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

 

                                                                                                              Equation 1

where sT, s₀ are positive constants.

The Maxwell’s equations for magnetic field are implemented by

 or                                                                                   Equation 2

 

                                                                                                                                   Equation 3

were

 

                                                                                                            Equation 4

With

 

                                                                                                       Equation 5

 

and

                                                                     Equation 6

The equation of momentum with variable viscosity is

 

                                                                                                                                               Equation 7

 

                                                                              The heat equation with internal heating Q is

 

                                                                                                                                          Equation 8

 

The conservation of mass equation is

                                                                                                                                                                             Equation 9

The state equation is

                                                        Equation 10

 

Here  is the velocity, p is the pressure, t is the time,  is the magnetic induction and  is the intensity of magnetic field,  is the magnetization,  is the magnetic permeability of vacuum, C_V,H is the specific heat capacity at constant volume and magnetic field per unit mass,  and k_t 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.

The undisturbed quiescent state

                                                                                                                                     Equation 11

 

                               Equation 12

 

                                                                                                   Equation 13

 

                                                                                Equation 14

 

                                           Equation 15

 

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 z. 

To study the stability of the quiescent state and perturb the relevant variables in the corresponding governing equations with framework of the linear theory

 

                                       , , T = , ,  Equation 16

                                                                Let the components of     

be perturbed the magnetization and magnetic field, respectively.

 Using these in Equation 2,Equation 6, linearizing, we obtain

 

                                                                                                                                                   Equation 17

 

                                                                                                                                                  Equation 18

 

                                                                                                                                     Equation 19

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 p by operating two times of curl on the resulting equations and linearizing together with , then gives

 

                             Equation 20

 

As before, using Equation 16 ,Equation 8 and applying basic state solutions, and linearizing, we obtain

 

                                                                                                     Equation 21

                                                                      Where

Finally, Equation 2,Equation 3, after using Equation 16 together with Equation 17,Equation 19, yields (after neglecting primes)

 

                                                                  Equation 22

 

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)

 

                                                                                  Equation 23

 

 Substituting into Equation 20,Equation 22, we get

 

                                     Equation 24

 

            Equation 25

 

                                                              Equation 26

 

Thus, Equation 24,Equation 26 are the governing linearized perturbation equations and they are non- dimensional zed using the following quantities:

 

                                                                                

                                                   Equation 27

 

 After using Equation 25 in Equation 22, Equation 24, we obtain (ignoring the asterisks)

 

                                 Equation 28

 

                                                                                 Equation 29

 Where .

We set

                                       Equation 30

Here,  denoted as the growth rate, which is complex frequency.

Substituting into Equation 26 ,Equation 28, we obtain

 

       Equation 31

 

                                Equation 32

 

                                                                              Equation 33

 

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

 

                                                                     Equation 34

 

After linearizing the equations for balancing the surface tension gradient with shear stress at the free surfaces (Pearson 1958), we have

 

 ;                                       Equation 35

where  is the surface tension and  ,  are the shear stress Using                                         Equation 37 yields

 

                                       Equation 36

 

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

 

 ;                         Equation 37

 

Using Equation 39 and non-dimensionlizing the equations, we get

 

  and                                                        Equation 38

where,  denoted as the Marangoni number .

Upper boundary free ferromagnetic at fixed heat flux is

 

      Equation 39

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:

 

                          Equation 40

 

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

 

                                                                                             Equation 42

 

                                                                                               Equation 43

From Equation 43,Equation 45 have a non-trivial solution if

 

                                                                                Equation 44

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: ,                                           Equation 45

 

 (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   corresponding wavenumber a with  and . The inner products concerned in the equations are assessed analytically in order to keep away from the errors in numerical integration. The reveals the computations that the convergence in resulting Ma_c crucially depends on . 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 2

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  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

Table 1 Comparison of (Mac, ac) and (Rtc, ac) for Rm =d=0
	Nield [18]				Present analysis
	Lower insulating		Lower conducting		Lower insulating		Lower Conducting
Bi	Mac	ac	Rc	ac	Mac	ac	Rc	ac
0	79.607	1.993	669	2.086	79.6067	1.9929	668.998	2.0856
0.01	79.991	1.997	670.38	2.089	79.9913	1.9966	670.381	2.0888
0.1	83.427	2.028	682.36	2.117	83.4267	2.0281	682.36	2.1162
0.2	87.195	2.06	694.78	2.144	87.1951	2.0603	694.779	2.1437
0.5	98.256	2.142	727.42	2.212	98.2562	2.1423	727.422	2.2116
1	116.127	2.246	770.57	2.293	116.127	2.2462	770.57	2.2928
2	150.679	2.386	831.27	2.393	150.679	2.3864	831.27	2.3926
5	250.598	2.598	925.51	2.519	250.598	2.5978	925.51	2.519
10	413.44	2.743	989.49	2.589	413.44	2.7426	989.492	2.5889
20	736	2.852	1036.3	2.632	736	2.8524	1036.3	2.6323
50	1699.62	2.941	1072.19	2.661	1699.62	2.9406	1072.19	2.6615
100	3303.83	2.976	1085.9	2.672	3303.83	2.9755	1085.9	2.6718
1000	32170.1	3.01	1099.12	2.681	32170.1	3.0101	1099.12	2.6813
1010	32.073´1010	3.014	1100.65	2.682	32.073´1010	3.0141	1100.65	2.6823

 

Table 2 Comparison of (Rtc , ac ) for Ma = Rm = d = 0 (lower and upper insulating case)
Bi	Sparrow et al. [19] Rc ac		Present study Rc ac
0	320	0	320	8.72918´10-17
0.01	338.905	0.58	338.905	0.5831
0.03	353.176	0.76	353.158	0.7623
0.1	381.665	1.015	381.665	1.0151
0.3	428.29	1.3	428.29	1.2992
1	513.792	1.64	513.79	1.6438
3	619.666	1.92	619.666	1.9211
10	725.15	2.11	725.148	2.1055
30	780.24	2.18	780.2238	2.176
100	804.973	2.2	804.973	2.2029
¥	816.748	2.21	816.746	2.2147

 

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 Bi Figure 2, d Figure 3 representing that their result is to increase the stability region. Besides, decrease the stability of the region by increasing  Ns Figure 4,  M ₃  Figure 5 Rm Figure 6 and R_t  Figure 7

Figure 1 Ma against a for R1= Rm =50, NS=5,  d = 0.5and M3 = 1

 

Figure 2  Ma against a for Rt = Rm = 50 Ns = Bi = 5 and M3 = 1

 

Figure 3 Ma against a for Rt = Rm = 50, Bi = 5, d = 0.5 and M3 = 1

 

Figure 4 Ma against a for Rt = Rm = 50, Bi = 5, d = 0.5 and Ns = 5
Figure 5 Ma against a for Rt  50, Bi = Ns = 5, d = 0.5 and M3 = 1

 

Figure 6 Ma against a for Rm = 50, Bi = Ns = 5, d = 0.5 and M3 = 1

 

Figure 7Mac against Bi for different Rt when Rm = 50, Ns = 5, d = 0.5 and M3 = 1

 

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 . Clearly, the results of lower insulating case are advancing the FTC compared to lower conducting. A further reveal that an increase in  is to delay the onset of FTC. This may be owing to the fact that with an increase in, the boundary of free surface departs from good conductor of heat and hence there is an increase in Ma_c.

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.

Figure 8  Mac against d for different Rt when Rm = 50 Ns = 5 Bi = 5 and M3 = 1

 

Figure 9  Mac against Ns for different Rt when Rm = 50, d = 0.5, Bi = 5 and M3 = 1

 

Figure 10 Mac against M 3 for different Rt when Rm = 50 Ns = 5 Bi = 5 and d = 0.5

 

Figure 11 Mac against Rmc for different Rt when M3 = 1 Ns = 5 Bi = 5 and d = 0.5

The effect of increase in nonlinearity of fluid magnetization (i.e. M₃ ) is shown in Figure 10 for different when ,  and . From the figure, it is seen that an increase in M₃ 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 . For , the case corresponds to only the surface tension force are in effect. The amount of is associated to the importance of buoyancy gravitational force. It is observed that increase in leads to decrease and signifying that the FF carries more heat efficiency than the ordinary viscous fluid case. This due to an increase the destabilizing the coupled magnetic and surface tension forces with increasing buoyancy gravitational force, , thus the more easily for fluid flow in the system.

 

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  , M₃ are having stabilizing effect on the system.

 

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.

 

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