# Fully non-homogeneous problem of two-dimensional second grade fluids

## TL;DR

This paper investigates the existence and uniqueness of solutions for a two-dimensional second grade fluid model with fully non-homogeneous boundary conditions, extending previous work by addressing boundary conditions on the curl of the velocity field.

## Contribution

It introduces new boundary conditions for the curl of the velocity, and develops a fixed-point approach to prove existence and uniqueness of solutions in this non-homogeneous setting.

## Key findings

- Existence of solutions established using fixed-point arguments.
- Uniqueness proved under additional boundary regularity conditions.
- Extended the analysis of second grade fluids to fully non-homogeneous boundary conditions.

## Abstract

This article studies the solutions of a two-dimensional grade-two fluid model with a fully non-homogeneous boundary condition for velocity u. Compared to problems with a homogeneous or tangential boundary condition, studied by many authors , we must add a boundary condition, otherwise the problem is no longer well-posed. We propose two conditions on z = curl (u -- $\alpha$$\Delta$u), which differ according to the regularity of z, on the portion of $\partial$$\Omega$ where $\alpha$u. n \textless{} 0. Following the approach of V. Girault and L.R. Scott in the tangential boundary case, we split the problem into a system with a generalized Stokes problem and a transport problem. But, compared to the study of these authors, we are now led to solve transport problems with boundary conditions. In two previous articles, we studied these transport problems. The results obtained in these articles allow us, by a fixed-point argument, to establish existence of the solutions for the fully non-homogeneous grade-two problem. Uniqueness requires the boundary condition with z in H 1 .