Effects of gravity on natural oscillations of sessile drops

Abstract

Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$ . Three types of $\alpha$ – $Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$ . As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$ . This implies that the ‘walking’ drop instability reported in previous literature does not exist.

Walking droplets have been halted

“…Conversely, as Bo increases, it is found that the lowest mode of a drop with α > 90 • gradually switches from mode {2, 0} to {1, 2}, with the consequence that for sufficiently high Bo the mode {1, 2} is the lowest mode regardless of the contact angle. As observed by Zhang et al (2023), this result implies that for large drops (and movable CL), non-axisymmetric oscillation will more likely manifest in large drops than in smaller ones.…”

“…The system of governing equations, free-surface and CL conditions can be turned into an eigenvalue problem through the normal mode expansion ψ(x, t) = φ(r, z)e iλt e ilϕ and η(s, ϕ, t) = y(s)e iλt e ilϕ , where λ is the unknown complex eigenvalue. Zhang et al (2023) tackle such a problem through a BEM, which can deal with arbitrary drop geometry, with the advantage of reducing the two-dimensional problem to a boundary integral equation. The modes are then classified by the pair {n, l}, where the latter is the azimuthal wavenumber and the former indicates the number of vertical layers of the perturbation, which is related to the polar wavenumber of spherical harmonics introduced in (1.1) as n = (k − l)/2 + 1.…”

“…The key contribution of the paper by Zhang et al (2023) lies in a systematic investigation of the effects of gravity on the axisymmetric and non-axisymmetric oscillations of sessile drops over a wide range of static contact angles α ∈ [30 • -150 • ] and Bond numbers Bo = ρgl 2 * /σ ∈ [0-10], with the characteristic length scale l * = [3 ṽ/(2π)] 1/3 based on the the drop volume ṽ. It is shown that the frequency of zonal modes decreases with gravity at small contact angle, but increases when the contact angle exceeds a certain critical value, with larger α being more sensitive to the effects of gravity (in agreement with the numerical simulations of Sakakeeny & Ling (2021)).…”

“…The appeal of the approach by Zhang et al (2023) comes from the effectiveness in solving the small-amplitude dynamics of inviscid drops of arbitrary shape. These results can be of valuable help in several applications, including inkjet printing, additive manufacturing, printed electronic components and circuits, forensic bloodstains, colloidal aggregation and biological flows such as plants transpiration, among many others.…”

“…As the drop shape differs from that of the spherical cap, however, there are no general theoretical models for the natural frequencies, owing to the difficulties in the geometrical treatment of the flattened drop configuration and the motion of the three-phase CL. Thus, the effect of gravity on the natural frequencies and the corresponding modes of a sessile drop had not been tackled until the recent paper of Zhang, Zhou & Ding (2023), where the corresponding eigenvalue problem is solved numerically through a general boundary element method (BEM) for different Bond numbers, CL conditions (either free or pinned) and static equilibrium contact angle.…”