Mathematical Modeling of Capillary Flows
Minisimposio at SIMAI
2008, Rome, September 15-19, 2008Workshop on: Liquid Penetrant
Testing, Messina, June 6, 2008
Locandina in pdf
Selected List of Papers
Mathematical and Numerical
Modeling of Liquids Dynamics in a Horizontal Capillary. Joint work with G.
Cavaccini, V. Pianese, S.
Iacono, and A. Jannelli, Presented at the
ICCMSE 2006. Abstract. 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 limita- tions 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.
A Two Immiscible
Liquids Penetration Model for Surface-Driven Capillary
Flows. Joint work with S.
Iacono, A. Jannelli,
G. Cavaccini, and V. Pianese, Presented at the
ICIAM 2007. PAMM: Proc. Appl. Math. Mech. 7 (2007) 2150003-2150004. DOI 10.1002/pamm.200700151 Abstract. We have derived, within the one-dimensional approximation and for a cylindrical capillary section, a two immiscible liquids penetration model for surface-driven capillary flows. Several test cases were considered where water was always in front of the other liquids. For the sake of brevity we report only the numerical results for two of the mentioned test cases: namely, ethanol and water, and mixture and water. The comparison between the two cases shows how the mixture, having a higher surface tension and a lower viscosity with respect to the ethanol, reaches a dipper distance inside the capillary.
One-dimensional Mathematical and Numerical
Modeling of Liquids Dynamics in a Horizontal Capillary. Joint
work with G. Cavaccini, V. Pianese, S. Iacono, and A. Jannelli.
J. Comput. Meth. Sci. Eng., 9 (2009) 3-16. Abstract.
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.
Liquid Dynamics in
a Horizontal Capillary: Extended Similarity Analysis. Joint
work with S. Iacono, A. Jannelli, G. Cavaccini, and V. Pianese,
to appear: Communications to SIMAI Congress, 2009. Abstract. 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.
Ill and Well-Posed
One-Dimensional Models of
Liquid Dynamics in a Horizontal Capillary. Joint
work with A. Jannelli, to appear: Communications to SIMAI Congress,
2009. Abstract. In this paper, we report a mathematical and numerical study of liquid dynamics models in a horizontal capillary. In particular, we prove that the classical model is ill-posed at initial time, and we present two different approaches in order to overcome this ill-posedness. By numerical viewpoint, we apply an adaptive strategy based on an one-step one-method approach, and we compare the obtained numerical approximations with suitable asymptotic solutions.
Liquid Penetrant
Testing: Industrial Process. Joint
work with G. Caturano, G. Cavaccini, A. Ciliberto, and V. Pianese,
to appear: Communications to SIMAI Congress, 2009. Abstract. Non-destructive inspection is an area of great interest and importance for the avia- tion industry. These kind of controls are routinely applied in various phases of design, development and production of manufactured. The present paper describes the whole process of inspection with liquid penetrant and the methodology adopted by Alenia for quality control of this process. Among the various methods of non-destructive testing, liquid penetrant is mainly applicable to metallic materials and detects discontinuities of size greater than 0.01 mm, provided that they are open on the surface. Due to broad typology of defects found and materials tested by this method, it is difficult to define a unique class of optimal parameters that describe the process. In this work we report an analysis of the different diagnostic target, according to the process parameters. Finally, we discuss the modeling activities developed within the last years.
Probability of
Detection for Penetrant Testing in
Industrial Environment. Joint
work with G. Caturano, G. Cavaccini, A. Ciliberto, and V. Pianese,
to appear: Communications to SIMAI Congress, 2009. Abstract. Introduced at the end of 60's by NASA, Probability of Detection (PoD) is becoming more and more one of the main approach in order to assess, quantitatively, the general detection capabilities of a Non Destructive Inspection process. In spite of its importance, PoD can be elaborated in a variety of ways and can lead to some misinterpretations. Alenia Aeronautica assessed a specific approach for liquid penetrant inspection that is strictly connected to the estimation of the inspection sensitivity and it can be aimed at various targets, such as: inspection procedure validation, evaluation of personnel proficiency, comparative analysis of penetrant inspection processing materials, equipment and procedures, and evaluation of automated inspection systems. To this purpose, PoD is conceived as the probability, at a fixed confidence level, to detect a discontinuity belonging to a predefined class. Experimental PoD curves are obtained by processing metallic samples with defects generated and developed under controlled conditions.
Extended Scaling Invariance of
One-dimensional Models of Liquid Dynamics in
a Horizontal Capillary. Joint work with S. Iacono, A. Jannelli, G. Cavaccini, and V. Pianese. Math. Meth. Appl. Sci., 35 (2012) 935-942.
Abstract.
In this paper, we consider the adaptive numerical solution of one-dimensional models of liquid dynamics in a horizontal
capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity: 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. Therefore, in order to solve a capillary problem effectively, it
would be advisable to apply an adaptive numerical method.
Here, we show how an extended scaling invariance that can be used to define a family of solutions from a computed
one. In the viscous case, the similarity transformation involves solutions of liquids with different density, surface tension,
viscosity, and capillary radii, whereas in the inviscid case, we can generate a family of solutions for the same liquid and
capillaries with different radii. With our study, we are able to prove that the monitor function, used in the adaptive
algorithm, is invariant with respect to the considered scaling group. It follows, from this particular results, that all the
solutions within the generated family verify the adaptive criteria used for the computed one. Moreover, all the solutions
have the same order of accuracy even if the maximum value of the step size varies under the action of the scaling group.
An Analytical and Numerical Study of Liquid Dynamics in a one-dimensional Capillary under Entrapped Gas Action. Joint work with S. Iacono. Math. Meth. Appl. Sci., (2014).
Abstract. 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. Due to 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 1D 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 modeling, goes on presenting the 1D 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 behavior 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.
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