Marangoni Convection in Spherical Shells
- Paper number
IAC-05-A2.4.07
- Author
Mr. Pravin Subramanian, Rutgers University, United States
- Coauthor
Mr. Abdelfattah Zebib, Rutgers University, United States
- Year
2005
- Abstract
Hollow spherical shells used as laser targets in inertial confinement fusion (ICF) experiments are made by microencapsulation. In one phase of manufacturing, the spherical shells contain a solvent (fluorobenzene, FB) and a solute (polystyrene, PAMS) in a water-FB environment. As the solvent evaporates it leaves behind the desired hardened plastic spherical shells, 1-2 mm in diameter. Perfect sphericity is demanded for efficient fusion ignition. It is proposed that Marangoni instabilities driven by surface tension dependence on the FB concentration (c) might be the source of observed surface roughness (buoyant forces are negligible in this micro-scale problem). Here we model this drying process, investigate conditions for incipient instabilities, and compute nonlinear axisymmetric and three-dimensional convection. The inner radius of the shell is taken stress-free and impermeable. At the receding outer radius r2(t) the FB evaporates with an assumed constant mass transfer coefficient. The non-dimensional equations along with the nonlinear boundary conditions are solved through a coordinate transformation, a second-order finite volume approach and Crank-Nicholson time marching. We first consider the diffusive state and determine r2(t) and c(r,t). Instability of the diffusive state is then considered in the limit of small Capillary number (Ca) which is a measure of the deviation of surface tension from its average value and thus the magnitude of surface deviation from sphericity. To leading order, the outer interface recedes in a convective state as it does in the diffusive state. Linear stability analysis assuming viscosity dependence on c and normal mode decomposition in surface harmonics leads to a partial differential system in (r,t) with time-dependent coefficients. We have performed frozen-time, quasi-steady-state calculations to determine the critical Reynolds number and degree of surface harmonics (linear convection is independent of the azimuthal wavenumber). We have also calculated maximum growth rates of perturbations by solving the initial boundary value problem with random initial conditions. We compute nonlinear, time-dependent, infinite Schmidt number convection by a second order accurate finite volume technique. Preferred supercritical patterns are investigated in the relevant parameters space and with various initial conditions. Companion O(Ca) surface deformations are determined and our results compared with available experiments.
- Abstract document
- Manuscript document
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