On the modeling of inhomogeneous incompressible fluid-like bodies

We use a thermodynamic framework that has been recently put into place to describe the dissipative response of materials to develop a model for the response of inhomogeneous incompressible fluid-like bodies whose stored energy depends on gradients of the density. Without appealing to additional balance laws, or concepts such as interstitial working, we are able to obtain models for such bodies that do not need boundary conditions for the density in order to have well posed governing equations. After obtaining a model for the response of inhomogeneous incompressible fluid-like body, we solve a simple boundary value problem and introduce the notion of (weak) solutions to the equations governing the flow of the body that has been developed to illustrate the issues concerning boundary conditions.