The Three Pillars of State: Density, Pressure, and Viscosity

Density and Pressure

At every point in a continuous fluid field, we can define properties.

Density ($\rho$) is mass per unit volume:

$$\rho = \lim_{\delta \mathcal{V} \to \delta \mathcal{V}^*} \frac{\delta m}{\delta \mathcal{V}}$$

where $\delta \mathcal{V}^*$ is the smallest volume where the continuum assumption still holds.

Pressure ($p$) is the normal force per unit area exerted by the fluid:

$$p = \lim_{\delta A \to 0} \frac{\delta F_n}{\delta A}$$

Viscosity

Viscosity is a measure of a fluid’s resistance to deformation (its “thickness” or internal friction).

• Dynamic Viscosity ($\mu$): The proportionality constant relating shear stress ($\tau$) to the velocity gradient ($du/dy$) in a Newtonian fluid: $\tau = \mu \frac{du}{dy}$.
• Kinematic Viscosity ($\nu$): The ratio of dynamic viscosity to density:

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

Kinematic viscosity is often interpreted as the diffusivity of momentum.

Temperature Dependence (Sutherland’s Law)

For air, we calculate dynamic viscosity at a given temperature $T$ using Sutherland’s Law:

$$\mu = \mu_{ref} \left( \frac{T}{T_{ref}} \right)^{3/2} \frac{T_{ref} + S}{T + S}$$

Where for air:

• $\mu_{ref} = 1.716 \times 10^{-5} \text{ kg/(m}\cdot\text{s)}$
• $T_{ref} = 273.15 \text{ K}$
• $S = 110.4 \text{ K}$

Recap in one minute

• Density ($\rho$) is mass per volume.
• Pressure ($p$) is normal force per area.
• Dynamic viscosity ($\mu$) relates shear stress to velocity gradients.
• Kinematic viscosity ($\nu = \mu/\rho$) relates to momentum diffusion.
• Gas viscosity increases with temperature (Sutherland’s Law), while liquid viscosity decreases.

Check your understanding

1. What is the difference between dynamic and kinematic viscosity?

   Show answer

   Dynamic viscosity ($\mu$) represents internal friction. Kinematic viscosity ($\nu = \mu/\rho$) also incorporates density and represents the diffusion rate of momentum in the fluid.

2. If you heat up a tank of water and a tank of air, what happens to their respective viscosities?

   Show answer

   The water’s viscosity will decrease, while the air’s viscosity will increase.

Your action step

Write a simple Python script to plot Sutherland’s Law for air from 200 K to 1000 K.