## Abstract

A nonlinear (energy) stability analysis is performed for a magnetized ferrofluid layer heated from below, in the stress-free boundary case. By introducing a suitable generalized energy functional, a rigorous nonlinear stability result is derived for a thermoconvective magnetized ferrofluid. The mathematical emphasis is on how to control the nonlinear terms caused by the magnetic body and inertia forces. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of the linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in the case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, *M*_{3}, on subcritical instability region has also been analysed. It is shown that with the increase of magnetic parameter, *M*_{3}, the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and the magnetic forces in the nonlinear energy stability analysis.

## 1. Introduction

Conventional hydrodynamic stability theory is mainly concerned with the determination of critical values of Rayleigh number, demarcating a region of stability from that of instability. The potentials of the linear theory of stability and the energy method are complementary to each other in the sense that linear theory gives the conditions under which hydrodynamic systems are definitely unstable. It cannot with certainty conclude stability. On the other hand, energy theory gives the conditions under which hydrodynamic systems are definitely stable. It cannot with certainty conclude instability. Suffering from its basic assumptions, the validity of the linearized stability theory becomes questionable. Hence, the nonlinear approach becomes inevitable to investigate the effects of finite disturbances. Indeed, there are cases for which the two theories coincide or nearly coincide. In these cases, the energy method gives sufficient conditions for stability, which are physically reasonable as well as the ranges of the stability parameters for which subcritical instabilities are possible.

The oldest method of nonlinear stability analysis that can deal with finite disturbances is the energy method, originated by Reynolds (1895) and Orr (1907), and then later Serrin (1959) and Joseph (1965, 1966, 1976) reformulated the energy method. Despite the success of this classical energy method in several stability problems, there is some scepticism about its ongoing indiscriminate use. Situations have been encountered, for example, in the magnetic Bénard problem (Rionero 1968; Galdi 1985) and the rotating Bénard problem (Galdi & Straughan 1985), where the classical energy theory did not produce the expected results. Rapid improvements of the classical energy theory have been made in the recent years (see Galdi & Padula 1990; Straughan 2004), and the Lyapunov direct method employed by Galdi (1985), Galdi & Straughan (1985), Rionero & Mulone (1988), Mulone & Rionero (1989), Galdi & Padula (1990) and Qin & Kaloni (1995) appears to have been the most successful one. Galdi & Straughan (1986) have considered a modified model problem of Drazin and Reid exhibiting sharp conditional stability, and showed that the instability may also occur from perturbations that, initially, are outside ball B of a suitable function space. It is now generally believed that this generalized energy method is definitely superior to the classical energy method. A nonlinear stability analysis of non-magnetic fluids using generalized energy stability theory has been considered by many authors (Guo *et al*. 1994; Guo & Kaloni 1995; Straughan & Walker 1996; Kaloni & Qiao 1997*a*,*b*, 2001; Straughan 1998, 2001; Payne & Straughan 2000).

Ferrofluids—suspensions of magnetic nanoparticles—exhibit a specific feature of the magnetic control of their physical parameters and flows appearing in such fluids. This magnetic control can be achieved by means of moderate magnetic fields with strength of the order of 10 mT. This sort of magnetic control also enables the design of a wide variety of technical applications such as the use of magnetic forces for basic research in fluid dynamics. One of the major applications of ferrofluid is its use in medical fields such as the transport of drugs to an injured site and the removal of tumours from the body. Other applications include the following: used as coolant and in devices like rotating shaft seals; loudspeakers; printers; sealed motors; acoustic devices, etc. The field of ferrofluid research is approximately 40 years old. Rosensweig (1985) has given an authoritative introduction to the research on magnetic liquids in his monograph, and the study of the effect of magnetization yields interesting information. The theory of convective instability of ferrofluids began with Finlayson (1970) and was interestingly continued by Lalas & Carmi (1971), Shliomis (1974), Schwab *et al*. (1983), Stiles & Kagan (1990), Blennerhassett *et al*. (1991), Venkatasubramanian & Kaloni (1994) and Sunil *et al*. (2005*a*,*b*, 2006).

The purpose of the present paper is to study the nonlinear stability analysis of magnetized ferrofluid heated from below via a generalized energy method. When buoyancy magnetization is absent (non-ferrofluid), there is a coincidence between the nonlinear and linear stability results. This in turn implies exclusion of the occurrence of subcritical instability. For a convection problem in magnetized ferrofluids, the linear critical magnetic thermal Rayleigh number is found to be higher in value than the nonlinear (energy) critical magnetic thermal Rayleigh number, which shows the possibility of the existence of subcritical instability. In this paper, we also examine the exact form of coupling between the buoyancy and the magnetic forces. The comparison of the results obtained by the energy method and the linear stability analysis, respectively, has been discussed finally. The mechanism of developing stability bounds in a ferrofluid is also important in material processing owing to its applications to the possibility of producing various new materials. This problem, to the best of our knowledge, has not been investigated yet.

## 2. Mathematical formulation of the problem

Here, we consider an infinite horizontal layer of thickness *d* of an electrically non-conducting incompressible thin ferrofluid with constant viscosity heated from below. The fluid is assumed to occupy the layer with gravity acting in the negative *z*-direction and the magnetic field, , acts outside the layer.

The equations governing the flow of an incompressible magnetized ferrofluid (using the Boussinesq approximation) are given as follows (Finlayson 1970; Rosensweig 1985):

Mass balance(2.1)Momentum balance(2.2)Temperature equation(2.3)Maxwell's equations in the magnetostatic limit are (2.4)The magnetization has the relationship(2.5)where *ρ*, *ρ*_{0}, ** q**,

*t*,

*p*,

*μ*,

*μ*

_{0},

**,**

*M***,**

*B**κ*

_{m},

*C*

_{0}and

*α*are the fluid density, reference density, filter velocity, time, pressure, viscosity, magnetic permeability of vacuum, magnetization, magnetic induction, thermal conductivity, specific heat at constant pressure and coefficient of thermal expansion, respectively. The subscripts m and f refer to the fluid–solid mixture and the fluid, respectively.

*T*

_{a}is the average temperature given by , where and are the constant average temperatures of the lower and upper surfaces of the layer, , and . The susceptibility and the pyromagnetic coefficients are defined, respectively, byThe basic state is assumed to be quiescent and is given by

(2.6)where the subscript ‘b’ denotes the basic state.

The nonlinear equations for the perturbations , and , which represent velocity, density, pressure, temperature, magnetic field intensity and magnetization, respectively, are given by(2.7a)(2.7b)(2.7c)(2.7d)where ; ; (by equation (2.4)_{2}); and *ϕ*′ is the perturbed magnetic potential.

The boundary conditions are(2.8)and ** q**′,

*θ*and

*ϕ*′ satisfy a plane tiling periodicity.

In §3, we develop a nonlinear energy stability analysis for equations (2.7*a*)–(2.7*d*) and (2.8).

## 3. Nonlinear stability analysis

To investigate the nonlinear stability analysis, the governing equations (2.7*a*)–(2.7*d*) in non-dimensional form (dropping asterisk) can be written as(3.1a)(3.1b)(3.1c)(3.1d)where the following non-dimensional quantities and parameters are introduced:where *R* is the convectional Rayleigh number; *N* is the magnetic Rayleigh number; and *M*_{1} is a ratio of the magnetic-to-gravitational forces. The parameter *M*_{3} measures the departure of linearity in the magnetic equation of state, and values from one (*M*_{0}=*ΧH*_{0}) to higher values are possible for the usual equations of state.

On multiplying (3.1*a*) by ** q**, (3.1

*c*) by

*θ*, (3.1

*d*) by

*ϕ*and integrating over

*V*, we get (after using equation (3.1

*b*), the boundary conditions and the divergence theorem)(3.2a)(3.2b)(3.2c)where denotes the integration over

*V*; denotes the

*L*

^{2}(

*V*) norm; and

*V*denotes a typical periodicity cell.

To study the nonlinear stability of basic state (2.6), an *L*^{2} energy, *E*(*t*), is constructed using equations (3.2*a*)–(3.2*c*), and the evolution of *E*(*t*) is given by(3.3)where(3.4)(3.5)(3.6)(3.7)with *λ*_{1} and *λ*_{2} two positive coupling parameters.

We now define(3.8)where *H* is the space of admissible solutions.

Then, we require *m*<1 so that(3.9)with .

In order to dominate the nonlinear terms and for studying the (conditional) nonlinear stability, we now introduce the generalized energy functional as(3.10)where *b*_{0} is a positive coupling parameter to be chosen and the complementary energy *E*_{1}(*t*) is given by(3.11)

The evolution of *V*_{g}(*t*) is given by(3.12)where(3.13)(3.14)(3.15)

Now, we write some easily obtainable results from equation (3.1*d*) and recall the embedding theorems(3.16)where *C*^{*} is a computable positive constant depending on *V*, and its value was given by Galdi & Straughan (1985) and the statement proved by Adams (1975).

Therefore, from equation (3.13) using (3.6), (3.14) and (3.16), the Cauchy–Schwartz and the Young inequalities (Hardy *et al*. 1994), we haveChoosingand defining(3.17)it then follows easily that(3.18)We next estimate nonlinear terms *N*_{1} and *N*_{0}. With the help of (3.10), (3.16) and (3.17), we find(3.19)and (3.20)Using (3.17)–(3.20) in (3.12), we get(3.21)where(3.22)This last estimate enables us to prove the following theorem of conditional nonlinear stability criterion.

*Let*(3.23)(3.24)*with À given by equation* (*3.22*). *Then, there exists a positive constant K*^{*}, *such that*(3.25)

The hypothesis and inequality (3.21) ensures thatTherefore, from inequality (3.21) by a recursive argument, we obtain(3.26)Now, we prove that there exists , such that(3.27)By equation (3.10), using equations (3.6), (3.14) and (3.17), and by virtue of the Poincaré-type inequalities, we haveLet , such that(3.28)then we obtainAssuming

with *k*_{0} given by (3.28), from (3.26) and (3.27), we have(3.29)Upon integrating this last inequality, we deduce the theorem. ▪

Since in equation (3.10) does not contain the term , the kinetic energy term for magnetic potential, it is worthwhile checking as to what happens to as *t*→∞.

Using inequality (3.16)_{1}, we have(3.30)Thus, equation (3.10) and inequality (3.30) ensures the decay of , i.e. .

### (a) Variational problem

We now return to equation (3.8) and use calculus of variation to find the maximum problem at the critical argument *m*=1. The associated Euler–Lagrange equations after taking transformations (dropping caps) are(3.31a)(3.31b)(3.31c)where *p* is a Lagrange's multiplier introduced, since ** q** is solenoidal.

On taking curl curl of equation (3.31*a*) and then taking the third component of the resulting equation, we find(3.32)Now, we assume a plane tiling form(3.33)where and *a* being the wavenumber (Chandrasekhar 1981, pp. 106–114; Straughan 2001). The wavenumber is found a *posteriori* to be non-zero; so from equations (3.32), (3.31*b*) and (3.31*c*), we see that satisfy(3.34a)(3.34b)(3.34c)Thus, the exact solution to the equations (3.34*a*)–(3.34*c*) subject to boundary conditions(3.35)is written in the form(3.36)where *A*_{0}, *B*_{0} and *C*_{0} are constants. Substituting solution (3.36) into equations (3.34*a*)–(3.34*c*), we get the equations involving coefficients of *A*_{0}, *B*_{0} and *C*_{0}. For the existence of non-trivial solutions, the determinant of the coefficients of *A*_{0}, *B*_{0} and *C*_{0} must vanish. This determinant on simplification yields(3.37)where ; ; and .

Maximum values of *λ*_{1} and are determined by the conditions and , respectively, and are found to be(3.38)The classical results in respect of Newtonian fluids (non-magnetic fluid) can be obtained when *M*_{1}=0, which imply *λ*_{1}=1 and . Thus, *λ*_{1}=1 is the optimal value, i.e. value that maximizes *R*_{e} (Straughan 2004, pp 59–62). Since we are dealing with magnetized ferrofluid, for sufficiently large values of buoyancy magnetization (*M*_{1}) we have(3.39)Using (3.38) and (3.39) in equation (3.37), we have(3.40)For *M*_{1} being sufficiently large, we obtain the magnetic thermal Rayleigh number(3.41)As a function of *x*, *N*_{e} given by equation (3.41) attains its minimum when(3.42)The coefficients *P*_{0}, …, *P*_{5}, being quite lengthy, has not been included here and are evaluated during numerical calculations. The values of critical wavenumber in nonlinear stability results are determined numerically using Newton–Raphson method by the condition . With *x* determined as a solution of equation (3.42), equation (3.41) will give the required critical magnetic thermal Rayleigh number *N*_{ce}.

The critical wavenumber, *x*_{c}, and critical magnetic thermal Rayleigh number, *N*_{ce}, depend on the parameter *M*_{3}, taking the valuesand intermediate values for intermediate *M*_{3}. Here, we can rearrange equation (3.40) to demonstrate the interaction of the buoyancy and magnetic modes of instability.(3.43)When *M*_{3} is very large, (3.43) reduces to .

This result demonstrates a tight coupling between the buoyancy and the magnetic forces for nonlinear energy stability analysis, which is possible because each individual convective mechanism yields the same wavenumber. Thus, for stability, an increase in the forces due to one mechanism makes possible a proportional decrease in the forces due to the other mechanism. This holds only for special values of the parameters; otherwise equation (3.43) must be used.

For analysing the linear instability results, we return to the perturbed equations (2.7*a*)–(2.7*d*) neglecting the nonlinear terms. We again perform the standard stationary mode analysis and look for the solution of these equations in the form (3.33). The boundary conditions in the present case are the same as those in (3.35). Following the procedure as stated earlier in the energy stability case, we have(3.44)This is exactly eqn (20) from Finlayson (1970).

We again consider that magnetic thermal Rayleigh number depends on the parameter *M*_{3}. For *M*_{1} being sufficiently large, the critical magnetic thermal Rayleigh number, in the linear case, is defined as .

The interaction of the buoyancy and magnetic modes of instability has been demonstrated by Finlayson (1970). When *M*_{3} is very large, his result demonstrates a tight coupling between the buoyancy and the magnetic forces for linear instability analysis.

There are instances in which the two theories coincide. This is true for the classical Bénard problem. In the absence of magnetic parameters (*M*_{1}=0 and *M*_{3}=0), we obtaini.e. the linear instability boundary≡the nonlinear stability boundary.

Here, the energy method leads to the strong result that arbitrary subcritical instabilities are not possible, which is in good agreement with the previous published work (Joseph 1965, 1966). Thus, for lower values of magnetic parameters, this coincidence is immediately lost.

## 4. Discussion of results and conclusion

The critical wavenumbers, and *x*_{ce}, and critical magnetic thermal Rayleigh numbers, and *N*_{ce}, depends on *M*_{3}. The variation of and *x*_{ce} and and *N*_{ce} with various parameters are given in table 1 and the results are further illustrated in figure 1. Figure 1 represents the plot of critical magnetic thermal Rayleigh numbers and *N*_{ce} versus magnetic parameter *M*_{3}. This figure indicates that the magnetic parameter *M*_{3} has a destabilizing effect, because as *M*_{3} increases, the values of and *N*_{ce} decreases. We also note that the values of are always higher than those of *N*_{ce}, and this is quite understandable because the linear stability theory gives sufficient conditions for instability, while the energy stability theory gives the sufficient condition for stability. Thus, the difference between the values of and *N*_{ce} reveals that there is a band of Rayleigh numbers where subcritical instabilities may arise. We note that this band shrinks as *M*_{3} increases (table 1). We also demonstrate coupling between the buoyancy and the magnetic forces in nonlinear energy stability analysis as well as linear instability analysis. The exact form of the coupling suggested by (3.43) in nonlinear analysis have been determined numerically and compared with the results obtained by linear analysis (Finlayson 1970) for free–free boundaries. The comparison of the results is shown in table 2 and figures 2 and 3. We conclude that the values obtained for coupling are closer to those for tight coupling in nonlinear energy stability analysis when compared with linear instability analysis.

In conclusion, for the proposed model, we are able to derive a rigorous nonlinear energy stability result for a magnetized ferrofluid by performing a nonlinear energy stability (conditional) analysis. We derive a nonlinear stability threshold very close to the linear instability one. We also see that the magnetic mechanism alone can induce a subcritical region of instability. For a convection problem in magnetized ferrofluids, the linear critical magnetic thermal Rayleigh number is found to be higher in value than the nonlinear (energy) critical magnetic thermal Rayleigh number, which shows the possibility of the existence of subcritical instability. It is important to realize that this region shrinks as magnetization increases.

Since we examined the nonlinear stability as well as linear instability analysis around the basic state by introducing small perturbations; so at this point it is worthwhile saying that for large initial perturbations, the nonlinear stability Rayleigh number is exactly the same as that for linear instability, and also an early onset of convection occurs. Also, in non-magnetic fluids, the best possible result is verified in that we show that the global nonlinear stability Rayleigh number is exactly the same as that for linear instability.

## Acknowledgments

The authors are grateful to Prof. D. D. Joseph, University of Minnesota, Minneapolis, and Prof. P. N. Kaloni, University of Windsor, Canada, for providing the necessary literature. Also, they would like to thank the two anonymous referees for their remarks and suggestions, which really improved the work considerably.

## Footnotes

- Received March 20, 2007.
- Accepted September 17, 2007.

- © 2007 The Royal Society