Salvatore Iacono

On the moving boundary formulation for parabolic problems on unbounded domains

The aim of this paper is to propose an original numerical approach for parabolic problems whose governing equations are defined on unbounded domains. We are interested in studying the class of problems admitting invariance property to Lie group of scalings. Thanks to similarity analysis the parabolic problem can be transformed into an equivalent boundary value problem governed by an ordinary differential equation and defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996), pp. 1473–1483]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either a noniterative, or an iterative method. Finally, the solution of the parabolic problem is retrieved by applying the inverse map of similarity.

On the Translation Groups and Non-iterative Transformation Methods

The aim of this work is to point out that, within group invariance theory, some classes of boundary value problems governed by ordinary differential equations can be transformed to initial value problems. The interest in the numerical solution of (free) boundary problems arises because these are (always) often nonlinear problems. The theoretical content of this paper is original: results already available in literature are related to the use of scali ng or spiral groups of transformations; here we show how it is also possible to use the invariance with respect to two translation groups. As far as applications of the proposed approach are concerned, we solve two problems: a free boundary problem describing a rope configuration against an obstacle, whe re we compare the obtained numerical results with the exact solution, and a boundary problem modeling the fall of a parachutist, where we modify the classical formulation of the problem in order to prescribe the total falling time.

One-dimensional mathematical and numerical modeling of liquid dynamics in a horizontal capillary

This paper is concerned with a mathematical and numerical study of liquid dynamics in a horizontal capillary. We derive a two-liquids model for the prediction of capillary dynamics. This model takes into account the effects of real phenomena: like the outside flow action, or the entrapped gas inside a closed-end capillary. Moreover, the limitations of the one-dimensional model are clearly indicated. Finally, we report on several tests of interest: an academic test case that can be used to check available numerical methods, a test for decreasing values of the capillary radius, a simulation concerning a closed-end capillary, and two test cases for two liquids flow. In order to study the introduced mathematical model, our main tool, is a reliable one-step adaptive numerical approach based on a one-step one-method strategy.

An analytical and numerical study of liquid dynamics in a one-dimensional capillary under entrapped gas action

Capillary dynamics has been and is yet an important field of research, because of its very relevant role played as the core mechanism at the base of many applications. In this context, we are particularly interested in the liquid penetration inspection technique. Because of the obviously needed level of reliability involved with such a non-destructive test, this paper is devoted to study how the presence of an entrapped gas in a close-end capillary may affect the inspection outcome. This study is carried out through a one-dimensional ordinary differential model that despite its simplicity is able to point out quite well the capillary dynamics under the effect of an entrapped gas. The paper is divided into two main parts; the first starts from an introductory historical review of capillary flows modelling, goes on presenting the one-dimensional second order ordinary differential model, taking into account the presence of an entrapped gas and therefore ends by showing some numerical simulation results. The second part is devoted to the analytical study of the model by separating the initial transitory behaviour from the stationary one. Besides, these solutions are compared with the numerical ones, and finally, an expression is deduced for the threshold radius switching from a fully damped transitory to an oscillatory one. Copyright

Liquid Dynamics in a Horizontal Capillary: Extended Similarity Analysis

The topic of this study is an extended similarity analysis for a one-dimensional model of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity, that is the only driving force acting on it. Besides the smaller is the capillary radius the steeper becomes the initial transitory of the meniscus location derivative, and as a consequence the numerical solution to a prescribed accuracy becomes harder to achieve. Here, we show how an extended scaling invariance can be used to define a family of solutions from a computed one. The similarity transformation involves both geometric and physical feature of the model. As a result, density, surface tension, viscosity, and capillary radius are modified within the required invariance. Within our approach a target problem of practical interest can be solved numerically by solving a simpler transformed test case. The reference solution should be as accurate as possible, and therefore we suggest to use for it an adaptive numerical method. This study may be seen as a complement to the adaptive numerical solution of the considered initial value problems.

Numerical Simulation of Capillary Flows Through Molecular Dynamics

Capillary pore imbibition represents a very challenging field of research because of its crucial role in several physical phenomenona. This paper deals with capillary dynamics in connection with non destructive test technique used to evidence eventual defects present in mechanical parts whose perfect integrity is vital for the reliability of all the systems (car, planes, etc.) which they are part of. For the description of such phenomenona, essentially two different approaches are possible. The first approach consists in considering the fluid motion as a continuum to be described in terms of mathematical models governed by ordinary or partial differential equation. Subsequently, according to the prescribed initial and boundary conditions to these models, the solution can be worked out numerically through one of the available numerical techniques such as finite difference, finite elements, finite volumes, meshless methods, etc. Conversely, the second approach considers the fluid motion as an atomistic ensemble of discrete particles (atoms or molecules), whose macroscopic features are strongly determined by the inner interactions occurring among them. In this context, the two most important conceptual available tools are the Lattice Boltzmann theory, mainly used by physicists and the Molecular Dynamics, MD for the sequel, that jointly with its numerical treatment represents the topic of the present paper. Capillary imbibition represents a specific aspect of the wider area of wetting solid by liquids and its description in terms of MD has revealed to be a very promising approach for the description of the capillary flows alternative to continuous differential models. At the beginning, MD was used to simulate essentially the behavior of biological materials, but over the years MD has proved to be a powerful tool to model and simulate nanotechnology structures. In this paper large scale MD is used in order to model the intrusion of water into a defect of a few given solid materials. Finally numerical results have been obtained for cylindrical defects of Titanium or Aluminum both internally smooth or rough.