Guillaume Kasperski

Up to the unsteadiness of axisymmetric thermocapillary flows in a laterally heated liquid bridge

Axisymmetric flows of fluids of various Prandtl numbers, Pr, in a cylindrical and undeformable liquid bridge submitted to a lateral heating are studied in a zero gravity environment, up to their transitions to unsteadiness. The governing equations are solved numerically by a pseudospectral scheme. The results depend on Pr, in a nonmonotonic fashion, as, for instance, the threshold of the onset to unsteadiness. It has always been reached, except for our Pr=1 experiment, run up to a Marangoni number value Ma=106. The steady flows at very high Reynolds numbers (up to 107) offer a rich structure. A first incursion into unsteadiness shows different flow structure evolutions, depending on whether Pr is smaller or larger than one.

On the numerical treatment of viscous singularities in wall-confined thermocapillary convection

Confined thermocapillary flows often present a viscous singularity where free and solid surfaces meet. Any computational treatment of these flows must filter this singularity, either explicitly by modifying the boundary conditions of the problem, or implicitly by using finite precision methods. The effect of an explicit polynomial filtering is here examined for an axisymmetric side-heated liquid bridge, leading to two results: (i) thermally convective flows converge very slowly with the filtering scale; the numerical experiments must therefore be conducted carefully in order to get physically relevant results; (ii) stream-function and temperature are not suitable test variables for estimating this convergence.

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.

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.

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...