^{1}and Krishna Kumar

^{2}

### Abstract

Stability of the free surface of thin sheets of a metallic liquid on a vertically vibrating hot plate, in the presence of a uniform and small rigid body rotation about the vertical axis, is investigated. The inhomogeneity in the surface tension due to a uniform thermal gradient across the liquid sheet prefers subharmonic response, while the rigid body rotation prefers harmonic response at the fluid surface. The competition results in Marangoni and Coriolis forces acting as fine-tuning parameters in the selection of wave numbers corresponding to different instability tongues for subharmonic and harmonic responses of the fluid surface.Solutions corresponding to various pairs of tongues may be induced in a thin layer of metallic liquid at the onset of parametrically forced surface waves. These give rise to multicritical points involving standing waves of two or more different wave numbers. Bicritical points may involve both the solutions oscillating subharmonically, harmonically, or one oscillating subharmonically and the other harmonically with respect to the vertical forcing frequency. Two tricritical points involving different types of solutions are also possible in a thin layer of mercury. The effect of variation of the Galileo number on critical acceleration and wave number in very low Prandtl number liquids is also presented.

I. INTRODUCTION

II. HYDRODYNAMIC SYSTEM

III. LINEAR STABILITY ANALYSIS

IV. SURFACE WAVES IN THIN LAYERS OF MERCURY

V. SURFACE WAVES IN MOLTEN SODIUM

VI. MARANGONI EFFECT AND LOW-PRANDTL-NUMBER FLUIDS

VII. CONCLUSIONS

### Key Topics

- Surface waves
- 33.0
- Surface tension
- 29.0
- Free surface
- 25.0
- Coriolis effects
- 23.0
- Numerical solutions
- 13.0

## Figures

Surface waves at the onset in a thin layer of mercury (, , and , , and ). The top left picture shows instability zones for subharmonic (black region) and harmonic (region under dots), respectively. The lowest point of the lowest zone ( is not shown in the picture as it is pushed up) gives critical acceleration and the critical wave number of parametrically excited surface waves. The plot at the top right shows the reduced forcing acceleration at the onset of waves. The curves at the bottom left show the variation of critical amplitudes of the vertical velocity (dashed-dotted line) and vertical vorticity and those on bottom right show critical amplitudes of the surface deformation (solid line) and temperature field , respectively, corresponding to the response.

Surface waves at the onset in a thin layer of mercury (, , and , , and ). The top left picture shows instability zones for subharmonic (black region) and harmonic (region under dots), respectively. The lowest point of the lowest zone ( is not shown in the picture as it is pushed up) gives critical acceleration and the critical wave number of parametrically excited surface waves. The plot at the top right shows the reduced forcing acceleration at the onset of waves. The curves at the bottom left show the variation of critical amplitudes of the vertical velocity (dashed-dotted line) and vertical vorticity and those on bottom right show critical amplitudes of the surface deformation (solid line) and temperature field , respectively, corresponding to the response.

Effect of Marangoni and Coriolis forces on the Faraday instability. The first column shows the effect of purely Marangoni force. The top viewgraph shows harmonic (regions inside dots) and subharmonic (black regions) surface waves for in the absence of the Coriolis force. The lower viewgraph shows the least values of the forcing amplitude for the four lowest tongues as a function of the Marangoni number. The subharmonic solution (continuous line) are the only preferred solutions as the Marangoni effect becomes stronger. The second column shows the effect of the Coriolis force only on the excited waves. The top viewgraph displays the instability tongues for in the absence of the Marangoni effect. The lower viewgraph shows the possibility of only harmonic solutions (broken lines) with increasing values of . Fluid parameters (, , and ) are relevant for mercury at a dimensionless forcing frequency .

Effect of Marangoni and Coriolis forces on the Faraday instability. The first column shows the effect of purely Marangoni force. The top viewgraph shows harmonic (regions inside dots) and subharmonic (black regions) surface waves for in the absence of the Coriolis force. The lower viewgraph shows the least values of the forcing amplitude for the four lowest tongues as a function of the Marangoni number. The subharmonic solution (continuous line) are the only preferred solutions as the Marangoni effect becomes stronger. The second column shows the effect of the Coriolis force only on the excited waves. The top viewgraph displays the instability tongues for in the absence of the Marangoni effect. The lower viewgraph shows the possibility of only harmonic solutions (broken lines) with increasing values of . Fluid parameters (, , and ) are relevant for mercury at a dimensionless forcing frequency .

Possible response at the onset of surface instability in the parameter space. The surface waves synchronous with forcing are called harmonic (), and those oscillating at half the forcing are called the subharmonic response. Surface waves with a wave number equal to the even multiple of are denoted as , and those with wave numbers equal to the odd multiple of are denoted by *Sn* for , where is the wave number of the first subharmonic response . The asterisk, circle, plus, and diamond correspond to , , , and , respectively. Fluid parameters (, , and ) are relevant for mercury at a dimensionless forcing frequency . Controlling the small rotation rate and the Marangoni number can lead to a response corresponding to the response on the onset of the surface wave in a thin layer of mercury.

Possible response at the onset of surface instability in the parameter space. The surface waves synchronous with forcing are called harmonic (), and those oscillating at half the forcing are called the subharmonic response. Surface waves with a wave number equal to the even multiple of are denoted as , and those with wave numbers equal to the odd multiple of are denoted by *Sn* for , where is the wave number of the first subharmonic response . The asterisk, circle, plus, and diamond correspond to , , , and , respectively. Fluid parameters (, , and ) are relevant for mercury at a dimensionless forcing frequency . Controlling the small rotation rate and the Marangoni number can lead to a response corresponding to the response on the onset of the surface wave in a thin layer of mercury.

Variation of the reduced critical acceleration with a Marangoni number for different values of the Galileo number . Plots are arranged (clockwise starting from the top left) with decreasing values of . The solid and dashed curves represent the reduced critical acceleration, which is the minimum of the first subharmonic tongue, for the dimensionless forcing frequency and 350, respectively. Fluid parameters ( and ) are relevant for molten sodium.

Variation of the reduced critical acceleration with a Marangoni number for different values of the Galileo number . Plots are arranged (clockwise starting from the top left) with decreasing values of . The solid and dashed curves represent the reduced critical acceleration, which is the minimum of the first subharmonic tongue, for the dimensionless forcing frequency and 350, respectively. Fluid parameters ( and ) are relevant for molten sodium.

Variation of the nondimensional critical wave number with a Marangoni number for different values of the Galileo number . Plots are arranged (clockwise from the top left) with decreasing values of the . The solid and dashed curves represent the dimensionless critical wave number for the dimensionless forcing frequency and 350, respectively. Fluid parameters are as in Fig. 4.

Variation of the nondimensional critical wave number with a Marangoni number for different values of the Galileo number . Plots are arranged (clockwise from the top left) with decreasing values of the . The solid and dashed curves represent the dimensionless critical wave number for the dimensionless forcing frequency and 350, respectively. Fluid parameters are as in Fig. 4.

Influence of the Prandtl number on the critical forcing amplitude and wave number for different values of the Marangoni numbers . The parameters are , , , and .

Influence of the Prandtl number on the critical forcing amplitude and wave number for different values of the Marangoni numbers . The parameters are , , , and .

Influence of the Marangoni number on the critical value of the reduced acceleration for different dimensionless forcing frequencies. For a relatively larger value of the Marangoni number, the critical acceleration shows a minimum (see bottom right). The parameters are , , , and .

Influence of the Marangoni number on the critical value of the reduced acceleration for different dimensionless forcing frequencies. For a relatively larger value of the Marangoni number, the critical acceleration shows a minimum (see bottom right). The parameters are , , , and .

Influence of the Marangoni number on the critical wave number for different values of forcing frequencies. always increases with increasing . The parameters are , , , and .

Influence of the Marangoni number on the critical wave number for different values of forcing frequencies. always increases with increasing . The parameters are , , , and .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content