3 Hydrodynamics
185
Fig. 3.12 Micro-tubestream beam and energy equation
the shape and position of microelement flow tube generally change with time,
unless the position of the flow tube is fixed. For the constant incompressible
flow, if the cross sections 1-1 and 2-2 (perpendicular to the streamline) are
taken along the microelement flow tube, the continuity equation and energy
equation of microelement flow can be obtained according to the conservation
law of mass and energy. Namely
z 1 +
p 1
γ
+
u 2
1
2g
= z 2 +
p 2
γ
+
u 2
2
2g
+ h w1−2
u 1 d A 1 = u 2 d A 2
where z 1 , p 1, and u 1 are the location, pressure, and velocity of the crosssection 1-1; z 2 , p 2, and u 2 are the location, pressure, and velocity of the crosssection 2-2; h w1-2 is the mechanical energy loss of the cross-section 1-2. The
above continuous equation shows that the flow rate of incompressible liquid
keeps constant along the same microelement flow beam. The energy equation
shows that along the same microelement beam, the total head of 1-1 crosssection element flow is equal to the total head of 2-2 cross-section element
flow plus the mechanical energy loss of 1-2 cross-section element flow.
Any actual liquid flow with a boundary is called a total flow. Obviously,
the total flow can be regarded as liquid flow consisting of numerous multimicron beams. The section that is perpendicular to the streamline is called
the cross section. Obviously for the total flow, if the streamline is a parallel
straight line, the cross section is flat (as shown in Fig. 3.13), otherwise it is
a curved surface. The volume of water passing through the section per unit
185
Fig. 3.12 Micro-tubestream beam and energy equation
the shape and position of microelement flow tube generally change with time,
unless the position of the flow tube is fixed. For the constant incompressible
flow, if the cross sections 1-1 and 2-2 (perpendicular to the streamline) are
taken along the microelement flow tube, the continuity equation and energy
equation of microelement flow can be obtained according to the conservation
law of mass and energy. Namely
z 1 +
p 1
γ
+
u 2
1
2g
= z 2 +
p 2
γ
+
u 2
2
2g
+ h w1−2
u 1 d A 1 = u 2 d A 2
where z 1 , p 1, and u 1 are the location, pressure, and velocity of the crosssection 1-1; z 2 , p 2, and u 2 are the location, pressure, and velocity of the crosssection 2-2; h w1-2 is the mechanical energy loss of the cross-section 1-2. The
above continuous equation shows that the flow rate of incompressible liquid
keeps constant along the same microelement flow beam. The energy equation
shows that along the same microelement beam, the total head of 1-1 crosssection element flow is equal to the total head of 2-2 cross-section element
flow plus the mechanical energy loss of 1-2 cross-section element flow.
Any actual liquid flow with a boundary is called a total flow. Obviously,
the total flow can be regarded as liquid flow consisting of numerous multimicron beams. The section that is perpendicular to the streamline is called
the cross section. Obviously for the total flow, if the streamline is a parallel
straight line, the cross section is flat (as shown in Fig. 3.13), otherwise it is
a curved surface. The volume of water passing through the section per unit