# Finite-amplitude ferro-convection and electro-convection in a rotating fluid

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## Abstract

The onset of Rayleigh–Bénard convection in a horizontal ferrofluid layer subjected to rotation with respect to the vertical axis is theoretically investigated in the paper. Using a truncated Fourier series representation, an analytically intractable fifth-order Lorenz model that has two quadratic nonlinearities is derived and then reduced to the analytically tractable Landau equation with cubic nonlinearity using the method of multi-scales. The critical Rayleigh number of the linear stability theory and that of the energy stability method is drawn from other works and compared with that obtained by the weakly nonlinear stability method reported in the paper. It is found that the critical Rayleigh number obtained by the two nonlinear theories predicts subcritical motions. Further, the results on electro-convection in a rotating fluid layer are extracted from the corresponding problem of ferro-convection by establishing an one-to-one correspondence between the governing equations of the two problems.

## Keywords

Ferro-convection Electro-convection Energy stability Rayleigh–Bénard Lorenz model## 1 Introduction

The theory of rotating Newtonian fluids and Rayleigh–Bénard convection in such fluids is a well-studied topic [1, 2, 3, 4, 5, 6]. The corresponding problem in ferrofluids (called rotating ferro-convection) and in dielectric fluids (called rotating electro-convection) can be interesting for reasons more than one.

Ferrofluids are synthesized suspensions of micron-sized magnetic particles in a carrier liquid of suitable choice. The carrier liquids generally used are synthetic oil, water, kerosene, or such other liquids. The suspended magnetic particles that possess a fixed magnetic moment are coated with a surfactant, such as oleic acid, to avoid coagulation. In the presence of an external magnetic field, the resulting orientation of the magnetic particles leads to a net magnetization of the fluid which depends on both the applied magnetic field and the temperature of the fluid. The applied magnetic field exerts a force on the fluid which is known as the Kelvin force. When the magnetic field strength is very high, the magnetization achieves its saturation value and the particles get aligned with the applied field. The magnetization generally depends on the magnetic field, temperature, and density of the ferrofluid. Hence, a local variation in magnetic field/temperature may give rise to thermo-magnetic convection which is equivalent to the natural convection occurring due to variation in temperature only. This phenomenon has attracted the interest of many researchers in past five decades and has widespread scientific applications [7, 8, 9, 10, 11, 12, 13, 14, 15].

The linear and nonlinear ferro-convection in nonrotating fluids occupying a very shallow enclosure has now been extensively studied [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. The linear and nonlinear problems of rotating ferro-convection have also attracted attention of many [40, 41, 42, 43, 44, 45]. The nonlinear stability analysis of rotating ferro-convection reported above has been studied using weakly nonlinear stability analysis (based on Lorenz model or Ginzburg–Landau equation) or by energy stability analysis.

The work on nonlinear, rotating electro-convection is nonexistent, though there are few works on nonlinear electro-convection in the absence of rotation (Siddheshwar and Radhakrishna [46] and reference therein). The one-to-one correspondence between the linear problems of ferro-convection and electro-convection lead Siddheshwar [47, 48] to take a unified approach at studying both.

- 1.
Recognizing that linear and nonlinear electro-convection problems in rotating and nonrotating systems have an analogy with the corresponding problems of ferro-convection.

- 2.
Comparing the critical Rayleigh numbers of linear stability, weakly nonlinear stability, and energy stability of rotating ferro-convection as well as rotating electro-convection - existence of subcritical motions.

- 3.
Connecting the analytically intractable Lorenz model of nonrotating and rotating ferro-convection with the corresponding Landau model that is analytically tractable.

- 4.
Analyzing whether the analytical solution of the Landau equation provides insights into the nature of the amplitude of convection in transient and steady-state regimes.

## 2 Problem formulation for the rotating ferro-convection

*x*,

*y*,

*z*), where

*z*-axis is assumed in such a way that the gravitational force, \({\mathbf{g }}\), is taken as \({\mathbf{g }} = -g_0 \hat{{\mathbf{k }}}\). The layer is subjected to rotation with respect to the

*z*-axis with a constant angular speed \(\varvec{\Omega }\). The motion described here occurs in a way as it appears to an observer at rest in a frame rotating about the same axis and with the same angular velocity (uniformly rotating frame of reference). An external magnetic field \({\mathbf{H }}\) is applied in the

*z*-direction. An adverse temperature gradient across the fluid layer is maintained by imposing a uniform temperature difference \(\varDelta T\) between the plates. Schematic of the same is described in Fig. 1.

For mathematical tractability, all physical quantities are assumed to be independent of the horizontal co-ordinate, *y*. Thus, the study pertains to two-dimensional ferro-convection confined within a horizontal ferrofluid layer. The height, *d*, and breadth, *b*, are such that when \(d/b<< 1\), and hence, there is no effect of the lateral walls (*x*) on the dynamics in the bulk of the ferromagnetic fluid.

### 2.1 Governing equations

*t*is time. The subscripts

*I*and

*R*refer to inertial and rotating frames of reference and \({\mathbf{r }}\) is the radius vector. The operator \(\dfrac{{\mathrm{D}}}{{\mathrm{D}}t}\) in Eq. (1) is given by \(\dfrac{{\mathrm{D}}}{{\mathrm{D}}t}=\dfrac{\partial }{\partial t}+ {\mathbf{q }}\cdot \varvec{\nabla }\). The second and third terms in Eq. (1) are centrifugal and Coriolis forces that occur due to rotation. Since the rotation is uniform, the Euler force is neglected in Eq. (1). The centrifugal force can be expressed as the gradient of a scalar quantity:

*R*are

*P*, can be written as \(P=p-\dfrac{\rho _0}{2}|\varvec{\Omega }|^2 |{\mathbf{r }}|^2\), the sum of the hydro-static pressure (

*p*) and that due to the centrifugal force. The physical quantities \(\rho _0\), \(\mu _0\), \(\mu \), \(\kappa \), \({\mathbf{M }}\), \({\mathbf{H }}\), and

*T*represent the static density, magnetic permeability, effective viscosity, thermal diffusivity, magnetization, magnetic field, and temperature, respectively. In writing Eq. (4), the Boussinesq approximation is assumed to be valid. The state equation may be written as

*K*and \(\chi \) are the pyromagnetic and magnetic susceptibility coefficients, \(M_0=M(T_0, H_0)\) is the reference magnetization, and \(H_0 \hat{{\mathbf{k }}}\) is the applied uniform vertical magnetic field.

We note from Eq. (7)\(_1\) that we can write \({\mathbf{H }} = {\mathbf {\nabla }} \phi \), where \(\phi \) is the magnetic potential.

### 2.2 Boundary conditions

*w*is the \(z\)-components of velocity vector, \({\mathbf{q }}\).

*k*is dimensionless wave number. Since isothermal boundary condition is assumed for temperature and when \(\chi \rightarrow \infty \) at the boundary, we obtain \(D\phi =0\) (see Finlayson [16]) at \(z=0\) and

*d*.

There are umpteen number of instances in which idealized boundary conditions have been considered with the sole purpose of seeking qualitative results (see Finlayson [16], Suslov [51], Laroze et al. [38], Rahman and Suslov [39], and certain references therein). The present paper is one of them. The analysis is exact and trustworthy in this case. We, however, note that the problem is best done with realistic boundary condition but the price we pay for this is that the analysis, especially the nonlinear one, is fraught with errors due to mandatory numerical procedure imposed by it. This study is at the present time is contemplated to be completed in the nearest future.

### 2.3 Basic state solution

*b*represents the basic state.

## 3 Stability analysis

### 3.1 Perturbation equations

*z*-direction, and \(\phi _z\) is the differentiation of \(\phi \) with respect to

*z*.

### 3.2 Nondimensionalization

*d*for length, \(d^2/\kappa \) for time, \(\kappa /d\) for velocity, \(\beta d\) for temperature, \(\beta d^2 K/(1+\chi )\) for magnetic scalar potential, and \(\mu \kappa /d^2\) for the pressure and the stress tensor. Using these scales in Eqs. (18)–(20) and simplifying, we get the dimensionless governing equations for the perturbations in the following form:

*curl*on Eq. (22) to get the vorticity equation:

*z*-component of vorticity which describes the local spinning motion of the continuum point. The term, \( \frac{\partial (\theta , \phi _z)}{\partial (x,z)}\), represents the Jacobian term and is defined as

## 4 Lorenz model

*u*,

*v*,

*w*, \(\theta \), and \(\phi \):

*t*, \(\delta ^2 = k^2+\pi ^2\), \(\xi _1 = \dfrac{\delta _M^2 + k^2 M_1 M_3}{\delta ^2 \delta _M^2}\), and \(\xi _2 = \dfrac{\pi k^2 M_1 M_3}{\delta ^2 \delta _M^2}\).

## 5 Steady finite-amplitude convection and subcritical motions

*k*, as a function of \(\text {Ta}\), \(M_1\), \(M_3\), and \(\text {Pr}\).

## 6 Landau amplitude equation for small amplitude convective motion

### 6.1 First-order system

### 6.2 Second-order system

### 6.3 Third-order system

*L*is the electric number. It is quite clear that Eqs. (49)–(52) can be obtained from Eqs. (21)–(24) by replacing \(M_1\) and \(M_3\) by

*L*and unity, respectively. Using this one-to-one correspondence between the ferro-convection and the electro-convection problems, we can easily obtain the Rayleigh numbers of the linear stability, weakly nonlinear stability, and energy stability analyses of rotating electro-convection by using Eqs. (32), (31), and (33), respectively.

## 7 Results and discussion

From the figure, it is obvious that for the present problem subcritical motions are possible. Further, it is apparent that the energy stability analysis yields the least critical Rayleigh number. A look at the critical wave number plot in Fig. 2 reveals that for the considered range of parameter values, the critical wave number varies in tandem with its corresponding Rayleigh number.

- 1.
Linearized Landau equation and

- 2.
The cubic Landau equation.

## 8 Conclusion

- 1
The present study leads to the conclusion that the critical value of the Rayleigh number of infinitesimal amplitude perturbation and those by the energy stability and weakly nonlinear stability analyses are not the same. The latter two instabilities are, in fact, subcritical in nature. These results clearly point to the fact that in so far as rotating Rayleigh–Bénard convection in ferromagnetic liquids or dielectric liquids is concerned, the linear stability analysis is inadequate and so is the weakly nonlinear stability analysis.

- 2
The present paper is a good example of treating rotating ferro-convection and electro-convection in a unified way. This would mean that there is really no need to study the electro-convection in isolation as the results can be got from the ferro-convection problem by considering the electric number in place of the buoyancy magnetization parameter and by taking a value of unity for the non-buoyancy magnetization parameter.

- 3
The weakly nonlinear stability analysis leads to the analytically intractable five-dimensional Lorenz model (with two quadratic nonlinearities) and this can be reduced to the one-dimensional Landau model (with a cubic nonlinearity) that is analytically tractable. The latter arises due to a local nonlinear stability analysis. This procedure of reduction gives us a physically realistic bounded solution of the Landau equation and thus we thereby come to know that some results of a weakly nonlinear stability analysis can, in fact, be obtained from a local nonlinear stability analysis without the need to pursue a numerical method.

## Notes

### Acknowledgements

The authors are grateful to their respective institutes of working for their encouragement. We thank the reviewer for the most valuable comments that helped us to modify the paper to the present form.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Chandrasekhar S (1981) Hydrodynamic and hydromagnetic stability. Dover, New YorkzbMATHGoogle Scholar
- 2.Getling AV (2001) Rayleigh–Bénard convection: structures and dynamics. World Scientific Press, SingaporezbMATHGoogle Scholar
- 3.Bhattacharjee JK (1987) Convection and chaos in fluids. World Scientific Press, SingaporezbMATHCrossRefGoogle Scholar
- 4.Greenspan HP (1969) The theory of rotating fluids. Cambridge University Press, LondonzbMATHGoogle Scholar
- 5.Veronis G (1966) Motions at subcritical values of the Rayleigh number in a rotating fluid. J Fluid Mech 24(3):545–554MathSciNetCrossRefGoogle Scholar
- 6.Straughan B (2013) The energy method, stability, and nonlinear convection. Springer, BerlinzbMATHGoogle Scholar
- 7.Rosensweig RE (1985) Ferrohydrodynamics. Cambridge University Press, CambridgeGoogle Scholar
- 8.Bashtovoy VG, Berkovsky BN, Vislovich AN (1988) Introduction to thermodynamics of magnetic fluids. Hemisphere, WashingtonGoogle Scholar
- 9.Nakatsuka K, Jeyadevan B, Neveu S, Koganezawa H (2002) The magnetic fluid for heat transfer applications. J Magn Magn Mater 252:360–362CrossRefGoogle Scholar
- 10.Shliomis MI (1974) Magnetic fluids. Sov Phys Usp 17:153–169CrossRefGoogle Scholar
- 11.Dubina SH, Wedgewood LW (2016) A Brownian dynamics study on ferrofluid colloidal dispersions using an iterative constraint method to satisfy Maxwell’s equations. Phys Fluids 28:072001CrossRefGoogle Scholar
- 12.Singh C, Das AK, Das PK (2016) Flow restrictive and shear reducing effect of magnetization relaxation in ferrofluid cavity flow. Phys Fluids 28:087103CrossRefGoogle Scholar
- 13.Jackson BA, Terhune KJ, King LB (2017) Ionic liquid ferrofluid interface deformation and spray onset under electric and magnetic stresses. Phys Fluids 29:064105CrossRefGoogle Scholar
- 14.Hassan MR, Zhang J, Wang C (2018) Deformation of a ferrofluid droplet in simple shear flows under uniform magnetic fields. Phys Fluids 30:092002CrossRefGoogle Scholar
- 15.Ahmed A, Fleck BA, Waghmare PR (2018) Maximum spreading of a ferrofluid droplet under the effect of magnetic field. Phys Fluids 30:077102CrossRefGoogle Scholar
- 16.Finlayson BA (1970) Convective instability of ferromagnetic fluids. J Fluid Mech 40:753–767zbMATHCrossRefGoogle Scholar
- 17.Lalas DP, Carmi S (1971) Thermoconvective stability of ferrofluids. Phys Fluids 14:436–437CrossRefGoogle Scholar
- 18.Schwab L, Hildebrandt U, Stierstadt K (1983) Magnetic Bénard convection. J Magn Magn Mater 39:113–114CrossRefGoogle Scholar
- 19.Stiles PJ, Kagan MJ (1990) Thermo-convective instability of a horizontal layer of ferrofluid in a strong vertical magnetic field. J Magn Magn Mater 85:196–198CrossRefGoogle Scholar
- 20.Siddheshwar PG (1993) Rayleigh–Bénard convection in a ferromagnetic fluid with second sound. Jpn Soc Mag Fluids 25:32–36Google Scholar
- 21.Siddheshwar PG (1995) Convective instability of ferromagnetic fluids bounded by fluid permeable magnetic boundaries. J Magn Magn Mater 149(1–2):148–150CrossRefGoogle Scholar
- 22.Sekhar GN, Rudraiah N (1991) Convection in magnetic fluids with internal heat generation. Trans ASME J Heat Trans 113:122–127CrossRefGoogle Scholar
- 23.Siddheshwar PG, Abraham A (2003) Effect of time-periodic boundary temperatures/body force on Rayleigh–Bénard convection in a ferromagnetic fluid. Acta Mech. 161:131–150zbMATHGoogle Scholar
- 24.Bajaj R, Malik SK (1995) Pattern formation in ferrofluids. J Magn Magn Mater 149(1–2):158–161CrossRefGoogle Scholar
- 25.Bajaj R, Malik SK (1997) Convective instability and pattern formation in magnetic fluids. J Math Anal Appl 207(1):172–191MathSciNetzbMATHCrossRefGoogle Scholar
- 26.Blennerhassett PJ, Lin F, Stiles PJ (1991) Heat transfer through strongly magnetized ferrofluids. Proc R Soc A 433:165–177zbMATHCrossRefGoogle Scholar
- 27.Abdullah AA, Lindsay KA (1991) Bénard convection in a non-linear magnetic fluid under the influence of a non-vertical magnetic field. Continuum Mech Thermodyn 3(1):13–25MathSciNetzbMATHCrossRefGoogle Scholar
- 28.Zebib A (1996) Thermal convection in a magnetic fluid. J Fluid Mech 321:121–136zbMATHCrossRefGoogle Scholar
- 29.Gotoh K, Yamada M (1982) Thermal convection in a horizontal layer of magnetic fluids. J Phys Soc Jpn 51:3042–3048CrossRefGoogle Scholar
- 30.Luo WL, Du T, Huang J (1999) Novel convective instability in a magnetic fluid. Phys Rev Lett 82(20):4134–4137CrossRefGoogle Scholar
- 31.Russell CL, Blennerhassett PJ, Stiles PJ (1999) Supercritical analysis of strongly nonlinear vortices in magnetized ferrofluids. Proc R Soc Lond A 455:23–67MathSciNetzbMATHCrossRefGoogle Scholar
- 32.Schwab L, Stierstadt K (1987) Field-induced wavevector-selection by magnetic Bénard-convection. J Magn Magn Mater 65(2–3):315–316CrossRefGoogle Scholar
- 33.Schwab L (1990) Thermal convection in ferrofluids under a free surface. J Magn Magn Mater 85(1–3):199–202CrossRefGoogle Scholar
- 34.Schwab L, Hildebrandt U, Stierstadt K (1983) Magnetic Bénard convection. J Magn Magn Mater 39(1–2):113–114CrossRefGoogle Scholar
- 35.Sunil Mahajan A (2008) A nonlinear stability analysis for magnetized ferrofluid heated from below. Proc R Soc A 464:83–98MathSciNetzbMATHCrossRefGoogle Scholar
- 36.Tangthieng C, Finlayson BA, Maulbetsch J, Cader T (1999) Heat transfer enhancement in ferrofluids subjected to steady magnetic fields. J Magn Magn Mater 201:252–255CrossRefGoogle Scholar
- 37.Snyder SM, Cader T, Finlayson BA (2003) Finite element model of magnetoconvection of a ferrofluid. J Magn Magn Mater 262:269–279CrossRefGoogle Scholar
- 38.Laroze D, Siddheshwar PG, Pleiner H (2013) Chaotic convection in a ferrofluid. Commun Nonlinear Sci Numer Simul 18(9):2436–2447MathSciNetzbMATHCrossRefGoogle Scholar
- 39.Rahman H, Suslov SA (2015) Thermomagnetic convection in a layer of ferrofluid placed in a uniform oblique external magnetic field. J Fluid Mech 764:316–348MathSciNetCrossRefGoogle Scholar
- 40.Gupta MD, Gupta AS (1979) Convective instability of a layer of ferromagnetic fluid rotating about a vertical axis. Int J Eng Sci 17:271–277zbMATHCrossRefGoogle Scholar
- 41.Venkatasubramanian S, Kaloni PN (1994) Effects of rotation on the thermoconvective instability of a horizontal layer of ferrofluids. Int J Eng Sci 32:237–256zbMATHCrossRefGoogle Scholar
- 42.Aurenhammer GK, Brand HR (2000) Thermal convection in a rotating layer of a magnetic fluid. Eur Phys J B 16:157–168CrossRefGoogle Scholar
- 43.Sunil Mahajan A (2008) A nonlinear stability analysis for rotating magnetized ferrofluid heated from below. Appl Math Comput 204:299–310MathSciNetzbMATHGoogle Scholar
- 44.Kaloni PN, Lou JX (2004) Weakly nonlinear instability of a ferromagnetic fluid rotating about a vertical axis. J Magn Magn Mater 284:54–68CrossRefGoogle Scholar
- 45.Nanjundappa CE, Shivakumara IS, Prakash HN (2014) Effect of Coriolis force on thermomagnetic convection in a ferrofluid saturating porous medium: a weakly nonlinear stability analysis. J Magn Magn Mater 370:140–149CrossRefGoogle Scholar
- 46.Siddheshwar PG, Radhakrishna D (2012) Linear and nonlinear electro-convection under AC electric field. Commun Nonlinear Sci Numer Simul 17:2883–2895MathSciNetzbMATHCrossRefGoogle Scholar
- 47.Siddheshwar PG (2002) Ferrohydrodynamic and electrohydrodynamic instability in Newtonian liquids: an analogy. East West J Math Spec Vol Comput Math Model 2:143–146Google Scholar
- 48.Siddheshwar PG (2002) Oscillatory convection in ferromagnetic, dielectric and viscoelastic liquids. Int J Mod Phys B 16:2629–2635CrossRefGoogle Scholar
- 49.Siddheshwar PG, Kanchana C (2017) Unicellular unsteady Rayleigh–Bénard convection in Newtonian liquids and Newtonian nanoliquids occupying enclosures: new findings. Int J Mech Sci 131–132:1061–1072CrossRefGoogle Scholar
- 50.Dey P, Suslov SA (2016) Thermomagnetic instabilities in a vertical layer of ferrofluid: nonlinear analysis away from a critical point. Fluid Dyn Res 48(6):061404MathSciNetCrossRefGoogle Scholar
- 51.Suslov SA (2008) Thermomagnetic convection in a vertical layer of ferromagnetic fluid. Phys Fluids 24:084101zbMATHCrossRefGoogle Scholar