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EQUATION OF CONTINUITY.

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...fluid on which attention is fixed. The relation between the two kinds of differentiation with respect to time This refers to the difference between tracking a specific "particle" of fluid as it moves (material derivative) versus looking at a fixed point in space as fluid passes through it (local derivative). is expressed by:

This relationship must be clearly understood. However, in a large class of important problems that we will deal with in the following sections, the distinction between these two types of differentiation practically disappears. Whenever the motion is very small, the terms u d/dx, and so on, become less important, and ultimately D/Dt is equal to d/dt.

238.

We must also express the condition that matter is neither created nor destroyed within the fluid. Suppose α, β, and γ original: "α, β, γ" representing the Greek letters alpha, beta, and gamma. are the edges of a small rectangular parallelepiped A six-sided solid object where every face is a parallelogram; essentially a slanted or standard 3D box. aligned with the coordinate axes. The amount of matter that passes out of this space in a small interval of time dt, in excess of what enters it, is:

This must be equal to the actual loss of mass sustained within that space, or:

Therefore:

This is the so-called equation of continuity. When the density (ρ) is constant (regarding both time and space), the equation takes this simpler form:

In problems related to sound, the velocities and the changes in density are usually treated as very small quantities. If we set ρ = ρ₀ (1+s)—where s is called the condensation A measure of the relative change in density of a fluid, used frequently in acoustics. and is considered small—and ignore the products of small terms like u ds/dx, we find:

In specific cases, these equations become even simpler. For an incompressible fluid A fluid whose density does not change even under pressure, such as liquid water in many conditions. where the motion is entirely parallel to the xy plane:

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