A drop of radius R 0 impacts a rigid surface with radius of curvature R 2 (see Figure 2(b)). Several values of the curvature parameter are shown: from left to right, , , , , and .
Axisymmetric sessile drop of density ρ and surface tension σ resting on a surface with radius of curvature R 2. Without gravity, the drop would be spherical with radius R 0, under gravitational force g it deforms to a shape given by R = R(θ) in spherical coordinates. The drop shape conforms to that of the substrate over the area 0 ⩽ θ ⩽ α.
(a) The static profiles of a liquid drop with on a flat surface. The three profiles are the sum of the first 50 spherical harmonic modes obtained by minimizing the surface and gravitational potential energy of a drop constrained in different ways, by averaging the reaction force over: the contact area (Eq. (19)) (solid line), the contact area rim (dashed line), and the center of the contact area (dashed-dotted line). We see that even for an O(1) Bond number, the averaging method provides a good approximation to the actual drop shape, which has a perfectly flat base. (b) The static profile of a drop obtained from the first 50 spherical harmonics using the averaging method (20) for several values of : (dotted line), (dashed-dotted line), (solid line) and (dashed line).
The dependence of the coefficients A m and D m from Eq. (31) on the scaled Ohnesorge number . Curves for m = 2, 4, 10, 40 (triangles, circles, dashed-dotted, and dashed lines, respectively) are shown, together with the limiting curves for m → ∞ (solid lines) corresponding to planar surface capillary waves.
The dependence of the dissipation coefficient C D in Eq. (34) on the Ohnesorge number .
Comparison of the nondimensional contact time as a function of the Weber number for and , obtained with our quasi-static model (38) (solid line), the simplified model (40) (dashed line), and numerical simulation of the first 250 spherical harmonic modes (42) (dashed-dotted line). The predictions of Gopinath and Koch14 (■), Foote13 (▼), and Okumura17 (horizontal line) are included for the sake of comparison.
The dependence of the nondimensional contact time on the rescaled Weber number . Results of the numerical model (42) for several values () of the curvature parameter follow a single curve (solid line). The analytic expression (43) (dashed line) is shown for the sake of comparison.
The effects of gravity on the nondimensional contact time as a function of the Weber number . The results of the numerical model (42) (dashed-dotted line), quasi-static model (38) (solid line), and the analytical expression (44) (dashed line), all for are plotted, together with the experimental results of Okumura et al., for (▼) and (■). For reference, the result of the numerical model (42) for (i.e., no gravity) is also shown (•).
The dependence of the coefficient of restitution C R on the Weber number with (a) and without (b) gravity, for a drop impacting a flat substrate (). Results of the quasi-static model (38) (solid lines) and the full numerical model (42) (points) are shown for four values of the Ohnesorge number : (•), (■), (▼), and (▲).
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