C. Delcarte

STABILITY OF THE AXISYMMETRIC BUOYANT-CAPILLARY FLOWS IN A LATERALLY HEATED LIQUID BRIDGE

The axisymmetric steady-states solutions of buoyant-capillary flows in a cylindrical liquid bridge are calculated by means of a pseudo-spectral method. The free surface is undeformable and laterally heated. The working fluid is a liquid metal, with a Prandtl number value Pr=0.01. Particular care was taken to preserve the physical regularity in our model, by writing appropriate flux boundary conditions. The location and nature of the bifurcations undergone by the flows are investigated in the space of the dimensionless numbers (Marangoni, Ma∈[0,600]; Rayleigh, Ra∈[0,5×104]). Saddle-node and Hopf bifurcations are found. By analyzing the steady state structures and the energy budgets, the saddle-node bifurcations are observed to play a determinant role. Only two sets of stable steady-states, connected by saddle-nodes, are allowed by the coupling of buoyancy and capillarity. Most of the solutions of the explored part of the (Ma, Ra) plane belong to these states.

Transient Rayleigh-Bénard-Marangoni solutal convection

Solutal driven flow is studied for a binary solution submitted to solvent evaporation at the upper free surface. Evaporation induces an increase in the solute concentration close to the free surface and solutal gradients may induce a convective flow driven by buoyancy and/or surface tension. This problem is studied numerically, using several assumptions deduced from previous experiments on polymer solutions. The stability of the system is investigated as a function of the solutal Rayleigh and Marangoni numbers, the evaporative flux and the Schmidt number. The sensitivity of the thresholds to initial perturbations is analyzed. The effect of viscosity variation during drying is also investigated. At last numerical simulations are presented to study the competition between buoyancy and Marangoni effects in the nonlinear regime.

Stability study of the floating zone with respect to the Prandtl number value

Thermocapillary convection in a laterally heated liquid bridge is studied numerically, using a Chebyshev spectral method. The stability of the axisymmetric basic state, with respect to 3D perturbations, is characterized over a large range of Prandtl number values (Pr∊[10−3,102]), thanks to the choice of a sufficiently sharp regularizing function of the vorticity singularities. Criteria are established to ensure the compatibility of this function with mass conservation, and to choose a correct model of the lateral heat flux in order to reach a spectral convergence of the results. First 3D nonlinear spectral computations are presented.

Sensitivity of the liquid bridge hydrodynamics to local capillary contributions

In the usual models of thermocapillary flows, a vorticity singularity occurs at the contact free surface–solid boundaries. The steady axisymmetric hydrodynamics of the side-heated liquid bridge of molten metal is addressed here for its sensitivity to the size δ of a length scale explicitly introduced to regularize the problem. By linear stability analysis of the flows, various stable steady states are predicted: The already known steady states which are reflection-symmetric about the mid-plane, but also others which do not possess this property. The thresholds in Ma of the associated bifurcations are strongly dependent on δ, and converge with δ→0 towards low values. Published data give these results some physical relevance.

Sensitivity of diffusive-convective transition to the initial conditions in a transient Bénard-Marangoni problem

Sensitivity of a transient Benard-Marangoni problem is studied using stochastic models to simulate the uncertainties of thermal initial conditions. Using different assumptions, three probabilistic models are developed and compared. Statistics are performed on flow velocities and temperatures. Transitions are examined with respect to the stochastic models.Copyright

Thermocapillary Flows and Vorticity Singularity

The usual modelling of thermocapillarity introduces a vorticity singularity along the contact of free surfaces with solid boundaries. The liquid bridge hydrodynamics is adressed here, for its sensitivity to the size of a filtering length, δ, introduced by an explicit regularization of the singularity. Systematically following the convergence of the numerical results with δ shows that the stability properties of the axisymmetric flows, and their bifurcation maps, are correctly identified provided that this length scale is small enough. Although the singularity treatment is localized near the boundaries, the flow stability is controlled by an accumulation of mechanical power in the vicinity of the mid plane of the liquid bridge, that is far from the boundaries.

About the ellipticity of the Chebyshev-Gauss-Radau discrete Laplacian with Neumann condition

The Chebyshev-Gauss-Radau discrete version of the polar-diffusion operator, 1r@?@?rr@?@?r-k^2r^2,k being the azimuthal wave number, presents complex conjugate eigenvalues, with negative real parts, when it is associated with a Neumann boundary condition imposed at r=1. It is shown that this ellipticity marginal violation of the original continuous problem is genuine and not due to some round-off error amplification. This situation, which does not lead per se to any particular computational difficulty, is taken here as an opportunity to numerically check the sensitivity of the quoted ellipticity to slight changes in the mesh. A particular mapping is chosen for that purpose. The impact of this option on the ellipticity and on the numerical accuracy of a computed flow is evaluated.

Stress singularities in confined thermocapillary convection

Axisymmetric thermocapillary convection is studied in a laterally heated liquid bridge. In this configuration, as in other wall-confined thermocapillary convection problems, a viscous singularity appears at the junction of the free and solid surfaces which any numerical approach of the problem must filter, either explicitly by smoothing the boundary conditions, or implicitly by using finite-precision discretization methods. Our approach is to filter the singularity explicitly, and to study the convergence properties of the solutions with the filter’s characteristics, which cannot easily be done when using finite-precision methods. Results show both quantitative (on scales) and qualitative (on symmetry properties) effects of the filter. Based on observations of the problems encountered when treating the laterally heated case, we propose to compare the results supplied by different numerical approaches on a simple half-zone model. This is an important step to pass before running oscillatory and/or 3D comput...

Application of the Adjoint Method to the Localization of Sources of Thermocapillary Instabilities

Axisymmetric thermocapillary convection is studied in a laterally heated liquidi bridge. The explicite treatment of the viscous stress singularity, localized at the junction of the free and solid surfaces, turns out to significantly influence the flow structure: multiple solutions are exhibited in zero gravity. In order to find the localization of the sources of this thermocapillary instability, a method based on the numerical solution of the system of adjoint equations is used.