Transition from dripping to jetting

We consider the critical Weber number (We_c ≡ ρV₀²D/σ) at which the transition from dripping to jetting occurs when a Newtonian liquid of density ρ and surface tension σ is injected with a velocity V₀ through a tube of diameter D downward into stagnant air, under gravity g. We extend Taylor's (1959) model for the recession speed of a free edge, and obtain in the inviscid limit an exact solution which includes gravity and inertia effects. This solution provides a criterion for the transition which is shown to occur at a critical Weber number We_c = 4Bo_o/Bo [1 + K Bo_oBo - ((1 + K Bo_oBo)² - 1)^1/2]², where Bo and Bo_o are the Bond numbers (Bo ≡ (ρgD²/2σ)]^1/2), respectively based on the inside and outside diameter of the tube, and K is a constant equal to 0.37 for the case of water injected in air This critical Weber number is shown to be in good agreement with existing experimental values as well as with new measurements performed over a wide range of Bond numbers.