Movement of liquid and gas

Содержание и основные понятия

For a steady flow of uncompressable fluid, the volume passing through different sections of the tube per second is the same. Speed increases are accompanied by changes in pressure and height.
GEF · 10 grade
PIF · Baseline
Interactive model
flow
flow rate
pressure
viscosity
Turbulence
01
The physical picture
What happens and what boundaries the model has.
Main message
For a steady flow of uncompressable fluid, the volume passing through different sections of the tube per second is the same. Speed increases are accompanied by changes in pressure and height.
In a narrow section, the flow rate is higher.
Bernulli's law expresses the preservation of the mechanical energy of a unit of volume.
Viscosity and turbulence result in losses that are not taken into account by the ideal model.
How to reason
First, name the system and the interaction, then select the values, and only then write down the law. The formula works within the model's assumptions.
02
Laws and values
Symbols should be read as a relationship of values, not as a recipe for substitution.
S
1
v
1
=
S
2
v
2
S_1v_1=S_2v_2
S
1
​
v
1
​
=
S
2
​
v
2
​
Continuity
p
+
ρ
v
2
2
+
ρ
g
h
=
c
o
n
s
t
p+\frac{\rho v^2}{2}+\rho gh=\mathrm{const}
p
+
2
ρ
v
2
​
+
ρ
g
h
=
const
Bernulli
Review of the situation
In the constriction of the horizontal pipe, the speed increases, and the static pressure of the ideal fluid decreases.
04
Links and application
In the real world
In the constriction of the horizontal pipe, the speed increases, and the static pressure of the ideal fluid decreases.
Model verification
Compare the prediction with observation, evaluate the units, the order of magnitude and the influence of the factors not taken into account. Disparity does not always mean an error of law: often idealization is no longer sufficient.
C
What needs to be understood
Outcome 1
Explain the physical meaning of the topic "Voice and gas motion" through the model, not just the formula.
Outcome 2
Clear and correctly use terms: flow, flow, pressure.
Outcome 3
Indicate the conditions for the applicability of the law, verify units and limit cases.
T
Terms of the topic
The definition should indicate the physical meaning, not a substitute for a domestic synonym.
flow
Amount of liquid, substance or other value transported through the surface over the selected time interval.
flow rate
The volume or mass of liquid passing through the cross section of the flow per unit time.
pressure
The ratio of normal force to the surface area on which it operates.
viscosity
The property of liquid or gas resists the relative movement of adjacent layers.
Turbulence
Current mode with irregular pulses of speed and intense mixing.
F
Origin and conclusion of the formulas
The fundamental law cannot be "proven" by algebras: for each record, it is stated whether it is a definition, an experimental law, or a model effect.
S
1
v
1
=
S
2
v
2
S_1v_1=S_2v_2
S
1
​
v
1
​
=
S
2
​
v
2
​
model effect
Continuousity.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
Δ
m
1
=
ρ
1
S
1
v
1
Δ
t
\Delta m_1=\rho_1S_1v_1\Delta t
Δ
m
1
​
=
ρ
1
​
S
1
​
v
1
​
Δ
t
The first section passes through the time Δt the mass density of 1 and the volume S1v1 Δt.
ρ
1
S
1
v
1
=
ρ
2
S
2
v
2
\rho_1S_1v_1=\rho_2S_2v_2
ρ
1
​
S
1
​
v
1
​
=
ρ
2
​
S
2
​
v
2
​
Maintaining the mass in the steady stream equates mass spending through sections.
ρ
1
=
ρ
2
⇒
S
1
v
1
=
S
2
v
2
\rho_1=\rho_2\quad\Rightarrow\quad S_1v_1=S_2v_2
ρ
1
​
=
ρ
2
​
⇒
S
1
​
v
1
​
=
S
2
​
v
2
​
For uncompressable fluid, density is reduced.
Conditions:
The set single-dimensional flow of non-compressable liquid without leakages and springs between sections; v is the mean cross-sectional velocity.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
p
+
ρ
v
2
2
+
ρ
g
h
=
c
o
n
s
t
p+\frac{\rho v^2}{2}+\rho gh=\mathrm{const}
p
+
2
ρ
v
2
​
+
ρ
g
h
=
const
model effect
Bernully.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
A
p
=
p
1
Δ
V
−
p
2
Δ
V
A_p=p_1\Delta V-p_2\Delta V
A
p
​
=
p
1
​
Δ
V
−
p
2
​
Δ
V
Pressure forces work on the same fluid volume at the entrance and exit of the section.
A
p
=
Δ
⁣
(
m
v
2
2
+
m
g
h
)
A_p=\Delta\!\left(\frac{mv^2}{2}+mgh\right)
A
p
​
=
Δ
(
2
m
v
2
​
+
m
g
h
)
According to the energy theory, this work changes the kinetic and gravitational potential energy volume.
p
1
+
ρ
v
1
2
2
+
ρ
g
h
1
=
p
2
+
ρ
v
2
2
2
+
ρ
g
h
2
p_1+\frac{\rho v_1^2}{2}+\rho gh_1=p_2+\frac{\rho v_2^2}{2}+\rho gh_2
p
1
​
+
2
ρ
v
1
2
​
​
+
ρ
g
h
1
​
=
p
2
​
+
2
ρ
v
2
2
​
​
+
ρ
g
h
2
​
A sub-set of m = ΔV and a division of ΔV give a constant sum along the current line.
Conditions:
The installed flow of an ideal uncompressed fluid along one current line; no pumps, turbines and visible heat losses.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
!
Model boundaries and typical errors
Error: Forget model condition
You can't put the formula on any situation just because it's a coincidence of letters. The key condition of this topic is that in a narrow section, the flow rate is higher.
How to Check the Mind
Bernulli's law expresses the preservation of the mechanical energy of the unit of volume. Then check the units, the direction of the vectors and the behaviour of the result at zero or very large parameter.
Status of the topic
Bernulli's equation and continuity refer to a deeper level. They are released for a steady flow of ideal uncompressable fluid along the current line. Viscosity, turbulence, pumps and energy losses require additional members; the ideal formula cannot be applied to a real pipe without verification.
Σ
Conclusion on the topic
Movement of liquid and gas
For a steady flow of uncompressable fluid, the volume passing through different sections of the tube per second is the same. Speed increases are accompanied by changes in pressure and height.
In a narrow section, the flow rate is higher.
Bernulli's law expresses the preservation of the mechanical energy of a unit of volume.
Any numerical result shall be accompanied by a unit, a sign and an explanation of its physical meaning.
The content is consistent with the FRP on Physics of the OS/OSS 2025 and the subject matter of the current lines of the PNP: I. M. Peryškin and co-authors (7–9), G. J. Mäkishev and coauthors (10–11), and with the lines "Engineers of the Future" and V. A. Kasyanov. The original texts are not reproduced, the text of the textbooks is not reproduced.