Constitutive equations describe the behaviour of a material subjected to certain loading conditions.
Constitutive equations in fluid mechanics:
Surface stresses (\sigma) on an element arise from a combination of viscous friction and pressure p which are described by the constitutive relations:
\sigma_{xx} = p + \lambda \nabla \cdot \mathbf{v} + 2\mu \frac{\partial u }{\partial x} , \quad \sigma_{xy} = \mu \left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right)
\sigma_{yy} = p + \lambda \nabla \cdot \mathbf{v} + 2\mu \frac{\partial v}{\partial y} , \quad \sigma_{yz} = \mu \left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \right)
\sigma_{zz} = p + \lambda \nabla \cdot \mathbf{v} + 2\mu \frac{\partial w}{\partial z} , \quad \sigma_{xz} = \mu \left(\frac{\partial u}{\partial y} + \frac{\partial w}{\partial x} \right)
where \mu and \lambda are the coefficients of dynamic and bulk viscosity respectively.
The expressions given above assumes that the relationship between stress and velocity gradients is

isotropic (properties of the fluid have no preferred direction)

linear (valid for Newtonian fluids)
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