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 with a uniformly volumetric heat source strength, , and in the occurrence of perpendicular magnetic field . The surfaces are maintained the constant temperatures at and . The gravity, , acting downward direction, where is the z-direction of unit vector. The stream of Bénard-Marangoni convection for thermocapillary force, (surface tension force), is given by
where The Maxwell’s equations for magnetic field are implemented by or were
With and The equation of momentum with variable viscosity is
Equation 7
The heat equation with internal heating
Equation 8 The conservation of mass equation
is
The state equation is Here is the velocity k is the thermal conductivity, is the magnetic
susceptibility and is the pyromagnetic co-efficient, is the thermal
expansion coefficient, is the density at , , , and the last term of Equation
7 denotes as is the rate of strain
tensor. The fluid is assumed to be incompressible having variable viscosity.
Experimentally, it has been demonstrated that the magnetic viscosity has got
exponential variation, with respect to magnetic field Rosenwieg
et al. (1969) As a first
approximation, for small field variation, linear variation of magnetic
viscosity has been used in the form , where is the variation
coefficient of magnetic field dependent viscosity and is considered to be
isotropic Vaidyanathan et al. (2002), is taken as viscosity
of the fluid when the applied magnetic field is absent._{t}The undisturbed quiescent state
Here we note that, , , are distributed
parabolically with the height of the FF layer due to the existence of
volumetric heat source, . However, for , the distributions
of basic state are linear in 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, , where is the variation of
viscosity coefficient. Since the first approximation of small (linear) field
variation in magneto-viscosity has been used. Substituting Equation 16 in Equation
7
and applying the basic state solutions, removing the pressure 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 is expanded in
following form by assuming the normal mode hypothesis (separation of variables)
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, denoted as the growth rate, which is complex frequency. Substituting into Equation 26 ,Equation 28, we obtain
The boundary conditions for these equations are
After linearizing the equations for balancing the surface tension gradient with shear stress at the free surfaces (Pearson 1958), we have ; where is the surface tension
and , are the shear stress Using Equation 37
yields 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 and where, denoted as the Marangoni number . 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 can be expanded upon the complete set:
Substitute in Equation 32 Equation 32, multiplying the
resulting equations respectively by , , and carrying out the integration by parts from z = 0 to z = 1
and using Equation 35 and Equation 41 we obtain Equation 41
From Equation 43,Equation 45 have a non-trivial solution if
Were
,
, Where The eigenvalue is extracted from Equation 45. A trivial function , can be considered to
satisfy the boundary conditions Equation 35, Equation 36 and Equation 41
by selecting the trial functions as , (i) For lower insulating case: , (ii) For lower conducting case: Heres denoted as the second kind Tchebyshev’ polynomials, such
that satisfy Equation 35, Equation 36 and Equation 41 except, namely and however the residual
from this equations is incorporated as a residual from Eqs. Equation 32,Equation 34 4. NUMERICAL RESULTS AND DISCUSSION It may be illustrated that Equation
41 with w = 0 leads to the Marangoni
number . The presented results for i = j = 8_{
the} order at which the convergence is attained, in general. The critical Marangoni number is determined by
corresponding critical wavenumber . The results thus attained for different and are existing graphically in
Figure 2 and also in Table 1 and Table 2To 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 and are very close to the existing values of Nield (1964) for and Sparrow
et al. (1964) for under the limiting condition in Table 1 and Table 2, respectively. The comparisons of calculated present results
agree well with results of previous numerical investigations are given in Table 1 and Table 2. It is evident that
the results are in good agreement between the present and published previously.
This validates the applicability and exactness of the method applied in solving
the convective instability problem considered
Figure 1,Figure 6 illustrates the
neutral stability curves corresponding for different , , , , and as well as different
bounding surfaces (surfaces of lower insulating and lower conducting). The
neutral stability curves are concave growing for each of these surfaces and the
curves of lower conducting case lie above lower insulating surfaces. The
neutral stability curves shift growing with
increasing
In Figure 8,Figure 11 analogous to solid
curves are corresponding to lower conducting and dotted curves corresponding to
lower insulating. The plot of against for various for , and Figure 7 It shows that the value of lower conducting and lower insulating by increasing
in The effect of MFD viscosity parameter on the onset of FTC in a FF layer is presented in Figure 9 for fixed , and . It is viewed that increases with increasing indicating its effect is to stabilize the system. That is, the effect of increasing is to delay the FTC in the existence of magnetic field. To explore the effect of strength of dimensionless internal heat source on the measure for the onset of FTC, the variation of is displayed against for , and . in Figure 9. It is seen that decreases quite hastily first and then quite gradually monotonically with representing the influence of increasing internal heating is to decrease and thus destabilize the system. In particular, it is seen that the curves of coalesce for various physical parameters as the strength of internal heating is increased.
The
effect of increase in nonlinearity of fluid magnetization (i.e. M_{3}_{ }is to decrease and thus increase in
the magnetization has destabilizing effects on the system but this effect is
very insignificant.The locus plot of against for various for , and Figure 11. It shows that they are bridging the space
between lower insulating and conducting cases by increasing in 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 and decrease with an increase in . The increase in , and decrease in ,
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