Surface tension driven oscillatory instability in a rotating fluid layer

1969 ◽  
Vol 39 (1) ◽  
pp. 49-55 ◽  
Author(s):  
G. A. McConaghy ◽  
B. A. Finlayson
Keyword(s):  
Surface Tension ◽  
Convective Instability ◽  
Marangoni Number ◽  
Fluid Layer ◽  
Oscillatory Instability ◽  
Taylor Number ◽  
Rotating Fluid ◽  
Asymptotic Equations ◽  
Critical Marangoni Number ◽  
Prandtl Numbers

Oscillatory convective instability is shown to occur in a rotating fluid layer when convection is caused by surface-tension gradients at a free surface. The asymptotic equations, valid when the Taylor number approaches infinity, are solved analytically, and the critical Marangoni number is evaluated numerically. Fluids with Prandtl numbers above 0·201 will exhibit only stationary instability. Fluids with smaller Prandtl numbers will exhibit oscillatory instability with the critical Marangoni number varying as M0T½ where M0 depends on the Prandtl number and T is the Taylor number.

The Galerkin method applied to convective instability problems

1968 ◽  
Vol 33 (1) ◽  
pp. 201-208 ◽  
Author(s):  
Bruce A. Finlayson
Keyword(s):  
Exact Solution ◽  
Time Dependence ◽  
Galerkin Method ◽  
Convective Instability ◽  
Fluid Layer ◽  
Oscillatory Instability ◽  
Rotating Fluid ◽  
Exponential Time ◽  
The Stability ◽  
The Galerkin Method

The Galerkin method is applied in a new way to problems of stationary and oscillatory convective instability. By retaining the time derivatives in the equations rather than assuming an exponential time-dependence, the exact solution is approximated by the solution to a set of ordinary differential equations in time. Computations are simplified because the stability of this set of equations can be determined without finding the detailed solution. Furthermore, both stationary and oscillatory instability can be studied by means of the same trial functions. Previous studies which have treated only stationary instability by the Galerkin method can now be extended easily to include oscillatory instability. The method is illustrated for convective instability of a rotating fluid layer transferring heat.

The influence of Coriolis force on surface-tension-driven convection

1966 ◽  
Vol 26 (4) ◽  
pp. 807-818 ◽  
Author(s):  
A. Vidal ◽  
Andreas Acrivos
Keyword(s):  
Surface Tension ◽  
Flow Pattern ◽  
Coriolis Force ◽  
Fluid Layer ◽  
Functional Relation ◽  
Wave Number ◽  
Neutral Stability ◽  
Uniform Rotation ◽  
Critical Marangoni Number ◽  
Deep Layers

The effect of uniform rotation on surface-tension-driven convection in an evaporating fluid layer is considered both theoretically and experimentally. The theoretical analysis follows the usual small-disturbance approach of perturbation theory and leads, at the neutral state, to a functional relation between the Marangoni and Taylor numbers which is then computed numerically. In addition, it is shown analytically that, in the limit of rapid rotation, the velocity and temperature fluctuations are confined to a thin Ekman layer near the surface, and that Mc = 4·42T½ and ac = 0·5T¼, where Mc and ac are, respectively, the critical Marangoni number and the critical wave number for neutral stability, and T is the Taylor number.The experimental part deals primarily with the flow pattern of a 50% solution of ethyl ether in n-heptane evaporating into still air. In this case, the convective flow is surface-tension-driven and its structure was observed using schlieren optics. In the absence of rotation, the flow shows a remarkable cellular pattern when the layer is shallow, but when the depth of the layer is increased the pattern quickly becomes highly irregular. In contrast, for T > 103, a cellular structure is always observed even for deep layers, a result which is attributable to the stabilizing effect of the Coriolis force. A further increase in T leaves the flow pattern unchanged except that the size of the cells is found to decrease as T−¼ which is in agreement with the results of the linear stability analysis.

The effect of shear and stratification on the stability of a rotating fluid layer

1973 ◽  
Vol 59 (2) ◽  
pp. 369-390 ◽  
Author(s):  
A. R. Brunsvold ◽  
C. M. Vest
Keyword(s):  
Stratified Flow ◽  
Reynolds Numbers ◽  
Travelling Wave ◽  
Taylor Number ◽  
Critical Parameters ◽  
Rotating Fluid ◽  
Critical Mode ◽  
Shear Rates ◽  
Prandtl Numbers ◽  
The Stability

The stability of a layer of Newtonian fluid confined between two horizontal disks which rotate with different angular velocities is studied. Both isothermal and adversely stratified fluids are considered for small shear rates at low to moderate Taylor numbers. The linearized formulation of the stability problem is given a finite-difference representation, and the resulting algebraic eigenvalue problem is solved using efficient numerical techniques. The critical parameters and disturbance orientations are determined as a function of the Taylor number for the isothermal flow, and for the stratified flow for Prandtl numbers of 0·025, 1·0 and 6·0.At high Taylor numbers, the unstratified fluid flows in Ekman-like layers near the disks, and two modes of instability are noted: the viscous-type ‘class A’ travelling wave, whose existence depends on Coriolis forces, and the inflexional ‘class B’ mode, which is nearly stationary with respect to the nearer bounding disk. As the Taylor number is decreased, the Ekman layers coalesce to form a fully developed flow. In this regime there is a Taylor number below which the class A waves are always damped. The critical Reynolds number for the class B waves increases rapidly as the Taylor number approaches zero.For Prandtl numbers of 1·0 and 6·0, the adversely stratified flow exhibits two distinct types of instability: convective and dynamical. At low Reynolds numbers, a stationary mode associated with Bénard convection in a rotating fluid is critical. It is stabilized and given orientation by the shear. At higher Reynolds numbers, the critical mode is a travelling wave of the nature of either the class A or class B waves, depending upon the Taylor number. For a Prandtl number of 0·025, the critical mode resembles oscillatory convection at small Reynolds numbers and a class A wave at larger shear rates.

On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity

1964 ◽  
Vol 19 (3) ◽  
pp. 321-340 ◽  
Author(s):  
L. E. Scriven ◽  
C. V. Sternling
Keyword(s):  
Surface Tension ◽  
Marangoni Number ◽  
Wave Number ◽  
Simple Criterion ◽  
Surface Viscosity ◽  
Unstable Surface ◽  
Critical Marangoni Number ◽  
Liquid Pools ◽  
Limiting Case ◽  
Shape Deformations

The onset of steady, cellular convection driven by surface tension gradients on a thin layer of liquid is examined in an extension of Pearson's (1958) stability analysis. By accounting for the possibility of shape deformations of the free surface it is found that there is no critical Marangoni number for the onset of stationary instability and that the limiting case of ‘zero wave-number’ is always unstable. Surface viscosity of a Newtonian interface is found to inhibit stationary instability. A simple criterion is found for distinguishing visually the dominant force, buoyancy or surface tension, in cellular convection in liquid pools.

Marangoni convection in an ewline Oldroyd-B fluid layer with ewline throughflow

2007 ◽  
Vol 85 (9) ◽  
pp. 947-955 ◽  
Author(s):  
S Saravanan
Keyword(s):  
Approximate Solution ◽  
Marangoni Number ◽  
Marangoni Convection ◽  
Fluid Layer ◽  
Linear Analysis ◽  
Wave Number ◽  
Retardation Time ◽  
Viscoelastic Parameters ◽  
Critical Marangoni Number ◽  
The Galerkin Method

The onset of Marangoni convection in a horizontal Oldroyd-B fluid layer in the presence of a vertical throughflow is determined by linear analysis. We find an approximate solution to the corresponding eigenvalue problem using the Galerkin method. The effects of viscoelastic parameters on the critical Marangoni number, wave number, and frequency are discussed. The study also reveals the existence of a critical retardation time for which the oscillatory motion reaches its maximum strength. This study has possible implications in microgravity situations. PACS No.: 47.20.Gv

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