Incompressible Flow - Derivation

Derivation

The fundamental requirement for incompressible flow is that the density, is constant within an infinitesimal volume, dV, which moves at the velocity of the fluid, v. Mathematically, this constraint implies that the material derivative (discussed below) of the density must vanish to ensure incompressible flow. Before introducing this constraint, we must apply the conservation of mass to generate the necessary relations. The mass is calculated by a volume integral of the density, :

The conservation of mass requires that the time derivative of the mass inside a control volume be equal to the mass flux, J, across its boundaries. 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. Now, using the divergence theorem we can derive the relationship between the flux and the partial time derivative of the density:

therefore:

The partial derivative of the density with respect to time need not vanish to ensure incompressible flow. When we speak of the partial derivative of the density with respect to time, we are referring to this rate of change within a control volume of fixed position. By allowing the partial time derivative of the density to be non-zero, we are not restricting ourselves to incompressible fluids because the density is allowed to change as observed from a fixed position as fluid flows through the control volume. This approach maintains generality, and not requiring that the partial time derivative of the density vanishes illustrates that compressible fluids can still undergo incompressible flow. What we are interested in now is the change in density of a control volume which moves along with the fluid velocity, v. The flux is related to the fluid velocity through the following function:

So that the conservation of mass implies that:

The previous relation (where we have used the appropriate product rule) is known as the continuity equation. 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) = v), 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 fluid 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 fluid velocity vanishes.