Extended scaling invariance of one‐dimensional models of liquid dynamics in a horizontal capillary

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. Copyright © 2012 John Wiley & Sons, Ltd.

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

“…The interested reader can find a similar table related to the so‐called polydimethylsiloxane (PDMS) silicone oils in Fazio et al .…”

“…We used the liquid parameters listed in [35] and reported, for the reader convenience, in Table I. The interested reader can find a similar table related to the so-called polydimethylsiloxane (PDMS) silicone oils in Fazio et al [38].…”

“…For instance, the scaling analysis presented in the previous section cannot be carried on for the adimensional model because a value of R appearing on the first two terms on the right-hand side of our model is replaced by the Reynolds number. Moreover, all dimensional parameters of different penetrant liquids can be taken from tables available in the literature, as for instance in the recent survey by Kornev and Neimark [22] or in Fazio et al [14], whereas in order to correlate a real liquid to the adimensional model we need to find out a characteristic mean velocity U and this is not available within the relevant literature.…”

“…By imposing the zero discriminant condition on the characteristic polynomial associated to (14), an equation in two variables, namely, the capillary radius and length can be easily worked out, provided that we keep on maintaining constant all the remaining parameters. Such a condition is just represented by the following…”