Control volume, continuum and viscosity

Control volume

A control volume is a fixed region of space, of chosen shape and size, through whose boundary (the control surface) fluid may flow in and out. Instead of following a particular mass (system approach), we study what enters and leaves the region. It is the basis of the Eulerian approach, used to write the continuity, momentum and energy equations for flow through pipes, nozzles, turbines and bends.

Continuum

Fluids are made of molecules with empty spaces between them. The continuum concept ignores this and treats the fluid as continuously distributed matter, so properties such as density, pressure and velocity are smooth functions of position and time and can be differentiated. It is valid when the smallest length of interest is much larger than the mean free path of the molecules (Knudsen number $\ll1$). It fails for rarefied gases at very high altitude.

Viscosity

Viscosity is the property of a fluid by which it resists relative motion between its layers. By Newton's law of viscosity, the shear stress is proportional to the velocity gradient:

$$\tau=\mu\frac{du}{dy}$$

where $\mu$ is the dynamic viscosity (unit Pa·s = N·s/m²; 1 poise = 0.1 Pa·s). The kinematic viscosity is $\nu=\mu/\rho$ (m²/s; 1 stokes = $10^{-4}\ \text{m}^2/\text{s}$). Viscosity of liquids decreases and that of gases increases with temperature.