
English: 
Hello welcome to my talk, All about Fluids. This talk is the first part of
the talk on potential flows with the focuses on the theory of potential flow.
in this talk I will tell you what are irrotational flows, and where we
can often see the irrotational flow.
it has been proven by numerous examples that the fluid motion is universally
governed by Navier-Stokes equation. Navier-Stokes equation is comprised of
a continuity equation and three momentum equations.  Here the vector form

English: 
Hello welcome to my talk All about Fluids, this talk is the first part of
the talk on potential flows with the focuses on the theory of potential flow.
in this talk I will tell you what are irrotational flows, and where we
can often see the irrotational flow.
it has been proven by numerous examples that the fluid motion is universally
governed by navier-stokes equation. navier-stocks equation is comprised of
a continuity equation and three momentum equations.  Here the vector form

English: 
of the momentum equation is given. in this equation we can see the
substantial derivative is used for calculating the total acceleration of
fluid: the local acceleration, the convective acceleration, the forces
acting on the fluid include body force, for instance, the fluid gravitational
force; the surface force due to the pressure gradient and the viscous force.
Navier-Stokes equation has four partial differential equations: one from the
continuity equation and three from the momentum equations for the velocity
components. The governing equation could solve for four unknowns: they are three

English: 
of the momentum equation is given. in this equation we can see the
substantial derivative is used for calculating the total acceleration of
fluid: the local acceleration, the convective acceleration, the forces
acting on the fluid include body force, for instance, the fluid gravitational
force; the surface force due to the pressure gradient and the viscous force.
Navier-Stokes equation has four partial differential equations: one from the
continuity equation and three from the momentum equations for the velocity
components. The governing equation could solve for four unknowns: they are three

English: 
velocity components: u, v and w; and the pressure p. Therefore the dynamical
system for incompressible flows is perfect.
However, solving navier-stocks equation is proven to be too difficult, if it is not
impossible, because, there are partial differential equations; there are
nonlinear partial differential equations; the practical flows mostly involve in
complicated turbulence, with eddies from very large, similar size as the fluid domain
of interest, all the way down to the very small length scale, Kolmogorov length scale.

English: 
velocity components: u, v and w; and the pressure p. Therefore, the dynamical
system for incompressible flows is perfect.
However, solving Navier-Stokes equation is proven to be too difficult, if it is not
impossible, because, there are partial differential equations; there are
nonlinear partial differential equations; the practical flows mostly involve in
complicated turbulence, with eddies from very large, similar size as the fluid domain
of interest, all the way down to the very small length scale, Kolmogorov length scale.

English: 
for many practical problems it might not be necessary to solve the full
Navier-Stokes equation. Instead, for some flows, simplified method can be
used to solve the problem. One famous class of the simplifications to
Navier-Stokes equation is the potential flow theory, in which the flow is assumed
as irrotational. Nowadays, the strict mathematical methods have
been well advanced and established for such flows.
for instance, the boundary element method (BEM) for wave-structure interaction.
The blade element method, another BEM method for fluid-blade interaction, the

English: 
for many practical problems it might not be necessary to solve the full
navier-stokes equation. Instead, for some flows, simplified method can be
used to solve the problem. One famous class of the simplifications to
navier-stokes equation is the potential flow theory, in which the flow is assumed
as irrotational. Nowadays, the strict mathematical methods have
been well advanced and established for
such flows.
for instance, the boundary element method (BEM) for wave-structure interaction.
The blade element method, another BEM method for fluid-blade interaction, the

English: 
methods based on the potential flow Theory find a lot of practical
applications. that's because: they are reliable; they
are fast and accurate in many applications, for instance, the boundary
element method is an industry standard for studying ocean platforms under
wave actions.  it is generally accepted that the potential flow is under the
following three assumptions: the flow is irrotational, this is a must assumption
for potential flows, since this assumption could lead to a velocity potential function.
the mathematical equation for this assumption is the curl of velocity

English: 
methods based on the potential flow Theory find a lot of practical
applications. that's because: they are reliable; they
are fast and accurate in many applications, for instance, the boundary
element method is an industry standard for studying ocean platforms under
wave actions.  it is generally accepted that the potential flow is under the
following three assumptions: the flow is irrotational, this is a must assumption
for potential flows, since this assumption could lead to a velocity potential function.
the mathematical equation for this assumption is the curl of velocity

English: 
vector is zero; the flow is incompressible, this is acceptable for many practical
problems, for which the flow speed is not very large. The mathematical equation for
incompressible flow is given by this equation. However it must be pointed out
that the compressible flows can be irrotational as well, but its dynamics
would be much more complicated; the third assumption is the flow is
inviscid. this assumption is acceptable when the corresponding Reynolds number is
very large as we see in many practical problems. The mathematical expression for

English: 
vector is zero; the flow is incompressible,
this is acceptable for many practical
problems, for which the flow speed is not very large. The mathematical equation for
incompressible flow is given by this equation. However it must be pointed out
that the compressible flows can be irrotational as well, but its dynamics
would be much more complicated; the third assumption is the flow is
inviscid. this assumption is acceptable when the corresponding Reynolds number is
very large as we see in many practical problems. The mathematical expression for

English: 
inviscid flow is given by this, the fluid viscosity coefficient MU is zero.
Then what is the potential flow? strictly speaking, a potential flow is an
irrotational flow, and no more hypotheses are needed for most practical
problems, an irrotational  flow may mean an inviscid flow, but they are not equal.
I will make a talk on this topic in another talk
therefore the hypotheses of an  incompressible and inviscid flow are
generally needed to make the potential flow simpler and all these hypotheses
have worked very well for many practical problems.
then what is an irrotational flow? the definition of irrotational flow is that

English: 
inviscid flow is given by this, the fluid viscosity coefficient MU is zero.
Then what is the potential flow? strictly speaking, a potential flow is an
irrotational flow, and no more hypotheses are needed for most practical
problems, an irrotational flow may mean an inviscid flow, but they are not equal.
I will make a talk on this topic in another talk.
Therefore the hypotheses of an incompressible and inviscid flow are
generally needed to make the potential flow simpler and all these hypotheses
have worked very well for many practical problems.
then what is an irrotational flow? the definition of irrotational flow is that

English: 
the circulation of fluid velocity vector on arbitrary closed curve is zero.
the curve C here and then the circulation GAMMA is given by this.
if the circulation GAMMA is 0, that means the flow is irrotational
using Stokes theorem, we can have this mathematical equation, so this changes the
circulation to the integration on the surface enclosed by the closed curve.
this equation would lead to another definition for the irrotational flow:
The curl of velocity vector is zero.

English: 
the circulation of fluid velocity vector on arbitrary closed curve is zero.
the curve C here and then the circulation  GAMMA is given by this.
if the circulation GAMMA is 0,  that means the flow is irrotational
using Stokes theorem, we can have this mathematical equation, so this changes the
circulation to the integration on the surface enclosed by the closed curve.
this equation would lead to another definition for the irrotational flow:
The curl of velocity vector is zero.

English: 
in this slide the physical meaning for irrotational flow is explained.
Irrotational flow can be understood that the fluid element, the very small
fluid element rotates around its own axis, 
when moving along the path, while the
irrotational flow means that the flow element does not rotate around its own
axis, when moving along its path.
A more intuitive understanding of the irrotational flow can be analogous to a
Ferris Wheel, on which the capsules can be taken as the fluid element and
the capsules are hinged on the rotating wheel . when the wheel rotates around the

English: 
in this slide the physical meaning for irrotational flow is explained.
Irrotational flow can be understood that the fluid element, the very small
fluid element rotates around its own axis, when moving along the path, while the
irrotational flow means that the flow element does not rotate around its own
axis, when moving along its path.
A more intuitive understanding of the irrotational flow can be analogous to a
Ferris Wheel, on which the capsules can be taken as the fluid element and
the capsules are hinged on the rotating wheel. when the wheel rotates around the

English: 
axis, the capsules keep upright positions all the time, people in the
capsules can fell there is no rotating motion: the capsules move very
similar to the irrotational flow.
the captures move with the wheel but not rotate around their own axes, since each
capsule has same speed as the speed of the hinge, so this is very similar to
the irrotational flow. A contrast is the horrible rotating wheel, on which the
capsules are wielded on the rotating wheel and when the wheel
rotates, the capsules move rigidly with the wheel. It would be in an

English: 
axis, the capsules keep upright positions all the time, people in the
capsules can fell there is no rotating motion: the capsules moves very
similar to the irrotational flow.
the captures move with the wheel but not rotate around their own axes, since each
capsule has same speed as the speed of the hinge, so this is very similar to
the irrotational flow. A contrast is the
horrible rotating wheel, on which the
capsules are wielded on the rotating wheel and when the wheel
rotates, the capsules move rigidly with the wheel. It would be in an

English: 
upside-down position when it reaches the top. we can see that the velocities are
different for different operations on the capsule and the differences of the
velocities on the capsule means a rotating motion around its own axis.
This is very similar to the fluid rotational motion
In reality, how we can tell which
is rotational or irrotational flow?  Generally we have two ways to check
whether the flow is irrotational or not: the first one is for checking whether
the curl of the flow velocity vector is zero. we need to check every point in

English: 
upside-down position when it reaches the 
top. we can see that the velocities are
different for different operations on the capsule and the differences of the
velocities on the capsule means a rotating motion around its own axis.
This is very similar to the fluid rotational motion
In reality, how we can tell which
is rotational or irrotational flow?  generally we have two ways to check
whether the flow is irrotational or not: the first one is for checking whether
the curl of the flow velocity vector is zero. we need to check every point in

English: 
the fluid domain to see whether the curl of the velocity vector is zero. If not
then it is rotation flow. The second way is to make closed curve,
any closed curve in the fluid if you like, and the check whether its
circulation is 0. if it is not,  then it is rotational flow. for instance, in
the Couette flow in the plot, we can draw a rectangle and check whether the
circulation is zero. Obviously, for the Couette flow, the circulation on the
rectangle is not zero, hence the Couette flow is rotational.

English: 
the fluid domain to see whether the curl of the velocity vector is zero. If not
then it is rotation flow. The second way is to make closed curve,
any closed curve in the fluid if you like, and the check whether its
circulation is 0. if it is not, then it is rotational flow. for instance, in
the Couette flow in the plot, we can draw a rectangle and check whether the
circulation is zero. Obviously, for the Couette flow, the circulation on the
rectangle is not zero, hence the Couette flow is rotational.

English: 
now we look at where we can often see the rotational and the irrotational flow.
for a uniform flow past a glass plate, due to the fluid viscosity near the
plate, there will be a layer where the fluid velocity increases from zero to
the same velocity as that of the uniform flow. This layer is called the viscous
boundary layer. If we extend the flow field more above the boundary layer, we
can see the velocity distribution like this, and we can easily check in the
boundary layer, the circulation is non zero here, so the flow in the boundary

English: 
now we look at where we can often see the rotational and the irrotational flow.
for a uniform flow past a glass plate, due to the fluid viscosity near the
plate, there will be a layer where the fluid velocity increases from zero to
the same velocity as that of the uniform flow. This layer is called the viscous
boundary layer. If we extend the flow field more above the boundary layer, we
can see the velocity distribution like this, and we can easily check in the
boundary layer, the circulation is non zero here, so the flow in the boundary

English: 
layer is rotational and outside the boundary layer, the circulation of the
flow velocity vector is zero, hence it is irrotational flow.  the other
example is the flow coming from the large area into a contracted area. In the
large area because the boundary is far away the main flow field, so the majority
of the flow in the large area is irrotational, and when the flow coming to
the contact area, because of the viscosity of the fluid, the flow might

English: 
layer is rotational and outside the boundary layer, the circulation of the
flow velocity vector is zero, hence it is irrotational flow.  the other
example is the flow coming from the large area into a contracted area. In the
large area because the boundary is far away the main flow field, so the majority
of the flow in the large area is irrotational, and when the flow coming to
the contact area, because of the viscosity of the fluid, the flow might

English: 
become rotational as it shown here . and when the flow come out from the
contracted area, the fluid might become irrotational again.  however this is a
simple example, where the flow is irrotational or
rotational even in the contracted area is
totally depending on the Reynolds number.
If Reynolds number is very large and then 
the flow even in the constructed area
could be irrotational. Another example is for the flow passing an aerofoil,
the flow above the upper surface of the aerofoil has a viscous boundary layer,
within the boundary layer, the flow is rotational and outflow is irrotational.

English: 
become rotational as it shown here. and when the flow come out from the
contracted area, the fluid might become irrotational again. However, this is a
simple example, where the flow is irrotational or
rotational even in the contracted area is totally depending on the Reynolds number.
If Reynolds number is very large and then the flow even in the constructed area
could be irrotational. Another example is for the flow passing an aerofoil,
the flow above the upper surface of the aerofoil has a viscous boundary layer,
within the boundary layer, the flow is rotational and outflow is irrotational.

English: 
in this slide we are looking at the famous example, the flow in a horizontal
pipe,  so this is a very famous example in many textbooks. when the Reynolds
number is less than 2300, the flow in the pipe would be laminar, and an
analytical solution can be found for such a simple flow.
In the steady state, the flow has a parabolic velocity profile, and such a case the flow is
rotational. We can easily check using the method presented previously. so this

English: 
in this slide we are looking at the famous example, the flow in a horizontal
pipe, so this is a very famous example in many textbooks. when the Reynolds
number is less than 2300, the flow in the pipe would be laminar, and an
analytical solution can be found for such a simple flow.
In the steady state, the flow has a parabolic velocity profile, and such a case the flow is
rotational. We can easily check using the method presented previously. so this

English: 
is the case we can take as the boundary layer is extended from the wall to the
central line of the pipe.  If we look at the development of the flow in the pipe
we can see at the inlet of the pipe, a large portion of the flow is
irrotational as you can see here.  and when the flow moves into the pipe, the fluid
boundary layers increase with the irrotational portion decreasing accordingly.
we can see red lines here.
In the fully developed flow, its velocity profile is parabolic, and
the flow here is rotational.
In the book of Schlichting and Gersten, ‘Boundary Layer Theory’, on page 95, it is

English: 
is the case we can take as the boundary layer is extended from the wall to the
central line of the pipe.  If we look at the development of the flow in the pipe
we can see at the inlet of the pipe, a large portion of the flow is
irrotational as you can see here.  and when the flow moves into the pipe, the fluid
boundary layers increase with the irrotational portion decreasing accordingly.
we can see red lines here.
In the fully developed flow, its velocity profile is parabolic, and
the flow here is rotational.
In the book of Schlichting and Gersten, Boundary Layer Theory, on page 95, it is

English: 
suggested the solutions of Navier-Stokes equation in the entire flow field can be
divided into two parts and we can see the flow passing an aerofoil. The outer
flow, which can be regarded as an irrotational flow, and the flow can
be studied using the potential flow theory, and the flow within the thin layer
close to the body or wall, as well as the wake behind the body, where the fluid
friction is important and therefore the Navier-Stokes equation must be solved.
Prandtl introduced the frictional layer as the boundary layer in 1904,

English: 
suggested the solutions of navier stokes equation in the entire flow field can be
divided into two parts and we can see the flow passing an aerofoil. The outer
flow, which can be regarded as an irrotational flow, and the flow can
be studied using the potential flow theory, and the flow within the thin layer
close to the body or wall, as well as the wake behind the body, where the fluid
friction is important and therefore the navier-stokes equation must be solved.
Prandtl introduced the frictional layer as the boundary layer in 1904,

English: 
the concept of the boundary layer is very important for solving many practical
problem, since the potential flow Theory cannot deal with the problem of flow
friction, while are serving the Navier-Stokes equation in the entire
fluid domain is difficult. nowadays the boundary layer
model is still very important in CFD modelling, which is very challenging for
many practical complicated problems. For the flow in a horizontal pipe when
the Reynolds number is large, say 10 million, and then the flow profile would
be a shape like this, so the flow in the pipe is fully turbulent,

English: 
the concept of the boundary layer is very important for solving many practical
problem, since the potential flow Theory cannot deal with the problem of flow
friction, while are serving the navier-stokes equation in the entire
fluid domain is difficult. nowadays the boundary layer
model is still very important in CFD modeling, which is very challenging for
many practical complicated problems. For the flow in a horizontal pipe when
the Reynolds number is large, say 10 million, and then the flow profile would
be a shape like this, so the flow in the pipe is fully turbulent ,

English: 
the majority of the flow field is irrotational and the small areas near
the pipe walls are rotational regions here and here. so when the Reynolds number
becomes larger, the boundary layer will become smaller. As a result of that, the entire
flow in the pipe can be approximated by
an irrotational flow.
for the irrotational flow the curl of the velocity vector is zero. this means this
determinant is zero. and this corresponding to an equation as this.
To make sure the equation satisfied, all the components along three
directions must be zero. hence we have three partial differential equations for

English: 
the majority of the flow field is irrotational and the small areas near
the pipe walls are rotational regions here and here. so when the Reynolds number
becomes larger, the boundary layer will become smaller. As a result of that, the entire
flow in the pipe can be approximated by an irrotational flow.
for the irrotational flow the curl of the velocity vector is zero. this means this
determinant is zero. and this corresponding to an equation as this.
To make sure the equation satisfied, all the components along three
directions must be zero. hence we have three partial differential equations for

English: 
the fluid velocity components as these. So based on the hypothesis of an irrotational
flow, three velocity component are no longer independent each other.
they linked with these 3 equations and it seems we can solve the flow field
using these 3 partial differential equations. however if we look at the 2D
irrotational flow, from the zero curl of the velocity vector, we can only
get the one partial differential equation as this. so for 2d irrotational
flow, there is only one equation for two velocity components, obviously we need
another equation for completing the system. superficially we can see there is

English: 
the fluid velocity components as these. So based on the hypothesis of an irrotational
flow, three velocity components are no longer independent each other.
they linked with these 3 equations and it seems we can solve the flow field
using these 3 partial differential equations. However if we look at the 2D
irrotational flow, from the zero curl of the velocity vector, we can only
get the one partial differential equation as this. so for 2D irrotational
flow, there is only one equation for two velocity components, obviously we need
another equation for completing the system. superficially we can see there is

English: 
a conflict in 2d and 3d cases, since the 3d case seems a closed system, while
the 2d case is not. Is this true or why we have such a conflict for 2d and
3d cases?
the answer is that the 3d cases, the three equations derived from zero
curl of velocity vector are not independent each other, because we can
use any two equations of these to derive the third one.
for instance, we take the first two equations, and then differentiate the
first equation with regard to x and the second equation with regard to y,

English: 
a conflict in 2d and 3d cases, since the 3d case seems a closed system, while
the 2d case is not. Is this true or why we have such a conflict for 2d and
3d cases?
the answer is that the 3d cases, the three equations derived from zero
curl of velocity vector are not independent each other, because we can
use any two equations of these to derive the third one.
for instance, we take the first two equations,  and then differentiate the
first equation with regard to x and the second equation with regard to y,

English: 
we have equations as these. equalling these two equations, we have an
equation as this. and then we have an equation like this.
so we can get an equation like this one. When we take it a special case for this
equation, C=0, then we have the exactly same equation for the third
equation. so this explains these three equations from the assumption of an
irrotational flow are not independent each other, and that's why we need
another equation for completing the dynamical system in 3d cases. this is the

English: 
we have equations as these. equalling these two equations, we have an
equation as this. and then we have an equation like this.
so we can get an equation like this one. When we take it a special case for this
equation, C=0, then we have the exactly same equation for the third
equation. so this explains these three equations from the assumption of an
irrotational flow are not independent each other, and that's why we need
another equation for completing the dynamical system in 3d cases. this is the

English: 
same as in 2D case, and it will be seen that the additional equation is
the continuity equation.
for the potential flow, the hypothesis of an
irrotational flow means a guarantee of
velocity potential function PHI, and the low the velocity vector can be simply
given as this.
so if the velocity vector is calculated from the velocity potential function,
then its curl is guaranteed to be zero, since we have equation as this.
in the Cartesian coordinate, three velocity
components are given as u, v and w,

English: 
same as in 2D case, and it will be seen that the additional equation is
the continuity equation.
for the potential flow, the hypothesis of an irrotational flow means a guarantee of
velocity potential function PHI, and the low the velocity vector can be simply
given as this.
so if the velocity vector is calculated from the velocity potential function,
then its curl is guaranteed to be zero, since we have equation as this.
in the Cartesian coordinate, three velocity components are given as u, v and w,

English: 
calculated using the velocity potential function as this. for the potential flow in most
applications, the fluid is also assumed as incompressible. this is true for many
practical applications. for the incompressible flow, its mathematical
expression is given by: the divergence of the velocity vector is zero. applying the
potential function to the continuity equation, it leads to the Laplace equation
in details the Laplace equation can be written in this form
as such, the problem with three velocity components degraded into one

English: 
calculated using the velocity potential function as this. for the potential flow in most
applications, the fluid is also assumed as incompressible. this is true for many
practical applications. for the incompressible flow, its mathematical
expression is given by: the divergence of the velocity vector is zero. applying the
potential function to the continuity equation, it leads to the Laplace equation
in details the Laplace equation can be written in this form.
as such, the problem with three velocity components degraded into one

English: 
scalar potential function, PHI. in 2D, the Laplace equation can be written in this
To solve the Laplace equation, the partial differential equation, we need to specify the relevant
boundary conditions. it is interesting to notice that we haven't used
Navier-Stokes equation or Euler equation in the irrotational flow yet. So here we
only apply the conditions of an irrotational flow and the incompressible
flow. the equation for the velocity potential function can be already
established: does this mean we can get the flow field without applying Navier-Stokes
equation or Euler equation?  the answer is NO, since when we solve the potential

English: 
scalar potential function, PHI. in 2D, the Laplace equation can be written in this
To solve the Laplace equation, the partial differential equation, we need to specify the relevant
boundary conditions. it is interesting to notice that we haven't used
navier-stokes equation or Euler equation
in the irrotational flow yet. So here we
only apply the conditions of an irrotational flow and the incompressible
flow. the equation for the velocity potential function can be already
established: does this mean we can get the flow field without applying navier-stokes
equation or Euler equation?  the answer is NO, since when we solve the potential

English: 
function, we need to specify both kinematic and dynamic boundary
conditions, but the dynamics boundary condition must be based on Navier-Stokes
or Euler equation, or more specifically, the Bernoulli's equation
which is the integral form of Euler equation
we have two different types of the boundary conditions: kinematic boundary
condition and dynamic boundary condition. the kinematic boundary
condition specifies the velocity of the fluid on the boundary, such as the non-
penetration condition as the normal velocity of the body surface and the
fluid motion are same, given by this. In irrotational flows, the no-slip condition

English: 
function, we need to specify both kinematic and dynamic boundary
conditions, but the dynamics boundary condition must be based on navier-stokes
or Euler equation, or more specifically, the Bernoulli's equation
which is the integral form of Euler equation
we have two different types of the boundary conditions: kinematic boundary
condition and dynamic boundary condition. the kinematic boundary
condition specifies the velocity of the fluid on the boundary, such as the non
penetration condition as the normal velocity of the body surface and the
fluid motion are same, given by this. In irrotational flows, the no-slip condition

English: 
is not applicable. Here n is the normal vector with the body motion
velocity, for a fixed boundary, V is zero. the second boundary condition is the
dynamic boundary precondition, for which the force acting on the boundary is
specified. however for irrotational flows, there might be fewer conditions to
impose, since there are no shear stresses within the flow. the most applied
dynamic boundary condition would be the free surface condition, for which the
pressure at the water side at the free surface must be same as the air pressure

English: 
is not applicable. Here n is the normal vector with the body motion
velocity, for a fixed boundary, V is zero. the second boundary condition is the
dynamic boundary precondition, for which the force acting on the boundary is
specified. however for irrotational flows, there might be fewer conditions to
impose, since there are no shear stresses within the flow. the most applied
dynamic boundary condition would be the free surface condition, for which the
pressure at the water side at the free surface must be same as the air pressure

English: 
at the free surface.  In this slide a simple example for the flow in a
horizontal pipe is considered. now we assume the flow in the pipe
is irrotational and we will compare the irrotational flow and the viscous
flow in the horizontal pipe.
based on the problem analysis in the horizontal pipe, we can see the velocity
component v and w are zero, so the Laplace equation is reduced to this
simple equation, since the velocity component v and w are 0 and
the solution for the velocity potential function is given by this. so the

English: 
at the free surface.  In this slide a simple example for the flow in a
horizontal pipe is considered. now we assume the flow in the pipe
is irrotational and we will compare the irrotational flow and the viscous
flow in the horizontal pipe.
based on the problem analysis in the horizontal pipe, we can see the velocity
component v and w are zero, so the Laplace equation is reduced to this
simple equation, since the velocity component v and w are 0 and
the solution for the velocity potential function is given by this. so the

English: 
velocity component u can be calculated as this.  plot the irrotational
flow profile, and compare to that of viscous flow. we can see the irrotational
flow is the uniform flow in the pipe, while the laminar viscous flow is parabolic
the third assumption for the potential flow is inviscid flow, thus Navier-Stokes
equation is reduced to Euler equation, with the fluid viscosity coefficient MU
assuming to be zero. if we take the body force at the fluid gravitational force
for irrotational flows, we can use the velocity potential function to

English: 
velocity component u can be calculate as this.  plot the irrotational
flow profile, and compare to that of viscous flow. we can see the irrotational
flow is the uniform flow in the pipe, while the laminar viscous flow is parabolic
the third assumption for the potential flow is inviscid flow, thus navier-stokes
equation is reduced to Euler equation, 
with the fluid viscosity coefficient MU
assuming to be zero. if we take the body force at the fluid gravitational force
for irrotational flows, we can use the velocity potential function to

English: 
replace the velocity ,and the use the vector calculus identity here, and we
have this and this, so put all these together we have an equation as this
based on the Euler equation, the irrotational flow and the conservative
body force, we can obtain the Bernoulli's equation, which is an integral form of
Euler equation as this. this is an unsteady Bernoulli's equation for
irrotational flows, and this steady Bernoulli 
 equation can be very useful in

English: 
replace the velocity, and the use the vector calculus identity here, and we
have this and this, so put all these together we have an equation as this
based on the Euler equation, the irrotational flow and the conservative
body force, we can obtain the Bernoulli's equation, which is an integral form of
Euler equation as this. this is an unsteady Bernoulli's equation for
irrotational flows, and this steady Bernoulli equation can be very useful in

English: 
some applications, for instance in ocean wave theory. this unsteady Bernoulli's
equation can be also written as this. if for a steady flow a conventional
Bernoulli's equation can be obtained as this
so we can see there might be different forms for Bernoulli's equation, however
as a basic requirement, the Bernoulli's equation is satisfied under the
following combined condition: 1, 2 and 3 as: the flow should be inviscid
flow and this is necessary for general form of Bernoulli's equation; the body
force must be conservative, for instance the fluid gravitational

English: 
some applications, for instance in ocean wave theory. this unsteady Bernoulli's
equation can be also written as this. if for a steady flow a conventional
Bernoulli's equation can be obtained as this
so we can see there might be different forms for Bernoulli's equation, however
as a basic requirement, the Bernoulli's equation is satisfied under the
following combined condition: 1, 2 and 3 as: the flow should be inviscid
Flow, and this is necessary for general form of Bernoulli's equation; the body
force must be conservative, for instance the fluid gravitational

English: 
force is a conservative force; the flow is irrotational flow for both steady and
unsteady flows or on the flow streamlines, this only for steady flows.
and we have shown this in the talk of Euler equation.
Similar to the irrotational
flows, in which we introduce a velocity potential function. here if an
incompressible flow is studied, we can introduce a stream function vector PSI,
as this, so the velocity vector is calculated as the curl of the stream
function, but because this stream function, the divergence of velocity vector is 0.
this is an incompressible flow.

English: 
force is a conservative force; the flow is irrotational flow for both steady and
unsteady flows or on the flow streamlines, this only for steady flows.
and we have shown this in the talk of Euler equation.
Similar to the irrotational
flows, in which we introduce a velocity potential function. here if an
incompressible flow is studied, we can introduce a stream function vector PSI,
as this, so the velocity vector is calculated as the curl of the stream
function, but because this stream function, the divergence of velocity vector is 0.
this is an incompressible flow.

English: 
it can be seen the stream function is still
a vector with 3 independent components, thus, generally, the stream function does
not provide  much advantage over the velocity vector in most cases. however
stream functions can be very useful in 2d cases.
Generally for the stream function, it should have three components as the normal vector as this,
however, in 2d we take the velocity component w as 0, this means that the stream
function should have only one component as this, the scalar psi to define the

English: 
it can be seen the stream function is still
a vector with 3 independent components, thus, generally, the stream function does
not provide much advantage over the velocity vector in most cases. however
stream functions can be very useful in 2D cases.
Generally, for the stream function, it should have three components as the normal vector as this,
however, in 2d we take the velocity component w as 0, this means that the stream
function should have only one component as this, the scalar psi to define the

English: 
velocity component u and v as these equations. Now consider the streamline
equation, given by this and write equation into this. combine these together
we can get the equation as this. this means the differential of the stream
function on the streamlines is always zero, it means the stream function is a
constant on a given stream line, and this is the main reason why it is called as
the stream function.
another interesting fact is that in 2D, the dot product of the gradient of

English: 
velocity component u and v as these equations. Now consider the streamline
equation, given by this and write equation into this. combine these together
we can get the equation as this. this means the differential of the stream
function on the streamlines is always zero, it means the stream function is a
constant on a given stream line, and this is the main reason why it is called as
the stream function
another interesting fact is that in 2D,  the dot product of the gradient of

English: 
the potential and the stream functions are zero as this, this means that the
fluid stream lines are perpendicular to the equipotential lines as we have seen
in many examples

English: 
the potential and the stream functions are zero as this,  this means that the
fluid stream lines are perpendicular to the equipotential lines as we have seen
in many examples
