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# Fluid with No Shearing

## Stress in Motion The

Bernoullis Equation
Derivation of Bernoullis equations from Newtons 2nd Law
and the constraints
Understanding of stagnation, static, dynamic, and
hydrostatic pressures
Applications of Bernoullis equations
Flow velocity measurement devices using Bernoullis
equations

## Static, stagnation, dynamic, and

total pressure
1
2
p + V + z = PT = constant along a streamline
2

## If elevation effects are neglected, the stagnation pressure

pressure obtainable along a given streamline.
Total pressure remains constant along a streamline.

, is the largest

## Stagnation streamline and the

stagnation point
If elevation effects are neglected, the stagnation
pressure (p+1/2V2) is the largest pressure obtainable
alomg a given streamline. It represents the conversion
of all the kinetic energy into a pressure rise.

Video 3.7

## Measurement of static and

stagnation pressures
(1)(3)

(1)(2)

Pitot tube
Incompressible
V2
p
+z+
=C
2g
g

V2
p
+z+
=C
2g
g

V1
p1 V2
p2
+ z1 +
=
+ z2 +
2g
g 2 g
g
2

V1
p1 p2
+ =
2

V1 =

2( p2 p1 )

2( pstagnation pstatic )
2

## Pitot static tube

Pitot tube
Subsonic compressible flow
0.3<M<1
Assume the flow is decelerate and compressed from the freestream state isentropically

2 pstatic
V=
1 static

cp

pstagnation 1

static

cv

V
2 pstagnation

M= =

c
1 pstatic

pstatic
c=
= RT
static

Pitot tube
Supersonic compressible flow
For supersonice flow (M>1), the streamline terminating at the Pitot
tubes stagnation point crosses the bow shock in front of the Pitot
tube.
Fluid traveling along this streamline is first decelerated
nonisentropically to a subsonic speed and the decelerated
isentropically to zero velocity at the stagnation point.

Re>400

## The Pitot tube

1
2
1
p3 = p + V
2

p
=

3
4
2
2
p4 = p1 = p
i.e.

Applications of Bernoullis
equation
In general, an increase in
velocity is accompanied by a
decrease in pressure
The wing of an airplane
generates a lift.
Cavitation in a flow of liquid,
p<pv (vapor pressure)
Flow velocity measurements

## The collapse of bubbles =>

transient pressure variations =>
damage of the surface in the
cavitation

Flowrate measurement

## Devices to measure flow rates in pipes and conduits

Place restrictions within the pipe
Basic principles: an increase in velocity causes a decrease in
pressure

1
1
p1 + v12 = p2 + v22
2
2

## for a horizontal, steady, inviscid, and incompressible flow

Assuming the velocity profiles are uniform,
Q=A1V1=A2V2, then
Q = A2

2( p1 p2 )
A 2
1 2
A1

( A2<A1)

pipe

## Various flow meters

are governed by the
Bernoullis and
continuity equations.

## Example 3.11 Venturi Meter

Kerosene (SG = 0.85) flows through the Venturi meter shown in
Figure E3.11 with flowrates between 0.005 and 0.050 m3/s.
Determine the range in pressure difference, p1 p2, needed to
measure these flowrates.

## Example 3.11 Solution (1/2)

For steady, inviscid, and incompressible flow, the relationship
between flowrate and pressure

Q 2 1 ( A2 A1 )
2( p1 p2 )
p1 p2 =
Q = A2
2
2 A2
1 ( A2 A1 )2

## The density of the flowing fluid

= SGH2O = 0.85(1000 kg/m3 ) = 850 kg/m3
The area ratio
A2 /A1 (D2 / D1 )2 (0.006m / 0.10m)2 = 0.36

Eq.3.20

## Example 3.11 Solution (2/2)

The pressure difference for the smallest flowrate is

)(

p1 p2 = 0.005m / s 850kg / m

(1 0.36 )
2

2 2

2 ( / 4)(0.06m)

## = 1160 N/m2 = 1.16 kPa

The pressure difference for the largest flowrate is

p1 p2 = 0.05 2 (850)

(1 0.36 )
2

2 2

2 ( / 4)(0.06m)

## = 1.16 105 N / m 2 = 116kPa

1.16kPa p1 p2 116kPa

## The sluice gate

Devices to measure
flowrates in open channels
Bernoulli equation

Continuity equation

Since p1=p2=0,
(potential energy change)

## The sharp crested weir

Q ( the flowrate of liquid over the top of the weir
plate)=Q(pw, b, H)
The average velocity across the top of the weir
(2gh)1/2

Q = c1Hb 2 gH = c1b 2 g H

2
3

## The energy line and the hydraulic

The energy line represents the total head
available to the fluid

p V2
+
+ z = H for steady,
2g

## inviscid and incompressible flow.

The elevation of EL is
obtained by measuring the
stagnation pressure with a
pitot tube

## The energy line and the hydraulic

The elevation of the hydraulic grade line (HGL) shows
the piezometric head (p/+z) by the static pressure tap
connected to the piezometer tube

## The energy line and the hydraulic

The distance from the pipe to the HGL indicates the
pressure within the pipe. If the pipe lies below the HGL,
the pressure is positive.

Example 3.14
Energy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E3.14 through a
hose of constant diameter. A small hole is found in the hose at
location (1) as indicate. When the siphon is used, will water leak out
of the hose, or will air leak into the hose?

## Example 3.14 Solution

Whether air will leak into or water will leak out of the hose depends
on whether the pressure within the hose at (1) is less than or
greater than atmospheric.
With the assumption of steady, incompressible, inviscid flow it follows
that the total head is constant, thus, the energy line is horizontal.
Since the hose diameter is constant, it follows from the continuity
equation (AV=constant) that the water velocity in the hose is constant
throughout.
Thus the hydraulic grade line is constant distance, V2/2g.
Since the pressure at the end of the hose is atmospheric, it follows
that the hydraulic grade line is at the same elevation as the end of
the hose outlet.
The fluid within the hose at any point above the hydraulic grade line
will be at less than atmospheric pressure.
Thus, air will leak into the hose through the hole at point (1).

## Restrictions on Use of the Bernoulli

Equation
Compressibility effects
The assumption of incompressibility is
reasonable for most liquid flows.
In certain instances, the assumption introduce
considerable errors for gases.
To account for compressibility effects

dp

1 2
+ V + gz = c
2

Rotational effects

Compressibility effects
For isothermal flow of perfect gas
2

dp 1 2
RT p1 V2
v
1
RT + V + gz = constant
+ z1 +
ln =
+ Z2
p 2
2g
g p2 2 g
For isentropic flow of perfect gas the density and
pressure are related by P / k =Ct, where k
=Specific heat ratio
1
k

1
k

1 2
C P dP + V + gz = constant
2
2
2
k p1 v1
k p2 v2

+ + gz1 =
+ + gz 2
k 1 1 2
k 1 2 2

Compressibility effects
To find the pressure ratio as a function of Mach number
Speed of sound

## The upstream Mach number

Compressible flow

Incompressible flow

M a1 = V1 / c1 = V1 / kRT1

1
p2 p1 k + 1 2
= 1 +
M a1 1

p1
2

p2 p1 k 2
= M a1
p1
2

Compressibility effects
k

1
p2 p1 k + 1 2
= 1 +
M a1 1

p1
2

For unsteady flows, V=V(s,t) along a streamline
The streamwise acceleration is deonted by

V
V
as =
+V
t
s
For unsteady, incompressible, inviscid flows

## Example 3.16 Unsteady Flow UTube

An incompressible, inviscid liquid is
placed in a vertical, constant diameter
U-tube as indicated in Figure E3.16.
When released from the nonequilibrium position shown, the liquid
column will oscillate at a specific
frequency. Determine this frequency.

## Example 3.16 Solution

Let points (1) and (2) be at the air-water interface of
the two columns of the tube and z=0 correspond to
the equilibrium position of the interface.
Hence , p1=p2=0, z1=-z, z2=z, v1=v2=v

s2

s1

V
V
ds =
t
t

s2

s1

dV
ds =l
dt

dV
( z ) = l
+ Z
dt
dz
= g
V=
dt
2
d Z 2g
2 +
z=0
dt
l

## The total length of the liquid colum

Liquid oscillation

= 2g / l

Rotational Effects
In general, the Bernoulli constant varies from streamline to
streamline
A rotational flow: use the Bernoulli equation is restricted to flow
along a streamline
An irrotational flow: Bernoulli equation both along and across a
streamline are valid.
More details in Ch. 6.

## Ex: Bernoulli Equation in

Translating Reference Frame
A light plane flies at 150 km/hr in standard air at an
altitude of 1000 m. Determine the stagnation pressure at
the leading edge of the wing. At a certain point close to
the wing, the air speed relative to the wing is 60 m/s.
Compute the pressure at this point.
The content of the passage is primarily focused on explaining the principles and applications of Bernoulli's equation in fluid dynamics, including various devices and methods for measuring flow rates and pressures. It also includes several examples and solutions to problems related to these concepts.
multiple_questions_with_answers

the passage contains multiple mathematical questions and their corresponding answers, such as the calculation of pressure differences in a venturi meter, the determination of whether air will leak into or water will leak out of a hose, and the calculation of the frequency of oscillation in a u-tube. each example is followed by a detailed solution.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
You are on page 1of 33

# Fluid with No Shearing

## Stress in Motion The

Bernoullis Equation
Derivation of Bernoullis equations from Newtons 2nd Law
and the constraints
Understanding of stagnation, static, dynamic, and
hydrostatic pressures
Applications of Bernoullis equations
Flow velocity measurement devices using Bernoullis
equations

## Static, stagnation, dynamic, and

total pressure
1
2
p + V + z = PT = constant along a streamline
2

## If elevation effects are neglected, the stagnation pressure

pressure obtainable along a given streamline.
Total pressure remains constant along a streamline.

, is the largest

## Stagnation streamline and the

stagnation point
If elevation effects are neglected, the stagnation
pressure (p+1/2V2) is the largest pressure obtainable
alomg a given streamline. It represents the conversion
of all the kinetic energy into a pressure rise.

Video 3.7

## Measurement of static and

stagnation pressures
(1)(3)

(1)(2)

Pitot tube
Incompressible
V2
p
+z+
=C
2g
g

V2
p
+z+
=C
2g
g

V1
p1 V2
p2
+ z1 +
=
+ z2 +
2g
g 2 g
g
2

V1
p1 p2
+ =
2

V1 =

2( p2 p1 )

2( pstagnation pstatic )
2

## Pitot static tube

Pitot tube
Subsonic compressible flow
0.3<M<1
Assume the flow is decelerate and compressed from the freestream state isentropically

2 pstatic
V=
1 static

cp

pstagnation 1

static

cv

V
2 pstagnation

M= =

c
1 pstatic

pstatic
c=
= RT
static

Pitot tube
Supersonic compressible flow
For supersonice flow (M>1), the streamline terminating at the Pitot
tubes stagnation point crosses the bow shock in front of the Pitot
tube.
Fluid traveling along this streamline is first decelerated
nonisentropically to a subsonic speed and the decelerated
isentropically to zero velocity at the stagnation point.

Re>400

## The Pitot tube

1
2
1
p3 = p + V
2

p
=

3
4
2
2
p4 = p1 = p
i.e.

Applications of Bernoullis
equation
In general, an increase in
velocity is accompanied by a
decrease in pressure
The wing of an airplane
generates a lift.
Cavitation in a flow of liquid,
p<pv (vapor pressure)
Flow velocity measurements

## The collapse of bubbles =>

transient pressure variations =>
damage of the surface in the
cavitation

Flowrate measurement

## Devices to measure flow rates in pipes and conduits

Place restrictions within the pipe
Basic principles: an increase in velocity causes a decrease in
pressure

1
1
p1 + v12 = p2 + v22
2
2

## for a horizontal, steady, inviscid, and incompressible flow

Assuming the velocity profiles are uniform,
Q=A1V1=A2V2, then
Q = A2

2( p1 p2 )
A 2
1 2
A1

( A2<A1)

pipe

## Various flow meters

are governed by the
Bernoullis and
continuity equations.

## Example 3.11 Venturi Meter

Kerosene (SG = 0.85) flows through the Venturi meter shown in
Figure E3.11 with flowrates between 0.005 and 0.050 m3/s.
Determine the range in pressure difference, p1 p2, needed to
measure these flowrates.

## Example 3.11 Solution (1/2)

For steady, inviscid, and incompressible flow, the relationship
between flowrate and pressure

Q 2 1 ( A2 A1 )
2( p1 p2 )
p1 p2 =
Q = A2
2
2 A2
1 ( A2 A1 )2

## The density of the flowing fluid

= SGH2O = 0.85(1000 kg/m3 ) = 850 kg/m3
The area ratio
A2 /A1 (D2 / D1 )2 (0.006m / 0.10m)2 = 0.36

Eq.3.20

## Example 3.11 Solution (2/2)

The pressure difference for the smallest flowrate is

)(

p1 p2 = 0.005m / s 850kg / m

(1 0.36 )
2

2 2

2 ( / 4)(0.06m)

## = 1160 N/m2 = 1.16 kPa

The pressure difference for the largest flowrate is

p1 p2 = 0.05 2 (850)

(1 0.36 )
2

2 2

2 ( / 4)(0.06m)

## = 1.16 105 N / m 2 = 116kPa

1.16kPa p1 p2 116kPa

## The sluice gate

Devices to measure
flowrates in open channels
Bernoulli equation

Continuity equation

Since p1=p2=0,
(potential energy change)

## The sharp crested weir

Q ( the flowrate of liquid over the top of the weir
plate)=Q(pw, b, H)
The average velocity across the top of the weir
(2gh)1/2

Q = c1Hb 2 gH = c1b 2 g H

2
3

## The energy line and the hydraulic

The energy line represents the total head
available to the fluid

p V2
+
+ z = H for steady,
2g

## inviscid and incompressible flow.

The elevation of EL is
obtained by measuring the
stagnation pressure with a
pitot tube

## The energy line and the hydraulic

The elevation of the hydraulic grade line (HGL) shows
the piezometric head (p/+z) by the static pressure tap
connected to the piezometer tube

## The energy line and the hydraulic

The distance from the pipe to the HGL indicates the
pressure within the pipe. If the pipe lies below the HGL,
the pressure is positive.

Example 3.14
Energy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E3.14 through a
hose of constant diameter. A small hole is found in the hose at
location (1) as indicate. When the siphon is used, will water leak out
of the hose, or will air leak into the hose?

## Example 3.14 Solution

Whether air will leak into or water will leak out of the hose depends
on whether the pressure within the hose at (1) is less than or
greater than atmospheric.
With the assumption of steady, incompressible, inviscid flow it follows
that the total head is constant, thus, the energy line is horizontal.
Since the hose diameter is constant, it follows from the continuity
equation (AV=constant) that the water velocity in the hose is constant
throughout.
Thus the hydraulic grade line is constant distance, V2/2g.
Since the pressure at the end of the hose is atmospheric, it follows
that the hydraulic grade line is at the same elevation as the end of
the hose outlet.
The fluid within the hose at any point above the hydraulic grade line
will be at less than atmospheric pressure.
Thus, air will leak into the hose through the hole at point (1).

## Restrictions on Use of the Bernoulli

Equation
Compressibility effects
The assumption of incompressibility is
reasonable for most liquid flows.
In certain instances, the assumption introduce
considerable errors for gases.
To account for compressibility effects

dp

1 2
+ V + gz = c
2

Rotational effects

Compressibility effects
For isothermal flow of perfect gas
2

dp 1 2
RT p1 V2
v
1
RT + V + gz = constant
+ z1 +
ln =
+ Z2
p 2
2g
g p2 2 g
For isentropic flow of perfect gas the density and
pressure are related by P / k =Ct, where k
=Specific heat ratio
1
k

1
k

1 2
C P dP + V + gz = constant
2
2
2
k p1 v1
k p2 v2

+ + gz1 =
+ + gz 2
k 1 1 2
k 1 2 2

Compressibility effects
To find the pressure ratio as a function of Mach number
Speed of sound

## The upstream Mach number

Compressible flow

Incompressible flow

M a1 = V1 / c1 = V1 / kRT1

1
p2 p1 k + 1 2
= 1 +
M a1 1

p1
2

p2 p1 k 2
= M a1
p1
2

Compressibility effects
k

1
p2 p1 k + 1 2
= 1 +
M a1 1

p1
2

For unsteady flows, V=V(s,t) along a streamline
The streamwise acceleration is deonted by

V
V
as =
+V
t
s
For unsteady, incompressible, inviscid flows

## Example 3.16 Unsteady Flow UTube

An incompressible, inviscid liquid is
placed in a vertical, constant diameter
U-tube as indicated in Figure E3.16.
When released from the nonequilibrium position shown, the liquid
column will oscillate at a specific
frequency. Determine this frequency.

## Example 3.16 Solution

Let points (1) and (2) be at the air-water interface of
the two columns of the tube and z=0 correspond to
the equilibrium position of the interface.
Hence , p1=p2=0, z1=-z, z2=z, v1=v2=v

s2

s1

V
V
ds =
t
t

s2

s1

dV
ds =l
dt

dV
( z ) = l
+ Z
dt
dz
= g
V=
dt
2
d Z 2g
2 +
z=0
dt
l

## The total length of the liquid colum

Liquid oscillation

= 2g / l

Rotational Effects
In general, the Bernoulli constant varies from streamline to
streamline
A rotational flow: use the Bernoulli equation is restricted to flow
along a streamline
An irrotational flow: Bernoulli equation both along and across a
streamline are valid.
More details in Ch. 6.

## Ex: Bernoulli Equation in

Translating Reference Frame
A light plane flies at 150 km/hr in standard air at an
altitude of 1000 m. Determine the stagnation pressure at
the leading edge of the wing. At a certain point close to
the wing, the air speed relative to the wing is 60 m/s.
Compute the pressure at this point.
user
The content of the passage is primarily focused on explaining the principles and applications of Bernoulli's equation in fluid dynamics, including various devices and methods for measuring flow rates and pressures. It also includes several examples and solutions to problems related to these concepts.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

The passage contains multiple mathematical questions and their corresponding answers, such as the calculation of pressure differences in a Venturi meter, the determination of whether air will leak into or water will leak out of a hose, and the calculation of the frequency of oscillation in a U-tube. Each example is followed by a detailed solution.