Properties of Fluids

Mass Density

$$\rho = \frac{\text{Mass}}{\text{Volume}} = \frac{m}{v}$$

At a point:

$$\rho = \lim_{\delta v \to 0} \frac{\delta m}{\delta v}$$

For liquids

Varies very slightly with temperature (negligible in calculations).

Example: Water

• $100\,\text{kgm}^{-3}$ - at $4\degree \text{C}$
• $995.7\,\text{kgm}^{-3}$ - at $30\degree \text{C}$

For gases

Highly dependent on pressure & temperature.

Specific Weight / Unit Weight

$$\omega = \gamma = \frac{\text{Weight}}{\text{Volume}} = \frac{w}{v}$$

Relative Density / Specific Density

$$s = \sigma = \frac{\text{Density of the substance}}{\text{Density of standard substance}}$$

For solids and liquids, water is the standard substance.

Pressure

A force is exerted on all surfaces in contact with a fluid. A scalar.

$$P = \frac{\text{Normal Force}}{\text{Area}} = \frac{F}{A}$$

Vapour Pressure

Vaporisation is when evaporation happens at the free surface of a liquid.

Vapour Pressure is the pressure due to liquid vapour just above the free surface of the liquid. Increases with temperature.

A liquid boils when: $\text{vapour pressure} = \text{external pressure on the liquid}$ $$

Bulk Modulus

$$k = \frac{\text{Change in pressure}}{\text{Change in volume, per volume}} = -\cfrac{\Delta p}{\frac{\Delta v}{v}} = -v \frac{\text{d}p}{\text{d}v}$$

In terms of the density:

$$k = \rho \frac{\text{d}p}{\text{d}\rho}$$

High bulk modulus means hard to compress.

Surface Tension

$$\sigma = \frac{\text{Tensile Force}}{\text{length}} = \frac{F}{L}$$

Negligible in many applications. Considered in small-scale applications. Causes capillary effect.

Viscosity

The force resisting the flow of a liquid.

In liquids, viscosity is mainly caused by inter-molecular attraction. Decreases slightly with temperature.

In gases, mainly due to momentum exchange between molecules. Increases with temperature.

Newton’s law of viscosity

In straight & parallel flow, the shear stress $\tau$ (as in $\frac{F}{A}$) between adjacent layers is proportional to the velocity gradient perpendicular to the layers.

$$\tau \propto \frac{\delta v}{\delta y} (= \text{velocity gradient})$$

As $\delta y \to 0$, $$

$$\tau = \mu \frac{\partial v}{\partial y}$$

Coefficient of dynamic viscosity

Above, $\mu$ is coefficient of dynamic viscosity or coefficient of absolute viscosity or coefficient of viscosity.$$

Fluids can be divided into 2 types:

• $\mu$ is a constant: Newtonian fluid
• $\mu$ is not a constant: Non-newtonian fluid (not focused on for s1)

Coefficient of kinematic viscosity

$$v = \frac{\mu}{\rho}$$