Capillary Free Surface Research Articles

Simulation and multi-objective optimization to improve the final shape and process efficiency of a laser-based material accumulation process

Common goals of modern production processes are precision and efficiency. Typically, they are conflicting and cannot be optimized at the same time. Multi-objective optimization methods are able to compute a set of good parameters, from which a decision maker can make a choice for practical situations. For complex processes, the use of physical experiments and/or extensive process simulations can be too costly or even unfeasible, so the use of surrogate models based on few simulations is a good alternative.In this work, we present an integrated framework to find optimal process parameters for a laser-based material accumulation process (thermal upsetting) using a combination of meta-heuristic optimization models and finite element simulations. In order to effectively simulate the coupled system of heat equation with solid-liquid phase transitions and melt flow with capillary free surface in three space dimensions for a wide range of process parameters, we introduce a new coupled numerical 3d finite element method. We use a multi-objective optimization method based on surrogate models. Thus, with only few direct simulations necessary, we are able to select Pareto sets of process parameters which can be used to optimize three or six different performance measures.

Existence and static stability of a capillary free surface appearing in a dewetted Bridgman process. Part II.

This paper presents six theoretical results concerning the existence and static stability of a capillary free surface appearing in a dewetted Bridgman crystal growth technique. The results are obtained in an axis-symmetric 2D model for semiconductors for which 𝜃𝜃𝑐𝑐+𝛼𝛼𝑒𝑒>𝜋𝜋 (where:𝜃𝜃𝑐𝑐- wetting angle and 𝛼𝛼𝑒𝑒- growth angle). Numerical results are presented in case of GaSb semiconductor growth. The reported results can help, the practical crystal growers, in better understanding the dependence of the free surface shape and size on the pressure difference across the free surface and the right choice of crystal size, pressure difference and thermal conditions for the growth process.

Existence and static stability of a capillary free surface appearing in a dewetted Bridgman process. Part I.

This paper presents six theoretical results concerning the existence and static stability of a capillary free surface appearing in a dewetted Bridgman crystal growth technique. The results are obtained in an axis-symmetric 2D model for semiconductors for which 𝜃𝜃𝑐𝑐+𝛼𝛼𝑒𝑒<𝜋𝜋 (where:𝜃𝜃𝑐𝑐- wetting angle and 𝛼𝛼𝑒𝑒- growth angle). Numerical results are presented in case of InSb semiconductor growth. The reported results can help, the practical crystal growers, in better understanding the dependence of the free surface shape and size on the pressure difference across the free surface and prepare the appropriate seed size, and thermal conditions before seeding the growth process.

Interface deformations due to counter-rotating vortices: Viscous versus elastic media.

Capillary forces determine the shape of a liquid interface. Although often not considered, elastic solids with a free surface are also subjected to surface forces and these become important for materials of low Young's modulus. Here we consider two equivalent problems where a capillary free surface deforms due to vortices: (i) in a steady viscous flow [solved by Jeong and Moffatt, J. Fluid Mech. 241, 1 (1992)], and (ii) in an elastic medium. The equations of linear incompressible elasticity and viscous flow are strictly identical, and the two-dimensional problems that we consider are solved using complex variable methods. Despite the similarity, the kinematics of the free surface is very different for the viscous and elastic cases. We show for the present problem that these kinematics result in displacement and velocity fields of different topology. Unexpectedly, the resulting surface deflections are even of opposite sign.

Study of Wetting in an Asymmetrical Vane-Wall Gap in Propellant Tanks

Wetting of an asymmetrical vane-wall gap geometry is investigated. The geometry is common in vane-type liquid propellant management devices. Critical wetting conditions are determined by applying the method by Concus and Finn (Concus, P., and Finn, R., On Capillary Free Surfaces in the Absence of Gravity, Acta Mathematica, Vol. 132, 1974, pp. 177-198). The critical wetting condition is expressed in terms of critical contact angle as a function of gap size, with the vane thickness and obliquity angle as relevant parameters. It is found that the increase of both the vane thickness and the obliquity angle improves the critical wetting conditions. The analytical results are confirmed with Surface Evolver numerical computations, which also provide graphical descriptions of capillary surfaces. Results from drop tower experiments confirm the analysis and also reveal that the advance rate of the meniscus tip decreases with the gap size when other parameters are fixed.

Discontinuous behavior of liquids between parallel and tilted plates

Discontinuous behavior of liquids between parallel and tilted plates in the absence of gravity is discussed. A principal finding, derived mathematically from the classical Young–Laplace–Gauss formulation for capillary free surfaces, is that in a large range of configurations liquid bridges between parallel plates are unstable with respect to small, even infinitesimal, tilting of one of the plates. Under a computationally based hypothesis of uniqueness of spherical bridges in a wedge, it is shown that the same discontinuous behavior prevails for all but very particular circumstances. The various liquid configurations, which form the basis for an experiment on board the Space Station Mir, are characterized and illustrated.

Drop-tower experiments for capillary surfaces in an exotic container

Low-gravity drop-tower experiments are carried out for an 'exotic' rotationally-symmetric container, which admits an entire continuum of distinct equilibrium symmetric capillary free surfaces. It is found that an initial equilibrium planer interface, a member of the continuum, will reorient toward a non-symmetric interface, as predicted by recent mathematical theory.

Exotic containers for capillary surfaces

In this paper we discuss ‘exotic’ rotationally symmetric containers that admit an entire continuum of distinct equilibrium capillary free surfaces. The paper extends earlier work to a larger class of parameters and clarifies and simplifies the governing differential equations, while expressing them in a parametric form appropriate for numerical integration. A unified presentation suitable for both zero and non-zero gravity is given. Solutions for the container shapes are depicted graphically along with members of the free-surface continuum, and comments are given concerning possible physical experiments.

On existence criteria for capillary free surfaces without gravity

On existence criteria for capillary free surfaces without gravity

On Capillary Free Surfaces without Gravity

The non-parametric capillary problem in the absence of gravity can be replaced by a Dirichlet problem for a quasilmnear equation of mixed type by introducing a stream function. We prove a maximum principle for an expression depending on the gradient of the stream function and, furthermore, regularity properties of C^1(\bar {\Omega} -solutions in domains \Omega with corners.

Analytical and computational studies of capillary free surfaces

Analytical and computational studies of capillary free surfaces

Height estimates for exterior problems of capillarity type

This work concerns boundary value problems for a class of nonlinear equations modeled on the physical equations for a capillary free surface in a gravitational field. The results consist principally of estimates for the height of a solution in an exterior domain. Structure conditions reflecting the nonlinearity of the mean curvature operator are imposed on a class of symmetric variational operators in terms of the Legendre transform of the variational integrand. Estimates are found for the boundary height of a rotationally symmetric solution in the exterior of a ball of radius R. These estimates, which are valid for any R9 are shown to be asymptotically exact as R tends to zero or infinity. This provides a proof of the asymptotic behavior of the boundary height which previously has been derived by a formal perturbation method. An asymptotic characterization of the solution in a neighborhood of the boundary is also given. For a general domain estimates are obtained from a maximum principle due to Finn in which the symmetric solutions serve as comparison functions. l Preliminaries* We begin by formulating the standard capillarity problem on an exterior domain n dimensional space; the physical case is n = 2. Let Ω c Rn be an exterior domain whose boundary, Σ, is a compact C1 hypersurface. When the generalized (vertical) cylinder Σ x R is immersed in an infinite reservoir of fluid, the action of capillarity gives rise to a free surface of static equilibrium in the outside of the cylinder. Let the height of this capillary free surface, assumed nonparametric over Ω, be given by the scalar function u(x)f x — (xlf - , xn) eΩ. Physical principles assert that the equilibrium free surface minimizes the functional

On the existence of capillary free surfaces in the absence of gravity

On the existence of capillary free surfaces in the absence of gravity

On capillary free surfaces in the absence of gravity

On capillary free surfaces in the absence of gravity

On capillary free surfaces in a gravitational field

On capillary free surfaces in a gravitational field