# Definition of incompressible flow

As all knows incompressible flow doesn't exist in reality, its an assumption introduced to simplify governing equations. We can not apply this assumption straightforward. Generally Mach number(M<0.3 for incompressible flow), density variation(zero density variation) and divergence of velocity (is equal to zero for incompressible flow) are the common criterion to define flow as incompressible flow. It is observed that in case of heat transfer problem (such as natural convection) density varies, which violates the last two criterion. Is it possible to define incompressible flow assumption which includes heat transfer process also (means density variation)?

• "As all knows, incompressible flows doesn't exist in reality": unless we're being extremely pedantic, much of the water flowing through plumbing is incompressible, because isothermal liquids have extremely small compressibilities. Apr 16 '13 at 15:43
• @GeoffOxberry The speed of sound in water is about 1.5 km/s. Water jets cutters have a nozzle velocity up to about 1 km/s, justifying a compressible formulation. It doesn't make sense to say that a material is incompressible; instead we can only say that it can be modeled as incompressible within a stated regime. Apr 16 '13 at 21:35
• @JedBrown: We talk about incompressible materials all the time in thermodynamics. The compressibility of water around room temperature is on the order of 1e-10 inverse Pascals up to around 100 MPa. A jet cutter can reach pressures of 700 MPa. Household plumbing and cooling water in chemical plants probably doesn't exceed 1 MPa, and I would be very surprised if it exceeded 10 MPa, because most plumbing in chemical plants is designed for velocities of 3-5 m/s, hence the qualifier "much of". Of course it's condition-dependent. Apr 16 '13 at 22:11
• @GeoffOxberry We seem to be saying the same thing: the material is accurately modeled as incompressible within a regime. The regime is implicit in many discussions, but we need that context to make the statement. Apr 17 '13 at 3:44
• @JedBrown: Yes. The gist of my remark was to point out that "incompressible flow conditions" are quite common. To quote George Box, "All models are wrong. Some are useful." Incompressible flow happens to be a useful model to the point where saying "it does not exist in reality" doesn't make sense unless we're trying to be pedantic. Apr 17 '13 at 19:07

Others have pointed out the Boussinesq approximation (note that it is different from Boussinesq for water waves), but you can also go a step further and allow for large density variation without going to a fully compressible formulation. This is called an "anelastic" model, and retains essentially the same computational structure as incompressible flow. For a nice introduction, see

To add to John's answer, it is very, very common in low-speed flows with small density variations to use the Boussinesq Approximation to approximate the density variation due to temperature or dilute species concentration. This approximates the density variation as a linear function of the temperature and, therefore, removes the variable density from the governing equations.

Incompressibility is ONLY defined as the velocity field being solenoidal. Incompressibility DOES NOT mean that density variation must be zero. From the continuity equation, the requirement that the velocity field have zero divergence only requires that the material derivative of density be zero. That is, the density of a material fluid particle must be constant. This is not the same as requiring that the density be spatially constant.

Compressibility is defined as $$\tau =- { dv \over v dp}$$ So if you substitute $v = {1 \over \rho}$ we get $$d\rho= -\rho \tau dp = - \rho^2 \tau VdV$$ using (Euler equation). Therefore, you can crudely say that effect of compressiblity increases when the velocity, $V$ is sufficiently high. In regimes when natural convection dominates over forced convection, the velocity or more precisely Richardson number is of the order < O(1). That being the incompressible domain. You may use incompressible NS equations, with Bousinessque approximation for natural convection.

Here

K.R. RAJAGOPAL, M. RUZICKA, and A.R. SRINIVASA, Math. Models Methods Appl. Sci. 06, 1157 (1996). ON THE OBERBECK-BOUSSINESQ APPROXIMATION. http://dx.doi.org/10.1142/S0218202596000481

you may find Boussinesq approximation derived using perturbation technique. Criterion stating when this approximation is valid is formulated there.

• Hi Jan, thanks for the answer! Do you mind editing to reflect the title and author? Even though DOIs are "permanent", the URL I'm being redirected to at worldscientific.com isn't loading properly :( May 11 '13 at 17:49