The common approach to the modelling of spray impact is to treat the phenomenon as a simple superposition of single drop impact events [1]. The main input for such model formulation is obtained either from experimental [2,3] or theoretical [4,5,6] studies of the impact of a single drop onto a dry wall, onto a uniform, undisturbed liquid film or into a deep pool [7]. However, in [8] it was shown that this conventional approach is not universal in the description of the spray impact and that in the case of relatively dense sprays, the interaction of crowns (Fig. 1) and the oscillations of the liquid-wall film must be taken into account. For example, these interactions result in the emerging of uprising jets during spray impingement of the diesel spray (see Fig. 2). In the study of spray impact we have chosen the following strategy of the modelling: 1. Description (experimental and theoretical) of single dropimpact. Determining of the parameters influencing the splash. 2. Description of the interaction of two drops on the wall surface. 3. Determining of the parameters of the single drop impacts influencing the dynamics of the film formed on the wall. Characterization of the film: the time averaged thickness, the time averaged velocity and its fluctuations. 4. Description of the influence of the oscillating motion of the film on the outcome from a single drop impact. Single drop impact onto a wetted wall—The motion of a kinematic discontinuity in the liquid film on the wall due to the drop impact, the formation of the uprising jet at this kinematic discontinuity and its elevation are analyzed. The theory [4] for the propagation of the kinematic discontinuity is generalized for the case of arbitrary velocity vectors in the inner and outer liquid films on the wall. Next, the mass, momentum balance and Bernoulli equations at the base of the crown are considered to obtain the velocity and the thickness of the jet on the wall. An analytical solution for the crown shape is obtained in the asymptotic case of such high impact velocities that the surface tension and the viscosity effects can be neglected in comparison to inertial effects. The edge of the crown is described by the motion of a rim, formed due to the surface tension. The theoretical predictions of the height of the crown are compared with experiments. The agreement is rather good in spite of the fact that no adjustable parameters are used (see Fig. 3). Three different cases are considered: normal axisymmetric impact of a single drop, oblique impact of a single drop, and impact and interaction of two drops. Next, two new parameters of single drop impact influencing the dynamics of the film formed due to the polydisperse spray impact are identified. The first one is associated with the relative presence of the crown on the film surface and allowing one to estimate the probability of crown interactions. The second parameter is associated with the axial momentum in the plane of the wall. Time-averaged film motion—The theory of the creation of the film by spray can be subdivided into three main parts: 1. The characterization of the spray, particularly definition of the flux vectors of scalar properties (number flux vector, volume flux vector, etc.) and the momentum flux tensor. 2. Boundary conditions at the time-averaged spray/film boundary. 3. Dynamics of the film motion on the wall. The mass and momentum equations of the film are formulated accounting for the volume flux of the spray, the dynamic pressure, and the time-averaged stress vector at the film “free” surface caused by the inertia of the spray. The inertial terms of the liquid in the film contains of the inertia of the time-averaged motion and the inertia of film oscillations. These oscillations are modelled as an ensemble of the radial flows in the film associated with the single drop impacts. The probability of the crown interactions is also taken into account. Jetting at the film surface due to impingement of a dense spray—Here we consider impact of such dense sprays that the probability of single crown to propagate without interaction with another crown is very small. The non-uniformities in the dynamic pressure in such sprays yields the significant fluctuations in the film velocity leading to the shocks and jetting (as in the case of the diesel spray impact shown in Fig. 2). We describe the statistically averaged distribution of drop impacts around a given drop assuming that all the impacting drops are distributed randomly in space and in time. The statistically averaged dynamic pressure around given drop is not uniform either in the time or in the radial direction. The self-similar solution for the statistically averaged radial velocity in the film and its thickness (Fig. 4) is obtained. The characteristic time of the instant of shock is estimated. The theoretical predictions of the jets diameter agree with the experimental data in the order of the magnitude.
An Implanted Magnetic Microfluidic Pump for In Vivo Bone Remodeling Applications
