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Uncertainty Quantification at the Molecular–Continuum Model Interface ^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

## 3. Problem Description

#### 3.1. Viscosity Calculation with Non-Equilibrium Molecular Dynamics (NEMD)

`mdFoam`solver [35,36] developed in OpenFOAM [37]. A popular TIP4P/2005 water model was chosen for the analysis, as it gives the best prediction of the viscosity for the complete range of temperatures. As previously mentioned, the value of the potential well is assumed to be uncertain, $\u03f5\left(\xi \right)$, where $\xi \sim \mathcal{U}({\u03f5}_{\mathrm{min}},{\u03f5}_{\mathrm{max}})$; the support of the distribution is based on the force-fields of other water models from the TIP4P family, ${\u03f5}_{\mathrm{min}}\approx 0.84\times {\u03f5}_{OO}$ from TIP4P [38] and ${\u03f5}_{\mathrm{max}}\approx 1.14\times {\u03f5}_{OO}$ from TIP4P/Ice [39], where ${\u03f5}_{OO}=1.287\times {10}^{-21}$ J and ${\sigma}_{OO}=3.1589$ Å are the values of the TIP4P/2005 model introduced by Abascal and Vega [28]. The solid–liquid interaction is modelled following the common Lorentz–Berthelot mixing rule [40], based on the LJ potential of silicon from Ritos et al. [41].

#### 3.2. Non-Intrusive Uncertainty Quantification in Multi-Scale Modelling

## 4. Results and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CFD | Computational fluid dynamics |

LJ | Lennard-Jones |

MD | Molecular dynamics |

NEMD | Non-equilibrium molecular dynamics |

NISP | Non-intrusive spectral projection |

PC | Polynomial chaos |

Probability density function | |

UQ | Uncertainty quantification |

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**Figure 1.**Schematic representation of viscosity measurement in non-equilibrium molecular dynamics (NEMD) modelling.

**Figure 2.**Workflow for uncertainty quantification in multi-scale simulation. QoI: quantity of interest.

**Figure 3.**Statistics derived from surrogate MD model. (

**a**) Expected value; (

**b**) Standard deviation. UQ: uncertainty quantification.

**Figure 5.**(

**a**) PDFs as a function of streamwise position. PDF for each x is normalised with respect to its maximum value; (

**b**) PDFs for three selected positions.

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**MDPI and ACS Style**

Zimoń, M.J.; Sawko, R.; Emerson, D.R.; Thompson, C.
Uncertainty Quantification at the Molecular–Continuum Model Interface . *Fluids* **2017**, *2*, 12.
https://doi.org/10.3390/fluids2010012

**AMA Style**

Zimoń MJ, Sawko R, Emerson DR, Thompson C.
Uncertainty Quantification at the Molecular–Continuum Model Interface . *Fluids*. 2017; 2(1):12.
https://doi.org/10.3390/fluids2010012

**Chicago/Turabian Style**

Zimoń, Małgorzata J., Robert Sawko, David R. Emerson, and Christopher Thompson.
2017. "Uncertainty Quantification at the Molecular–Continuum Model Interface " *Fluids* 2, no. 1: 12.
https://doi.org/10.3390/fluids2010012