Machine-learning-based non-Newtonian fluid model with molecular fidelity

We introduce a machine-learning-based framework for constructing continuum a non-Newtonian fluid dynamics model directly from a microscale description. Dumbbell polymer solutions are used as examples to demonstrate the essential ideas. To faithfully retain molecular fidelity, we establish a micro-macro correspondence via a set of encoders for the microscale polymer configurations and their macroscale counterparts, a set of nonlinear conformation tensors. The dynamics of these conformation tensors can be derived from the microscale model, and the relevant terms can be parametrized using machine learning. The final model, named the deep non-Newtonian model ($\mathrm{Dee}\mathrm{P}{\mathrm{N}}^2$), takes the form of conventional non-Newtonian fluid dynamics models, with a generalized form of the objective tensor derivative that retains the microscale interpretations. Both the formulation of the dynamic equation and the neural network representation rigorously preserve the rotational invariance, which ensures the admissibility of the constructed model. Numerical results demonstrate the accuracy of $\mathrm{Dee}\mathrm{P}{\mathrm{N}}^2$ where models based on empirical closures show limitations.

Figure 1

Quasiequilibrium relaxation process of a dumbbell suspension obtained from direct MD simulation, the present $\mathrm{Dee}\mathrm{P}{\mathrm{N}}^2$, canonical Hookean, and FENE-P model. Left: Encoder function $g\left(r\right)$; middle: evolution of ${\mathit{\tau }}_{\mathrm{p}}$; right: evolution of ${\mathbf{c}}_1=〈\mathbf{r}{\mathbf{r}}^T〉$. Two parameter sets of the FENE-P model are examined: (1) the initial conditions of both ${\mathbf{c}}_1$ and ${\mathit{\tau }}_{\mathrm{p}}$ are consistent with MD and (2) the initial and final conditions of ${\mathbf{c}}_1$ are consistent with MD. The Hookean model parameters are set following (1).

Figure 2

Evolution of the reverse Poiseuille flow of a dumbbell suspension obtained from MD and various models. Left: velocity profiles at $t=20,60,220$; right: velocity evolution at $y=6$ and $y=14$. The parameters of the Hookean and FENE-P model are chosen such that the equilibrium bond length matches the MD results.

Figure 3

The micro-macro correspondence during the evolution of the reverse Poiseuille flow of the dumbbell suspension presented in Fig. 2 with the same line scheme. (a) Evolution of ${\mathbf{c}}_1$ at $y=6$; (b, c) Normal stress difference ${{\mathit{\tau }}_{\mathrm{p}}}_{xx}-{{\mathit{\tau }}_{\mathrm{p}}}_{yy}$ and shear stress ${{\mathit{\tau }}_{\mathrm{p}}}_{xy}$ at $y=6$ (upper lines) and $y=14$ (lower lines). (d) Shear-rate-dependent viscosity. The predictions by Hookean model show large deviations from the MD results and are not shown in (a), (c), and (d) for visualization purposes.

Figure 4

The effectiveness of the objective tensor derivative constructed by (15). The additional source term plays a vital role in the accurate modeling of the fluid systems. The model that uses the canonical upper-convected derivative shows apparent deviations from the MD results for the evolution of the velocities (left) and ${\mathbf{c}}_1$ (right) at $y=6$.

Figure 5

Comparison of the time evolution of reverse Poiseuille flow of a three-bead suspension obtained from direction MD simulation, $\mathrm{Dee}\mathrm{P}{\mathrm{N}}^2$, and Hookean model. The parameters of the Hookean model are chosen such that the polymer average bond length matches the MD results. Left: Velocity at $y=6$ and 14; right: average value of $cos\theta$ at $y=6$.