Weakly nonlinear instability of a Newtonian liquid jet

Abstract

A weakly nonlinear stability analysis of an axisymmetric Newtonian liquid jet is presented. The calculation is based on a small-amplitude perturbation method and performed to second order in the perturbation parameter. The obtained solution includes terms derived from a polynomial approximation of a viscous contribution containing products of Bessel functions with different arguments. The use of such an approximation is not needed in the inviscid case and the planar case, since the equations of those problems can be solved in an exact form. The developed model depends on three dimensionless parameters: the initial perturbation amplitude, the perturbation wavenumber and the liquid Ohnesorge number, the latter being the dimensionless liquid viscosity. The influence of the approximate terms was shown to be relatively small for a large range of Ohnesorge numbers so that they can be ignored. This simplification provides a jet model as simple to use as the previous ones, but taking into account the liquid viscosity and the cylindrical geometry. The jet model is used to reveal the effect of both the wavenumber and the Ohnesorge number on the formation of satellite drops, which is known as a nonlinear effect. Results are found in good agreement with direct numerical simulations and forced liquid jet experiments for wavenumbers lower than a threshold value. Satellite drop formation is retarded with increasing Ohnesorge number and wavenumber, as expected by the damping and size effects of viscosity. The threshold number corresponds to the maximum wavenumber for which satellite drop formation is predicted before jet breakup, and for which volume conservation is satisfied within a certain amount. The volume conservation criterion is imposed to ensure that the conclusions inferred by our model are safe.

Drop dynamics of viscoelastic filaments

“…It has been known for 50 years [11,12] that capillary jets of viscoelastic polymer solutions exhibit the peculiar morphology called beads-on-a-string (BOAS). The instability and the initial sinusoidal growth has been reported [13][14][15] and has also been observed in the stretching of capillary bridges using extensional rheometers such as Capillary Breakup Extensional Rheometer (CaBER) [7,[16][17][18][19][20][21]. These studies have consistently evidenced the linear viscous-capillary thinning, the exponential polymeric thinning [22] and the existence of drops attached to a thin filament [2,[23][24][25][26][27][28], depending on the fluid properties.…”

Weakly nonlinear shape oscillations of inviscid drops

“…the volume is not accurately conserved (Renoult et al. 2018). The relative volume deviation is determined analytically.…”

“…As a consequence, the non-dimensional drop volume may deviate from its exact value of V s = 4π/3, i.e. the volume is not accurately conserved (Renoult et al 2018). The relative volume deviation is determined analytically.…”

Non-linear shape oscillations of a viscous liquid drop

“…This is a Poisson equation for the modified pressure P 21 = p 21 + v 2 1 /2, found in the cylindrical formulation for a jet in [11]. The structure of the solution in terms of its dependency on the polar angular coordinate and on time is determined by v 2 1 , and by the first-order term products on the right-hand side of equation (39).…”