# The Natural Breakup Length of a Steady Capillary Jet: Application to Serial Femtosecond Crystallography

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## Abstract

**:**

## 1. Introduction

## 2. The Breakup Length of Both Ballistic and Flow-Focused Jets

#### Linear Global Instability vs. Nonlinear Breakup Length

## 3. Scaling Law

#### A Simplified Approach

**v**is the liquid velocity vector) implies that the axial distance ${l}_{z}$ from the breakup point along which the jet velocity ${V}_{j}$ is perturbed (Figure 2) verifies ${v}_{z1}/{l}_{z}\sim {v}_{r1}/{D}_{j}$, where ${v}_{r1}$ and ${v}_{z1}$ represent the radial and axial perturbation velocities, respectively. Due to the convective character [36] of the jetting regime, the perturbation produced by the breakup travels only a few few jet diameters in the upstream direction, i.e., ${l}_{z}\sim {D}_{j}$, which implies that ${v}_{r1}\sim {v}_{z1}$. This is so because ${V}_{j}$ is nearly constant around the breakup region. This means that the perturbation radial and axial kinetic energies are commensurate with each other, i.e., $\rho {v}_{r1}^{2}\sim \rho {v}_{z1}^{2}$, and the radial and axial viscous stresses are also of the same order of magnitude, $\mu {v}_{r1}/{D}_{j}\sim \mu {v}_{z1}/{l}_{z}$. These results allow us to retain only the radial kinetic energy and viscous stress in the balance of energy described above.

## 4. Experimental Validation

^{®}water) was injected with a precision syringe pump into the air at the atmospheric pressure through an orifice of diameter $D=250$ $\mathsf{\mu}$m. The jet length was determined using a long-time exposure imaging (Canon EOS 2000D) with a telecentric lens (e.g., 0.9× CobaltTL from Edmund Scientific (Barrington, NJ, USA), plus C-Mount adaptor) to avoid spatial distortion. We assumed that the jet ends at the steady position of the sharp transition from the visually undisturbed jet to the larger diameter blurred region (droplets).

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Ballistic jet issued from an orifice with approximately the same diameter as that of the jet. L represents the axial length of the cylindrical ejector (it may be either the thickness of a plate or the length of a tube). (

**b**) Flow-focused capillary jet emitted from an orifice of diameter D. H is the distance from the front of feeding capillary tube with inner diameter ${D}_{1}$ to the inner face of the plate with thickness L where the orifice of diameter D is made. In flow focusing, $\Delta P$ is the pressure drop along the gas streamlines in the discharge orifice. Note that the diameter ${\mathcal{D}}_{G}\equiv {\left[8\rho {Q}^{2}/({\pi}^{2}\Delta P)\right]}^{1/4}$ is actually not a geometric parameter, since it is obtained from the operating conditions (see text) and cannot be marked in the figure. Despite this, it is assumed equivalent to the diameter $2R$ of the ejector (either a tube or a plate) in ballistic jets.

**Figure 3.**Distance ${L}_{j}\left(t\right)$ of the jet front position from the feeding tube exit as a function of time for different levels $\ell =$10, 11, and 13 of spatial discretization ($\mathrm{Basilisk}$, [34]). The time is measured in terms of the capillary time ${t}_{0}={(\rho {R}^{3}/\sigma )}^{1/2}$, where R is the tube radius. The density and viscosity of the gas environment are 1000 and 100 times smaller than those of the liquid domain, respectively.

**Figure 4.**The critical jet length ${L}^{*}$ obtained from the global instability analysis and compared to the average intact jet length L. The results were obtained as a function of the flow rate Q for a water ballistic capillary jet issuing from a PFA tube with inner diameter ${D}_{1}=0.25$ mm.

**Figure 5.**(

**a**) A short water microjet emitted with $Q=8.2$ $\mathsf{\mu}$L/min and ${G}_{o}=10.4$ mg/min. The focused capillary meniscus from whose apex the jet issues can be observed through the translucent plastic nozzle. The jet length is approximately 105 $\mathsf{\mu}$m. (

**b**) A long microjet of a mixture of water/glycerol (20/80 v/v %) emitted with $Q=20$ $\mathsf{\mu}$L/min and ${G}_{o}=7$ mg/min. In the two experiments, the jet was emitted with Ejector 4.

**Figure 6.**Values of We and Ca in our experiments. The grey and blue symbols correspond to ballistic and flow-focused jets, respectively. The red line corresponds to the predicted convective-to-absolute instability transition for a cylindrical capillary jet in a vacuum [39]. The black line is a guide to the eye. The upper cloud of points corresponds to the liquid with the highest viscosity.

**Figure 7.**Breakup length ${L}_{j}/{d}_{\sigma}$ as a function of $\zeta $ obtained from both experiments and numerical simulations (symbols) and calculated from (4) (line). The legend indicates the discharge orifice dimensions, the liquid, and the environment (vacuum or air). The legend also indicates the point corresponding to the nanojet [37]. The inset shows ${L}_{j}$ as a function of Q. The properties of the liquids used are given in Table 1.

**Figure 8.**Probability density function of the logarithmic errors around the scaling law (4) for ${\alpha}_{\rho}=15.015$ and ${\alpha}_{\mu}=0.53$ (symbols), and the corresponding normal distribution with zero average and variance equal to 0.0225 (line).

Liquid | $\mathit{\rho}$ (kg·m${}^{-3}$) | $\mathit{\sigma}$ (N·m${}^{-1}$) | $\mathit{\mu}$ (Pa·s) |
---|---|---|---|

water (22 ${}^{\circ}$C) | 1000 | 0.072 | 0.001 |

water/ethanol (65/35 v/v %) (20 ${}^{\circ}$C) | 943 | 0.035 | 0.0026 |

ethanol (22 ${}^{\circ}$C) | 795 | 0.023 | 0.00125 |

water/glycerol (20/80 v/v %) (22 ${}^{\circ}$C) | 1217 | 0.065 | 0.0914 |

Ejector | Orifice Shape | Dimensions ($\mathsf{\mu}$m) | ${\mathit{D}}_{1}$ ($\mathsf{\mu}$m) |
---|---|---|---|

1 | slit | 15×45 | 30 |

2 | slit | 20×60 | 30 |

3 | round | 30 | 30 |

4 | round | 50 | 50 |

5 | round | 75 | 75 |

6 | round | 70 | 100 |

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**MDPI and ACS Style**

Gañán-Calvo, A.M.; Chapman, H.N.; Heymann, M.; Wiedorn, M.O.; Knoska, J.; Gañán-Riesco, B.; López-Herrera, J.M.; Cruz-Mazo, F.; Herrada, M.A.; Montanero, J.M.; Bajt, S. The Natural Breakup Length of a Steady Capillary Jet: Application to Serial Femtosecond Crystallography. *Crystals* **2021**, *11*, 990.
https://doi.org/10.3390/cryst11080990

**AMA Style**

Gañán-Calvo AM, Chapman HN, Heymann M, Wiedorn MO, Knoska J, Gañán-Riesco B, López-Herrera JM, Cruz-Mazo F, Herrada MA, Montanero JM, Bajt S. The Natural Breakup Length of a Steady Capillary Jet: Application to Serial Femtosecond Crystallography. *Crystals*. 2021; 11(8):990.
https://doi.org/10.3390/cryst11080990

**Chicago/Turabian Style**

Gañán-Calvo, Alfonso M., Henry N. Chapman, Michael Heymann, Max O. Wiedorn, Juraj Knoska, Braulio Gañán-Riesco, José M. López-Herrera, Francisco Cruz-Mazo, Miguel A. Herrada, José M. Montanero, and Saša Bajt. 2021. "The Natural Breakup Length of a Steady Capillary Jet: Application to Serial Femtosecond Crystallography" *Crystals* 11, no. 8: 990.
https://doi.org/10.3390/cryst11080990