Living Reference Work Entry

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

pp 1-70

Date: Latest Version

Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

  • Josef MálekAffiliated withFaculty of Mathematics and Physics, Mathematical Institute, Charles University in Prague Email author 
  • , Vít PrůšaAffiliated withFaculty of Mathematics and Physics, Mathematical Institute, Charles University in Prague


The chapter starts with overview of the derivation of the balance equations for mass, momentum, angular momentum, and total energy, which is followed by a detailed discussion of the concept of entropy and entropy production. While the balance laws are universal for any continuous medium, the particular behavior of the material of interest must be described by an extra set of material-specific equations. These equations relating, for example, the Cauchy stress tensor and the kinematical quantities are called the constitutive relations. The core part of the chapter is devoted to the presentation of a modern thermodynamically based phenomenological theory of constitutive relations. The key feature of the theory is that the constitutive relations stem from the choice of two scalar quantities, the internal energy and the entropy production. This is tantamount to the proposition that the material behavior is fully characterized by the way it stores the energy and produces the entropy. The general theory is documented by several examples of increasing complexity. It is shown how to derive the constitutive relations for compressible and incompressible viscous heat-conducting fluids (Navier–Stokes–Fourier fluid), Korteweg fluids, and compressible and incompressible heat-conducting viscoelastic fluids (Oldroyd-B and Maxwell fluid).


Continuum mechanics, Constitutive relations, Thermodynamics