The Incompressibility Assumption
Table of Contents
All materials, whether gas, liquid or solid exhibit some change in volume when subjected to a compressive stress. The degree of compressibility is measured by a bulk modulus of elasticity, E, defined as either E=δp/ (δρ/ρ ), or E=δp/(-δV/V), where δp is a change in pressure and δρ or δV is the corresponding change in density or specific volume. Since δp/δρ =c2, where c is the adiabatic speed of sound, another expression for E is E =ρc2. In liquids and solids E is typically a large number so that density and volume changes are generally very small unless exceptionally large pressures are applied.
If an incompressible assumption is made in which densities are assumed to remain constant, it is important to know under what conditions that assumption is likely to be valid. There are, in fact, two conditions that must be satisfied before compressibility effects can be ignored. Let us define “incompressibility” as a good approximation when the ratio δ ρ/ρ is much smaller than unity. To determine the conditions for this approximation we must estimate the magnitude of changes in density.
In steady flow, the maximum change in pressure can be estimated from Bernoulli’s relation to be δp=ρu2. Combining this with the above relations for the bulk modulus, we see that the corresponding change in density is δρ/ρ = u2/c2.
Thus, the assumption of incompressibility requires that fluid speed be small compared to the speed of sound,
(1) $latex \displaystyle u\ll c.$
In unsteady flow another condition must also be satisfied. If a significant change in velocity, u, occurs over a time interval t and distance l, then momentum considerations (for an inviscid fluid) require a corresponding pressure change of order δp = ρul/t . Since changes in density are related to changes in pressure through the square of the sound speed, δp=c2δρ , this relation becomes δρ/ρ = (u/c)l/(ct).
Comparing with expression (1), we see that the factor multiplying (u/c) must also be much less than one.
(2) $latex 1\ll ct$
Physically, this condition says that the distance traveled by a sound wave in the time interval t must be much larger than the distance l, so that the propagation of pressure signals in the fluid can be considered nearly instantaneous compared to the time interval over which the flow changes significantly.
An example of why both conditions are required can be found in the collapse of a vapor bubble. During the collapse process the surrounding liquid can be treated as an incompressible fluid because the collapse velocity is much less than the speed of sound. However, at the instant the bubble vanishes, all the fluid momentum rushing toward the point of collapse must be stopped. If this really happened instantaneously, the collapse pressure would be enormous, i.e., much larger than what is actually observed. Since a sound signal requires time to travel out from the collapse point to signal incoming fluid that it must stop, Condition Two is violated (i.e., l > ct ). An accurate numerical model of the collapse process, one capable of predicting the correct pressure transients, requires the addition of a bulk compressibility in the liquid.