# What is Incompressible Flow?

#1

In fluid dynamics, incompressible flow refers to a flow in which the density remains constant in any fluid parcel, i.e. any infinitesimal volume of fluid moving in the flow. This type of flow is also referred to as isochoric flow, from the Greek isos-choros (ἴσος-χώρος) which means “same space/area”. It is important to underline the difference between an “incompressible flow” and an “incompressible fluid”: while the first is a characteristic of the flow, the second is a characteristic of the material. All fluids are apriori compressible but many are considered to be incompressible because the density variation is negligible for common applications. Incompressibility is a feature exhibited by any fluid under certain conditions. In Table 1, a set of different cases is shown, to highlight the independency between the usual definition of the fluid and the characteristics of the flow.

\begin{array}{|l|l|l|} \hline & \text{Water} & \text{Air} \\\ \hline \text{Incompressible Flow} & \text{Flow in a river} & \text{Low Mach applications} \\\ & \text{Submarines hydrodynamics} & \text{(e.g. car aerodynamics)}\\\ \hline \text{Compressible Flow} & \text{Water Hammer} & \text{High Mach applications} \\\ & \text{Submarine explodes} & \text{(e.g. supersonic flight)}\\\ \hline \end{array}

Table 1: Examples of flow compressibility for different fluids.

## Continuity Equation

### Flow in a Pipe

Let’s consider a fluid flow in a closed pipe. Consider now the volume of the flow contained between two sections of the pipe, as shown in figure 1. The continuity equation, in a Eulerian framework, is the analysis of the mass transfer across the boundary sections of this volume.

Figure 1: Fluid flow in a pipe

Since no mass creation or deletion can occur within the pipe, the variation of mass contained in the domain can be written as:

\Delta m = \Delta m_1 - \Delta m_2 \tag{1}

where m is the total mass in the domain, \Delta m_1 is the inward mass flux and \Delta m_2 is the outward mass flow. The total mass can be written as:

m = \rho V_e \tag{2}

where \rho is the fluid density and V_e is the volume of the examined portion of the pipe. Under the hypothesis of incompressible flow (\rho constant) and non-deformable pipe (V_e constant), equation (2) implies that m is constant, so:

\Delta m = \Delta m_1 - \Delta m_2=0 \tag{3}

By developing equation (3), we obtain:

\rho \Delta V_1 = \rho \Delta V_2 \tag{4}
\rho \Delta t A_1 u_1 = \rho \Delta t A_2 u_2 \tag{5}

where A_i is the area of the i-th section, \Delta t is the time interval, and u_1 and u_2 are the inward and outward velocities respectively. Finally, the continuity equation can be rewritten as:

A_1 u_1 =A_2 u_2=F \tag{6}

for any couple of sections, where F is the fluid flow rate.
This analysis is valid for real pipes (with solid walls), but also “virtual” pipes, i.e. groups of streamlines with no mass exchange across the lateral surface, which is always the case for laminar flows. “Virtual” pipes can also be seen as streams which do not diffuse or mix with the rest of the flow.
The principle of conservation of the fluid flow rate is why tap jets decrease their section while falling (figure 2): the falling liquid accelerates due to gravity, thus the section of the flow decreases proportionally.
In this section, we have focused on real or virtual pipes of finite dimension, but we can also shrink the pipe to an infinitesimal dimension until it basically coincides to a single streamline. This way, we could extend the principle of conservation of the fluid flow rate to any continuous problem, as shown in the following paragraph.

Figure 2: Flow rate conservation for tap flows ^3

### Differential Formulation

The same procedure done for the flow in a pipe can be done for a fluid flow in any given domain. The first step consists of imposing the conservation of the mass, i.e. that the variation of the mass in the domain is equal to the net mass flow across the boundaries. This means that no mass creation or deletion occurs in the domain. Thus, the conservation equation can be written as:

\cfrac{\partial m}{\partial t}=-\int_S \left(u\rho \right) \cdot dS \tag{7}

where m is the mass of the domain, (u\rho) is the mass flux across the boundaries, and S is the boundary of the domain.
By using the Gauss’ theorem (or divergence theorem), the surface integral can be rewritten as a volumetric integral defined in the domain \Omega:

\cfrac{\partial m}{\partial t}=-\int_S \left( \nabla \cdot \left( u\rho \right) \right) \: d\Omega \tag{8}

The mass of the fluid can be obtained by integrating the density, thus equation (8) can be rewritten as follows:

\cfrac{\partial}{\partial t}\int_{\Omega}\rho \: d\Omega=-\int_S \left(\nabla \cdot \left(u\rho \right) \right) \: d\Omega \tag{9}

Hence, the conservation equation can be reduced to:

\cfrac{\partial \rho}{\partial t}=-\nabla \cdot \left(u\rho \right) \tag{10}

which is valid for both compressible and incompressible flows.
Let’s now impose the incompressibility hypothesis. Incompressibility requires any fluid particle to have a constant density, which is mathematically expressed by the material derivative of the density:

\cfrac{D \rho}{Dt}=0 \tag{11}

We can now rewrite equation (10) in terms of material derivatives:

\cfrac{D \rho}{Dt}-u\cdot \nabla \rho + \nabla \cdot (u\rho)=0 \tag{12}
\cfrac{D \rho}{Dt}-u\cdot \nabla \rho + \rho \nabla \cdot u + u \cdot \nabla \rho=0 \tag{13}

By replacing the incompressibility condition (equation (11)) in the new form of the continuity equation (equation (13)) we obtain:

\nabla \cdot u =0 \tag{14}

Equation (14) shows that, for incompressible flows, the incompressibility condition is respected for any divergence-free velocity field. Thus, the imposition of a constraint on the density (equation (11)) is equivalent to the imposition of a constraint on the divergence of the velocity field.

## Dependency on the material parameters

The conditions for which an incompressible flow occurs vary depending on the material properties of the fluid. Any fluid can be considered as incompressible under certain conditions on velocity and pressure. The tendency of a fluid to change volume when subjected to an external load depends on the material itself. For the sake of simplicity, let’s consider the common case of an elastic isotropic material$^4$. The volumetric stiffness can be expressed as:

K=\rho \cfrac{\partial p}{\partial \rho} \tag{15}

where p is the pressure, \rho is the density, and K is the bulk modulus. The bulk modulus measures the stiffness of a material to volumetric loads or, in other words, the pressure required to produce a reference volumetric strain. The higher K is, the higher the pressure variations must be to induce a compressible flow. Table II reports the bulk modulus for some materials; from this table we deduce that water will behave as a compressible fluid only if huge variations in the pressure occur (e.g. water hammer), while air requires low variations in the pressure to be modelled as an incompressible fluid.

\begin{array}{|l|l|} \hline \text{Material} & \text{Bulk modulus} \\\ \hline \text{Diamond} & \text{442 GPa} \\\ \hline \text{Stainless Steel} & \text{163 GPa}\\\ \hline \text{Glass} & \text{33-55 GPa} \\\ \hline \text{Mercury} & \text{28.5 GPa}\\\ \hline \text{Glycerin} & \text{4.35 GPa}\\\ \hline \text{Water} & \text{2.2 GPa}\\\ \hline \text{Methanol} & \text{823 MPa (at 20$^\circ$C and 1atm)}\\\ \hline \text{Air} & \text{0.142 MPa}\\\ \hline \end{array}

Table 2: Examples of bulk modulus for different materials ^5

In more practical terms, water will require a pressure gradient 15000 times higher than air to exhibit the same density variation (animation 1), thus it is more likely to behave as an incompressible flow in common applications.

Animation 1: Different weight needed to induce a 10% variation in density in an 1 cm^2 piston with different materials.

The volumetric stiffness can also be expressed through the Poisson ratio (\nu), which is directly linked to the bulk modulus ^6:

\nu = \cfrac{1}{2}-\cfrac{E}{6K} \tag{16}

## Related topics

• Compressible flow
• Linear elasticity
• Navier-Stokes equations
• Compressibility
• Mass transfer
• Boussinesq approximation

## Resources

^8: Hung, N.-X., Bordas, S. P. A. and Hung, N.-D. (2009), Addressing volumetric locking and instabilities by selective integration in smoothed finite elements. Commun. Numer. Meth. Engng., 25: 19–34. doi:10.1002/cnm.1098