The impact of a drop on liquid surfaces is studied experimentally and theoretically in the region of the fully developed splashing. In order to reveal the influence of viscosity and target liquid depth on the resulting flow patterns, the experiments were carried out with water and 70% glycerol-water solution, and for different target liquid depths. Based on the experimental observations, a dynamic model of the central jet formation at the cavity collapse is developed. This model predicts an emergence of a liquid flow up into the central jet and simultaneously a small flow velocity downward and allows us to evaluate the velocities of these two flows. A theoretical model for the cavity submergence is presented. This model gives the constant velocity of the cavity submergence which is half the initial drop impact velocity. Analytical solution for the gravity-capillary cavity collapse has been derived and provides a good fit to the experimental results. Theoretical analysis and experiments have shown that the maximum cavity radius and the cavity collapse time depend on both the Froude number and the dimensionless capillary length.
Young’s modulus is a fundamental physical parameter that determines not only the mechanic but also the electronic properties of a solid thin film. In here, we show that the elastic modulus is not a constant as that of conventional treatment but varies with film thickness. Scaling behavior is found based on the theoretical analysis of the free energy of surface-to-volume ratio of the film and results of the elastic modulus measurement. It has been shown that there exists some film thickness hb when the surface energy of the film comes into play. The hb is inverse proportional to the bulk Young’s modulus and depends strongly on the in-plain strain ε0 as ε0−2. For Si nanofilms, the variation of dimensionless elastic modulus Ψ=E/Ebulk with the dimensionless film thickness η=h/hb can be represented in the following form: Ψ=η0.226. The present investigation illustrates the importance of the effect of dimensionality on the basic parameter of a thin film as well as providing important implications for electronic devices, in particular for the Si-based strained nanodevices.
The theoretical models for the deformation of a liquid drop impinging on a solid flat surface at the initial and late stages are proposed. It was found that at the initial stage of the drop impact, the thickness of the emerging film decreases rapidly along its radius r, as r −6 , that is similar to the splash jet induced by the blunt-body impact on the liquid surface. The thickness of the film levels off with time due to the viscous force, and the late stage of the drop spreading is controlled by the action of viscous and capillary forces. The influence of the capillary forces is localized in the vicinity of the triple line, and it causes the formation of the thick border ͑blob͒ on the edge of the spreading drop. An analytical solution of the model in viscous limit reveals that the minimum film thickness scales as Re −2/5 and the drop maximum radius in its maximum extension as Re 1/5. The analytical solution for the dynamics of the blob mass growth is also obtained. The kinetic energy of the drop at its maximum extension remains greater than zero in the drop-spreading process even accounting for viscous effect.
We present a theoretical analysis of the vibrating resonator in cantilever and bridge configurations operating as ultrasensitive mass sensors. An exact solution of the problem has been obtained. For the small mass ratio, the asymptotic solutions (which relate the frequency shift, mass ratio, and position of the attached particle) have been derived. It has been shown that the mass and position of the attached particle for the cantilever configuration can be unambiguously resolved by the use of three consecutive resonant frequencies. For the bridge configuration, the particle mass can be deduced by using only two measured resonant frequencies.
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