Flux of momentum
A ρ1 v 1 dA 1 v 1
into the
=
(2.19)
1
control volume
The amount of momentum in the control volume is ∀ ρv dV so
Time rate of
∂
{ ∀ ρv dV}
(2.20)
change of momentum =
∂t
in the control volume
At the upstream section, the force acting on the differential area dA1 of the control volume is p1
dA1 where p1 is the pressure from the upstream fluid on the differential area. The total force in
the x-direction at section 1 is A1 p1 dA 1 . Similarly, at section 2, the total force is A2 p 2 dA 2 .
There is a fluid shear stress τo acting along the interface between the water and the bed and
banks. The shear on the control volume is in a direction opposite to the direction of flow and
results in a force -τoP dx where τo is the average shear stress on the interface area, P is the
average wetted perimeter and dx is the length of the control volume. The term P dx is the
interface area.
The body force component acting in the x-direction is denoted Fb and will be discussed in a
subsequent section. The statement of conservation of momentum in the x-direction for the
control volume is:
∂
A ρ 2 v 2 dA 2 - A ρ1 v 1 dA 1 + ∂t ∀ ρv dV = A p1 dA 1 - A p 2 dA 2 - L τ o P dx + Fb
2
2
(2.21)
2
1
1
2
Again, as with the conservation of mass equation, it is convenient to use average velocities
instead of point velocities. We define a momentum coefficient β so that when average velocities
are used instead of point velocities, the correct momentum flux is considered.
The momentum coefficient for incompressible fluids is:
1
β=
2
(2.22)
A v dA
V2 A
For steady incompressible flow, Equation 2.21 is combined with Equation 2.22 to give
ρβ 2 V22 A 2 - ρβ1 V12 A 1 =
p1 dA 1 -
p 2 dA 2 - L τ o P dx + Fb
(2.23)
A1
A2
The pressure force and shear force terms on the right-hand side of Equation 2.23 are usually
abbreviated as Σ Fx so:
Fx =
p1 dA 1 -
p 2 dA 2 - L τo P dx + Fb
(2.24)
A1
A2
2.7
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