OSCILLATORY POROUS MEDIUM FERROCONVECTION IN A VISCOELASTIC MAGNETIC FLUID WITH NON-CLASSICAL HEAT CONDUCTION

The classical stability analysis is used to examine the combined effect of viscoelasticity and the second sound on the onset of porous medium ferroconvection. The fluid and solid matrix are assumed to be in local thermal equilibrium. Considering the boundary conditions appropriate for an analytical approach, the critical values pertaining to both stationary and oscillatory instabilities are obtained by means of the normal mode analysis. It is observed that the oscillatory mode of instability is preferred to the stationary mode of instability. It is shown that the oscillatory porous medium ferroconvection is advanced through the magnetic forces, nonlinearity in magnetization, stress relaxation due to viscoelasticity, and the second sound. On the other hand, it is observed that the presence of strain retardation and porous medium delays the onset of oscillatory porous medium ferroconvection. The dual nature of the Prandtl number on the Rayleigh number with respect to the Cattaneo number is also delineated. The effect of various parameters on the size of the convection cell and the frequency of oscillations is also discussed. This problem may have possible implications for technological applications wherein viscoelastic magnetic


INTRODUCTION
Ferroconvection is a transfer of heat from one place to another in ferromagnetic liquids and its importance is due to the applications suggested by several authors [1][2][3] and many more.Ferrofluids, also known as magnetic fluids, are colloidal suspensions of nanosized ferromagnetic particles stably dispersed in organic or non-organic carrier fluids such as kerosene, water, and hydrocarbon.When exposed to an external magnetic field, they behave paramagnetically with susceptibility usually large for magnetic liquids [4].Ferrofluids have commercial applications like vacuum feed-throughs for manufacturing semi-conductors [5].Ferrofluid is also used in taking the drug in a human body to a target site by applying a magnetic field [6].However, we can find many applications in different fields [7].Finlayson [8] studied the convective instability of ferromagnetic fluids due to Bénard in the presence of a uniform vertical magnetic field and explained the thermomechanical interaction concept of ferromagnetic fluids.Lalas and Carmi [9] studied the thermoconvective stability of ferrofluids in the absence of buoyancy effects.Non-Darcy ferroconvection problem with gravity modulation using regular perturbation has been addressed by Nisha Mary and Maruthamanikandan [10].Darcy-Brinkman ferroconvection with temperature-dependent viscosity has been studied by Soya Mathew and Maruthamanikandan [11] and thermorheological and magnetorheological effects on Marangoni-ferroconvection with internal heat generation has been investigated by Maruthamanikandan et al. [12].Effect of MFD viscosity on ferroconvection in a fluid saturated porous medium with variable gravity has been examined by Vidya Shree et al. [13].
A good amount of attention is honoured to Rayleigh-Bénard convection (RBC) problems in Newtonian liquids with respect to heat transfer and other engineering applications as referred above.On the other hand, at shallow depths of the reservoirs, oil sands contain waxy crude which are viscoelastic fluids.They exhibit both liquid and solid properties and have many applications to the nuclear, petroleum, and chemical industries.They also have applications in cooling electronic devices, crystal growth, and material processes.In the study of viscoelastic fluids, the rheological equation involves one or two relaxation times (Bird et al. [14] and Joseph [15]) and also oscillatory convection is witnessed which is not noticed in Newtonian fluids.The Oldroyd model [16] is used for describing the viscoelastic properties of dilute polymers.The fact that principle of exchange of stabilities is not valid was shown by Green [17].Recently, the onset of Darcy-Brinkman convection in a binary viscoelastic fluid saturated porous layer has been addressed studied by Swamy et al. [18].
The equation governing temperature (heat transport equation) in classical theory assumes a parabolic-type partial differential equation that admits thermal signals at an infinite speed, which is unrealistic.The new theories modified the classical Fourier's law of heat conduction and hence contain a hyperbolic-type heat transport equation that admits the thermal signals at a finite speed.As per this theory, heat propagates as a wave phenomenon rather than a diffusion phenomenon and the wavelike thermal disturbance is referred to as second sound (Chandrasekharaiah [19]).Gurtin and EEJP. 2 (2023) Pipkin [20] investigated a general principle of thermal conduction in nonlinear analysis, including memories, a concept having a finite propagation speed.Straughan and Franchi [21] addressed the Bénard advection problem when the Maxwell-Cattaneo heat flow law is utilized in place of the ordinary Fourier theory of thermal conductivity.Soya Mathew and Maruthamanikandan [22] investigated oscillatory porous medium ferroconvection with Maxwell-Cattaneo law of heat conduction where they showed that the oscillatory mode of convection is preferred to stationary mode for large values of Prandtl and second sound parameter.
Under these conditions, the present paper is dedicated to examining convective instability in a Cattaneo viscoelastic ferrofluid saturated sparsely packed porous medium.The influence of various parameters is explored that perhaps direct us to oscillatory convection.T T T Δ = − (see Fig. 1).It is assumed that at a quiescent state the temperature varies linearly across the depth.When the magnitude of T Δ become larger than the critical one, thermal convection will set in due to buoyancy force.
The fluid layer is exposed to a magnetic field 0 H → acting parallel to the vertical z-axis and the gravity force acting vertically downwards.We assume that Oldroyd's model is sufficient to characterize the viscoelastic behaviour which is simple enough to be tractable analytically.The governing equations supporting the Boussinesq approximation are written as follows.
is the fluid velocity, 0 ρ is the reference density, ε is the porosity, t is the time, p is the pressure, g → is the acceleration due to gravity, ρ is the fluid density, f μ is the dynamic viscosity, f μ is the effective viscosity, k is the permeability of the porous medium, H → is the magnetic field, B → is the magnetic induction, T is the temperature, 0 μ is the magnetic permeability, M → is the magnetization, 1 k is the thermal conductivity, α is the thermal expansion coefficient, , C is the specific heat at constant volume and magnetic field, m χ is the magnetic susceptibility, m K is the pyromagnetic coefficient, Q → is the heat flux, τ is a constant with the dimensions of time and 1 2 q Maxwell's equations for a non-conducting fluid with no displacement currents become (Finlayson [8]) Equations characterizing the basic state are introduced in the form 1 0 , (0, 0, 0) , ( where . The solution pertaining to the basic state reads 0 [1 ] b z . (2.12)

STABILITY ANALYSIS
We shall obtain the dimensionless equations following the small perturbation stability analysis enveloping normal modes (Finlayson [8], Soya Mathew and Maruthamanikandan [11]).The perturbed state equations involving infinitesimally small perturbations are where the primes indicate perturbed quantities.The perturbed governing linearized equations take the form We take divergence on both sides of equation (3.4) and substitute in equation 3).The resulting system of linearized perturbed equations are as follows where with ' φ being the magnetic potential.
The normal mode solution is accessible and the same has the form where l and m are respectively the wave numbers in the x and y directions and σ is the growth rate.Substitution of (3.9) into equations (3.6) to (3.8) leads to we obtain the following dimensionless equations (asterisks are neglected for simplicity) where ( )( ) . The parameter 2 M is neglected as it is of very small order (Finlayson   [8]).When 1 λ = , we obtain the following equations is the inverse Darcy number, Cattaneo number and is the non-buoyancy-magnetization parameter.The boundary conditions encompassing free and isothermal surfaces are [8]).

Stationary Instability
As for the stationary mode, equations (3.17) - (3.19) turn out to be the following with 1 2 , A A and 3 A being constants, is taken into consideration.On applying the solvability condition, we obtain where the superscript 'st' stands for stationary convection.Equation (3.23) exactly coincides with that obtained by Soya Mathew and Maruthamanikandan [22] and Soya Mathew et al. [23] followed by the corresponding deductions.

Oscillatory Instability
The dimensionless equations concerning the overstable motion are ) (3.26) On applying the solvability condition, we

RESULTS AND DISCUSSION
The study is concerned with porous medium ferroconvection in a viscoelastic magnetic fluid with non-classical heat conduction.We have obtained the conditions for both stationary and oscillatory convection using linear theory, which is based on the normal mode technique.The thermal Rayleigh number R, characterising the stability of the system, is obtained as a function of the different parameters of the study.The eigenvalue expression and the associated critical numbers are determined by using MATHEMATICA software.As we can observe from the expression (3.23) stationary Rayleigh is independent of the viscoelastic parameters as obtained by Soya Mathew and Maruthamanikandan [22].Also, if the Cattaneo number is taken below the threshold value, then only stationary convection occurs [23].Hence, for stationary convection, viscoelastic fluid behaves same as Newtonian fluid.Rayleigh number for oscillatory mode is obtained as a function of Prandtl number, Cattaneo number, magnetic, viscoelastic and porous parameters.
In   From Figures 6 and 7, we witness the dual nature of the Prandtl number Pr depending on the Cattaneo number G. If the Cattaneo number G is above the threshold value, then on increasing Pr there is a decrease in osc c R as noticed in the work of Nagouda and Pranesh [25] and if the Cattaneo number G is below the threshold value, then on increasing Pr there is an increase in osc c R as noticed in the work of Swamy et al. [18].In Fig. 8   For Newtonian ferromagnetic fluid ( ) . ., 0 and 0 i e F F = =

EEJP. 2 (2023)
Naseer Ahmed, S. Maruthamanikandan, et al.    ω also decreases.Hence, we can conclude that from Figs. 12 through 20 that the frequency c ω of oscillatory ferroconvective instability is sensitive to all the parameters of the study.On the other hand, wave number depicts the size and shape of the convection cell.From Tables 2 through 10, it follows that convection cell size is also sensitive to the all the parameters of the study at hand.Indeed, the convection cell size is enlarged with an increase in   It is worth mentioning that for Newtonian fluids only stationary convection is possible, but due to the presence of second sound, oscillatory instability is preferred to stationary stability as pointed out by Straughan [24].CONCLUSIONS 1.The system is destabilized through the presence of magnetic forces caused by the magnetization of ferrofluids.2. Nonlinearity in magnetization is shown to destabilize the system.3. Viscoelastic relaxation and second sound are shown to destabilize the system.4. Viscoelastic retardation, inverse Darcy number and Brinkman number are shown to stabilize the system. 5. Prandtl number destabilizes as well as stabilizes the system depending on the over and below threshold values of the Cattaneo number respectively.6. Critical wavenumber and frequency of oscillatory motions are calculated as functions of all the parameters of the problem.They are shown to be sensitive to all the parameters of the problem.

Figure 1 .
Figure 1.Physical Configuration 2. MATHEMATICAL FORMULATION Let us consider an incompressible Cattaneo viscoelastic ferromagnetic fluid saturated porous medium confined between the two surfaces of non-finite length horizontally of finite thickness d.We consider Oldroyd's model to characterize the viscoelastic behaviour which is a non-Newtonian one.The lower surface at 2 z d = − and upper surface at 2 z d = are maintained at temperatures T 1 and T 0 respectively with 1 20) through (3.22) along with the boundary conditions embrace an eigenvalue problem with R being an eigenvalue.The straightforward solution ( )

2 R
are computed by means of the MATHEMATICA software mathematical package.

Fig. 2 critical 3 critical 1 F 2 F
Rayleigh number c R is expressed as a function of magnetic Rayleigh number N by keeping all other parameters as constant by fixing their values as N increases, c R decreases and hence the system is destabilized.We observe that oscillatory convection is preferred to stationary convection as osc c R is less than st c R and hence the principle of exchange of instabilities is not valid.In Fig. Rayleigh number c R is expressed as a function of the magnetic Rayleigh number N by varying 1 F and keeping all other parameters as constant by fixing their values as 2 0.3, increases, the osc c R value decreases which indicates that the stress relaxation parameter 1 F hastens the oscillatory ferroconvection.In Fig. 4 critical Rayleigh number c R is expressed as a function of the magnetic Rayleigh number N by varying and keeping all other parameters as constant by fixing their values as 1 1.5, As there is an increase in the values of 2 F , we notice that there is an increase in osc c R which indicates that the strain retardation parameter 2 F slows down the onset of oscillatory ferroconvection.

Figure 6
Figure 6.Plot of osc c R versus N with variation in Pr with 1 1 2 3 1.5, 0.3, 5, 3, 0.06 3. F F Da G and M − = = = Λ= = = critical Rayleigh number osc c R is expressed as a function of N by varying Pr and keeping all other parameters as constant by fixing their values as 1 0, .e., in the absence of viscoelastic parameters.In this case also there is a decrease in osc c R as we increase Pr which again clearly suggests that the dual nature of Pr is only due to the presence of the Cattaneo number.

Figure 8 .
Figure 8. Plot of osc c R

Fig. 9 critical 1 1Oscillatory ferroconvection is delayed because as 1 Da
Rayleigh number osc c R is expressed as a function of magnetic Rayleigh number N by varying 1 Da − and keeping all other parameters as constant by fixing their values as − is increased, there is an increase in the values of osc c R .The reason for this is the increase in 1 Da − will decrease the porous medium permeability and hence the convective instability is impeded.

Figure 9 In Fig. 10 critical
Figure 9. Plot of osc c R

3 2 ,
and F M Λ and the opposite is found to be true with respect to an increase in the rest of the parameters.

Table 1 .
Critical values of the Rayleigh number and wave number with

Table 2 .
Critical values of the wave number varying with 1

Table 3 .
Critical values of the wave number varying with 2

Table 4 .
Critical values of the wave number varying with G by fixing

Table 5 .
Critical values of the wave number varying with Pr by fixing

In the absence of Second Sound)Table 6 .
Critical values of the wave number varying with Pr by fixing

Table 7 .
Critical values of the wave number c

Table 8 .
Critical values of the wave number varying with

Table 9 .
Critical values of the wave number varying with Λ by fixing

Table 10 .
Critical values of the wave number varying with 3