Compressibility and the Bulk modulus

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All materials, whether solids, liquids or gases, are compressible, i.e. the volume V of a given mass will be reduced to V - dV when a force is exerted uniformly all over its surface. If the force per unit area of surface increases from p to p + dp, the relationship between change of pressure and change of volume depends on the bulk modulus of the material.

Bulk modulus (K) = (change in pressure) / (volumetric strain)

Volumetric strain is the change in volume divided by the original volume. Therefore,

(change in volume) / (original volume) = (change in pressure) / (bulk modulus)

i.e., -dV/V = dp/K

Negative sign for dV indicates the volume decreases as pressure increases.

In the limit, as dp tends to 0,

K = -V dp/dV à 1

Considering unit mass of substance, V = 1/r à 2

Differentiating,

Vdr + rdV = 0

dV = - (V/r)dr à3

putting the value of dV from equn.3 to equn.1,

K = - V dp / (-(V/r)dr)

i.e. K = rdp/dr

The concept of the bulk modulus is mainly applied to liquids, since for gases the compressibility is so great that the value of K is not a constant.

The relationship between pressure and mass density is more conveniently found from the characteristic equation of gas.

For liquids, the changes in pressure occurring in many fluid mechanics problems are not sufficiently great to cause appreciable changes in density. It is therefore usual to ignore such changes and consider liquids as incompressible.

Gases may also be treated as incompressible if the pressure changes are very small, but usually compressibility cannot be ignored. In general, compressibility becomes important when the velocity of the fluid exceeds about one-fifth of the velocity of a pressure wave (velocity of sound) in the fluid.

Typical values of Bulk Modulus:

K = 2.05 x 10⁹ N/m² for water

K = 1.62 x 10⁹ N/m² for oil.