Impulse-induced liquid jets from bubbles with arbitrary contact angles

This paper investigates the relationship between the contact angle of a spherical bubble attached to a tube submerged in a container and the jet speed induced by an impulsive acceleration at its base. While it has been well established that bubble geometry strongly influences the ejection speeds of liquid jets, mathematical studies of liquid jets with arbitrary bubble shapes remain limited. In this work, we derive a pressure impulse in the small-cavity limit as a tractable integral of classical Legendre functions. It is shown that the jet speed can be divided into two components: (i) the velocity induced by the hydrostatic pressure impulse distribution created by the curvature of the bubble, and (ii) the velocity induced by the distribution of the submersion of the tube in a container. This decomposition reveals that an optimal bubble curvature emerges only when the tube is submerged: the optimality is absent for non-submerged configurations, where the jet speed increases monotonically with bubble depth. Experiments confirm this non-monotonicity and quantitatively support the predicted shift of the optimal geometry with submersion depth.

💡 Research Summary

This paper investigates how the contact angle of a spherical bubble attached to the tip of a tube influences the speed of a liquid jet generated by an impulsive acceleration at the bottom of a container. The authors formulate the problem in terms of a pressure‑impulse field Π that satisfies Laplace’s equation in the liquid domain, with mixed Dirichlet–Neumann boundary conditions on the bubble surface, the container wall, and the free surface at the tube entrance. By non‑dimensionalising lengths with the tube radius R and introducing the submersion depth h (the distance the tube tip is below the free surface), the governing problem is reduced to a three‑dimensional axisymmetric Laplace problem.

In the limit of a small bubble (λ = R/r → 0, where r is the container radius), the side‑wall condition disappears and the domain becomes a semi‑infinite half‑space outside a spherical cavity. The authors adopt toroidal coordinates (α, β) – a classic transformation introduced by Lebedev (1965) – which maps the spherical bubble surface to a constant‑β line (β = π + θ) and the plane z = 0 (the tube entrance) to β = 2π. In these coordinates the Laplace equation separates, and the pressure impulse can be expressed as a superposition

Π∞(ξ,z) = Π_f(ξ,z) + h Π_g(ξ,z),

where Π_f represents the contribution from the curvature of the bubble alone (no submersion) and Π_g captures the additional pressure redistribution caused by the tube’s immersion depth h. Both Π_f and Π_g are written as integrals over the spectral parameter τ involving Legendre functions of the first kind P_{‑½ + iτ}(cosh α). Explicitly,

f_θ(α,β) = 2√(cosh α − cos β) ∫₀^∞