KAHIBARO
Table of Contents
What Compressibility Means
Compressibility describes how easily a substance changes its volume when pressure is applied. If a fluid becomes much smaller in volume when squeezed, it is highly compressible. If its volume changes only a little, it is weakly compressible.
In fluid mechanics, this idea is important because liquids and gases behave very differently. Gases are usually highly compressible, while most liquids are nearly incompressible under ordinary conditions. This is why air in a syringe can be compressed noticeably, but water in the same syringe resists volume change strongly.
Compressibility connects pressure change to volume change. If pressure increases, the volume of a fluid usually decreases.
A Simple Physical Picture
Imagine a fluid made of many particles. In a gas, the particles are relatively far apart, so squeezing the gas can reduce the empty space between them. In a liquid, particles are already much closer together, so there is much less room to compress the fluid.
This microscopic picture explains why gases change volume easily and liquids usually do not.
Quantifying Compressibility
To measure compressibility, physicists use the fractional change in volume produced by a change in pressure. A common quantity is the compressibility $\kappa$:
$$
\kappa = -\frac{1}{V}\frac{dV}{dP}
$$
Here, $V$ is volume and $P$ is pressure. The minus sign is included because increasing pressure usually decreases volume, so $dV/dP$ is negative.
If $\kappa$ is large, the fluid compresses easily. If $\kappa$ is small, the fluid resists compression.
Important definition:
$$
\kappa = -\frac{1}{V}\frac{dV}{dP}
$$
A larger $\kappa$ means greater compressibility.
Bulk Modulus
Instead of compressibility, it is often convenient to use the bulk modulus, usually written as $B$ or $K$. It tells how strongly a fluid resists compression.
$$
B = -V\frac{dP}{dV}
$$
The bulk modulus is the reciprocal of compressibility:
$$
B = \frac{1}{\kappa}
$$
A large bulk modulus means the fluid is hard to compress. A small bulk modulus means it is easy to compress.
Relationship between bulk modulus and compressibility:
$$
B = \frac{1}{\kappa}
$$
High bulk modulus, low compressibility.
Low bulk modulus, high compressibility.
Liquids and Gases
The difference between liquids and gases can be summarized clearly.
| Fluid type | Compressibility | Volume change under pressure |
|---|---|---|
| Gas | High | Large |
| Liquid | Very low | Small |
For many introductory problems, liquids such as water are treated as incompressible. This does not mean their volume never changes. It means the change is so small that it can often be ignored.
Gases, however, usually cannot be treated this way, because pressure changes can strongly affect their volume.
Small Volume Changes
If the pressure change $\Delta P$ is not too large, the volume change $\Delta V$ can be approximated using the bulk modulus:
$$
\Delta P = -B\frac{\Delta V}{V}
$$
or equivalently,
$$
\frac{\Delta V}{V} = -\frac{\Delta P}{B}
$$
This equation shows that the fractional change in volume is proportional to the pressure change.
If $B$ is very large, then $\Delta V/V$ is very small. That is the case for most liquids.
For small changes,
$$
\frac{\Delta V}{V} = -\frac{\Delta P}{B}
$$
This is one of the most useful working equations for compressibility.
Why Compressibility Matters
Compressibility affects many physical situations. In hydraulics, liquids are useful because their volume changes very little, allowing pressure to be transmitted effectively. In gases, compressibility is essential for understanding pumps, engines, sound propagation, and atmospheric behavior.
Compressibility also matters when pressure changes are very large. Even liquids that seem incompressible in daily life can show measurable compression under high pressure.
Everyday Examples
A bicycle pump works by compressing air, not by compressing a liquid. When the piston moves inward, the pressure of the trapped air rises and its volume falls. This is possible because air is compressible.
Water in a sealed metal container changes volume only a tiny amount when pressure is applied. That is why water is often modeled as incompressible in simple fluid calculations.
Summary
Compressibility tells how much a fluid's volume changes when pressure changes. Gases are generally highly compressible, while liquids are only slightly compressible. The compressibility is defined by
$$
\kappa = -\frac{1}{V}\frac{dV}{dP}
$$
and its reciprocal is the bulk modulus,
$$
B = \frac{1}{\kappa}
$$
A fluid with large compressibility changes volume easily. A fluid with large bulk modulus strongly resists compression.