Constitutive relation: overview

A constitutive relation is an equation expressing the relation between two physical quantities (usually one kinetic and the other kinematic) with reference to a material. For example, Hooke’s law F = -kx states that the force needed to compress a particular spring by a certain distance varies linearly with that distance.

For an incompressible, viscous Newtonian fluid, George Stokes proposed that the constitutive relation must satisfy the following hypotheses:

  1. The stress tensor of a fluid at rest is the pressure exerted by the fluid (hydrostatic).
  2. The shear stress tensor is a linear function of the deformation tensor (velocity gradient).
  3. The shear stress tensor is zero if the flow involves no shearing of the fluid-body.
  4. The fluid is isotropic.


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