Control volume

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In fluid mechanics, a control volume is a mathematical abstraction often employed by scientists and mathematicians in the process of creating mathematical models of physical processes. Generally, a control volume can be thought of as an arbitrary volume in which the mass and the enclosed energy of the fluid remains constant. As fluid moves, this implies that the mass that enters the control volume is the same amount that the one that leaves it. The same rule applies to the energy.

1 Overview
2 Substantive derivative
3 References
4 See also
5 External links

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.

One can then argue that since the physical laws behave in a certain way on a particular control volume, they behave the same way on all such volumes, since that particular control volume was not special in any way. 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.

In fluid mechanics the constitutive equations (Navier-Stokes equations) are by nature integrals. They therefore apply on volumes. Finding forms of the equation that are independent of the control volumes allows simplification of the integral signs.

For understanding the substantive derivative, we might do the following simple derivation:

Assuming that a control volume is filled with fluids and has the pressure $p = p (x, y, z, t)$ .

At first, we take the total differential

$dp=\frac{\partial{p}}{\partial{t}}dt+\frac{\partial{p}}{\partial{x}}dx+\frac{\partial{p}}{\partial{y}}dy+\frac{\partial{p}}{\partial{z}}dz$

The rate of pressure change is

$\frac{dp}{dt}=\frac{\partial{p}}{\partial{t}}+\frac{\partial{p}}{\partial{x}}\frac{dx}{dt}+\frac{\partial{p}}{\partial{y}}\frac{dy}{dt}+\frac{\partial{p}}{\partial{z}}\frac{dz}{dt}$

Hence,

$\frac{dp}{dt}=\frac{\partial{p}}{\partial{t}}+{\mid}v_{x}{\mid}\frac{\partial{p}}{\partial{x}}+{\mid}v_{y}{\mid}\frac{\partial{p}}{\partial{y}}+{\mid}v_{z}{\mid}\frac{\partial{p}}{\partial{z}}$

$\frac{D}{Dt}=\frac{\partial}{\partial t}+{\mathbf v}\cdot\nabla$

therefore,

$\frac{dp}{dt}=\frac{\partial{p}}{\partial{t}}+{\mid}v_{x}{\mid}\frac{\partial{p}}{\partial{x}}+{\mid}v_{y}{\mid}\frac{\partial{p}}{\partial{y}}+{\mid}v_{z}{\mid}\frac{\partial{p}}{\partial{z}}=\frac{Dp}{Dt}$