# A numerical model based on the curvilinear coordinate system for the MAC method simplified

## TL;DR

This paper introduces a simplified MAC numerical method using curvilinear coordinates and spline-based edge geometry to efficiently simulate incompressible fluid flows without free surfaces, validated through standard benchmark problems.

## Contribution

The paper presents a novel simplified MAC method with curvilinear coordinates and spline-based geometry for improved fluid flow simulation accuracy and efficiency.

## Key findings

- Accurate simulation of parallel plates, cavity, and atherosclerosis flows.
- Results agree well with existing literature.
- Demonstrates the method's applicability to complex geometries.

## Abstract

In this paper we developed a numerical methodology to study some incompressible fluid flows without free surface, using the curvilinear coordinate system and whose edge geometry is constructed via parametrized spline. First, we discussed the representation of the Navier-Stokes and continuity equations on the curvilinear coordinate system, along with the auxiliary conditions. Then, we presented the numerical method -- a simplified version of MAC (\textit{Marker and Cell}) method -- along with the discretization of the governing equations, which is carried out using the finite differences method and the implementation of the FOU (\textit{First Order Upwind}) scheme. Finally, we applied the numerical methodology to the parallel plates problem, lid-driven cavity problem and atherosclerosis problem, and then we compare the results obtained with those presented in the literature. Keywords: finite differences, simplified MAC, curvilinear coordinates, parallel plates, did-driven cavity, atherosclerosis.