Learning unknown physics of non-Newtonian fluids

We present a formulation of the physics-informed neural network (PINN) method for learning the effective viscosity of the generalized Newtonian fluid from measurements of velocity and pressure in time-dependent three-dimensional flows and apply it to estimating viscosity models of two non-Newtonian systems (polymer melts and suspensions of particles) in shear flow between two parallel plates using only velocity measurements from numerical simulations. The PINN-inferred viscosity models agree with empirical models for shear rates with large absolute values but deviate for shear rates near zero where empirical models have an unphysical singularity. We show that once the unknown physics is learned the PINN method can be used to solve the momentum conservation equation governing flow of non-Newtonian fluids.

Figure 1

Inverse and forward PINN solutions for the synthetic data generated from the analytical solution for a power-law fluid with $n=0.898$. (a) The average velocity profile from 100 runs for the PINN results and one run for the PINN forward model. (b) The average $\frac{du}{dy}$ of the 100 runs versus the average $\mu$. The width of the gray area corresponds to four standard deviations of $\mu$. (c) Average error in satisfying the ODE.

Figure 2

Inverse and forward PINN model results for polymer chains of length $N=2$. (a) Resulting velocity profiles. (b) Resulting viscosity profile. (c) Error in satisfying the ODE. Simulation data are from [17].

Figure 3

Inverse PINN results for polymer chains of length $N=25$. (a) Resulting velocity profiles. (b) Resulting viscosity profile. (c) Error in satisfying the ODE. Simulation data are from [17].

Figure 4

Inverse PINN results for suspensions with average volume fraction ${\varphi }_a=0.2$ (a)–(c) and 0.4 (d)–(f). Panels (c) and (f) show the residuals of Eq. (10) for the two ${\varphi }_a$ values after the PINN-estimated velocity and viscosity are substituted into it. The inverse PINN model is trained with three hidden layers that each have 100 nodes. In the direct suspension simulations, $\mu \left(u_y\right)$ is estimated using the Eilers model and $u\left(y\right)$ and $\varphi \left(y\right)$ measurements.

Figure 5

(a) Final suspension local volume fraction profiles $\varphi \left(y\right)$ in steady state. (b) Suspension viscosities learned from the PINN model as a function of the local volume fraction. Results are compared with the Eilers [23, 24], Krieger [25], and Boyer et al. [26] models fitted to the data. Filled symbols represent points that occur in the range $0h\le y\le 0.85h$, and empty symbols are in the range $0.85h\le y\le h$, to denote the deviations that occur from the theoretical values in the middle of the channel.

Figure 6

Viscosity of (left) the polymer melt with $N=2$ and (right) suspension with $\varphi =0.2$ obtained from the PINN method, direct numerical simulations (denoted as “Simulation”), and Eq. (15) (denoted as “$\mu$ Eqn.”).