Mechanics of Fluids

20 Fundamental concepts
1.5 COMPRESSIBILITY
All matter is to some extent compressible. That is to say, a change in the
pressure applied to a certain amount of a substance always produces some
change in its volume. Although the compressibility of different substances
varies widely, the proportionate change in volume of a particular material
that does not change its phase (e.g. from liquid to solid) during the compression
is directly related to the change in the pressure.
Bulk modulus of The degree of compressibility of a substance is characterized by the bulk
elasticity modulus of elasticity, K, which is defined by the equation
Compressibility
K =− δp
δV/V
(1.7)
Here δp represents a small increase in pressure applied to the material and
δV the corresponding small increase in the original volume V. Since a rise in
pressure always causes a decrease in volume, δV is always negative, and the
minus sign is included in the equation to give a positive value of K.AsδV/V
is simply a ratio of two volumes it is dimensionless and thus K has the same
dimensional formula as pressure. In the limit, as δp → 0, eqn 1.7 becomes
K =−V(∂p/∂V). As the density ϱ is given by mass/volume = m/V
so K may also be expressed as
dϱ = d(m/V) =− m dV
dV =−ϱ
V2 V
K = ϱ(∂p/∂ϱ) (1.8)
The reciprocal of bulk modulus is sometimes termed the compressibility.
The value of the bulk modulus, K, depends on the relation between pressure
and density applicable to the conditions under which the compression
takes place. Two sets of conditions are especially important. If the compression
occurs while the temperature is kept constant, the value of K is
the isothermal bulk modulus. On the other hand, if no heat is added to or
taken from the fluid during the compression, and there is no friction, the
corresponding value of K is the isentropic bulk modulus. The ratio of the
isentropic to the isothermal bulk modulus is γ , the ratio of the specific heat
capacity at constant pressure to that at constant volume. For liquids the
value of γ is practically unity, so the isentropic and isothermal bulk moduli
are almost identical. Except in work of high accuracy it is not usual to
distinguish between the bulk moduli of a liquid.
For liquids the bulk modulus is very high, so the change of density with
increase of pressure is very small even for the largest pressure changes
encountered. Accordingly, the density of a liquid can normally be regarded as
constant, and the analysis of problems involving liquids is thereby simplified.
In circumstances where changes of pressure are either very large or very sudden,
however – as in water hammer (see Section 12.3) – the compressibility
of liquids must be taken into account.