^{1,a)}, B. D. Todd

^{2,b)}and Stefan Luding

^{1,c)}

### Abstract

Non-equilibrium molecular dynamics simulations of an atomic fluid under shear flow, planar elongational flow, and a combination of shear and elongational flow are unified consistently with a tensorial model over a wide range of strain rates. A model is presented that predicts the pressure tensor for a non-Newtonian bulk fluid under a homogeneous planar flow field. The model provides a quantitative description of the strain-thinning viscosity, pressure dilatancy, deviatoric viscoelastic lagging, and out-of-flow-plane pressure anisotropy. The non-equilibrium pressure tensor is completely described through these four quantities and can be calculated as a function of the equilibrium material constants and the velocity gradient. This constitutive framework in terms of invariants of the pressure tensor departs from the conventional description that deals with an orientation-dependent description of shear stresses and normal stresses. The present model makes it possible to predict the full pressure tensor for a simple fluid under various types of flows without having to produce these flow types explicitly in a simulation or experiment.

I. INTRODUCTION

II. SIMULATION DETAILS

III. NEWTONIAN PRESSURE TENSOR

IV. RESULTS AND DISCUSSION

A. Conventional description and an existing model

B. A new tensorial description

1. Rotating the pressure tensor to its principal orientation

2. Description of the non-Newtonian pressure tensor

3. Constitutive model for non-Newtonian fluid

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Tensor methods
- 131.0
- Shear flows
- 44.0
- Eigenvalues
- 24.0
- Non Newtonian flows
- 18.0
- Viscosity
- 17.0

## Figures

NEMD simulation data of the shear stress (a) and first normal stress difference (b) for a fluid under shear (PCF), elongation (PEF), and planar mixed flow (PMF) compared to the prediction of the second-order fluid (SOF) model (ρ = 0.8442, T = 0.722).

NEMD simulation data of the shear stress (a) and first normal stress difference (b) for a fluid under shear (PCF), elongation (PEF), and planar mixed flow (PMF) compared to the prediction of the second-order fluid (SOF) model (ρ = 0.8442, T = 0.722).

Viscosity of a WCA fluid under shear (PCF), elongation (PEF), and mixed flow (PMF) at state point ρ = 0.8442, T = 0.722. The data are fitted with a Carreau model, and the equilibrium viscosity η0 and a kinetic theory prediction are shown at s = 0. The inset contains the same data as a semi-log plot.

Viscosity of a WCA fluid under shear (PCF), elongation (PEF), and mixed flow (PMF) at state point ρ = 0.8442, T = 0.722. The data are fitted with a Carreau model, and the equilibrium viscosity η0 and a kinetic theory prediction are shown at s = 0. The inset contains the same data as a semi-log plot.

Pressure (a) and eigenvalues of the deviatoric pressure tensor (b) as a function of the magnitude of the strain rate tensor (ρ = 0.8442, T = 0.722). The inset in (a) gives the proportionality between p and s 3/2 in a logarithmic graph, error bars give the standard error and the dashed line is the fit of the pressure dilatancy.

Pressure (a) and eigenvalues of the deviatoric pressure tensor (b) as a function of the magnitude of the strain rate tensor (ρ = 0.8442, T = 0.722). The inset in (a) gives the proportionality between p and s 3/2 in a logarithmic graph, error bars give the standard error and the dashed line is the fit of the pressure dilatancy.

The parameter a as a function of the magnitude of the strain rate tensor (ρ = 0.8442, T = 0.722). The inset shows the normalized anisotropy of the pressure tensor.

The parameter a as a function of the magnitude of the strain rate tensor (ρ = 0.8442, T = 0.722). The inset shows the normalized anisotropy of the pressure tensor.

Principal orientation angles (a) as a function of the magnitude of the strain rate tensor (ρ = 0.8442, T = 0.722), the inset shows the lag angle. In (b), the lag angle is scaled by the vorticity ω, such that the data collapse onto a single profile.

Principal orientation angles (a) as a function of the magnitude of the strain rate tensor (ρ = 0.8442, T = 0.722), the inset shows the lag angle. In (b), the lag angle is scaled by the vorticity ω, such that the data collapse onto a single profile.

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