Marangoni Wave Research Articles

Corrigendum: Longitudinal modulation of Marangoni wave patterns in thin film heated from below: instabilities and control

Corrigendum: Longitudinal modulation of Marangoni wave patterns in thin film heated from below: instabilities and control

Resonance damping of gravity–capillary waves on water covered with a visco-elastic film of finite thickness: A reappraisal

A new approach to the problem of damping of gravity–capillary waves (GCW) on water covered with a layer of viscous liquid (a film) of finite thickness with two elastic boundaries is developed. It is shown that the rotational component of GCW can be described formally as a “forced” longitudinal or Marangoni wave (MW), and the potential component of GCW plays a role of the “external force.” The resonance-like excitation of the forced MW is demonstrated when the GCW and MW frequencies and wave numbers are approximately close to each other. For a film that is thinner than the viscous boundary layers in film, a single forced MW exists that is located within the boundary layer beneath the water surface. For a thick film, the forced MW is characterized by the existence of two spatially separated MW modes: one is localized in the boundary layer below the upper, air–film interface and another within the boundary layers in the vicinity of the water–film interface. Then, at different elasticities of the interfaces, a double peak dependence of the GCW damping coefficient on wave number can occur due to the resonance with the two forced MW modes. The dependence of the damping coefficient on film thickness is characterized by a strong maximum appearing when the film and boundary layer thickness values are comparable to each other. The developed theory is consistent with existing numerical studies and experiment.

Longitudinal Modulation of Marangoni Wave Patterns in Thin Film Heated From Below: Instabilities and Control

Non-linear Marangoni waves, which are generated by the long-wave oscillatory instability of the conductive state in a thin liquid film heated from below in the case of a deformable free surface and a substrate of very low conductivity, are considered. Previously, the investigation of traveling Marangoni waves was restricted to the analysis of the bifurcation and stability with respect to disturbances with strongly different wave vectors. In the present article, for the first time, the modulational instability of traveling waves is investigated. We derive the amplitude equation for the modulated traveling wave, which describes non-linear interaction of the main convective pattern with the perturbations with slightly different wavenumbers. The amplitude equation differs from the conventional complex Ginzburg–Landau equation as it contains an additional term of the local liquid level rise. Linear stability analysis reveals two modulational instability modes: the amplitude modulational and the phase modulational (Benjamin–Feir) ones. It is shown that traveling rolls are stable against the longitudinal modulation for the uncontrolled convection. We also investigate the influence of the non-linear feedback control, which was applied previously to eliminate subcritical excitation of traveling rolls. Computations reveal both the modulational modes under the non-linear feedback control. The obtained results show that the modulational instabilities significantly influence the region of parameters where the non-linear feedback control is efficient for stabilization of waves.

Nonlinear feedback control of Marangoni wave patterns in a thin film heated from below

We investigate the effect of nonlinear feedback control on the oscillatory mode of the long-wave Marangoni instability. Thin liquid film heated from below with a deformable free surface atop a substrate of low thermal conductivity is in our spotlight. We apply nonlinear feedback control to change the excitation of convection from subcritical to supercritical, or to stabilize a definite kind of convective pattern. Nonlinear interaction of traveling waves is examined by means of weakly nonlinear analysis within the amplitude equations. The cases of single wave, of pair of counter-propagating waves, and of three-dimensional two-wave patterns are considered. It is shown that by means of quadratic feedback control it is possible to replace a subcritical bifurcation by a supercritical one, as well as extend the stability region of traveling rolls and traveling rectangles and produce additional structures (traveling squares).

Linear-stability analysis of thermocapillary-buoyancy convection in annular two-layer system with a radial temperature gradient

In this paper, we present a set of linear-stability analysis results of the thermocapillary-buoyancy convection in an annular two-layer system of 5cSt silicone oil and HT-70 with a radial temperature gradient. The annular two-layer system is heated at the outer cylindrical wall and cooled at the inner wall. The effects of the depth ratio, the radius ratio and the aspect ratio of the annular pool on the critical conditions for the convection transition are analyzed. The flow destabilizing mechanism is identified. Results indicate that the hydrothermal wave, or the combined hydrothermal wave and three-dimensional oscillatory flow, should be responsible for the flow destabilization when the stable thermocapillary-buoyancy convection transits to the three-dimensional oscillatory convection. With the increase of the radius ratio of the annular pool, the critical Marangoni number, wave number and phase velocity all decrease. With the increase of the depth ratio η, the critical Marangoni number initially increases before a dramatic decrease. The maximum value of the critical Marangoni number locates in the range of η=0.625–0.833.

Marangoni convection in a variable viscosity fluid layer with feedback control

Feedback control was applied to the steady Marangoni convection in a horizontal layer of fluid with variable viscosity and free-slip at the lower boundary heated from below and cooled from above. Prediction for the onset of convection are obtained from the analysis by numerical technique. The effects of feedback control are studied by examining the critical Marangoni numbers and wave numbers. It is shown that the onset of Marangoni convection with variable viscosity can be delayed and the critical Marangoni number can be increased through the use of feedback control.

Feedback Control of the Marangoni-Bénard Convection in a Horizontal Fluid Layer with Internal Heat Generation

Feedback control is applied to the steady Marangoni-Bénard convection in a horizontal layer of fluid with internal heat generation heated from below and cooled from above. The analytical technique is used to obtain the close form analytical expression for the onset of Marangoni-Bénard convection when the lower boundary is conducting. The effects of feedback control are studied by examining the critical Marangoni numbers and wave numbers. It is shown that the critical Marangoni number decreases as the value of internal heat generation but the critical Marangoni number can be increased through the use of feedback control.

Stability of Thermocapillary Convection in Annular Pools with Low Prandtl Number Fluid

In order to understand the stability characteristics of thermocapillary convection in an annular pool with low Prandtl number (Pr = 0.011) fluid, a linear stability analysis had been performed. The annular pool was heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surface were adiabatic. The results show that the critical Marangoni number and the critical wave number decrease with the increase of the aspect ratio, which is defined as the ratio of the depth to the gap width of the annular pool. When the aspect ratio exceeds 0.12, the critical wave number keeps almost constant. As the radius ratio increases, the critical Marangoni number decreases slightly, while the critical wave number increases. In the gravity condition, the effect of the buoyancy on the critical Marangoni number and wave number increases gradually with the increase of the aspect ratio.

Effects of Control on the Onset of Marangoni-Benard Convection with Uniform Internal Heat Generation

The effect of control on the onset of Marangoni-B\'{e}nard convection in a horizontal layer of fluid with internal heat generation heated from below and cooled from above is investigated. The resulting eigenvalue problem is solved exactly. The effects of control are studied by examining the critical Marangoni numbers and wave numbers. It is found that the onset of Marangoni-B\'{e}nard convection with internal heat generation can be delayed through the use of control. Keywords: Marangoni-B\'{e}nard convection; control; internal heat generation.

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

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

Theoretical analysis of Marangoni instability of an evaporating droplet by energy method

Evaporation of a droplet is a process of heat and mass transports. When the droplet evaporates, heat is removed from the surface and the surface temperature is reduced thereby, resulting in a temperature gradient normal to the free surface. Such a situation is similar to that with a liquid layer heated from below. Thus, Marangoni instability as described by Pearson [J. Fluid Mech. 4 (1958) 489] may probably exist in an evaporating droplet, which is a very important and interesting phenomenon both from the academic and application points of view. In the present work, the energy method, which is appropriate for unsteady flows, is applied to investigate the stability of an evaporating droplet against disturbances of any amplitude. The results predicted by the present study possess similar trends with those acquired by the linear stability analysis. Both studies indicate that, as time proceeds, both the increase of the surface temperature reduction and the growth of the thermal boundary layer near the free surface are conducive to the onset of instability. The critical Marangoni number and wave number increase with the droplet initial temperature. For all the cases studied, the critical Marangoni numbers predicted by the energy method are smaller than those calculated by the linear stability analysis. However, the subcritical region between the Ma c's as calculated by these two methods is very narrow. This consistency indicates the validity of linear stability analysis as a first approach to the analysis of Marangoni instability of an evaporating droplet.

Generation of a Wave on a Liquid Film Flowing Down a Heated Inclined Plane. 2nd Report, Disturbance and Marangoni Waves.

An investigation was conducted into the generation of a disturbance wave on a liquid film flowing down a heated inclined plane. Theoretical analysis shows that the Marangoni effect due to the surface tension gradient contributes to reduction of the critical flow rate of the liquid on the generation of a disturbance wave. Theoretical results also suggest the existence of two boundaries for generation of a disturbance wave, at a fixed inclined angle. The lower boundary, however, could not be observed for thin water film because liquid film breakdown occurred before the lower critical flow rate was achieved. In this paper, silicone oil with high viscosity was used as the working fluid instead of water. Small waves were observed under the lower critical flow rate. The surface temperature gradient along a wave is one order of magnitude larger than that of a disturbance wave under the higher critical flow condition. We defined a wave under lower critical condition as a Marangoni wave. The measured velocity of a Marangoni wave is almost the same as the liquid surface velocity.

EFFECTS OF INTERNAL HEAT GENERATION AND VARIABLE VISCOSITY ON MARANGONI CONVECTION

The problem of Marangoni convection in a horizontal fluid layer with internal heat generation is solved using the sequential gradient-restoration algorithm (SGRA) developed for optimal control problems. A general discussion of the method is given. The effects of the temperature-dependent surface tension and viscosity are studied by examining the critical Marangoni numbers and wave numbers. The validity of the method is established by comparison with the work in the literature for the case of a linear temperature profile across the fluid layer.

Effects of Internal Heat Generation and Variable Viscosity on Marangoni Convection

The problem of Marangoni convection in a horizontal fluid layer with internal heat generation is solved using the sequential gradient-restoration algorithm (SGRA) developed for optimal control problems. A general discussion of the method is given. The effects of the temperature-dependent surface tension and viscosity are studied by examining the critical Marangoni numbers and wave numbers. The validity of the method is established by comparison with the work in the literature for the case of a linear temperature profile across the fluid layer.

Relaxation effects in monolayers and their contribution to water wave damping: II. The Marangoni phenomenon and gravity wave attenuation

The Marangoni wave theory is reviewed and extended to the short gravity water wave regime. This theoretical approach is verified by wave damping measurements in the Hamburg wind-wave tunnel performed in the presence of monomolecular hexadecanoic acid methyl ester, (Z)-9-octadecen-1-ol, and hexadecyltrimethylammonium bromide surface films. A good agreement of the theoretical with the experimental results is obtained, when performing the calculations with the help of dilational modulus and phase angle values, which were obtained from dynamic surface potential measurements over a wavy water surface (H. Hühnerfuss, P. A. Lange, and W. Walter, J. Colloid Interface Sci. 108, 430, 1985). Dilational modulus values determined by classical Langmuir-trough measurements are not suitable for determining wave damping ratios and give rise to too large values.

The Marangoni wave in ripples on an air-water interface covered by a spreading film

The Marangoni wave, brought forth as a solution of the characteristic equation for ripple motion on an air-water interface covered by a film, is shown to possess a condition of minimum damping when the loss angle φ is negative, and nears −45°. Experimental evidence of this phenomenon was observed in the waving of a tank filled with distilled water and covered with a film of oleic alcohol.