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A quasi-static model of drop impact
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View: Figures


Image of FIG. 1.
FIG. 1.

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 .

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

(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).

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

The dependence of the dissipation coefficient C D in Eq. (34) on the Ohnesorge number .

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

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 (•).

Image of FIG. 9.
FIG. 9.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A quasi-static model of drop impact