 |
Drop impact on a dry/wetted wall is of relevance to many industrial applications as well as to natural sciences. For some applications, such as spray coating or icing of plane wings, both the maximum spreading diameter and the residual thickness of the wall film are of paramount importance for the efficient design and optimisation of the different technologies.In this paper, we propose a modification to existing models for crown propagation and residual film thickness based on stagnation-point flow solutions. It is generally accepted that the position of the crown base exhibits a square-root dependence on time, whereby the constant of proportionality C is inversely proportional to the fourth root of the initial film thickness h_0. We introduce the following two modifications. First, we include the thinning of the initial film thickness, which is no longer considered to be constant. The evolution of the film thickness is obtained directly from the potential flow theory for stagnation-point flow. Second, the constant C is modified to include the momentum losses in the spreading lamella due to boundary layer effects. For the estimation of viscous losses, two different exact solutions of the Navier-Stokes equations are considered with different boundary conditions. The first analytical solution is based on the classical Hiemenz flow solution for droplet impingement on a dry wall. In the second approach, the classical Hiemenz solution is extended to orthogonal stagnation-point flow against a fluid film, resting against a plane wall. This second approach enables for the first time to evaluate the effect of sliding on the crown propagation due to the presence of a liquid wall film. A comparison of the two solutions for the standard and extended Hiemenz-flow is discussed for a few representative test cases. Both solutions lead to a significant improvement in the prediction of the crown propagation rate compared to inviscid models. The preliminary results show that the inclusion of sliding effects becomes increasingly important with increasing wall films thickness (δ = h_0/D_0 > 0.3), where the inception of viscous losses is temporally delayed, as confirmed by experiments. The advantages of the stagnation-point flow approach are twofold. First, it enables a smooth transition from the inertia-driven to the viscous-controlled regime of crown propagation. Second, it lays the foundation for modelling continuously the transition from low-viscosity to high-viscosity wall films and for assessing how the viscosity ratio affects the spreading rate of the crown.
