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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.

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This is a mathematical and numerical study of liquid dynamics in a horizontal capillary. We present a two-liquids model which takes into account the effects of real phenomena like the outside flow dynamics. Moreover, we report on results obtained by an adaptive numerical method. The two liquids penetration modelThe model considered here is of interest for the non-destructive control named "liquid penetrant testing" used in the production of airplane parts as well as in many industrial applications where the detection of open defects is of interest.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. With reference to Fig. 1, we consider a column of liquid 1, usually water, of fixed length 0 entrapped within a horizontal cylindrical capillary of radius R and length L. At the left end of the capillary we have a reservoir filled with a penetrant liquid 2 and the moving interface between the two liquids is denoted by . The governing equation is given by the second order ordinary differential equationwhere ρ 1 and ρ 2 are the densities, µ 1 and µ 2 are the dynamic viscosities of the two liquids, γ 1 and γ 12 are the surface free energy for the liquid 1-air and the liquid 1-liquid 2 interfaces, ϑ 1 and ϑ 12 are corresponding menisci contact angles, and t is time. We have taken into account here the coefficient of apparent mass c = O(1), introduced by Szekely et al. [1]. Moreover, in the case of a vertical capillary, the gravity action should be considered, by adding to the left hand side of equation (1) the termwhere the plus or mines sign have to be used when the liquid reservoir is over or below the cavity, respectively. Full details on the derivation of this model, as well as extensive numerical simulations, will be reported elsewhere. For an overview on capillary dynamics the interested reader is referred to the recent book by de Gennet et al. [2] and the references quoted therein.

The paper details recent results concerning the design of a new intake system for a 1.4 liter displacement ELASIS-FIAT engine. A classical approach, based on theoretical one-dimensional characterization of the whole system, is presented. This approach, however, requires a relevant number of geometrical information which are usually unavailable in the first phase of the design process. For this rason, a statistical analysis on a number of existing devices was also carried out to the aim of providing such initial data as a function of prescribed levels of pressure losses and noise emission for the device. The methodology allows then to define a base configuration of the system, to start the 1D analyses. The base geometry is further refined taking into account the layout constraints and the presence of resonators for the reduction of the noise emission. Experimental data collected on a prototype of the designed system have confirmed the robustness of the whole design procedure