# Capillary Thinning of Viscoelastic Threads of Unentangled Polymer Solutions

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## Abstract

**:**

## 1. Introduction

## 2. Capillary Thinning for the Oldroyd-B Model

## 3. Finite Extensibility Effects

#### 3.1. FENE-P Model

#### 3.2. Equations of Motion in the Entrance Zone

#### 3.3. Flow in the Thread and the Thinning Law

## 4. Discussion

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A thinning filament of radius ${a}_{0}$ and length ${L}_{f}$ connecting two semi-spherical droplets (a cross-section along the main axis is shown here); ${z}_{m}$ corresponds to the middle of the thread. ${L}_{t}\ll {L}_{f}$ is the size of the filament/droplet transition regions (shown in cyan).

**Figure 2.**The thread/droplet transition region; $r=a\left(z\right)$ defines the free surface in cylindrical coordinates $(z,r,\theta )$; ${a}_{0}$ and ${v}_{0}$ are the thread radius and the flow velocity at the end of the uniform thread region. Note that a longitudinal cross-section including the cylindrical symmetry axis $(z$) is shown here. Thus, the upper curve corresponds to the cylindrical angle $\theta =0$, the lower curve to $\theta ={180}^{\circ}$, and the r-axis coincides with the Cartesian ${x}_{1}$-axis. The inset clarifies the cylindrical coordinates used here.

**Figure 4.**Coordinates u and ${u}_{\perp}$ attached to point O of a generic streamline shown as thick curve. Note that some actual streamlines are depicted in Figure 3.

**Figure 7.**Time-dependence of the thread radius: ${a}_{0}$ (upper solid curve) and $ln{a}_{0}$ (lower curve) vs. $t/\tau $. The dashed line indicates the asymptotic exponential law, Equation (12).

**Figure 8.**The dependence $a\left(z\right)/{l}_{e}$ vs. $z/{l}_{e}$ in the transition zone: our data (crosses), the similarity solution for the bead-string structure obtained using a neo-Hookean elastic model [12] (solid line); ${l}_{e}={2}^{1/3}{a}_{0}$ is the elasto-capillary length.

**Figure 9.**The time dependence of the thread thickness $d=2{a}_{0}$. Red solid curves: theoretical results (cf. Section 3.3). Thick black curves: experimental data for semidilute aqueous solutions of Praestol-2540 with different concentrations from 125 to 1000 ppm (cf. Figure 5 of ref. [18]). Red circles indicate the onset of significant deviations between theoretical and experimental curves. The theoretical breakup points are indicated with red arrows for each red curve. For the two highest concentrations, we also indicate the reduced time $t/\tau $ (with $t=0$ corresponding to the theoretical breakup). Thin red dashed lines indicate the exponential asymptotic behavior at short times (according to Equation (12)).

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**MDPI and ACS Style**

Semenov, A.; Nyrkova, I.
Capillary Thinning of Viscoelastic Threads of Unentangled Polymer Solutions. *Polymers* **2022**, *14*, 4420.
https://doi.org/10.3390/polym14204420

**AMA Style**

Semenov A, Nyrkova I.
Capillary Thinning of Viscoelastic Threads of Unentangled Polymer Solutions. *Polymers*. 2022; 14(20):4420.
https://doi.org/10.3390/polym14204420

**Chicago/Turabian Style**

Semenov, Alexander, and Irina Nyrkova.
2022. "Capillary Thinning of Viscoelastic Threads of Unentangled Polymer Solutions" *Polymers* 14, no. 20: 4420.
https://doi.org/10.3390/polym14204420