In continuum mechanics and thermodynamics, a control volume (CV) is a mathematical abstraction employed in the process of creating mathematical models of physical processes.
In an inertial frame of reference, it is a fictitious region of a given volume fixed in space or moving with constant flow velocity through which the continuuum (a continuous medium such as gas, liquid or solid) flows.
[1] At steady state, a control volume can be thought of as an arbitrary volume in which the mass of the continuum remains constant.
At steady state, and in the absence of work and heat transfer, the energy within the control volume remains constant.
It is analogous to the classical mechanics concept of the free body diagram.
Typically, to understand how a given physical law applies to the system under consideration, one first begins by considering how it applies to a small, control volume, or "representative volume".
There is nothing special about a particular control volume, it simply represents a small part of the system to which physical laws can be easily applied.
This gives rise to what is termed a volumetric, or volume-wise formulation of the mathematical model.
In this way, the corresponding point-wise formulation of the mathematical model can be developed so it can describe the physical behaviour of an entire (and maybe more complex) system.
Finding forms of the equation that are independent of the control volumes allows simplification of the integral signs.
The control volumes can be stationary or they can move with an arbitrary velocity.
[2] Computations in continuum mechanics often require that the regular time derivation operator
Consider a bug that is moving through a volume where there is some scalar, e.g. pressure, that varies with time and position:
The last parenthesized expression is the substantive derivative of the scalar pressure.
Since the pressure p in this computation is an arbitrary scalar field, we may abstract it and write the substantive derivative operator as