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20
Onset of surfacetensiondriven Bénard convection
, 1995
"... below can exhibit a longwavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This longwavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestri ..."
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below can exhibit a longwavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This longwavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (< 0.017 cm for 0.102 cm 2 s −1 viscosity liquid), the system evolves to a strongly deformed longwavelength state which can take the form of a localized depression (‘dry spot’) or a localized elevation (‘high spot’), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (> 0.024 cm), the system forms only the hexagonal convection cells. A twolayer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the longwavelength instability are compared to our twolayer theory and to a onelayer theory that accounts for the upper gas layer solely with a heat transfer coefficient.
Marangoni–Bénard Convection with a Deformable Free Surface
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1998
"... Computations of Marangoni convection are usually performed in two or threedimensional domains with rigid boundaries. In two dimensions, allowing the free surface to deform can result in a solution set with a qualitatively different bifurcation structure. We describe a finiteelement technique for ..."
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Cited by 6 (4 self)
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Computations of Marangoni convection are usually performed in two or threedimensional domains with rigid boundaries. In two dimensions, allowing the free surface to deform can result in a solution set with a qualitatively different bifurcation structure. We describe a finiteelement technique for calculating bifurcations that arise due to thermal gradients in a twodimensional domain with a deformable free surface. The fluid is assumed to be Newtonian, to conform to the Boussinesq approximation, and to have a surface tension that varies linearly with temperature. An orthogonal mapping from the physical domain to a reference domain is employed, which is determined as the solution to a pair of elliptic partial differential equations. The mapping equations and the equilibrium equations for the velocity, pressure, and temperature fields and their appropriate nonlinear boundary conditions are discretized using the finiteelement method and solved simultaneously by Newton iteration. Contact angles other than 90 degrees are shown to disconnect the transcritical bifurcations to flows with an even number of cells in the expected manner. The loss of stability to single cell flows is associated with the breaking of a reflectional symmetry about the middle of the domain and therefore occurs at a pitchfork bifurcation point for contact angles both equal to, and less than, 90 degrees.
Evaporative instabilities in climbing films
 J. Fluid Mech
"... We consider flow in a thin film generated by partially submerging an inclined rigid plate in a reservoir of ethanol – or methanol–water solution and wetting its surface. Evaporation leads to concentration and surface tension gradients that drive flow up the plate. An experimental study indicates tha ..."
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We consider flow in a thin film generated by partially submerging an inclined rigid plate in a reservoir of ethanol – or methanol–water solution and wetting its surface. Evaporation leads to concentration and surface tension gradients that drive flow up the plate. An experimental study indicates that the climbing film is subject to two distinct instabilities. The first is a convective instability characterized by flattened convection rolls aligned in the direction of flow and accompanied by freesurface deformations; in the meniscus region, this instability gives rise to pronounced ridge structures aligned with the mean flow. The second instability, evident when the plate is nearly vertical, takes the form of transverse surface waves propagating up the plate. We demonstrate that the observed longitudinal rolls are driven by the combined influence of surface deformations and alcohol concentration gradients. Guided by the observation that the rolls are flattened, we develop a quasitwodimensional theoretical model for the instability of the film, based on lubrication theory, which includes the effects of gravity, capillarity and Marangoni stresses at the surface. We develop stability criteria for the film which are in qualitative agreement with our experimental observations. Our analysis yields an equation for the shape of the interface which is solved numerically and reproduces the salient features of the observed flows, including the slow lateral drift and merging of the ridges. 1.
Simulation of ThreeDimensional BénardMarangoni Flows Including Deformed Surfaces
, 2009
"... We present a coupled thermalfluid model for BénardMarangoni convection in a threedimensional fluid layer. The governing equations are derived in detail for two reasons: first, we do not assume a flat free surface as commonly done; and second, we prepare for the use of flexible discretizations. Th ..."
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We present a coupled thermalfluid model for BénardMarangoni convection in a threedimensional fluid layer. The governing equations are derived in detail for two reasons: first, we do not assume a flat free surface as commonly done; and second, we prepare for the use of flexible discretizations. The governing equations are discretized using spectral elements in space and an operator splitting approach in time. Since we are here primarily interested in steady state solutions, the focus is on the spatial discretization. The overall computational approach is very attractive to use for several reasons: (i) the solution can be expected to have a high degree of regularity, and rapid convergence can be expected; (ii) the spectral element decomposition automatically gives a convenient parameterization of the free surface that allows powerful results from differential geometry to easily be exploited; (iii) free surface deformation can readily be included; (iv) both normal and tangential stresses are conveniently accounted for through a single surface integral; (v) no differentiation of the surface tension is necessary in order to include thermocapillary effects (due to integrationbyparts twice); (vi) the geometry representation of the free surface need only be C 0 across element boundaries even though curvature effects are included. Threedimensional simulation results are presented, including the free surface deflection due to buoyancy and thermocapillary effects.
Twofluid MarangoniBénard convection with a deformable interface
"... Two immiscible fluid layers that are subjected to a temperature gradient perpendicular to their interface, exhibit a range of behaviors that is considerably richer than for the singlefluid case. We describe a numerical technique for calculating thermallydriven flows in two fluid layers which uses ..."
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Two immiscible fluid layers that are subjected to a temperature gradient perpendicular to their interface, exhibit a range of behaviors that is considerably richer than for the singlefluid case. We describe a numerical technique for calculating thermallydriven flows in two fluid layers which uses a simple technique based on a Landau transformation to map the physical domain into a reference domain, enabling the unknown location of the deformable interface to be determined. The coupled system of nonlinear partial differential equations, comprising mapping, continuity, momentum and energy equations and the appropriate boundary conditions, is solved using the finiteelement method in twodimensional domains. Numerical bifurcation techniques are used to investigate the multiplicity of the solution set. The case of heating from above is considered in some detail and the results of finiteelement computations are compared with linear stability calculations performed on unbounded domains. The principal advantages of the finiteelement approach are the ability to determine the effect of non90 degree contact angles (when the conducting solution no longer exists and traditional linear stability approaches fail), the ability to determine the role of finite aspect ratio domains and the relative volume fractions of the two fluids, and the capability of calculating the nonlinear development of flows beyond the critical temperature gradient.
Control of Marangoni Convection in a VariableViscosity Fluid Layer with Deformable
"... Abstract—The effectiveness of a proportional feedback control to suppress the Marangoni instability in a variableviscosity fluid layer with a deformable free upper surface is investigated. Viscosity variation and deformable free surface have destabilizing effects on the stability limit. The stabili ..."
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Abstract—The effectiveness of a proportional feedback control to suppress the Marangoni instability in a variableviscosity fluid layer with a deformable free upper surface is investigated. Viscosity variation and deformable free surface have destabilizing effects on the stability limit. The stability thresholds for the shortscale mode are strongly dependent on viscosity variation and controller gain while the stability thresholds for the longscale mode are greatly influenced by gravity and surface deformation. The feedback control strategy through thermal perturbation in the boundary data is shown effective in suppressing the Marangoni convection and delaying the onset of instability.
Pattern Formation in Solutal Convection: Vermiculated Rolls and Isolated Cells
"... Observations of the peculiar behaviour of a drink of liqueur topped with cream led us to perform experiments showing that the instability is a convection phenomenon that arises through destabilizing surfacetension forces. The convection is solutal: driven by gradients of concentration of a solute, ..."
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Observations of the peculiar behaviour of a drink of liqueur topped with cream led us to perform experiments showing that the instability is a convection phenomenon that arises through destabilizing surfacetension forces. The convection is solutal: driven by gradients of concentration of a solute, rather than by heat gradients as in the more commonly studied thermal convection. The convective patterns, vermiculated rolls and isolated cells, are quite unlike the usual planforms. They are associated with an elastic surface film, and the Marangoni number is high, characteristic of solutal convection. We have conducted further experiments that reproduce these patterns in simpler working fluids.
Effects of Control on the Onset of MarangoniBénard Convection with Uniform Internal Heat Generation 1
"... Abstract The effect of control on the onset of MarangoniBénard 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 cri ..."
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Abstract The effect of control on the onset of MarangoniBénard 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 MarangoniBénard convection with internal heat generation can be delayed through the use of control. Keywords MarangoniBénard convection, control, internal heat generation. 1
Instabilities and Spatiotemporal Chaos of Longwave Hexagon Patterns in Rotating Marangoni Convection
, 2002
"... We consider surfacetension driven convection in a rotating uid layer. For nearly insulating boundary conditions we derive a longwave equation for the convection planform. Using a Galerkin method and direct numerical simulations we study the stability of the steady hexagonal patterns with respec ..."
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We consider surfacetension driven convection in a rotating uid layer. For nearly insulating boundary conditions we derive a longwave equation for the convection planform. Using a Galerkin method and direct numerical simulations we study the stability of the steady hexagonal patterns with respect to general sideband instabilities. In the presence