Liquid bridge length scale based nondimensional groups for mapping transitions between regimes in capillary break-up experiments

Criteria to identify transitions between dynamic self-similar linear thinning regimes of liquid bridges are of utmost importance in order to accurately interpret results in capillary break-up rheometry. Currently available criteria encompass many experimental difficulties or rely on numerical approaches. Here, we introduce a different set of nondimensional groups, ${\text{Oh}}_{\text{L}}={\eta }_{\text{in}}/\sqrt{\gamma \rho L}$ and $a=R/L$, based on the experimentally relevant axial length scale of a liquid bridge $L$, for viscous-dominated fluids undergoing capillary break-up in air. This framework is further extended to encompass the effect of outer viscous fluids. As a result, we present a two-dimensional operating map in which the boundaries are set by fluid properties and a single geometrical parameter, related to the experimental configuration. This approach establishes guidelines to correctly interpret experimental data and identify transitions in capillary break-up experiments of liquid bridges surrounded by fluids of different viscosities.

Figure 1

(a) High-resolution images from CaBER experiments performed with surrounding immiscible outer liquids. The schematics highlights the differences in the expected inner and outer flow profiles for $p=18.7\gg 1$ and $p=0.002\ll 1$, respectively. (b) 2D operating map as a function of the dimensionless groups ${\text{Oh}}_{\text{L}}$ and $p$. The derived criteria are used to delimit the areas within which specific regimes and transitions can be experimentally detected. Circled green letters correspond to the capillary break-up experiments for various $L$, and inner and outer fluid viscosities of Fig. 2.

Figure 2

Minimum radius evolution ($R_{\text{min}}$) in time, with $t_{\text{b}}$ as the break-up time, for several fluids and geometrical conditions in CaBER experiments performed in air [(a), (b) (open circles)] and with an outer liquid [(b) (open squares), (c)]. Solid dots represent calculated $R_{\text{crit}}$. Thinning curves in A are shifted 10 ms apart for clarity. (a) Experimentally detectable regimes are determined by the set geometrical conditions. For the same fluid (${\eta }_{\text{in}}=0.365\text{Pa}\text{s}, \gamma =65\text{mN}\text{m}$), $R_{\text{crit}}$ via Eq. (2) shifts to larger radii and the experimental window ($R_{\text{up}}-R_{\text{res}}$) expands as the geometrical parameter $L$ increases (right to left: 480, 600, and $1000\mu \text{m}$). Consistently, the thinning dynamics appear to be solely V dominated for the lowest $L$ (open squares), whereas a region following the IV scaling is clearly distinguishable for the highest $L$ (open triangles). $R_{\text{crit}}$ predicted with Eq. (2) matches the deviation of the data from the predicted V scaling ($H=0.0709$, red solid line) towards the IV scaling ($H=0.0304$, orange dashed-dotted line). (b) Thinning curves of the same inner fluid (${\eta }_{\text{in}}=0.518\text{Pa}\text{s}, \gamma =63.7\text{mN}\text{m}$ and $30.0\text{mN}\text{m}$ for D and E, respectively) at comparable geometrical conditions show different behaviours depending on the outer medium viscosity (open circles $p=34435$, open squares $p=96$). For $p\gg 1$, the transition from V ($H=0.0709$, red solid line) to LV regime [$H\left(p\right)=0.02$ [31], blue dashed line] is detectable only if ${\eta }_{\text{out}}$ is sufficiently high to fulfill the condition $p_{\text{res}}\ll p\ll p_{\text{up}}$. (c) For $p=0.002, R_{\text{crit}}$ according to Eq. (4) correspond to the deviation from LSF ($H=0.5$, blue solid line) to the LV regime [$H\left(\text{p}\right)=0.04$ [10], blue dashed line].

Figure 3

(a) Capillary thinning evolution of a Newtonian fluid with ${\eta }_{\text{in}}=0.365\text{mPa}\text{s}$ in air and with $L_{\text{max}}=900\mu \text{m}$. $R_{\text{crit}}$ calculated from the linear evolution of $u$ (black solid dot, high-resolution image in the inset) overestimates the V to IV transition, which is instead correctly captured by $R_{\text{crit}}$ calculated incorporating the corrective factor $C$ in Eq. (2) (black solid star). The enlargement of the inset shows the visible difference between the ideal cylinder geometry (black dashed lines) and the actual profile (red solid lines). (b) Temporal evolution of $R\left(\text{z}\right)$ and $u_{\text{a}}\text{(z)}$ in a capillary break-up experiment of a glycerol-water mixture in air (${\eta }_{\text{in}}=0.365\text{mPa}\text{s}$) with $L_{\text{max}}=900\mu \text{m}$ ($-18.33\text{ms}<t-t_{\text{b}}<-6.33\text{ms}$ at $\mathrm{\Delta }t=0.66\text{ms}$). Inset: The deviation of the actual profile from the ideal cylinder geometry implicates a large difference at $L_{\text{max}}$ between $u_{\text{a}}=0.15\text{m/s}$ (red solid line) and $u=0.22\mathrm{m}/\mathrm{s}$ (black dashed line) calculated assuming a linear evolution of the axial velocity. (c) Direct comparison between the temporal evolution of $R_{\text{min}}$ (open symbols) and $L_{\text{max}}$ (solid symbols), determined from the experimental $u_{\text{a}}$ profiles. The characteristic length remains approximately constant throughout the complete bridge evolution for case A of Fig. 2 (top, red squares). Also for cases B (middle, orange circles) and E (bottom, blue triangles) $L_{\text{max}}$ remains constant in the V regime, up to the transitions to the asymmetrical thinning regimes (IV or LV).