The curved reach of the river shown in Figure 2.1 is rather complex to analyze in terms of

Newton's Second Law because of the curvature in the flow. Therefore, as a starting point, the

differential length of reach dx is isolated as a control volume.

For this control volume, shown in Figure 2.2, the pressure terms p1 and p2 are directed toward

the control volume in a direction normal to the Sections 1 and 2. The shear stress τo is exerted

along the interface between the water and the wetted perimeter and is acting in a direction

opposite to the axis x. The statement of conservation of linear momentum is:

Sum of the forces

Time rate of change

Flux of momentum

Flux of Momentum

acting on the fluid

=

+ of momentum in

- into the control

out of the control

in the control volume

the control volume

volume

volume

Figure 2.2. The control volume for conservation of linear momentum.

The terms in the statement are vectors so we must be concerned with direction as well as

Consider the conservation of momentum in the direction of flow (the x-direction in Figure 2.2). At

the outflow section (section 2), the flux of momentum out of the control volume through the

differential area dA2 is:

ρ2v2 dA2 v2

(2.17)

Here ρ2 v2 dA2 is the mass flux (mass per unit of time) and ρ2 v2 dA2 v2 is the momentum flux

through the area dA2.

Flux of momentum

A ρ 2 v 2 dA 2 v 2

(2.18)

out of the control

=

2

volume

Similarly, at the inflow section (section 1),

2.6

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