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In fluid mechanics, the fundamental laws that describe flow can be expressed for a control volume (CV), which is a volume fixed in space or moving with a certain velocity through which the fluid flows.

The CV formulation of the mass conservation principle in fluid mechanics can be expressed by the following equation:

(3.11)

The first term represents the rate of change of the mass in the CV, while the second term is the net rate of flux of mass across the bounding control surfaces (CS; Figure 3.6). Details on the derivation of Eq. (3.11) can be found in basic fluid mechanics textbooks [15].

In most hydraulics problems, it is convenient to assume incompressible flow, as well as uniform flow at each inflow or outflow section of the control surface, so that

(3.12)

Q_i represents all the flow rates exiting (positive) or entering (negative) the CV through the permeable surfaces. For stationary problems, the CV does not vary over time; therefore, the conservation of mass equation simply becomes

(3.13)

Figure 3.6 Control volume (CV) and bounding control surface (CS).

A direct application of the above equation is the case of a hydraulic junction (Figure 3.7) – the overall flow entering the junction equals the flow leaving the junction:

(3.14)

With this analogy, the law that applies in the flow in Eq. (3.14) is equivalent to Kirchhoff's current law applied to the nodes of electric circuits. Junctions in most of the hydraulic circuits are described in the following chapters, and the law that applies in Eq. (3.13) will be used frequently in describing the operation of the system.