Mathematical Modeling of Capillary Flows

Selected List of Papers

** 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.

** 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.

** 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.

** 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.

** 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.

** 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.

** 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.

** 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.

** 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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(c) by Riccardo Fazio Last modified: December 20, 2013