232
P. Liu
τdsP
pA+
(pA) ds
∂
∂
s
ds
W
Wsinθ
1
pA
θ
2
1
2
Fig. 3.60 Stress of micro segment liquid mass
according to Newton’s second law, the motion equation established along the
flow direction is
p A −
p +
∂ p
∂s
ds
A + γ Ads sin θ − τ w ds P = ρ Ads
dV
dt
After simplification, we get
1
g
∂ V
∂t
+
∂z
∂s
+
1
γ
∂ p
∂s
+
1
g
V
∂ V
∂s
+
4τ w
γ D
= 0
This equation is a differential equation of motion for one-dimensional
unsteady gradually varied flow. Substituting the resistance equation
4τ w
γ D =
dhw
ds of uniform flow into the above equation and integrating it from
section 1-1 to section 2-2, the total flow energy equation is obtained as
follows:
2
1
∂
∂s
z +
p
γ
+
V 2
2g
ds +
2
1
1
g
∂ V
∂t
ds +
2
1
dhw = 0
z 1 +
p 1
γ
+
V 2
1
2g
= z 2 +
p 2
γ
+
V 2
2
2g
+
2
1
1
g
∂ V
∂t
ds + hw 1−2