# Constitutive equations in fluid mechanics

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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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