2.13 Incompressible ﬂow
We can derive a set of equations for incompressible ﬂow that includes:
- mass conservation, Eq. (2.8 );
- momentum conservation, Eq. (2.19 );
- the material derivative for , Eq. (2.26 );
- the rate of deformation tensor, Eq. (2.33 );
- the Newtonian ﬂuid model, Eq. (2.41 ).
Combining Eq. (2.41 ) and Eq. (2.33 ) and using Eq. (2.36 ) for the deviatoric part of a tensor, gives an expression for stress:
An incompressible ﬂuid exhibits constant over time, i.e. for moving volumes of ﬂuid Combining the material derivative Eq. (2.14 ) and mass conservation Eq. (2.8 ) gives which results in the incompressibility condition
A homogeneous, incompressible material exhibits = constant uniformly throughout the entire ﬂuid. With that assumption, Eq. (2.45 ) can be written as
The identity and the incompressibility Eq. (2.46 ) yield the terms in Eq. (2.47 ).
Mass and momentum conservation, represented by Eq. (2.46 ) and Eq. (2.47 ) respectively, provide two equations — one scalar, one vector — for two ﬁelds, and . However, Eq. (2.46 ) cannot be solved in its own right since it provides only one equation for vector , containing 3 components.
A scalar equation, including both and , can be derived by taking the divergence of Eq. (2.47 ) and eliminating terms by substituting Eq. (2.46 ), noting that . For and that are constant and uniform, the equation is