Minimal formulation of the linear spatial analysis of capillary jets: Validity of the two-mode approach

A rigorous and complete formulation of the linear evolution of harmonically stimulated capillary jets should include infinitely many spatial modes to account for arbitrary exit conditions [J. Guerrero et al., J. Fluid Mech. 702, 354 (2012)]. However, it is not rare to find works in which only the downstream capillary dominant mode, the sole unstable one, is retained, with amplitude determined by the jet deformation at the exit. This procedure constitutes an oversimplification, unable to handle a flow rate perturbation without jet deformation at the exit (the most usual conditions). In spite of its decaying behavior, the other capillary mode (subdominant) must be included in what can be called a “minimal linear formulation.” Deformation and mean axial velocity amplitudes at the jet exit are the two relevant parameters to simultaneously find the amplitudes of both capillary modes. Only once these amplitudes are found, the calculation of the breakup length may be eventually simplified by disregarding the subdominant mode. Simple recipes are provided for predicting the breakup length, which are checked against our own numerical simulations. The agreement is better than in previous attempts in the literature. Besides, the limits of validity of the linear formulation are explored in terms of the exit velocity amplitude, the wave number, the Weber number, and the Ohnesorge number. Including the subdominant mode extends the range of amplitudes for which the linear model gives accurate predictions, the criterion for keeping this mode being that the breakup time must be shorter than a given formula. It has been generally assumed that the shortest intact length happens for the stimulation frequency with the highest growth rate. However, we show that this correlation is not strict because the amplitude of the dominant mode has a role in the breakup process and it depends on the stimulation frequency.

Figure 1

(a) Computational geometry, boundary conditions for the tracer, and typical profile of a jet just before the front-ligament pinching (FLP), with the refined quadtree mesh. The lower half of the figure is a reflection of the upper half, as the computations are axisymmetric. For this simulation it is $\text{We}=14.8,\text{Oh}=0.01,k_{\mathrm{temp}}=0.697,$ and $w_0=0.154$. (b) An example of rear-ligament pinching (RLP).

Figure 2

Breakup diagrams showing for each dimensionless time, $t$, the parts of the axis occupied by the liquid (filled) and by the air (empty). The simulation has been carried out for $\text{Oh}=0.01,k_{\mathrm{temp}}=0.697,w_0=0.154,$ and (a) $\text{We}=4$ or (b) $\text{We}=20$. To help its interpretation, in panel (a) two vertical segments are drawn for a specific instant, the lower one in $z$ belonging to the intact length of the jet and the upper one defining the limits of the most recently detached drop. All the previously detached drops have been erased for the sake of clarity. In the diagram (a), after the transient regime, eight breakup events (encircled) have been selected to calculate the mean breakup length and its associated dispersion. In diagram (b), examples of FLP and RLP are marked.

Figure 3

Numerical estimates of breakup lengths for different velocity perturbation profiles, all having the same flow rate. Breakups for front-ligament pinching (FLP) and rear-ligament pinching (RLP) are presented. The simulations are conducted for $w_0=0.154,\text{We}=14.8,\text{Oh}=0.01,$ and $k_{\mathrm{temp}}=0.697$. Deviations in the measured lengths over several breakup processes are estimated as $\pm 0.2$.

Figure 4

Dimensionless breakup length versus the amplitude of perturbation of the mean axial velocity, for $\mathrm{We}=14.8,\text{Oh}=0.01,$ and $k_{\mathrm{temp}}=0.697$, according to the linear theory with one mode and two modes and numerical simulations (front-ligament pinching, FLP, and rear-ligament pinching, RLP). Deviations in the measured lengths are smaller than the size of the symbols.

Figure 5

Dimensionless breakup length vs the dimensionless jet velocity, $\beta =\sqrt{\mathrm{We}}$. Symbols have the same meaning as in Fig. 4. Other parameters are fixed to $w_0=0.154,\text{Oh}=0.01,$ and $k_{\mathrm{temp}}=0.697$.

Figure 6

Amplitude of the dominant capillary mode per amplitude of velocity perturbation, $f_{\mathrm{d}}/w_0$, as a function of the temporal wave number $k_{\mathrm{temp}}=\omega /\beta$, for different values of the Weber and Ohnesorge numbers.

Figure 7

Breakup length as a function of the temporal wave number $k_{\mathrm{temp}}=\omega /\beta$, for different values of the amplitude of velocity perturbation, $w_0$. The remaining parameters are $\text{We}=14.8,\text{Oh}=0.01$. Lines are the linear theory predictions and symbols are numerical results for front and end pinching as in previous figures.

Figure 8

Breakup length as a function of the temporal wave number $k_{\mathrm{temp}}=\omega /\beta$, for different values of the Ohnesorge number. The remaining parameters are $\text{We}=14.8,w_0=0.154$. Lines are the linear theory predictions and symbols are numerical results for front and end pinching as in previous figures. Only the vertical dispersion bars exceeding the dimensions of the points are plotted.

Figure 9

Comparison between (i) numerical data and the linear theory extracted by Moallemi et al. [8], and (ii) numerical data and the two spatial linear theories (monomodal and bimodal) described in this work. As in Ref. [8], the dimensionless breakup length is represented vs $ln(2/3\delta )$, with $\delta$ being the mean axial velocity perturbation divided by the jet velocity, instead of $w_0$, which is made dimensionless with the capillary velocity.