#
Numerical Simulations of Polymer Solution Droplet Impact on Surfaces of Different Wettabilities^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Numerial Methods

**τ**

_{p}), and the governing equations can be written as follows:

_{0}impacting at an initial velocity of V

_{0}was simulated. To analyze droplet impact dynamics, we used the spreading factor, defined as the ratio between the spreading diameter and droplet initial diameter (D

_{0}). In addition, the time was made dimensionless using the kinematic time scale t

_{c}= D

_{0}/V

_{0}.

**A**was expressed as follows:

**A**).

**I**the identity tensor.

**V**

_{c}, describing the relative velocity at the free surface between the fluids, followed the equation below [33]:

_{f}and ${\varphi}_{f}$ represent cell surface area and mass flux, respectively, while the coefficient C

_{α}, set here to 1, defines the degree of compression at the interface. It is worth noting that the adoption of an interface compression scheme avoids the tedious geometrical reconstruction of the interface habitually done in implementation of the VOF method. In addition, the algebraic method used here can readily be extended to unstructured meshes. Further details of the numerical discretization schemes and techniques can be found in Reference [34].

_{cl}corresponds to the contact line velocity, was numerically approximated by taking the velocity within the first cell above the substrate. The effect of the surface hysteresis was accounted for by replacing the equilibrium contact angle θ

_{E}in Equation (9) by either the receding contact angle θ

_{R}or the advancing contact angle θ

_{A}according to the direction of the triple line velocity. By adopting this approach, our model became sensitive to the substrate hysteresis, which plays a significant role in simulating droplet impacts on surfaces with varied wettabilities. It is worth noting that unlike in Reference [36], the presented model did not rely heavily on experimental measurements of the dynamic contact angle evolution in order to match a given experiment.

^{−6}.

#### 2.2. Polymer Solution and Substrates Properties

## 3. Results

#### 3.1. Validation of Newtonian Droplet Impact Dynamics

#### 3.2. Impact of Polymer Solution Droplet

#### 3.2.1. Hydrophilic Surface

_{0}) function of the dimensionless time (t

_{c}= t V

_{0}/D

_{0}). We observed that during the kinematic phase (t

_{c}< 1), droplet spreading seemed to be independent of the fluid model (Figure 4). However, at subsequent phases of the two droplets’ spreading, we observed a much more marked difference between the two cases. Additionally, the viscoelastic liquid droplet displayed little oscillation while having a greater maximum spreading diameter, due to the dominance of the elastic normal stress over of the surface tension, which favored retraction. Interestingly, the liquid flowing out of the droplet center towards its periphery seemed to behave similarly to the filament thinning and stretching situation we reported in Reference [25].

#### 3.2.2. Superhydrophobic Surface

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Comparison between simulation and experiment of the dimensionless spreading diameter evolution of a droplet upon impact on a superhydrophobic substrate with hysteresis. VOF: volume of fluid.

**Figure 3.**A comparative simulation of a 1 mm diameter droplet impacting at 1 m/s on a hydrophilic substrate between (

**right**) a Newtonian solvent DEP and (

**left**) polymer solution, DEP + 2.5 wt. % PS.

**Figure 5.**Simulated comparison between (

**right**) the Newtonian solvent DEP and (

**left**) viscoelastic polymer solution, DEP + 2.5 wt. % PS, for the impact of a 1 mm diameter droplet at 1 m/s on the superhydrophobic substrate.

**Figure 6.**Impact of a 1 mm droplet of polymer solution, DEP + 2.5 wt. % PS, at 1 m/s on the superhydrophobic substrate.

**Figure 7.**Simulated spreading diameter of 1 mm droplets of a Newtonian (DEP) fluid and two polymer solutions, DEP + 0.25 wt. % and DEP + 2.5 wt. % PS, impacting at 1 m/s.

Liquids | Interfacial Surface Tension (mN/m) | Viscosity (mPa.s) |
---|---|---|

DEP | 37 | 14 |

DEP + 2.5% PS | 37 | 31 |

Substrate | Equilibrium CA (°) | Advancing CA (°) |
---|---|---|

Hydrophilic (H) | 74 | 90 |

Superhydrophobic (SH) | 154 | 162 |

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**MDPI and ACS Style**

Tembely, M.; Vadillo, D.; Soucemarianadin, A.; Dolatabadi, A.
Numerical Simulations of Polymer Solution Droplet Impact on Surfaces of Different Wettabilities. *Processes* **2019**, *7*, 798.
https://doi.org/10.3390/pr7110798

**AMA Style**

Tembely M, Vadillo D, Soucemarianadin A, Dolatabadi A.
Numerical Simulations of Polymer Solution Droplet Impact on Surfaces of Different Wettabilities. *Processes*. 2019; 7(11):798.
https://doi.org/10.3390/pr7110798

**Chicago/Turabian Style**

Tembely, Moussa, Damien Vadillo, Arthur Soucemarianadin, and Ali Dolatabadi.
2019. "Numerical Simulations of Polymer Solution Droplet Impact on Surfaces of Different Wettabilities" *Processes* 7, no. 11: 798.
https://doi.org/10.3390/pr7110798