Nonlinear parametric models of viscoelastic fluid flows*

Reduced-order models have been widely adopted in fluid mechanics, particularly in the context of Newtonian fluid flows. These models offer the ability to predict complex dynamics, such as instabilities and oscillations, at a considerably reduced computational cost. 

 In contrast, the reduced-order modeling of non-Newtonian viscoelastic fluid flows is relatively unstudied.  

 This work explores the use of the sparse identification of nonlinear dynamics (SINDy) algorithm to develop interpretable reduced-order models for a broad class of  viscoelastic flows. 

 In particular, we explore a benchmark oscillatory viscoelastic flow on the four-roll mill geometry using the classical Oldroyd-B fluid. 

 This flow exemplifies many canonical challenges associated with non-Newtonian flows, including transitions, asymmetries, instabilities, and bifurcations arising from the interplay of viscous and elastic forces, all of which require expensive computations to resolve fast timescales and long transients.  

 First, we demonstrate the effectiveness of our data-driven surrogate model in predicting the transient evolution on a simplified representation of the dynamical system. We then describe the ability of the reduced-order model to accurately reconstruct spatial flow field in a basis obtained via proper orthogonal decomposition. 

 Finally, we develop a fully parametric, nonlinear model that captures the dominant variations of the dynamics with the relevant nondimensional Weissenberg number. 

 This work illustrates the potential to reduce computational costs and improve design, optimization, and control of a large class of non-Newtonian fluid flows with modern machine learning and reduced-order modeling techniques.