
English: 
Hello welcome to my talk, All about Fluids. this talk is on an interesting
topic of whether viscose a fluid could form a potential flow?
this is the first part of the talk with the focus on the theoretical aspect,
relevant to the topic.
As you all know it very well for
potential flow, the fluid is required to be irrotational and inviscid. and
in many cases the flow is also assumed to be incompressible
the question is, 'do we need all these assumptions for

English: 
Hello welcome to my talk, All about Fluids. this talk is on an interesting
topic of whether viscose a fluid could form a potential flow?
this is the first part of the talk with the focus on the theoretical aspect,
relevant to the topic.
As you all know it very well for
potential flow, the fluid is required to be irrotational and inviscid. and
in many cases the flow is also assumed to be incompressible
the question is, 'do we need all these assumptions for

English: 
an irrotational flow?'  I will answer the question in the talk
let's start from the full Navier-Stokes equation
we can see in the full Navier-Stokes equation, in the left hand side, the
substantial derivative is used for calculating the flow acceleration
which consists of two parts: the local acceleration and the convective acceleration
in the right hand side, the force due to the flow pressure; the body force
acting on the fluid; the dissipation term due to the fluid viscosity and the fluid

English: 
an irrotational flow?'  I will answer the question in the talk
let's start from the full Navier-Stokes equation
we can see in the full Navier-Stokes equation, in the left hand side, the
substantial derivative is used for calculating the flow acceleration
which consists of two parts: the local acceleration and the convective acceleration
in the right hand side, the force due to the flow pressure; the body force
acting on the fluid; the dissipation term due to the fluid viscosity and the fluid

English: 
compressibility term. the last two terms are related to the fluid viscosity.
for an inviscid fluid, the fluid viscosity coefficient MU equals to zero, thus the
Navier-Stokes equation is degenerated to Euler equation
although Euler equation is not perfect, it is very useful for deriving the
Bernoulli's equation, which is actually an integral form of Euler equation under
the assumptions of an inviscid and steady flow
this is a conventional assumption for the inviscid flow

English: 
compressibility term. the last two terms are related to the fluid viscosity.
for an inviscid fluid, the fluid viscosity coefficient MU equals to zero, thus the
Navier-Stokes equation is degenerated to Euler equation
although Euler equation is not perfect, it is very useful for deriving the
Bernoulli's equation, which is actually an integral form of Euler equation under
the assumptions of an inviscid and steady flow
this is a conventional assumption for the inviscid flow

English: 
in this slide, I would like to clarify the concepts: an inviscid fluid and inviscid flow
and this clarification of these two
concepts is very important for irrotational flows
an inviscid fluid is a fluid with zero viscosity, that is, MU=0
This is an idea assumption since all fluids are viscous, however for many
practical applications, this assumption is acceptable because it could simplify
the fluid dynamics a great deal, while it still gives a very reliable and good
approximations to the problems. An inviscid flow is a flow for

English: 
in this slide, I would like to clarify the concepts: an inviscid fluid and inviscid flow
and this clarification of these two
concepts is very important for irrotational flows
an inviscid fluid is a fluid with zero viscosity, that is, MU=0
This is an idea assumption since all fluids are viscous, however for many
practical applications, this assumption is acceptable because it could simplify
the fluid dynamics a great deal, while it still gives a very reliable and good
approximations to the problems. An inviscid flow is a flow for

English: 
which the last two terms in the full Navier-Stokes equation vanish
Obviously, an inviscid fluid will cause an inviscid flow, however a
viscous fluid could also cause an inviscid flow, such as the uniform flows
and the incompressible irrotational flows,  you will see this clearly later in the talk
Conventionally, there are two basic hypotheses for a potential flow as the
following hypotheses: 1 and 2: the flow is an irrotational flow,  this
assumption would lead to a velocity potential function, and under such an

English: 
which the last two terms in the full Navier-Stokes equation vanish
Obviously, an inviscid fluid will cause an inviscid flow, however a
viscous fluid could also cause an inviscid flow, such as the uniform flows
and the incompressible irrotational flows, we will see this clearly later in the talk
Conventionally, there are two basic hypotheses for a potential flow as the
following hypotheses: 1 and 2: the flow is an irrotational flow, this
assumption would lead to a velocity potential function, and under such an

English: 
assumption, the three velocity components are no longer independently each other.
the mathematical expression for the irrotational flow is the curl of the
velocity vector is 0 or its circulation of a closed curve is 0,
the second assumption is inviscid fluid, MU=0, and
the corresponding fluid dynamics equation is Euler equation.
however in many practical problems, the flow is also assumed as incompressible
this assumption could simplify the problem so for finding the solution
easier. This assumption is made for the purpose of simplifying the problem

English: 
assumption,  the three velocity components are no longer independently each other.
the mathematical expression for the irrotational flow is the curl of the
velocity vector is 0 or its circulation of a closed curve is 0,
the second assumption is inviscid fluid, MU=0, and
the corresponding fluid dynamics equation is Euler equation.
however in many practical problems, the flow is also assumed as incompressible
this assumption could simplify the problem so for finding the solution
easier. This assumption is made for the purpose of simplifying the problem

English: 
rather than a pre-requisite,  because we have seen in many cases the compressible flows
can be potential flows too, so the question is: for a potential flow, do we need all
these assumptions?
in this slide I will show you what is the potential flow. in principle
potential flow is a irrotational flow, because this assumption could lead to the
velocity potential function and there should be no more assumptions needed for
a potential flow, then what is the irrotational flow?  the definition of a
potential flow, the circulation of fluid velocity vector on an arbitrary closed

English: 
rather than a pre-requisite, because we have seen in many cases the compressible flows
can be potential flows too, so the question is: for a potential flow, do we need all
these assumptions?
in this slide I will show you what is the potential flow. in principle
potential flow is a irrotational flow, because this assumption could lead to the
velocity potential function and there should be no more assumptions needed for
a potential flow, then what is the irrotational flow?  the definition of a
potential flow, the circulation of fluid velocity vector on an arbitrary closed

English: 
curve, is 0, for instance, for the arbitrary closed curve in the fluid, C,
here its circulation is calculated as this. if the circulation is zero, then the
flow is irrotational. Based on the Stokes theorem, we can convert
the integral on a closed curve to a surface of the enclosed curve and from
the zero circulation, we can obtain another definition of irrotational flow,
as this,  the curl of the velocity vector is zero
the physical definition of an irrotational flow is that the fluid elements do not
rotate around their own axes, while moving along the pathlines as this. this is

English: 
curve, is 0, for instance, for the arbitrary closed curve in the fluid, C,
here its circulation is calculated as this. if the circulation is zero, then the
flow is irrotational. Based on the Stokes theorem, we can convert
the integral on a closed curve to a surface of the enclosed curve and from
the zero circulation, we can obtain another definition of irrotational flow,
as this, the curl of the velocity vector is zero
the physical definition of an irrotational flow is that the fluid elements do not
rotate around their own axes, while moving along the pathlines as this. this is

English: 
similar to the famous Ferris wheel, on which the capsules are hinged to the
large rotating wheel. when the wheel rotates, the capsules keep their upright
position all the time, and the passengers in the capsules could not feel any
rotation. the velocities of the different positions on the capsule are
all same, same as the velocity at the hinge.
for a rotational flow, the fluid
elements rotate around their own axes, while moving along the pathline as this

English: 
similar to the famous Ferris wheel, on which the capsules are hinged to the
large rotating wheel. when the wheel rotates, the capsules keep their upright
position all the time, and the passengers in the capsules could not feel any
rotation. the velocities of the different positions on the capsule are
all same, same as the velocity at the hinge.
for a rotational flow, the fluid
elements rotate around their own axes, while moving along the pathline as this

English: 
in reality, how we can tell which flow 
is the rotational or irrotational flow.
Generally, there are two ways to check whether the flow is irrotational or not
the first one for checking whether the curl of the flow velocity vector is zero
the second way is to make a closed curve,
any closed curve in the fluid if you like, and check whether the
circulation is a zero.
here take the Couette flow as an example
its velocity distribution is a given as this
and the curl of the velocity vector given by this, it is not zero

English: 
in reality, how we can tell which flow is the rotational or irrotational flow.
Generally, there are two ways to check whether the flow is irrotational or not
the first one for checking whether the curl of the flow velocity vector is zero
the second way is to make a closed curve,
any closed curve in the fluid if you like, and check whether the
circulation is a zero.
here take the Couette flow as an example
its velocity distribution is a given as this
and the curl of the velocity vector given by this, it is not zero

English: 
and if we draw rectangle in the Couette flow and check whether its
circulation is zero. Obviously in this Couette flow,
its circulation is not zero, thus we can see both methods have shown the
Couette flow is rotational.  now we may have a question, where we can see the
rotational and irrotational flows in practical flows? here a uniform flow passes
a plate and it can be seen near the plate surface, there will be a boundary

English: 
and if we draw rectangle in the Couette flow and check whether its
circulation is zero. Obviously in this Couette flow,
its circulation is not  zero, thus we can see both methods have shown the
Couette flow is rotational.  now we may have a question, where we can see the
rotational and irrotational flows in practical flows? here a uniform flow passes
a plate and it can be seen near the plate surface, there will be a boundary

English: 
layer where the flow velocity increases from zero at the plate surface
due to the no-slip condition to the same velocity as that of the uniform
flow, this layer is called boundary layer. if we extend the flow to a large range of
the flow as this, so we can see in the boundary layer, we draw a rectangle and
we can see its circulation is not zero, thus the flow within the boundary layer
is rotational. and outside of the boundary layer,
we can see the circulation on a rectangle is zero
thus the flow outside the boundary layer is irrotational.

English: 
layer where the flow velocity increases from zero at the plate surface
due to the no-slip condition to the same velocity as that of the uniform
flow, this layer is called boundary layer. if we extend the flow to a large range of
the flow as this, so we can see in the  boundary layer, we draw a rectangle and
we can see its circulation is not zero, thus the flow within the boundary layer
is rotational. and outside of the boundary layer,
we can see the circulation on a rectangle is zero
thus the flow outside the boundary layer is irrotational.

English: 
Here we look at the famous
example, the flow in a horizontal pipe as this
so when the Reynolds number is small, such as, less than 2300, and the flow in
the pipe is laminar, and we can obtain the analytical solution for the fluid
velocity distribution as this. This flow is also called Hagen-Poiseuille flow,
and it can be seen such a flow is a rotational, however with the
fluid Reynolds number is increased, say 
to ten million, and the flow in the
pipe is fully turbulent and the flow velocity profile is

English: 
Here we look at the famous
example, the flow in a horizontal pipe as this
so when the Reynolds number is small, such as, less than 2300, and the flow in
the pipe is laminar, and we can obtain the analytical solution for the fluid
velocity distribution as this. This flow is also called Hagen-Poiseuille flow,
and it can be seen such a flow is a rotational, however with the
fluid Reynolds number is increased, say, to ten million, and the flow in the
pipe is fully turbulent and the flow velocity profile is

English: 
something like this, as such the large portion of the flow is irrotational, only
the regions near the pipe walls here and here are the viscous boundary layers
and flow within the boundary layer are rotational. When the Reynolds number
become very large and the boundary layer would become thinner, as such the entire
flow might be taken as an irrotational flow for a good approximation, this is a
good practice in many practical problems
for an irrotational flow, the zero curl of the velocity vector would lead to

English: 
something like this, as such the large portion of the flow is irrotational, only
the regions near the pipe walls here and here are the viscous boundary layers
and flow within the boundary layer are rotational. When the Reynolds number
become very large and the boundary layer would become thinner, as such the entire
flow might be taken as an irrotational flow for a good approximation, this is a
good practice in many practical problems
for an irrotational flow, the zero curl of the velocity vector would lead to

English: 
a velocity potential function, Phi and the fluid velocity vector of three
independent components are transformed 
to a scalar function, Phi, and this
assumption of irrotational flow could simplify the dynamic problem very much
in a Cartesian coordinate,  the three velocity components are calculated from
the velocity potential function as this, so if we obtain the velocity potential
function and then we can calculate the velocity components, use these three
equations

English: 
a velocity potential function, Phi and the fluid velocity vector of three
independent components are transformed to a scalar function, Phi, and this
assumption of irrotational flow could simplify the dynamic problem very much
in a Cartesian coordinate, the three velocity components are calculated from
the velocity potential function as this, so if we obtain the velocity potential
function and then we can calculate the velocity components, use these three equations,

English: 
if the flow is incompressible, the corresponding continuity equation could
lead to Laplace equation for the velocity potential function. we have seen
many applications by solving the Laplace equation in the practical problems
This is generally acceptable when the flow velocity is small, for instance,
less than 0.3 Mach number
if the flow is compressible, the continuity equation would lead to a partial
differential equation for the velocity potential function as this.
so we can see this partial differential equation for the velocity potential

English: 
if the flow is incompressible, the corresponding continuity equation could
lead to Laplace equation for the velocity potential function. we have seen
many applications by solving the Laplace equation in the practical problems
This is generally acceptable when the flow velocity is small, for instance,
less than 0.3 Mach number
if the flow is compressible, the continuity equation would lead to a partial
differential equation for the velocity potential function as this
so we can see this partial differential equation for the velocity potential

English: 
function is much more complicated than the Laplace equation. and also here we
have one more unknown, the fluid density, and we also need to solve the fluid
density in the dynamics. thus we need a one more equation for the
problem, for instance, we may use the idea gas equation
if the compressible flow is steady and the assumptions of an idea gas,
and small disturbance in the flow field as this, so we can get the partial
differential equation for the velocity potential function, Phi, as this. Here M

English: 
function is much more complicated than the Laplace equation. and also here we
have one more unknown, the fluid density, and we also need to solve the fluid
density in the dynamics. thus we need a one more equation for the
problem, for instance, we may use the idea
gas equation
if the compressible flow is steady and the assumptions of an idea gas,
and small disturbance in the flow field as this, so we can get the partial
differential equation for the velocity potential function, Phi, as this. Here M

English: 
is the Mach number. For more details how we can get this partial differential
equation for the velocity potential function can be found in the book, 'Modern
Compressible Flow' of John D Anderson.
from the full Navier-Stokes equation, we can see, to obtain an inviscid flow,
the last two terms must vanish. For this purpose, the simple
analyses are made for these two terms.
to make sure these equations are satisfied
the velocity vector gradient tensor or the velocity vector divergence are 0,
for instance, a uniform flow can satisfy this.

English: 
is the Mach number. For more details how we can get this partial differential
equation for the velocity potential function can be found in the book, 'Modern
Compressible Flow' of John D Anderson.
from the full Navier-Stokes equation, we can see, to obtain an inviscid flow,
the last two terms must vanish. For this purpose the simple
analyses are made for these two terms.
to make sure these equations are satisfied
the velocity vector gradient tensor or the velocity vector divergence are 0,
for instance, a uniform flow can satisfy this.

English: 
To ensure the term of this and this are both zero,
this can be satisfied in incompressible irrotational flow, since for
incompressible irrotational flows, we have this and this, however, for
compressible flows, these two terms are not zero.
Now we look at the last
two terms in the full Navier-Stokes equation for irrotational flows, the last
two terms can be written as this, and put these together, we have equations as this.
and for the convective acceleration
we can use the vector calculus identity, as this. Consider the flow is an irrotational

English: 
To ensure the term of this and this are both zero,
this can be satisfied in incompressible irrotational flow, since for
incompressible irrotational flows, we have this and this, however, for
compressible flows, these two terms are not zero.
Now we look at the last
two terms in the full Navier-Stokes equation for irrotational flows, the last
two terms can be written as this, and put these together, we have equations as this.
and for the convective acceleration
we can use the vector calculus identity, as this. Consider the flow is an irrotational

English: 
flow, so we can have this, if the fluid gravitational force is the only body
force, we can put all this together, we have the equation for the irrotational
flows as this.
so the integral form for the irrotational flow is given by this
this is the general Bernoulli's equation, under the only assumption of
irrotational flow, here we don't make the assumptions for inviscid flow or
incompressible flow. if the irrotational flow is incompressible, then
the term related to the fluid the viscosity disappears
so we have the conventional Bernoulli's equation for irrotational flow

English: 
flow, so we can have this, if the fluid gravitational force is the only body
force, we can put all this together, we have the equation for the irrotational
flows as this
so the integral form for the irrotational flow is given by this
this is the general Bernoulli's equation, under the only assumption of
irrotational flow, here we don't make the assumptions for inviscid flow or
incompressible flow. if the irrotational flow is incompressible, then
the term related to the fluid the viscosity disappears
so we have the conventional Bernoulli's equation for irrotational flow

English: 
this is the same equation as we seen in many textbooks,
however for deriving
this Bernoulli's equation, the assumption of an inviscid flow is not required.
and the Bernoulli's equation is automatically satisfied when the
irrotational flow is assumed to be incompressible

English: 
this is the same equation as we seen in many textbooks,
however for deriving
this Bernoulli's equation, the assumption of an inviscid flow is not required.
and the Bernoulli's equation is automatically satisfied when the
irrotational flow is assumed to be incompressible
