Two-dimensional equilibrium configurations in Korteweg fluids

TL;DR

The article tackles equilibrium configurations of two-dimensional Korteweg fluids of third grade by deriving a single nonlinear elliptic PDE from the mechanical equilibrium condition using thermodynamically consistent constitutive relations obtained via an extended Liu procedure. It analyzes both linear and nonlinear reductions and demonstrates numerical solutions of a Dirichlet boundary-value problem, producing density fields and contour plots as proof of concept. The key contribution is reducing an overdetermined 2D equilibrium system to a tractable scalar PDE and validating it with preliminary computations, which lays groundwork for higher-dimensional studies and experimental comparison. This work advances understanding of interface and capillarity effects in nonlocal fluid models and points to future investigations in 3D equilibria, stability, and temperature-field effects.

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.

Figures (2)

• Figure 1: Plot of the density $\rho$ (left) and contour plot (right). The values of the parameters are (from the top): ($m=1, \gamma=1$), ($m=1, \gamma=-1$), ($m=-1, \gamma=1$), ($m=-1, \gamma=-1$).
• Figure 2: Plot of the density $\rho$ (left) and contour plot (right). The values of the parameters are (from the top): ($m=1, n=-2$), ($m=1, n=-3$), ($m=1, n=1$), ($m=1,n=0$).