# Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature

## TL;DR

This paper develops a reduced basis model for Navier-Stokes equations in curved channels, enabling efficient computation of steady-state solutions across various geometries and viscosities, with applications to blood flow symmetry breaking.

## Contribution

It introduces an empirical interpolation-based reduced order model for Navier-Stokes in curved domains, demonstrating accurate and efficient solutions for parameter variations.

## Key findings

- Reduced order model accurately predicts steady states.
- Critical Reynolds number increases with wall curvature.
- Model applicable to blood flow symmetry studies.

## Abstract

We consider the Navier-Stokes equations in a channel with a narrowing and walls of varying curvature. By applying the empirical interpolation method to generate an affine parameter dependency, the offline-online procedure can be used to compute reduced order solutions for parameter variations. The reduced order space is computed from the steady-state snapshot solutions by a standard POD procedure. The model is discretised with high-order spectral element ansatz functions, resulting in 4752 degrees of freedom. The proposed reduced order model produces accurate approximations of steady-state solutions for a wide range of geometries and kinematic viscosity values. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the valve shape. Through our computational study, we found that the critical Reynolds number for the symmetry breaking increases as the wall curvature increases.