Two-dimensional equilibrium configurations in Korteweg fluids

TL;DR

This paper investigates equilibrium configurations in two-dimensional Korteweg fluids, deriving a nonlinear elliptic equation and exploring its analytical and numerical solutions for specific boundary conditions.

Contribution

It introduces a novel equilibrium equation for 2D Korteweg fluids derived via an extended Liu procedure, with analytical and numerical analysis.

Findings

Derived a nonlinear elliptic equation for equilibrium states

Provided analytical and numerical solutions for the equation

Presented preliminary numerical results for Dirichlet boundary conditions

Abstract

In this paper, after reviewing the form of the constitutive equations for a third grade Korteweg fluid, recently derived by means of an extended Liu procedure, an equilibrium problem is investigated. By considering a two--dimensional setting, it is derived a single nonlinear elliptic equation such that the equilibrium conditions are identically satisfied. Such an equation is discussed both analytically and numerically. Moreover, by considering a particular boundary value problem of Dirichlet type, some preliminary numerical solutions are presented.