Hydrothermal wave instability of thermocapillary driven convection in a plane layer subjected to a uniform magnetic field

0273-1177(95)00133-6

Adv. Space Res. Vol. 16, No. 7, pp. (7)55-(7)58, 1995 Ccpyright 63 1995 COSPAR Printedin Great Britain. All ri ts -cd 0273-1177/Y i@$9.50 + 0.00

HYDROTHERMAL WAVE INSTABILITY OF THERMOCAPILLARY DRIVEN CONVECTION IN A PLANE LAYER SUBJECTED TO A UNIFORM MAGNETIC FIELD J. Priede and G. Gerbeth Research Center RossendorJ P.O. Box 510119, D-01314 Dresden, Germany ABSTRACT Thermocapillary driven motion is considered in a horizontal electrically conducting fluid layer heated from the side and exposed to a magnetic field coplanar to the layer. The hydrothermal wave instability and its control by the magnetic field is studied by a linear stability analysis. The special assumption of disturbances traveling crosswise the basic flow allows an analytical solution of the problem. For a particular class of perturbations considered here, the critical Marangoni number and the wavelength of the most unstable mode increase directly with the strength of the applied magnetic field. INTRODUCTION Flows driven by thermally induced surface tension gradients present an important aspect of several material processing technologies like semiconductor crystal growth from melt, particularly, under conditions of microgravity. Since common semiconductor melts posses an electrical conductivity close to that of liquid metals magnetic field presents an attractive tool for a contactless control of the hydrodynamics, mainly aimed to suppress convective instabilities. If a horizontal temperature gradient is imposed along the liquid layer a surface tension gradient appears. Thus, the tangential balance in the surface tension is broken and as a result the surface is set into motion. Since the liquid posses viscosity the surface motion drives the flow in the bulk liquid. It was shown by Smith & Davis /l/ that the thermocapillary effect sets up a mechanism of hydrothermal instabilities in such dynamic layers. They showed that a coupled effect produced by both temperature and velocity gradients can give rise to an instability of hydrothermal waves propagating obliquely with respect to the basic flow. Our concern in this paper will be the study of the coplanar magnetic field influence on the hydrothermal wave instability of thermocapillary driven shear flow of an electrically conducting fluid. We confine our linear stability analysis to disturbances traveling crosswise the basic flow. Although such restriction imposed on the most unstable mode is n priord not validated, it nevertheless seems attractive to be considered as a first approximation of the full three dimensional stability problem, because of the possibility to obtain an analytic solution for the dispersion relation. FORMULATION

OF THE

PROBLEM

Consider an unbounded horizontal layer of viscous electrically conducting liquid of density p, kinematic viscosity Y, thermal conductivity 6 and electric conductivity 8. The layer having, at rest, depth d is bounded from below by a plane, perfectly thermally and electrically insulating plate and above by a free surface characterized by thermal conductance h per unit area. A constant temperature gradient p is imposed along the layer, and a steady shear flow is set up by viscous surface stress as a result of temperature dependence of surface tension T. The latter is assumed to change according to the linear law 7 = 70 - Y(T - To), (1) (7)55

(7156

I. Prkde and G. Gerbeth

where 7 = -dT/dT > 0 is the negative rate of change of surface tension with temperature. flow is subjected to a uniform magnetic field of induction B. The depth d is supposed to be enough so that buoyancy can be neglected when compared to the thermocapillary effects. surface tension is supposed to be high enough so that the free surface may be considered planar and nodeformable boundary.

The small The as a

On changing to a dimensionless form of both the governing equations and boundary conditions, the depth d is chosen as a length scale, and the time t, velocity field V, pressure field p, temperature difference T-To, and the induced electrostatic potential cpare referred to scales &/v, v/d, pv2/d2, /3d and Bv, respectively. The equations governing the fluid flow are the Navier-Stokes equation with an electromagnetic force term added, the incompressibility constraint, the energy equation and electric current continuity equation. As a result, the problem is characterized by the following dimensionless parameters: the Hartmann number Ha = Bdm, the Prandtl number Pr = V/K, the Reynolds number Re = r/3d2/pu2 and the the Biot number Bi = hdln. The Marangoni number, which is used in parallel to the Reynolds number, is simply the product of Reynolds and Prandtl numbers: Ma=RePr. The influence of the magnetic field on the steady basic state , maintaining zero mass flux through any vertical plane, parallel flow V = (C,O, 0) is characterized by an effective Hartmann number defined as Ha, = (eB* ez)Ha, where eB = B/]B] is a unit vector pointing in the direction of the imposed magnetic field, and Ha is the conventional Hartmann number introduced above. If the magnetic field is coplanar to the liquid layer defined by the relation (Ed . e,) = 0, then it has no influence on the basic flow, and consequently it remains the same as without magnetic field c(z)

=

&($+;--$,

Jo

=

Rex;,

T(?,r)=-s-PrRe(;+&&),

(2)

p(z) = (Ed - e,)Re

This is exactly the return flow, stability of which was considered by Smith & Davis [l] without magnetic field. It should be noted, that even though the magnetic field has no effect on the basic flow it does affect the disturbances and, consequently, the stability of the flow. We analyze the linear stability of the basic state (2)-(3) by applying infinitesimal disturbances in the form of plane waves traveling crosswise to the basic flow. RESULTS

AND

DISCUSSION

The neutral stability curves for Bi=O, P~0.01 and for different values of the Hartmann number are shown in Figure 1. If the magnetic field is absent (Ha=O), then the marginal Marangoni number decreases as k- ’ for large scale disturbances (k < 1) and increases as k3i2 for small scale disturbances (k >> 1). A coplanar magnetic field has a stabilizing effect within an intermediate range of wavenumbers, while short and longwave disturbances remain uninfluenced by the magnetic field. The frequency of the neutral disturbance mode tends to a constant value for large scale disturbances and increases as k2 for small scale disturbances. As it is seen in Figure 2, both the minima of the above neutral stability curves defining the critical value of the Marangoni number at which the first unstable disturbance mode appears and the wavelength of this most unstable disturbance increases for sufficiently strong magnetic field linearly with the Hartmann number. The dependences of critical Marangoni number and wavenumber on the Prandtl number at different Hartmann numbers, plotted in Figure 3, shows that the lower the Prandtl number the higher the Hartmann number required to affect the instability. Moreover, it is seen that within the range of Prandtl number influenced by the magnetic field, both the critical wave number and the corresponding Reynolds number (Ma/Pr) are almost independent of the Prandtl number. For sufficiently small Prandtl numbers, both the critical wavenumber and corresponding Marangoni number, change as Pr’f’ and are not affected by the magnetic field. In the case of low Prandtl numbers, presenting the main interest for discussion here, the mechanism

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Critical poinla -- --.

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Driven Convecti .

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.

Figure 1: The marginal Marangoni number (left) and wave frequency (right) versus wave number at different Hartmann numbers for Pr = 10S2 and Bi = 0. accounting for the calculated dependencies of critical parameters may be explained by the following physical considerations. Thus, as far as both boundaries are considered as adiabatic ones, there is no reference temperature specified over the depth of the layer. As a result, the wavelength of disturbance characterizes the effective thermal length scale rather than the depth of the layer. Obviously, the time scale of the instability is determined by the shortest one from viscous and thermal diffusion times over the corresponding characteristic length scales. Choosing the longest characteristic time as an effective one, would lead to diffusive smoothing of that disturbances with shorter relaxation time. If the magnetic field is absent the depth of the layer is the effective hydrodynamic length scale for longwave disturbances. If the wave is long enough providing the thermal diffusion time across it, which exceeds the viscous diffusion time over the depth of the layer, then the magnitude of temperature disturbances has to be determined solely by the advection of the basic temperature field by the velocity disturbances. Obviously, the heat diffusion cannot influence temperature disturbances occurring at such long distances over the time shorter than the characteristic diffusion time on these distances. As a result, the amplitude of temperature disturbances cannot depend on the wavelength of such long waves. Thus, the shorter the wavelength the higher the temperature gradient across the wave and, consequently, the lower the Reynolds number providing the balance between thermocapillary and viscous shear stresses at the free surface. The amplitude of temperature disturbance remains independent of the wavelength, until the limit is reached at which temperature disturbances begin to smooth because of heat diffusion. For adiabatic boundaries the critical limit is reached already at the length N Prm1i2, beyond which heat diffusion

0.1

1

Hartma~nOnulnber 1000 100

0.001 ti

0.1

1

100 1000 Hartma;~““mber

Figure 2: The critical Marangoni number (left) and wavenumber (right) depending on the Hartmann number at different Prandtl numbers. .usnI&,-E

1. Priede and G. Gerbetb

CO-58

0.1

0.01

le.06

1%0500001

0.001 001 0.1 1 Prandllnumber

10

Figure 3: The critical Marangoni number number at different Hartmann numbers.

100

1000

0.001 . ..’ . ..’ ’ **’ . . J ‘-1 * ’ *’ 18-06 le-050.0001 0.001 001 0.1 1 Prandtlnumber

(left) and wavenumber

(right) depending

- *.’ . * c’ * 10

100

low

on the Prandtl

across the wave becomes important for characteristic time determined by the viscous diffusion over the depth of the layer. This explains the scaling of critical wavelength for sufficiently small Prandtl numbers. It has to be stressed, that this scaling concerns only adiabatic boundary conditions. The action of the coplanar magnetic field tries to eliminate crosswise liquid motion. Thus, the application of a coplanar field leads to surviving of flow field disturbances occurring almost along the field lines either, those, which “minimally” crosses the field lines. This means that the first type of disturbances are stretched along the field lines, while the second type corresponds to the shortwave disturbances, which are dominated by the viscosity rather than by the magnetic field. The order of magnitude analysis of the corresponding disturbance equations shows, that all intermediate disturbances having wavelength larger than O(Hu-‘), but smaller than O(Ha), are confined within the boundary layer of thickness scaling as N (lEHs)-‘/*. Thus, the instability is influenced by the magnetic field if the corresponding critical wavelength occurs within the range affected by the magnetic field. This can occur for low Prandtl numbers, only when the magnetic field is strong enough such as: Ha - PT-~/~. Obviously if the condition above holds, then the amplitude of temperature disturbance begins already to reduce at the critical wavelength O(Ha) on account of electromagnetic confinement of velocity disturbances rather than because of diffuse smoothing. CONCLUSIONS We have shown that the magnetic field imposed coplanar to the liquid layer, but perpendicular to the basic flow has a stabilizing effect with respect to the hydrothermal wave disturbances propagating crosswise to the basic flow. For sufficiently strong magnetic field the critical Marangoni number and the wavelength of the most unstable mode increase linearly with the strength of the imposed magnetic field. It should be noted, that the disturbances traveling along the basic flow are not influenced by the magnetic field. Consequently, for increasing magnetic field, these disturbances will actually determine the stability. However, this is a problem of complete three dimensional analysis, which is presently studied by a full numerical approach. ACKNOWLEDGMENT Financial

support

from “German

Space Agency”

is gratefully

acknowledged.

REFERENCES

1. M.K. Smith, and S.H. Davis, Instabilities of dynamic thermocapillary Convective instabilities. J. Fluid Me&, vol. 132, 1983, pp. 119-144.

liquid layers.

Part

1.