://doi.org/10.1016/0017-9310(81)90057-0 Charles, S. W. (2002). The Preparation Of Magnetic Fluids. https://doi.org/10.1007/3-540-45646-5_1 Kassoy D. R. And Zebib, A. (1975). Variable Viscosity Effects On The Onset Of Convection In Porous Media", 18(12), 1649-1651. https://doi.org/10.1063/1.861083 Lebon, G. And Cloot, A. (1986). A Thermodynamical Modeling Of Fluid Flows Through Porous Media", Application Kassoy Dr, Zebib A. Variable Viscosity Effects On The Onset Of Convection In Porous Media To Natural Convection, International Journal Of Heat And Mass Transfer, 29(3), 381-389. https://doi.org/10.1016/0017-9310(86)90208-5 Nanjundappa, C. E. Ravisha, M. Lee, J. And Shivakumara, I. S. (2011). Penetrative Ferroconvection In A Porous Layer”, 243-257. https://doi.org/10.1007/s00707-010-0367-9 Nanjundappa, C. E. Shivakumara, I. S, And Prakash, H. N. (2012). Penetrative Ferroconvection Via Internal Heating In A Saturated Porous Layer With Constant Heat Flux At The Lower Boundary", Journal Of Magnetism Magnetic Materials, 324(9), 1670-1678. https://doi.org/10.1016/j.jmmm.2011.11.057 Nanjundappa, C. E. Shivakumara, I. S. Lee, J. And Ravisha, M. (2011). Effect Of Internal Heat Generation On The Onset Of Brinkman-Bénard Convection In A Ferrofluid Saturated Porous Layer", International Journal Of Thermal Sciences, 50(2), 160-168. https://doi.org/10.1016/j.ijthermalsci.2010.10.003 Nield, D. A. (1964). Surface Tension And Buoyancy Effects In Cellular Convection, Journal Of Fluid Mechanics, 19(3), 341-352. https://doi.org/10.1017/S0022112064000763 Nkurikiyimfura, I. Wanga, Y. And Pan, Z. (2013). Heat Transfer Enhancement By Magnetic Nanofluids-A Review", Renewable And Sustainable Energy Reviews, 548-561. https://doi.org/10.1016/j.rser.2012.12.039 Patil, P. R. And Vaidyanathan, G. (1981). Effect Of Variable Viscosity On The Setting Up Of Convection Currents In A Porous Medium", International Journal Of Engineering Sciences, 421-426. https://doi.org/10.1016/0020-7225(81)90062-8 Patil, P. R. And Vaidyanathan, G. (1982). Effect Of Variable Viscosity On Thermohaline Convection In A Porous Medium", Journal Of Hydrology, 147-161. https://doi.org/10.1016/0022-1694(82)90109-3 Rosensweig, R. E. (1985). Ferrohydrodynamics, https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/ferrohydrodynamics-by-r-e-rosensweig-cambridge-university-press-1985-344-pp-45/F9ED8D5FBD40AD7CFF5CB1D827ADA3CC Rosenwieg, R.E. Kaiser, R. Miskolczy, G. (1969). Viscosity Of Magnetic Fluid In A Magnetic Field" Journal Of Colloid Interface Science, 680-686. https://doi.org/10.1016/0021-9797(69)90220-3 Savitha, Y. L. Nanjundappa, C. E. And Shivakumara, I. S. (2021). Penetrative Brinkman Ferroconvection Via Internal Heating In High Porosity Anisotropic Porous Layer : Influence Of Boundaries", https://doi.org/10.1016/j.heliyon.2021.e06153 Shivakumara, I. S. Lee, J. And Nanjundappa, C. E. (2012). Onset Of Thermogravitational Convection In A Ferrofluid Layer With Temperature Dependent Viscosity", Asme Journal Of Heat Transfer, 134(1). https://doi.org/10.1115/1.4004758 Shliomis, M. L. (1974). Magnetic Fluids", 17(2), 153. https://doi.org/10.1070/PU1974v017n02ABEH004332 Sparrow, E. W. Goldstein, R. J. And Jonson, V. K. (1964). Thermal Instability In A Horizontal Fluid Layer : Effect of Boundary Conditions And Nonlinear Temperature Profile", Journal of Fluid Mechanics, 513-528. https://doi.org/10.1017/S0022112064000386 Vaidyanathan, G. Sekar, R. Ramanathan, A. (2002). Effect of Magnetic Field Dependent Viscosity On Ferroconvection In Rotating Porous Mediu, Indian Journal of Pure And Applied Mathematics, 159-165. https://doi.org/10.1016/S0304-8853(02)00355-4
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