Mathematically, this constraint implies that the material derivative (discussed below) of the density must vanish to ensure incompressible flow.
Mathematically, we can represent this constraint in terms of a surface integral: The negative sign in the above expression ensures that outward flow results in a decrease in the mass with respect to time, using the convention that the surface area vector points outward.
When we speak of the partial derivative of the density with respect to time, we refer to this rate of change within a control volume of fixed position.
This approach maintains generality, and not requiring that the partial time derivative of the density vanish illustrates that compressible fluids can still undergo incompressible flow.
What interests us is the change in density of a control volume that moves along with the flow velocity, u.
Now, we need the following relation about the total derivative of the density (where we apply the chain rule): So if we choose a control volume that is moving at the same rate as the fluid (i.e. (dx/dt, dy/dt, dz/dt) = u), then this expression simplifies to the material derivative: And so using the continuity equation derived above, we see that: A change in the density over time would imply that the fluid had either compressed or expanded (or that the mass contained in our constant volume, dV, had changed), which we have prohibited.
We must then require that the material derivative of the density vanishes, and equivalently (for non-zero density) so must the divergence of the flow velocity: And so beginning with the conservation of mass and the constraint that the density within a moving volume of fluid remains constant, it has been shown that an equivalent condition required for incompressible flow is that the divergence of the flow velocity vanishes.
In some fields, a measure of the incompressibility of a flow is the change in density as a result of the pressure variations.
But a solenoidal field, besides having a zero divergence, also has the additional connotation of having non-zero curl (i.e., rotational component).
As defined earlier, an incompressible (isochoric) flow is the one in which This is equivalent to saying that i.e. the material derivative of the density is zero.
For a flow to be accounted as bearing incompressibility, the accretion sum of these terms should vanish.
This implies that, From the continuity equation it follows that Thus homogeneous materials always undergo flow that is incompressible, but the converse is not true.
Some versions are described below: These methods make differing assumptions about the flow, but all take into account the general form of the constraint
The stringent nature of incompressible flow equations means that specific mathematical techniques have been devised to solve them.