## Abstract

Stability analysis of parametrically driven surface waves in liquid metals in the presence of a uniform vertical magnetic field is presented. Floquet analysis gives various subharmonic and harmonic instability zones. The magnetic field stabilizes the onset of parametrically excited surface waves. The minima of all the instability zones are raised by a different amount as the Chandrasekhar number is raised. The increase in the magnetic field leads to a series of bicritical points at a primary instability in thin layers of a liquid metal. The bicritical points involve one subharmonic and another harmonic solution of different wavenumbers. A tricritical point may also be triggered as a primary instability by tuning the magnetic field.

## 1. Introduction

Parametrically driven surface waves, known since the pioneering experiments of Faraday (1831), have attracted considerable interests (Zhang & Viñals 1997*a*,*b*; Mahr & Rehberg 1998; Müller 1998; Miles 1999; Pétrélis *et al*. 2000; Kudrolli *et al*. 2001; Arbell & Fineberg 2002; Porter & Silber 2002; Wagner *et al*. 2003; Westra *et al*. 2003). Thin layers of viscous fluids show an interesting possibility of a bicritical point (Kumar 1996; Cerda & Tirapegui 1997; Kumar *et al*. 2004; Mondal & Kumar 2004, 2006) at the onset of driven surface waves. They are also seen in the presence of a two-frequency vertical vibration (Edwards & Fauve 1994; Besson *et al*. 1996; Silber & Skeldon 1999). The magnetic field is known to have a stabilizing effect on convective instability (e.g. Chandrasekhar 1961) in liquid metals. The role of the horizontal magnetic field on surface waves in liquid metals was investigated recently by Ji *et al*. (2005). However, the effects of the magnetic field on *parametrically driven* surface waves in liquid metals are not investigated. The presence of the magnetic field increases dissipation in fluids due to Joule heating. The selection of wavenumber in thin layers of viscous fluids is strongly affected by the dissipation. The presence of the magnetic field is therefore likely to modify multicritical points, which occur in thin layers of fluids. This may significantly affect pattern selection close to the onset of driven surface waves.

In this paper, we investigate the stability of parametrically driven surface waves in thin layers of a liquid metal (e.g. mercury) subjected to a uniform vertical magnetic field. The magnetic field always stabilizes the onset of driven surface waves. The surface waves in thick (greater than or equal to 1 cm) layers of a liquid metal at a higher forcing frequency (greater than 10 Hz) are found to be subharmonic with sinusoidal forcing, if the forcing amplitude is raised above a threshold. The magnetic field leads to a series of bicritical points involving different wavenumbers and response frequencies in thin layers of fluids at low forcing frequencies. This leads to a series of jumps in critical wavenumber as a function of magnetic field. In addition, there is a possibility of a tricritical point as the primary instability in the presence of a small magnetic field. These may lead to the selection of interesting superlattice patterns and pattern dynamics.

## 2. Hydrodynamic system

We consider a laterally extended layer of thickness *d* of an incompressible liquid metal of density *ρ*, kinematic viscosity *ν* and magnetic diffusivity *λ*_{m} resting on a flat, rigid and perfectly conducting plate, which is subjected to a vertical sinusoidal oscillation of amplitude *ϵ* and angular frequency *ω* in the presence of a constant vertical magnetic field ** B**=

*B*

_{0}

**, where**

*λ***is a unit vector along the vertically upward direction.**

*λ*In a frame of reference fixed with the vibrating plate, the free surface of the fluid is initially flat, stationary and coincident with the *z*=0 plane. In this frame of reference, the effective gravity is *G*(*t*)=*g*−*ϵ* cos *ωt*. The basic state of rest has a time-dependent pressure *P*(*t*)=*P*_{0}−*ρG*(*t*)*z*, where *P*_{0} is the uniform atmospheric pressure. As the flat surface is deformed at the onset of surface waves, the free surface is located at *z*=*ζ*(**x**, *t*), where **x**≡(*x*, *y*) lies in the horizontal plane, and obeys the kinematic condition (Lamb 1932)(2.1)The magnetic field in the fluid is no longer uniform. The perturbed magnetic field ** b**(

**x**,

*z*,

*t*) in the fluid makes the total magnetic field

**=**

*B**B*

_{0}

**+**

*λ***. Rescaling all the length-scales by the fluid thickness**

*b**d*before surface instability, time-scales by the viscous diffusion time

*d*

^{2}/

*ν*and the perturbation in the magnetic field by the magnitude of the uniform external magnetic field

*B*

_{0}, we arrive at the following hydrodynamical equations (e.g. Meneguzzi

*et al*. 1987), which govern the dynamics in the bulk of the liquid at the onset of surface waves:(2.2)(2.3)(2.4)where

_{m}=

*ν*/

*λ*

_{m}is the magnetic Prandtl number and is the Chandrasekhar number. For most of the liquid metals (e.g. mercury), the magnetic Prandtl number (

_{m}≈10

^{−6}) is very small and may be neglected. We set

_{m}=0 in the rest of this paper. The induced magnetic field is slaved to the velocity field in this limit, and can be determined by the equation(2.5)The dynamic boundary conditions are determined by the stress tensor at the free surface. The dimensionless stress tensor now is the sum of the standard liquid stress and the Maxwell stress tensor . The perturbed part of the total stress tensor

_{ij}at the liquid surface is then given by(2.6)The dimensionless form of the effective gravity is , where is the Galileo number, is the dimensionless forcing amplitude and

*Ω*=

*ωd*

^{2}/

*ν*is the dimensionless forcing frequency. The jump in the normal stress at any point on the free surface is equal to the surface tension

*σ*times the curvature of the surface at that point. For the dimensionless linear problem, this translates to(2.7)where is the capillary number. Equations (2.6) and (2.7) lead to the expression for dimensionless form of the pressure at the free surface given as(2.8)Another equation for the pressure in the bulk of the liquid, computed from (2.2), is given by(2.9)Eliminating the pressure from equations (2.8) and (2.9) at the free surface (

*z*=0), we arrive at the following equation:(2.10)The tangential stresses

*Π*

_{xz}and

*Π*

_{yz}must also vanish at the free surface (

*z*=0). By taking partial differentiation of

*Π*

_{xz}and

*Π*

_{yz}with respect to

*x*and

*y*, respectively, adding them and using the equation of continuity (2.4) lead to the following condition:(2.11)The liquid rests on a rigid plate, located at

*z*=−1, which is assumed to be perfectly conducting. All the components of the velocity field vanish at the rigid plate. In addition, all the components of magnetic field (section 42 in Chandrasekhar 1961) should also vanish in the perfectly conducting plate. That is,(2.12)The free surface is actually the fluid–air interface. As the density and the viscosity of air are negligibly small in comparison to those of the liquid, the presence of air does not affect the dynamics in the liquid. Since air is non-conducting, no current can cross the boundary. Consequently, the current

*j*

_{3}normal to the free surface is zero. In addition, a normal magnetic field must be continuous at the free surface. This leads to the continuity of all components of the magnetic field

**across the free surface. However, they must vanish far away from the free surface. Therefore,(2.13)and**

*b***→0 as**

*b**z*→∞. The perturbative magnetic field

*b*_{air}, which exists in the non-conducting air (or vacuum) above the liquid metal, is derivable from a potential

*ϕ*(Chandrasekhar 1961). That is,

*b*_{air}=

*∇**ϕ*, where the potential

*ϕ*is the solution of the equation(2.14)

## 3. Floquet analysis

We express all the fields in the normal modes as e^{ik.x} of the horizontal plane, where is the horizontal wavenumber. The stability problem is analysed by the Floquet theory (e.g. Kumar & Tuckerman 1994). We expand the fields as(3.1)(3.2)(3.3)(3.4)where *μ*=*s*+i*αΩ* is the Floquet exponent, with *s* and *α* being real numbers. Inserting (3.4) in (2.14) leads to(3.5)where *D*≡d/d*z*. As the magnetic field *b*_{air} decays to zero as *z*→∞, the solution of (3.5) for each *n* is(3.6)where *A*_{n} can be fixed by matching magnetic fields at the free surface. As the vertical current *j*_{3} vanishes at the free surface for all **x**, the horizontal components of the magnetic field in the liquid metal at the free surface can be computed from the relations(3.7)Matching the horizontal components of the magnetic fields in the air and the liquid metal at the free surface (*z*=0) yields the following condition for each *n*:(3.8)while the continuity of the vertical components of the magnetic field across the free surface leads to(3.9)Elimination of *A*_{n} from (3.8) and (3.9) results in a single condition(3.10)which must be satisfied by *b*_{3n} at *z*=0 for each *n*. The linear stability problem then explicitly depends on the vertical coordinate *z* and time *t*. The relevant equations describing the complete linear stability of the horizontally extended fluid layer are(3.11)(3.12)(3.13)(3.14)(3.15)(3.16)(3.17)where . In the absence of any external magnetic field (=0), this problem reduces to the problem of stability of parametrically driven surface waves in viscous liquid (Kumar 1996).

The general solutions for the bulk equations (3.11) and (3.12) may be written as(3.18)(3.19)where(3.20)(3.21)Applying the six boundary conditions (3.13)–(3.16) on the general solutions (3.18) and (3.19) leads to a matrix equation for each value of *n* given by(3.22)where *Ψ*_{n} and *Φ*_{n} are the two column matrices, whose transpose are and , respectively. The operation by the inverse of the square matrix _{n}, which is 6×6 in size, on *Φ*_{n} yields *P*_{n}, *Q*_{n}, *R*_{n}, *S*_{n}, *C*_{n}, *D*_{n} in terms of *ζ*_{n} for each *n*. Finally, the application of the pressure-jump condition (3.17) at the free surface leads to the following recursion relation:(3.23)The stability of the free surface can be determined numerically by converting the recursion relation to an ordinary eigenvalue (Kumar & Tuckerman 1994) problem given by(3.24)The marginal stability boundaries are defined by the curves in the −*k* plane on which *s*(, *k*)=0. We compute marginal stability curves for harmonic (*α*=0) and subharmonic (*α*=1/2) solutions by setting *s*=0.

## 4. Results and discussions

Figure 1 shows the stability boundaries for mercury in the /−*k* plane. Tongue-like white regions bounded by dotted curves and black regions are instability zones for subharmonically and harmonically excited surface waves, respectively. A bicritical point occurs as a primary instability when the lowest point of two different tongues requires the same value of the forcing amplitude or the reduced forcing amplitude /. Two different wavenumbers are then excited simultaneously at the instability onset. Changing the magnetic field, which is measured by the Chandrasekhar number , for fixed values of all other external and fluid parameters can trigger a series of bicritical points as the primary instability. For a thin layer of mercury, a tricritical point can be excited as a primary instability (figure 1*e*) for =6.7. Thus, a small magnetic field may have a strong effect on the selection of critical wavenumbers at the instability onset. This is due to the sensitivity of the selection of wavenumbers on the total dissipation in the system. Even a small variation in the Chandrasekhar number selects completely different bicritical and tricritical points. The resulting surface patterns are likely to show drastic changes with the variation of the magnetic field in thin layers of mercury.

Figure 2 shows the variations of the minimum forcing amplitude for different instability zones. The minima of all the tongues increase with increasing value of . However, the minima of various tongues increase by a different amount with increasing . Consequently, the curves showing variation of the minima of different instability zones intersect each other. The point of intersection of two curves indicates the formation of a bicritical point at that particular . A point in the /−*Q* plane, where more than two curves meet, shows the formation of a multicritical point. Figure 2 shows the possibility of many bicritical points and one tricritical point at the primary instability when is raised for fixed values of all other parameters. Figure 3 shows the selection of critical wavenumber(s) as the Chandrasekhar number is varied. The critical wavenumber (solid curve) increases in steps at the bicritical points with increasing values of . The dashed line shows the variation of the wavenumber corresponding to the minimum of different tongues. The minima of all tongues vary smoothly and slowly with . However, the critical wavenumber shows large change due to jumps at the multicritical points. In the absence of the magnetic field (=0) or for small values of , the first subharmonic tongue is always excited (figure 1*a*,*d*). This clearly shows that the multicritical points are triggered due to the increase in the magnetic field only.

The bicritical points can also be formed by changing the forcing frequency, if the Chandrasekhar number is raised sufficiently. Figure 4 displays the effect of forcing frequency on the minimum forcing amplitude. The curves show the variation of the minimum of various tongues as a function of driving frequency. They intersect each other at the bicritical points. Figure 5 shows the effect of forcing frequency on the wavenumber corresponding to the minimum of various tongues. For a very small driving frequency, the wavenumber corresponds to a higher-order tongue. As the forcing frequency is increased, the critical wavenumber jumps from a higher value to a lower one at the bicritical points. The subharmonic response corresponding to the first tongue (SH1) is always excited at a higher driving frequency, as expected. Thus, the increase in the driving frequency along with the increase in the magnetic field brings strong competition in the selection of wavenumber in a thin layer of a liquid metal. In a thick liquid layer, a small magnetic field is unable to excite a wavenumber corresponding to a higher-order tongue. The free surface is excited subharmonically when the driving frequency is higher (greater than 10 Hz) for moderate values of . The critical forcing amplitude increases with , as seen in figures 1 and 2. The presence of a vertical magnetic field introduces additional damping in a liquid metal, which delays the appearance of forced waves.

The variation of the critical forcing amplitude as a function of driving frequency in a thin layer of mercury is displayed in figure 6. The plot shows a minimum in the curve, which is interesting. The driving frequency, for which the threshold is minimum, increases with increasing value of . Figure 7 shows the dispersion relation, i.e. the variation in critical wavenumber with *ω*. The critical wavenumber increases monotonically with the driving frequency. However, the wavenumber varies very little with variation of magnetic field. The magnetic field affects the selection of wavenumbers strongly in the vicinity of multicritical points only. The minimum in the threshold with respect to the driving frequency may be understood in the following way. The increase in the magnetic field does not change the wavenumber significantly away from any bicritical point, but increases the threshold. The lowering of driving frequency decreases the wavenumber, which becomes unfavourable in thin layers of the liquid metal. The large wavelength surface waves in shallow liquid layers experience more resistance from the rigid plate raising their threshold. The combined effect makes the threshold higher at a lower driving frequency. That is why the variation of the threshold with driving frequency shows a minimum. At higher driving frequencies, the rigid plate does not significantly affect the excitation of surface waves. Therefore, the curves showing variation of the threshold with driving frequency are almost parallel at higher frequencies.

## 5. Conclusions

We have investigated the effect of a uniform vertical magnetic field on parametrically driven surface waves. The magnetic field stabilizes the onset of driven surface waves in liquid metals. The minimum of different tongues increases differently. This leads to a series of multicritical points in a thin layer of the liquid. The critical wavenumber shows spontaneous jumps to higher values with increasing magnetic field. These jumps correspond to the appearance of bicritical points involving subharmonic and harmonic solutions of different wavenumbers simultaneously. Applying a uniform magnetic field may bring interesting pattern selection in a thin layer of a liquid metal. The magnetic field also facilitates the appearance of a bicritical point in relatively thick layers of a liquid metal.

## Footnotes

- Received July 18, 2006.
- Accepted October 26, 2006.

- © 2006 The Royal Society