Energetic variational approaches for non-Newtonian fluid systems

We consider the dominant equations for the motion of the non-Newtonian fluid in a domain from an energetic point of view. We apply an energetic variational approach and the first law of thermodynamics to derive the generalized compressible non-Newtonian fluid system. We also derive the generalized incompressible non-Newtonian fluid system by using an energetic variational approach. Moreover, we investigate the enthalpy, entropy, free energy, and conservation laws of our compressible fluid system. The key idea of making our mathematical models for non-Newtonian fluid flow is that we use the Riemannian metric induced by a flow map in the domain to study our energy densities for non-Newtonian fluid.