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La forma correcta de utilizar la
ecuación de Bernoulli
rEsumEn
La ecuación de Bernoulli es un caso determinado de un problema
del ujo de uido. Se deben cumplir algunas restricciones para
aplicar correctamente esta ecuación en particular. El ujo del
uido debe considerarse no viscoso, incompresible, estable e
irrotacional. Sin embargo, si el ujo del uido es rotacional, la
ecuación de Bernoulli aún puede aplicarse siempre y cuando los
puntos de interés estén en la misma línea de corriente del ujo.
Aquí, nos enfocaremos en demostrar que, en el caso de un ujo
de uido rotacional, los puntos de interés deben encontrarse
en la misma línea de corriente del ujo y por esta razón, sí se
podría continuar con el uso de la ecuación de Bernoulli. El
principio solo es aplicable a los ujos isentrópicos: cuando los
efectos de los procesos irreversibles (e.g. turbulencia, fricción) y
los procesos no adiabáticos (e.g. la radiación de calor, difusión
de masa) son pequeños y pueden despreciarse.
Palabras clave: ujo, Bernoulli, viscoso, incompresible,
estable, irrotacional, línea de corriente, isentrópico, irreversible,
turbulencia, fricción
aBstraCt
Bernoulli’s equation is a certain case of a uid ow problem.
Some restrictions must be met in order to correctly apply this
particular equation. e uid ow must be considered inviscid,
incompressible, steady and irrotational. However, if the uid
ow is rotational, Bernoulli`s equation can still be applicable
as long as the points of interest are on the same streamline
of the uid ow. Here, we will focus on demonstrate that,
in the case of a rotational uid ow, the points of interest
must be on the same streamline and because of that, it can
be proceeding with the usage of Bernoulli`s equation. e
principle is only applicable for isentropic uid ows: when the
eects of irreversible processes (e.g. turbulence, friction) and
non-adiabatic processes (e.g. heat radiation, mass diusion) are
small and can be neglected.
Key words: uid, Bernoulli, inviscid ow, incompressible,
steady, irrotational, streamline, isentropic, adiabatic, turbulence,
friction
L A. A
1 Universidad de San Martín de Porres,
Lima - Perú
larriola@usmp.pe
e correct way to use Bernoulli´s equation
Recibido: mayo 12 de 2019 | Revisado: junio 17 de 2019 | Aceptado: julio 11 de 2019
https://doi.org/10.24265/campus.2019.v24n28.01
| C | V. XX IV | N. 28 | PP. - | - | |
© Los autores. Este artículo es publicado por la Revista Campus de la Facultad de Ingeniería y Arquitectura de la Universidad
de San Martín de Porres. Este artículo se distribuye en los términos de la Licencia Creative Commons Atribución No-comercial
– Compartir-Igual 4.0 Internacional (https://creativecommons.org/licenses/ CC-BY), que permite el uso no comercial,
distribución y reproducción en cualquier medio siempre que la obra original sea debidamente citada. Para uso comercial
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Inroduction
Various forms of Bernoulli’s equation
can be modeled because of the existence of
various types of uid ow, and therefore
Bernoulli’s principle can be applied
too. Bernoulli’s principle states that,
an increase in the speed of a uid ow
occurs simultaneously with a decrease in
internal pressure. e principle is named
after Daniel Bernoulli published it in his
book Hydrodynamic in 1738. Although,
Bernoulli deduced that pressure decreases
when the ow speed increases, it was
Leonhard Euler who derived Bernoulli’s
equation in its usual form in 1752. All
in all, there is a correct way of using
Bernoulli`s equation with condence and
which is briey described to continuation.
e law that explained the
phenomenon from the energy
conservation point of view was found
in his Hydrodynamic work. Later, Euler
deduced an equation for an inviscid
ow (assuming that viscosity was
insignicant) from which Bernoulli’s
equation arises naturally when
considering a stationary case subjected
to a conservative gravitational eld.
Discussion
To arrive at Bernoulli`s equation,
certain assumptions had to be made
which limit us the level of applicability.
According to Euler equation Eq. 1,
dened by Anderson, Jr. (1989), it gives
the variation of pressure with respect to
speed variation, ignoring shear forces
(inviscid uid ow) and body forces
(weight of the air uid particle is ignored).
Only pressure forces were considered.
(1)
Integrating Eq. (1) by using a
limit integration and considering an
incompressible uid ow (change in
density is very small because of low
speed), will give us Bernoulli`s equation
applicable to points 1 and 2 which are on
the same streamline. However, if the ow
is uniform throughout the eld, then
the constant in Eq. (2) is the same for
all streamlines as dened by Anderson,
Jr. (1989).
= const along
streamline
(2)
e assumptions that were made
during the derivation of this equation
led us to some restrictions that must be
implemented in order to use Bernoulli`s
equation. But, rst of all, we must verify
if the ow eld in question is possible
to exist. is is done by verifying if
continuity equation is fullled.
1. Continuity Equation in its vector
form
e continuity equation states that,
“the net outow of mass through
the surface surrounding the volume
must be equal to the decrease of mass
within the volume” (Bertin and Smith
1998, p. 24). is is, when a uid is
in motion, it must move in such a way
that mass is conserved as it is stated
in Eq. 3 dened by Bertin and Smith
(1998):
(3)
3
INTRODUCTION
Various forms of Bernoulli's equation can be modeled because of the existence
of various types of fluid flow, and therefore Bernoulli's principle can be applied too.
Bernoulli's principle states that, an increase in the speed of a fluid flow occurs
simultaneously with a decrease in internal pressure. The principle is named after
Daniel Bernoulli published it in his book Hydrodynamic in 1738. Although, Bernoulli
deduced that pressure decreases when the flow speed increases, it was Leonhard
Euler who derived Bernoulli's equation in its usual form in 1752. All in all, there is a
correct way of using Bernoulli`s equation with confidence and which is briefly
described to continuation.
The law that explained the phenomenon from the energy conservation point of
view was found in his Hydrodynamic work. Later, Euler deduced an equation for an
inviscid flow (assuming that viscosity was insignificant) from which Bernoulli’s equation
arises naturally when considering a stationary case subjected to a conservative
gravitational field.
DISCUSSION
To arrive at Bernoulli`s equation, certain assumptions had to be made which
limit us the level of applicability. According to Euler equation Eq. 1, defined by
Anderson, Jr. (1989), it gives the variation of pressure with respect to speed variation,
ignoring shear forces (inviscid fluid flow) and body forces (weight of the air fluid particle
is ignored). Only pressure forces were considered.
= − (1)
Integrating Eq. (1) by using a limit integration and considering an incompressible fluid
flow (change in density is very small because of low speed), will give us Bernoulli`s
4
equation applicable to points 1 and 2 which are on the same streamline. However, if
the flow is uniform throughout the field, then the constant in Eq. (2) is the same for all
streamlines as defined by Anderson, Jr. (1989).
= const along streamline
(2)
The assumptions that were made during the derivation of this equation led us to some
restrictions that must be implemented in order to use Bernoulli`s equation. But, first of
all, we must verify if the flow field in question is possible to exist. This is done by
verifying if Continuity Equation is fulfilled.
1. Continuity Equation in its Vector Form
The continuity equation states that, “the net outflow of mass through the surface
surrounding the volume must be equal to the decrease of mass within the volume”
(Bertin and Smith 1998, p. 24). This is, when a fluid is in motion, it must move in
such a way that mass is conserved as it is stated in Eq. 3 defined by Bertin and
Smith (1998).
(3)
Where ρ is the fluid density, t is the time,
is the flow velocity vector field.
4
equation applicable to points 1 and 2 which are on the same streamline. However, if
the flow is uniform throughout the field, then the constant in Eq. (2) is the same for all
streamlines as defined by Anderson, Jr. (1989).
= const along streamline
(2)
The assumptions that were made during the derivation of this equation led us to some
restrictions that must be implemented in order to use Bernoulli`s equation. But, first of
all, we must verify if the flow field in question is possible to exist. This is done by
verifying if Continuity Equation is fulfilled.
1. Continuity Equation in its Vector Form
The continuity equation states that, “the net outflow of mass through the surface
surrounding the volume must be equal to the decrease of mass within the volume”
(Bertin and Smith 1998, p. 24). This is, when a fluid is in motion, it must move in
such a way that mass is conserved as it is stated in Eq. 3 defined by Bertin and
Smith (1998).
(3)
Where ρ is the fluid density, t is the time,
is the flow velocity vector field.
4
equation applicable to points 1 and 2 which are on the same streamline. However, if
the flow is uniform throughout the field, then the constant in Eq. (2) is the same for all
streamlines as defined by Anderson, Jr. (1989).
= const along streamline
(2)
The assumptions that were made during the derivation of this equation led us to some
restrictions that must be implemented in order to use Bernoulli`s equation. But, first of
all, we must verify if the flow field in question is possible to exist. This is done by
verifying if Continuity Equation is fulfilled.
1. Continuity Equation in its Vector Form
The continuity equation states that, “the net outflow of mass through the surface
surrounding the volume must be equal to the decrease of mass within the volume”
(Bertin and Smith 1998, p. 24). This is, when a fluid is in motion, it must move in
such a way that mass is conserved as it is stated in Eq. 3 defined by Bertin and
Smith (1998).
(3)
Where ρ is the fluid density, t is the time,
is the flow velocity vector field.
4
equation applicable to points 1 and 2 which are on the same streamline. However, if
the flow is uniform throughout the field, then the constant in Eq. (2) is the same for all
streamlines as defined by Anderson, Jr. (1989).
= const along streamline
(2)
The assumptions that were made during the derivation of this equation led us to some
restrictions that must be implemented in order to use Bernoulli`s equation. But, first of
all, we must verify if the flow field in question is possible to exist. This is done by
verifying if Continuity Equation is fulfilled.
1. Continuity Equation in its Vector Form
The continuity equation states that, “the net outflow of mass through the surface
surrounding the volume must be equal to the decrease of mass within the volume”
(Bertin and Smith 1998, p. 24). This is, when a fluid is in motion, it must move in
such a way that mass is conserved as it is stated in Eq. 3 defined by Bertin and
Smith (1998).
(3)
Where ρ is the fluid density, t is the time,
is the flow velocity vector field.
L A. A
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Where is the uid density, t is the
time, is the ow velocity vector eld.
2. Steady ow
To see further how mass conservation
places restrictions on the velocity eld,
consider a steady uid ow. at is, for
a relatively low speed ow, the pressure
variations are suciently small, and
because of this, the density change is
also small that can be assumed to be
constant and so, the density of the
uid ow does not vary with time.
3. Incompressible
Since density change is very small for
low velocity airows, it can be assumed
to be constant. One way to proof this,
is by verifying if we are dealing with
low velocity airows. As a rule of
thumb, if its mach number is lower
than 0,3 or has a velocity less than 300
ft/s or 100 m/s (or approximately 200
mph), then the velocity airow can
be assumed to be small and treated as
incompressible Anderson, (1989) and
Anderson (2003):
Where:
V is the ow velocity
a is the Speed of Sound
T is the temperature of the ow eld
γ is the ratio of the specic heats at
constant pressure and volume respectively
and has a value of γ = 1.4 for dry air.
R is the air constant for ideal gas and has
a value of:
4
equation applicable to points 1 and 2 which are on the same streamline. However, if
the flow is uniform throughout the field, then the constant in Eq. (2) is the same for all
streamlines as defined by Anderson, Jr. (1989).
= const along streamline
(2)
The assumptions that were made during the derivation of this equation led us to some
restrictions that must be implemented in order to use Bernoulli`s equation. But, first of
all, we must verify if the flow field in question is possible to exist. This is done by
verifying if Continuity Equation is fulfilled.
1. Continuity Equation in its Vector Form
The continuity equation states that, “the net outflow of mass through the surface
surrounding the volume must be equal to the decrease of mass within the volume”
(Bertin and Smith 1998, p. 24). This is, when a fluid is in motion, it must move in
such a way that mass is conserved as it is stated in Eq. 3 defined by Bertin and
Smith (1998).
(3)
Where ρ is the fluid density, t is the time,
is the flow velocity vector field.
5
2. Steady Flow
To see further how mass conservation places restrictions on the velocity field,
consider a steady fluid flow. That is, for a relatively low speed flow, the pressure
variations are sufficiently small, and because of this, the density change is also
small that can be assumed to be constant and so, the density of the fluid flow does
not vary with time.
= 0
3. Incompressible
Since density change is very small for low velocity airflows, it can be assumed to
be constant ( = ). One way to proof this, is by verifying if we are dealing with
low velocity airflows. As a rule of thumb, if its Mach Number is lower than 0,3 or has
a velocity less than 300 ft/s or 100m/s (or approximately 200 mph), then the velocity
airflow can be assumed to be small and treated as incompressible Anderson,
(1989) and Anderson (2003).
=
=
√
Where:
V is the flow velocity
a is the Speed of Sound
T is the temperature of the flow field
γ is the ratio of the specific heats at constant pressure and volume respectively and
has a value of γ = 1.4 for dry air.
5
2. Steady Flow
To see further how mass conservation places restrictions on the velocity field,
consider a steady fluid flow. That is, for a relatively low speed flow, the pressure
variations are sufficiently small, and because of this, the density change is also
small that can be assumed to be constant and so, the density of the fluid flow does
not vary with time.
= 0
3. Incompressible
Since density change is very small for low velocity airflows, it can be assumed to
be constant ( = ). One way to proof this, is by verifying if we are dealing with
low velocity airflows. As a rule of thumb, if its Mach Number is lower than 0,3 or has
a velocity less than 300 ft/s or 100m/s (or approximately 200 mph), then the velocity
airflow can be assumed to be small and treated as incompressible Anderson,
(1989) and Anderson (2003).
=
=
√
Where:
V is the flow velocity
a is the Speed of Sound
T is the temperature of the flow field
γ is the ratio of the specific heats at constant pressure and volume respectively and
has a value of γ = 1.4 for dry air.
6
R is the air constant for ideal gas and has a value of
Then, so far, the continuity equation reduces to:
or:
(4)
4. Remember that shear forces (friction) and body forces (gravity) were ignored
to get Eq. (1). But in the case where conservative body forces are considered like
gravity, Bernoulli`s equation would include an extra term ρgh as shown in Eq. (5).
In many applications of Bernoulli`s equation, the change in the term ρgh (change in
potential energy) along the streamline flow is so small in comparison to the other
terms that it can be ignored. For example, in the case of an aircraft in flight, the
change in height “h” along a streamline flow is so small that the term ρgh can be
disregarded.
(5)
= P
total
where
Static pressure = P
Dynamic pressure =
Total Pressure = Ptotal
6
R is the air constant for ideal gas and has a value of
Then, so far, the continuity equation reduces to:
or:
(4)
4. Remember that shear forces (friction) and body forces (gravity) were ignored
to get Eq. (1). But in the case where conservative body forces are considered like
gravity, Bernoulli`s equation would include an extra term ρgh as shown in Eq. (5).
In many applications of Bernoulli`s equation, the change in the term ρgh (change in
potential energy) along the streamline flow is so small in comparison to the other
terms that it can be ignored. For example, in the case of an aircraft in flight, the
change in height “h” along a streamline flow is so small that the term ρgh can be
disregarded.
(5)
= P
total
where
Static pressure = P
Dynamic pressure =
Total Pressure = Ptotal
en, so far, the continuity equation
reduces to:
or: (4)
4. Remember that shear forces
(friction) and body forces (gravity)
were ignored to get Eq. (1). But in
the case where conservative body
forces are considered like gravity,
Bernoulli`s equation would include
an extra term
ρgh as shown in Eq. (5).
In many applications of Bernoulli`s
equation, the change in the term
ρgh
(change in potential energy) along
the streamline ow is so small in
comparison to the other terms that it
can be ignored. For example, in the
case of an aircraft in ight, the change
in height “h” along a streamline ow
is so small that the term
ρgh can be
disregarded.
(5)
where
Static pressure = P
Dynamic pressure =
Total Pressure = Ptotal
5. Inviscid Flow
e product of viscosity times shear
velocity gradient denes the term
shear stress. We must understand
that, there are no real uids for
which viscosity is zero. But, there are
many real cases where this product is
suciently small that, the shear stress
term, can be ignored when compared
to other terms in the governing
equations as described by Bertin and
Smith (1998).
6
R is the air constant for ideal gas and has a value of
Then, so far, the continuity equation reduces to:
or:
(4)
4. Remember that shear forces (friction) and body forces (gravity) were ignored
to get Eq. (1). But in the case where conservative body forces are considered like
gravity, Bernoulli`s equation would include an extra term ρgh as shown in Eq. (5).
In many applications of Bernoulli`s equation, the change in the term ρgh (change in
potential energy) along the streamline flow is so small in comparison to the other
terms that it can be ignored. For example, in the case of an aircraft in flight, the
change in height “h” along a streamline flow is so small that the term ρgh can be
disregarded.
(5)
= P
total
where
Static pressure = P
Dynamic pressure =
Total Pressure = Ptotal
6
R is the air constant for ideal gas and has a value of
Then, so far, the continuity equation reduces to:
or:
(4)
4. Remember that shear forces (friction) and body forces (gravity) were ignored
to get Eq. (1). But in the case where conservative body forces are considered like
gravity, Bernoulli`s equation would include an extra term ρgh as shown in Eq. (5).
In many applications of Bernoulli`s equation, the change in the term ρgh (change in
potential energy) along the streamline flow is so small in comparison to the other
terms that it can be ignored. For example, in the case of an aircraft in flight, the
change in height “h” along a streamline flow is so small that the term ρgh can be
disregarded.
(5)
= P
total
where
Static pressure = P
Dynamic pressure =
Total Pressure = Ptotal
6
R is the air constant for ideal gas and has a value of
Then, so far, the continuity equation reduces to:
or:
(4)
4. Remember that shear forces (friction) and body forces (gravity) were ignored
to get Eq. (1). But in the case where conservative body forces are considered like
gravity, Bernoulli`s equation would include an extra term ρgh as shown in Eq. (5).
In many applications of Bernoulli`s equation, the change in the term ρgh (change in
potential energy) along the streamline flow is so small in comparison to the other
terms that it can be ignored. For example, in the case of an aircraft in flight, the
change in height “h” along a streamline flow is so small that the term ρgh can be
disregarded.
(5)
= P
total
where
Static pressure = P
Dynamic pressure =
Total Pressure = Ptotal
6
R is the air constant for ideal gas and has a value of
Then, so far, the continuity equation reduces to:
or:
(4)
4. Remember that shear forces (friction) and body forces (gravity) were ignored
to get Eq. (1). But in the case where conservative body forces are considered like
gravity, Bernoulli`s equation would include an extra term ρgh as shown in Eq. (5).
In many applications of Bernoulli`s equation, the change in the term ρgh (change in
potential energy) along the streamline flow is so small in comparison to the other
terms that it can be ignored. For example, in the case of an aircraft in flight, the
change in height “h” along a streamline flow is so small that the term ρgh can be
disregarded.
(5)
= P
total
where
Static pressure = P
Dynamic pressure =
Total Pressure = Ptotal
6
R is the air constant for ideal gas and has a value of
Then, so far, the continuity equation reduces to:
or:
(4)
4. Remember that shear forces (friction) and body forces (gravity) were ignored
to get Eq. (1). But in the case where conservative body forces are considered like
gravity, Bernoulli`s equation would include an extra term ρgh as shown in Eq. (5).
In many applications of Bernoulli`s equation, the change in the term ρgh (change in
potential energy) along the streamline flow is so small in comparison to the other
terms that it can be ignored. For example, in the case of an aircraft in flight, the
change in height “h” along a streamline flow is so small that the term ρgh can be
disregarded.
(5)
= P
total
where
Static pressure = P
Dynamic pressure =
Total Pressure = Ptotal
L B
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Eq. (13) shows that the given ow
velocity eld satisfy the continuity
equation. Second, we need to nd out if
the given ow velocity eld is rotational
or irrotational.
Evaluating the partial derivatives of
equations (8) and (9) yield equations
(14) and (15) respectively.
(14)
(15)
Substituting equations (14) and (15)
into Eq. (7) yields:
It is clear that, the given velocity ow
eld is rotational. So, that means that we
can still use Bernoulli`s eq. only if the two
given points are on the same streamline.
So, we need to identify the streamline by
using the 2D streamline Eq. (16) dened
by Bertin and Smith (1998):
(16)
Solving this equation for the given
velocity components shown in equations
(8) and (9) one nds that
and
Since this equation is a point function,
then
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω =
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω =
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω =
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω
=
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω =
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω =
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
8
First, we need to identify if continuity equation is satisfied, or in other words, if the
velocity flow field is possible to exist.
Continuity equation in a 2D form is:
+
= 0 (10)
Then,
= 2 −
(11)
and
= − 2
(12)
Substituting equations (11) and (12) into Eq. (10) yields:
2 − + − 2 = 0
(13)
Eq. (13) shows that the given flow velocity field satisfy the continuity equation.
Second, we need to find out if the given flow velocity field is rotational or irrotational.
Evaluating the partial derivatives of equations (8) and (9) yield equations (14) and
(15)
respectively.
ω =
−
= −2 (14)
= − (15)
8
First, we need to identify if continuity equation is satisfied, or in other words, if the
velocity flow field is possible to exist.
Continuity equation in a 2D form is:
+
= 0 (10)
Then,
= 2 −
(11)
and
= − 2
(12)
Substituting equations (11) and (12) into Eq. (10) yields:
2 − + − 2 = 0
(13)
Eq. (13) shows that the given flow velocity field satisfy the continuity equation.
Second, we need to find out if the given flow velocity field is rotational or irrotational.
Evaluating the partial derivatives of equations (8) and (9) yield equations (14) and
(15)
respectively.
ω =
−
= −2 (14)
= − (15)
8
First, we need to identify if continuity equation is satisfied, or in other words, if the
velocity flow field is possible to exist.
Continuity equation in a 2D form is:
+
= 0 (10)
Then,
= 2 −
(11)
and
= − 2
(12)
Substituting equations (11) and (12) into Eq. (10) yields:
2 − + − 2 = 0
(13)
Eq. (13) shows that the given flow velocity field satisfy the continuity equation.
Second, we need to find out if the given flow velocity field is rotational or irrotational.
Evaluating the partial derivatives of equations (8) and (9) yield equations (14) and
(15)
respectively.
ω =
−
= −2 (14)
= − (15)
8
First, we need to identify if continuity equation is satisfied, or in other words, if the
velocity flow field is possible to exist.
Continuity equation in a 2D form is:
+
= 0 (10)
Then,
= 2 −
(11)
and
= − 2
(12)
Substituting equations (11) and (12) into Eq. (10) yields:
2 − + − 2 = 0
(13)
Eq. (13) shows that the given flow velocity field satisfy the continuity equation.
Second, we need to find out if the given flow velocity field is rotational or irrotational.
Evaluating the partial derivatives of equations (8) and (9) yield equations (14) and
(15)
respectively.
ω =
−
= −2 (14)
= − (15)
8
First, we need to identify if continuity equation is satisfied, or in other words, if the
velocity flow field is possible to exist.
Continuity equation in a 2D form is:
+
= 0 (10)
Then,
= 2 −
(11)
and
= − 2
(12)
Substituting equations (11) and (12) into Eq. (10) yields:
2 − + − 2 = 0
(13)
Eq. (13) shows that the given flow velocity field satisfy the continuity equation.
Second, we need to find out if the given flow velocity field is rotational or irrotational.
Evaluating the partial derivatives of equations (8) and (9) yield equations (14) and
(15)
respectively.
ω =
−
= −2 (14)
= − (15)
8
First, we need to identify if continuity equation is satisfied, or in other words, if the
velocity flow field is possible to exist.
Continuity equation in a 2D form is:
+
= 0 (10)
Then,
= 2 −
(11)
and
= − 2
(12)
Substituting equations (11) and (12) into Eq. (10) yields:
2 − + − 2 = 0
(13)
Eq. (13) shows that the given flow velocity field satisfy the continuity equation.
Second, we need to find out if the given flow velocity field is rotational or irrotational.
Evaluating the partial derivatives of equations (8) and (9) yield equations (14) and
(15)
respectively.
ω =
−
= −2 (14)
= − (15)
8
First, we need to identify if continuity equation is satisfied, or in other words, if the
velocity flow field is possible to exist.
Continuity equation in a 2D form is:
+
= 0 (10)
Then,
= 2 −
(11)
and
= − 2
(12)
Substituting equations (11) and (12) into Eq. (10) yields:
2 − + − 2 = 0
(13)
Eq. (13) shows that the given flow velocity field satisfy the continuity equation.
Second, we need to find out if the given flow velocity field is rotational or irrotational.
Evaluating the partial derivatives of equations (8) and (9) yield equations (14) and
(15)
respectively.
ω =
−
= −2 (14)
= − (15)
9
Substituting equations (14) and (15) into Eq. (7) yields:
It is clear that, the given velocity flow field is rotational ( . So, that means that
we can still use Bernoulli`s eq. only if the two given points are on the same streamline.
So, we need to identify the streamline by using the 2D streamline Eq. (16) defined by
Bertin and Smith (1998):
(16)
Solving this equation for the given velocity components shown in equations (8) and
(9) one finds that
and
Since this equation is a point function, then
where
(17)
and
9
Substituting equations (14) and (15) into Eq. (7) yields:
It is clear that, the given velocity flow field is rotational ( . So, that means that
we can still use Bernoulli`s eq. only if the two given points are on the same streamline.
So, we need to identify the streamline by using the 2D streamline Eq. (16) defined by
Bertin and Smith (1998):
(16)
Solving this equation for the given velocity components shown in equations (8) and
(9) one finds that
and
Since this equation is a point function, then
where
(17)
and
9
Substituting equations (14) and (15) into Eq. (7) yields:
It is clear that, the given velocity flow field is rotational ( . So, that means that
we can still use Bernoulli`s eq. only if the two given points are on the same streamline.
So, we need to identify the streamline by using the 2D streamline Eq. (16) defined by
Bertin and Smith (1998):
(16)
Solving this equation for the given velocity components shown in equations (8) and
(9) one finds that
and
Since this equation is a point function, then
where
(17)
and
9
Substituting equations (14) and (15) into Eq. (7) yields:
It is clear that, the given velocity flow field is rotational ( . So, that means that
we can still use Bernoulli`s eq. only if the two given points are on the same streamline.
So, we need to identify the streamline by using the 2D streamline Eq. (16) defined by
Bertin and Smith (1998):
(16)
Solving this equation for the given velocity components shown in equations (8) and
(9) one finds that
and
Since this equation is a point function, then
where
(17)
and
9
Substituting equations (14) and (15) into Eq. (7) yields:
It is clear that, the given velocity flow field is rotational ( . So, that means that
we can still use Bernoulli`s eq. only if the two given points are on the same streamline.
So, we need to identify the streamline by using the 2D streamline Eq. (16) defined by
Bertin and Smith (1998):
(16)
Solving this equation for the given velocity components shown in equations (8) and
(9) one finds that
and
Since this equation is a point function, then
where
(17)
and
L A. A
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω =
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
6. Irrotational ow.
If the 2D ow contains no singularities,
then the Vorticity Vector in Eq. (6)
for irrotational ow must be zero as
dened by Bertin and Smith (1998).
(6)
or
(7)
If ω = 0 (irrotational ow), then the
constant in Eq. (2) is real in all the
uid ow. But if ω≠0 (rotational ow),
then, this constant is only real along a
streamline. Here, we present an example
of the correct way of using Bernoulli`s
equation.
Let`me consider a 2D velocity ow
eld at sea level and
dened by:
(8)
(9)
Where “ ” and “ ” are dened in m/s.
First, we need to identify if continuity
equation is satised, or in other words, if
the velocity ow eld is possible to exist.
Continuity equation in a 2D form is:
(10)
en, (11)
and (12)
Substituting equations (11) and (12)
into Eq. (10), yields:
(13)
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω =
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
7
5. Inviscid Flow: The product of viscosity times shear velocity gradient defines the
term shear stress, . We must understand that, there are no real fluids for which
viscosity is zero. But, there are many real cases where this product is sufficiently
small that, the shear stress term, can be ignored when compared to other terms in
the governing equations as described by Bertin and Smith (1998).
6. Irrotational flow. If the 2D flow contains no singularities, then the Vorticity Vector
in Eq. (6) for irrotational flow must be zero as defined by Bertin and Smith (1998).
=
(6)
= 0
or
ω =
(7)
If ω = 0 (irrotational flow), then the constant in Eq. (2) is real in all the fluid flow. But if
ω≠0 (rotational flow), then, this constant is only real along a streamline. Here, we
present an example of the correct way of using Bernoulli`s equation.
Let`s consider a 2D velocity flow field at sea level
and defined by:
(8)
(9)
Where “u” and “v ” are defined in m/s.
129
| C | V. XXIV | N. 28 | - | 2019 | | ISSN (): - | ISSN ( ): - |
where
(17)
and
(18)
Integrating Eq. (17) with respect to
“x” yields
(19)
Where is the respective stream
function
en, taking the derivate of Eq. (19)
with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
erefore; and the
streamline is:
(21)
Now, if we intend to use Bernoulli`s
equation, for example to nd the static
pressure dierence between two points
in the ow, we must be sure to have
these two points on the same streamline.
Consider these two points to be: (-1, 2)
and (2, 2). e coordinates of these two
points are dened in meters.
For point one (-1, 2), Eq. (21) results
in -4.
For point two (2, 2), Eq. (21) also
results in -4
en, these two points are on the same
streamline, so we can use Bernoulli`s
equation only between these two points
even though the ow is rotational. Now,
using Eq. (8) and (9)
For point one (-1, 2)
For point two (2, 2)
(negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
e static pressure dierence between
and which are located on the same
streamline in the uid ow is Pa.
9
Substituting equations (14) and (15) into Eq. (7) yields:
It is clear that, the given velocity flow field is rotational ( . So, that means that
we can still use Bernoulli`s eq. only if the two given points are on the same streamline.
So, we need to identify the streamline by using the 2D streamline Eq. (16) defined by
Bertin and Smith (1998):
(16)
Solving this equation for the given velocity components shown in equations (8) and
(9) one finds that
and
Since this equation is a point function, then
where
(17)
and
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
10
(18)
Integrating Eq. (17) with respect to “x” yields
(19)
Where is the respective stream function
Then, taking the derivate of Eq. (19) with respect to “y” yields
(20)
Replacing Eq. (20) into Eq. (18) yields
Therefore
And the streamline is:
(21)
Now, if we intend to use Bernoulli`s eq, for example to find the static pressure
difference between two points in the flow, we must be sure to have these two points
on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points
are defined in meters.
For point one (-1, 2), Eq. (21) results in -4.
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= 1- (-2) = 3 m/s
=
=
=
=
= 4.5 m/s
5.41 m/s
For point two (2, 2)
= 4 - 4 = 0 m/s
= 2 – 8 = - 6 m/s (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= 1- (-2) = 3 m/s
=
=
=
=
= 4.5 m/s
5.41 m/s
For point two (2, 2)
= 4 - 4 = 0 m/s
= 2 – 8 = - 6 m/s (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= 1- (-2) = 3 m/s
=
=
=
=
= 4.5 m/s
5.41 m/s
For point two (2, 2)
= 4 - 4 = 0 m/s
= 2 – 8 = - 6 m/s (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= ͳǦሺǦʹሻൌ͵Ȁ
=
=
=
=
= ͶǤͷȀ
ͷǤͶͳȀ
For point two (2, 2)
ൌͶǦͶൌͲȀ
= ʹͺൌǦȀ (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= ͳǦሺǦʹሻൌ͵Ȁ
=
=
=
=
= ͶǤͷȀ
ͷǤͶͳȀ
For point two (2, 2)
ൌͶǦͶൌͲȀ
=
ʹͺൌǦȀ (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= ͳǦሺǦʹሻൌ͵Ȁ
=
=
=
=
= ͶǤͷȀ
ͷǤͶͳȀ
For point two (2, 2)
ൌͶǦͶൌͲȀ
= ʹͺൌǦȀ (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= ͳǦሺǦʹሻൌ͵Ȁ
=
=
=
=
= ͶǤͷȀ
ͷǤͶͳȀ
For point two (2, 2)
ൌͶǦͶൌͲȀ
= ʹͺൌǦȀ (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= ͳǦሺǦʹሻൌ͵Ȁ
=
=
=
=
= ͶǤͷȀ
ͷǤͶͳȀ
For point two (2, 2)
ൌͶǦͶൌͲȀ
= ʹͺൌǦȀ (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= ͳǦሺǦʹሻൌ͵Ȁ
=
=
=
=
=
ͶǤͷȀ
ͷǤͶͳȀ
For point two (2, 2)
ൌͶǦͶൌͲȀ
= ʹͺൌǦȀ (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= 1- (-2) = 3 m/s
=
=
=
=
= 4.5 m/s
5.41 m/s
For point two (2, 2)
= 4 - 4 = 0 m/s
= 2 – 8 = - 6 m/s (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
11
For point two (2, 2), Eq. (21) also results in -4
Then, these two points are on the same streamline, so we can use Bernoulli`s equation
only between these two points even though the flow is rotational. Now, using Eq. (8)
and (9)
For point one (-1, 2)
= 1- (-2) = 3 m/s
=
=
=
=
= 4.5 m/s
5.41 m/s
For point two (2, 2)
= 4 - 4 = 0 m/s
= 2 – 8 = - 6 m/s (negative sign means opposite direction)
Using Bernoulli`s Eq. (2):
12
The static pressure difference between
and
which are located on the same
streamline in the fluid flow is Pa.
CONCLUSION
Throughout this paper, the correct way of using Bernoulli`s equation has been shown.
Initially, it has been presented some assumptions for which this equation is valid to
apply. These assumptions led to a set of restrictions that must be met in order to apply
correctly this equation. However, during the process of the application of Bernoulli`s
equation, the analyst has to be sure which restrictions apply for the particular case.
According to this, the results must be presented in a similar way as it was done in this
paper like the pressure difference between the two points on the same streamline.
REFERENCES
Anderson, Jr, J.D. (1989). Introduction to Flight. 3rd ed. New York: McGraw-Hill.
Anderson, Jr. J.D. (2003). Modern Compressible Flow with Historical Perspective. 2
nd
ed. New York: McGraw-Hill.
Bertin, J.J. and Smith, M.L. (1998). Aerodynamics for Engineer. 3rd ed. New Jersey:
Prentice Hall.
Luis A. Arriola*
Facultad de Ingeniería y Arquitectura
Escuela Profesional de Ciencias Aeronáuticas
12
The static pressure difference between
and
which are located on the same
streamline in the fluid flow is Pa.
CONCLUSION
Throughout this paper, the correct way of using Bernoulli`s equation has been shown.
Initially, it has been presented some assumptions for which this equation is valid to
apply. These assumptions led to a set of restrictions that must be met in order to apply
correctly this equation. However, during the process of the application of Bernoulli`s
equation, the analyst has to be sure which restrictions apply for the particular case.
According to this, the results must be presented in a similar way as it was done in this
paper like the pressure difference between the two points on the same streamline.
REFERENCES
Anderson, Jr, J.D. (1989). Introduction to Flight. 3rd ed. New York: McGraw-Hill.
Anderson, Jr. J.D. (2003). Modern Compressible Flow with Historical Perspective. 2
nd
ed. New York: McGraw-Hill.
Bertin, J.J. and Smith, M.L. (1998). Aerodynamics for Engineer. 3rd ed. New Jersey:
Prentice Hall.
Luis A. Arriola*
Facultad de Ingeniería y Arquitectura
Escuela Profesional de Ciencias Aeronáuticas
L B
130
| C | V. XXIV | N. 28 | - | 2019 | | ISSN (): - | ISSN ( ): - |
Conclusion
roughout this paper, the correct way
of using Bernoulli`s equation has been
shown. Initially, it has been presented
some assumptions for which this equation
is valid to apply. ese assumptions led
to a set of restrictions that must be met
in order to apply correctly this equation.
However, during the process of the
application of Bernoulli`s equation, the
analyst has to be sure which restrictions
apply for the particular case. According
to this, the results must be presented in
a similar way as it was done in this paper
like the pressure dierence between the
two points on the same streamline.
References
Anderson, Jr, J.D. (1989). Introduction to
Flight. 3rd ed. New York: McGraw-
Hill.
Anderson, Jr. J.D. (2003). Modern
Compressible Flow with Historical
Perspective. 2
nd
ed. New York:
McGraw-Hill.
Bertin, J.J. and Smith, M.L. (1998).
Aerodynamics for Engineer. 3rd ed.
New Jersey: Prentice Hall.
L A. A
