
English: 
Hello welcome to my talk All about Fluids.
This talk is on a comparison of using the
fluid element method and the Reynolds transport theorem for deriving
Navier-Stokes equation, and we will talk how we can derive Navier-Stokes
equation correctly.
The conventional methods for deriving Navier-Stokes equation include the fluid
element method and the using the Reynolds transport theorem.
the fluid element method
directly works on a small fluid element, such as the cube in the figure
Based on the conservation law, we can build the dynamic equation for

English: 
Hello welcome to my talk, All about Fluids.
this talk is on a comparison of using the
fluid element method and the Reynolds transport theorem for deriving
Navier-Stokes equation, and we will talk how we can derive Navier-Stokes
equation correctly
The conventional methods for deriving Navier-Stokes equation include the fluid
element method and the using the Reynolds transport theorem.
the fluid element method
directly works on a small fluid element, such as the cube in the figure
Based on the conservation law, we can build the dynamic equation for

English: 
the fluid element in a very intuitive manner, but the substantial derivative
must be used since the physical parameter f would be a function of
the independent spatial variables x, y, and z and time t. This derivation
is a very straightforward with the obvious physics, but the substantial
derivative can be a problem in the compressible flows.
the Reynolds transport theorem works on the whole fluid domain, examines the
overall behavior of the fluid. In the derivation the control volume cv is used

English: 
the fluid element in a very intuitive manner, but the substantial derivative
must be used since the physical parameter f would be a function of
the independent spatial variables x, y, and z and time t. This derivation
is a very straightforward with the obvious physics, but the substantial
derivative can be a problem in the compressible flows.
the Reynolds transport theorem works on the whole fluid domain, examines the
overall behavior of the fluid. In the derivation the control volume cv is used

English: 
thus the volume integral form of the physical parameter f. That is, the
quality of this physical parameter would be the function of time only. As such
the substantial derivative can be avoided. Using the conservation laws and
the Reynolds transport theorem,  the Navier-Stokes equation can be easily
derived, where the understanding of the Reynolds transport theorem would be
very important. In both derivation methods, two types of the force must be
considered: the body force, which is generally proportional to the fluid
volume.  For instance, the gravitational force per unit volume
rho*g.  And the surface force is a force of the surface integral over the

English: 
thus the volume integral form of the physical parameter f. That is, the
quality of this physical parameter would be the function of time only. As such
the substantial derivative can be avoided. Using the conservation laws and
the Reynolds transport theorem, the Navier-Stokes equation can be easily
derived, where the understanding of the Reynolds transport theorem would be
very important. In both derivation methods, two types of the force must be
considered: the body force, which is generally proportional to the fluid
volume. For instance, the gravitational force per unit volume
rho*g.  And the surface force is a force of the surface integral over the

English: 
stresses: the stresses due to the fluid pressure, fluid viscous stresses
and fluid compressibility
In the fluid element method, the derivation is made directly on the
physical parameter f. In the Eulerian representation, the physical parameter
f would be the function of the independent spatial variables x, y and
z and time.  thus its time rate of change is generally given in form of the
substantial derivative based on the rule of calculus, so we have the derivative f

English: 
stresses: the stresses due to the fluid pressure, fluid viscous stresses
and fluid compressibility
In the fluid element method, the derivation is made directly on the
physical parameter f. In the Eulerian representation, the physical parameter
f would be the function of the independent spatial variables x, y and
z and time. Thus its time rate of change is generally given in form of the
substantial derivative based on the rule of calculus, so we have the derivative f

English: 
with regard to time, given by this. and use velocity components, we have
this and this
here we can see the expression as this. It is the conventional form for the
substantial derivative.
Reynolds transport theorem works on the quality of the physical parameter
the volume integral of the physical parameter f, the quality I. As a result of
the volume integral, the quality of the physical parameter is the
function of time only, it is independent of the spatial variables x, y and z.
As such the time rate of change of the physical quality would be same as the

English: 
with regard to time, given by this. and use velocity components, we have
this... and this
here we can see the expression as this. It is the conventional form for the
substantial derivative.
Reynolds transport theorem works on the quality of the physical parameter
the volume integral of the physical parameter f, the quality I. As a result of
the volume integral, the quality of the physical parameter is the
function of time only, it is independent of the spatial variables x, y and z.
As such the time rate of change of the physical quality would be same as the

English: 
normal derivative, so no substantial derivative is needed.
And the Reynolds transport theorem calculates the time rate of the
change of the physical quality and given in this form.
how to derive the Reynolds transport theorem can be found in my talk,
'Reynolds transport theorem'
In the next few slides the continuity equation will be derived, use both
fluid element method and the Reynolds transport theorem
These two methods are used to show the difference between these two methods,
mainly on that the different ways can be used to derive the continuity equation

English: 
normal derivative, so no substantial derivative is needed.
And the Reynolds transport theorem calculates the time rate of the
change of the physical quality and given in this form.
how to derive the Reynolds transport theorem can be found in my talk
'Reynolds transport theorem'.
In the next few slides the continuity equation will be derived, use both
fluid element method and the Reynolds transport theorem
These two methods are used to show the difference between these two methods,
mainly on that the different ways can be used to derive the continuity equation

English: 
Here the fluid element method is used for deriving the continuity equation.
Consider a cube, it is the fluid element
For a time interval delta-t, the fluid mass in y direction flows in the cube from the
vertical surface delta-Sy at left hand side, and at the same time the fluid
mass flows out of the cube from the vertical surface delta-Sy at right
hand side. Therefore the net mass increment is the
difference between the mass flowing in less the mass flowing out
thus the mass increment can be calculated as this in y direction

English: 
Here the fluid element method is used for deriving the continuity equation.
Consider a cube, it is the fluid element
For a time interval delta-t, the fluid mass in y direction flows in the cube from the
vertical surface delta-Sy at left hand side, and at the same time the fluid
mass flows out of the cube from the vertical surface delta-Sy at right
hand side. Therefore, the net mass increment is the
difference between the mass flowing in less the mass flowing out,
thus the mass increment can be calculated as this in y direction

English: 
In a similar manner we can calculate the mass in x direction given as this and
in z-direction, given as this
So if we put all these mass increments together, so the total mass change is
given as this. At same time, since the volume of
the fluid element is fixed, delta V, given by delta x times delta y times delta z.
so the mass is given by this formula, its
increment would also cause the change of density as this. So put these two mass
increments together, we have the equation as this. And we can

English: 
In a similar manner we can calculate the mass in x direction given as this and
in z-direction, given as this.
So if we put all these mass increments together, so the total mass change is
given as this. At same time, since the volume of
the fluid element is fixed, delta V, given by delta x times delta y times delta z.
so the mass is given by this formula, its
increment would also cause the change of density as this. So put these two mass
increments together, we have the equation as this. And we can

English: 
write the equation as this. Here we must note the density differentiation
with regard to time is given by a partial differentiation, it is because
the density change here is due to the mass increment within the time interval
delta t only. So we can write this equation in a
vector form as this. This is the continuity equation
We can also use the Reynolds transport theorem to derive the continuity equation
by simply taking the physical parameter f as the fluid density

English: 
write the equation as this. Here we must note the density differentiation
with regard to time is given by a partial differentiation, it is because
the density change here is due to the mass increment within the time interval
delta t only. So we can write this equation in a
vector form as this. This is the continuity equation
We can also use the Reynolds transport theorem to derive the continuity equation
by simply taking the physical parameter f as the fluid density

English: 
It can be seen that the volume integral of the fluid density is actually
the total mass of the fluid.
in the control volume
the total mass is unchangeable. Thus its time rate of the fluid mass
must be 0, given as this. Applying the Reynolds transport theorem
in a form as this
we can obtain the integral form of the continuity equation, as this part
From this integral form of continuity equation
we can easily deduce the differential form of a continuity equation given as this
this is exactly same as that continuity
equation from the fluid element method.

English: 
It can be seen that the volume integral of the fluid density is actually
the total mass of the fluid.
in the control volume
the total mass is unchangeable. Thus its time rate of the fluid mass
must be 0, given as this. Applying the Reynolds transport theorem
in a form as this
we can obtain the integral form of the continuity equation, as this part
From this integral form of continuity equation
we can easily deduce the differential form of a continuity equation given as this
this is exactly same as that continuity
equation from the fluid element method.

English: 
In the next few slides the moment equation will be derived using the
fluid element method and the Reynolds transport theorem, respectively. And we
will see in deriving the Navier-Stokes equation we must use the substantial
derivative carefully.
In this slide, following Stokes, the total stress tensor is given as capital T
double arrow. This is a combination of the stresses
from the pressure and the compressibility contributions and here
the second viscous coefficient, LAMBDA given by this coefficient

English: 
In the next few slides the moment equation will be derived using the
fluid element method and the Reynolds transport theorem, respectively. And we
will see in deriving the Navier-Stokes equation we must use the substantial
derivative carefully.
In this slide, following Stokes, the total stress tensor is given as capital T
double arrow. This is a combination of the stresses
from the pressure and the compressibility contributions and here
the second viscous coefficient, LAMBDA given by this coefficient

English: 
and the viscous stress tensor is an asymmetrical stress tensor. It is derived
directly from the fluid motion, rather than the conventional artificial
construction of the symmetrical stress tensor.
If you would like to see more information about this asymmetrical stress tensor,
Please see my talk: 'Is Navier-Stokes equation correct?'
Using the summation expression for the divergence of the viscous stress tensor
and the definition of the stress tensor component, we can have this
and we can obtain the expression as this.
write it back to the vector form, it

English: 
and the viscous stress tensor is an asymmetrical stress tensor. It is derived
directly from the fluid motion, rather than the conventional artificial
construction of the symmetrical stress tensor.
If you would like to see more information about this asymmetrical stress tensor,
Please see my talk: 'Is Navier-Stokes equation correct?'
Using the summation expression for the divergence of the viscous stress tensor
and the definition of the stress tensor component, we can have this
and we can obtain the expression as this.
write it back to the vector form, it

English: 
is as this. So we can obtain the divergence of
the total stress tensor, given as in this form. This is a very
useful equation, we will use this in both derivation methods.
So on the vertical surface at the left hand and right hand sides,
the net surface forces in y direction is given and calculates as this
due to the total stress tensor component T_22, and here delta V for the fluid
element is given as delta x times delta y times delta z.

English: 
is as this. So we can obtain the divergence of
the total stress tensor, given as in this form. This is a very
useful equation,  we will use this in both derivation methods.
So on the vertical surface at the left hand and right hand sides,
the net surface forces in y direction is given and calculates as this
due to the total stress tensor component T_22, and here delta V for the fluid
element is given as delta x times delta y times delta z.

English: 
Similarly for the stress tensor component T_23, we can obtain the net force in y
direction for the stress tensor component T_23.
And the we can also calculate the net force in y direction due to the tensor
component T_21. So we put all these surface forces
together, we get the net surface force in y direction as this
Now for the dynamics of the motion of the fluid element,
the Cauchy's momentum equation is used. It should be noted this is a

English: 
Similarly for the stress tensor component T_23, we can obtain the net force in y
direction for the stress tensor component T_23.
And the we can also calculate the net force in y direction due to the tensor
component T_21. So we put all these surface forces
together, we get the net surface force in y direction as this
Now for the dynamics of the motion of the fluid element,
the Cauchy's momentum equation is used. It should be noted this is a

English: 
different from the actual Newton's second law of motion, but it is called
Newton's second law of motion in many textbooks, since it can be
superficially derived from this equation. This part is m*a_y,
and this part is the body force on the element and this is the surface force on
the element. However, we must remember in this case, for the fluid element, the mass
is not constant if the flow is compressible.
Now use the substantial derivative, we can write this equation further into
this form. Similarly we can derive the momentum
equations on x- and z- directions as these.

English: 
different from the actual Newton's second law of motion, but it is called
Newton's second law of motion in many textbooks, since it can be
superficially derived from this equation. This part is m*a_y,
and this part is the body force  on the element and this is the surface force on
the element. However, we must remember in this case, for the fluid element, the mass
is not constant if the flow is compressible.
Now use the substantial derivative, we can write this equation further into
this form. Similarly we can derive the momentum
equations on x- and z- directions as these.

English: 
Now put these together and apply the second viscous coefficient, we can obtain
the momentum equation in a vector form as this. Here we can see the
substantial derivative of the fluid acceleration given as this. And this part
is the divergence of the total stress tensor T double arrow.
For the momentum equation based on the Reynolds transport theorem, it is
based on the Newton's second law of motion, the time rate of the change of the
total momentum equals to the total body force and the total surface force together.

English: 
Now put these together and apply the second viscous coefficient, we can obtain
the momentum equation in a vector form as this. Here we can see the
substantial derivative of the fluid acceleration given as this. And this part
is the divergence of the total stress tensor T double arrow.
For the momentum equation based on the Reynolds transport theorem, it is
based on the Newton's second law of motion, the time rate of the change of the
total momentum equals to the total body force and the total surface force together.

English: 
Here we can see the time rate of change of the total momentum.
this is the total body force and this is the total surface force.
Why we can use the Newton's second law of motion directly here. It is because the overall
equality, i.e., the total momentum, is the function of time only. This is very
different from the individual physical parameter f, which is basically a
function of the spatial variables x, y, z and the time t.
so we can use the Gauss divergence theorem to change the total surface force

English: 
Here we can see the time rate of change of the total momentum.
this is the total body force and this is the total surface force.
Why we can use the Newton's second law of motion directly here. It is because the overall
equality, i.e., the total momentum, is the  function of time only. This is very
different from the individual physical parameter f, which is basically a
function of the spatial variables x, y, z and the time t.
so we can use the Gauss divergence theorem to change the total surface force

English: 
into a volume integral as this. And then we can apply the Reynolds transport theorem
so we have the integral momentum
equation for the fluid as this. And from this we can easily derive the
differential momentum equation for the fluid.
Again the expression for the divergence of the total stress tenor has been applied.
Now we can make some more mathematical manipulations on the momentum equation
from the Reynolds transport theorem. The vector form of the left-hand side
can be written in Einstein's summation as this

English: 
into a volume integral as this. And then we can apply the Reynolds transport theorem
so we have the integral momentum
equation for the fluid as this. And from this we can easily derive the
differential momentum equation for the fluid.
Again the expression for the divergence of the total stress tenor has been applied.
Now we can make some more mathematical manipulations on the momentum equation
from the Reynolds transport theorem. The vector form of the left-hand side
can be written in Einstein's summation as this

English: 
and  use the rule of calculus, you can write the terms as these.
So the first two terms from the first partial differentiation and the last
two terms are coming from the second partial differentiation. Re-order the
terms, we can obtain the expression as this. Now when we write these terms back
into the vector form, we have the expression as this, and we can see this
part vanishes automatically due to the continuity equation, so the left hand
side of the momentum equation is given by this. So we can obtain the equation as
this. So this is exactly same momentum

English: 
and use the rule of calculus, you can write the terms as these.
So the first two terms from the first partial differentiation and the last
two terms are coming from the second partial differentiation. Re-order the
terms, we can obtain the expression as this. Now when we write these terms back
into the vector form, we have the expression as this, and we can see this
part vanishes automatically due to the continuity equation, so the left hand
side of the momentum equation is given by this. So we can obtain the equation as
this. So this is exactly same momentum

English: 
equation as that from the fluid element method.
Now in this slide a discussion would be made for what is wrong when using
the Newton's second law of motion directly on the fluid element. We know
We know in spite of of the different form in original Newton's second law of motion
it is for the constant mass of the body. The universal Newton's second law of
motion should be the time rate of the change of the momentum equals to the
force applied on the body. So in real form, the Newton's second law of
motion should be given as this: this term is time rate of change of

English: 
equation as that from the fluid element method.
Now in this slide a discussion would be made for what is wrong when using
the Newton's second law of motion directly on the fluid element. We know
We know in spite of the different form in original Newton's second law of motion
it is for the constant mass of the body. The universal Newton's second law of
motion should be the time rate of the change of the momentum equals to the
force applied on the body. So in real form, the Newton's second law of
motion should be given as this: this term is time rate of change of

English: 
the momentum for the fluid element and this term is the body force acting on
the fluid element and this part is the surface force acting on the element.
so we cancel out the constant value delta V, we have the differential equation as
this. And similarly we can have the momentum equation in x direction and
z directions. As such if we put all these together, we have the momentum
equation as this. So we use the definition of the
substantial derivative, the momentum equation can be written as this. And re-order
the terms in the left hand side, we have the equation as this.

English: 
the momentum for the fluid element and this term is the body force acting on
the fluid element and this part is the surface force acting on the element.
so we cancel out the constant value delta V,  we have the differential equation as
this. And similarly we can have the momentum equation in x direction and
z directions. As such if we put all these together we have the momentum
equation as this. So we use the definition of the
substantial derivative, the momentum equation can be written as this. And re-order
the terms in the left hand side, we have the equation as this.

English: 
If we compare this momentum equation with the correct momentum equation we derived before,
we can see the extra term here and this term would be 0,
if the incompressible flows are considered. That  means if the Newton's second
law of motion is applied directly on the fluid element, the momentum
equation would be not correct for the compressible flows. This is a big question
for this term.
In this slide, we tried to make some explanations on why we cannot use the
Newton's second law of motion directly on the fluid element.
Howard Brenner explained the correctness of Navier-Stokes equation

English: 
If we compare this momentum equation with the correct momentum equation we derived before,
we can see the extra term here and this term would be 0,
if the incompressible flows are considered. That means if the Newton's second
law of motion is applied directly on the fluid element, the momentum
equation would be not correct for the compressible flows. This is a big question for this term.
In this slide, we tried to make some explanations on why we cannot use the
Newton's second law of motion directly on the fluid element.
Howard Brenner explained the correctness of Navier-Stokes equation

English: 
for compressible flows in his papers in 2005 titled 'Navier-Stokes revisited'
and in 2012 titled 'Beyond Navier-Stokes', he gave the definition
of the substantial derivative as this. Here the vector Vm is the Eulerian mass
transport velocity and for the fluid motion we should have two different
velocity vector Vm, the mass velocity and the vector Vv the volume velocity.
According to the book 'the Navier-Stokes
Problem in the 21st century',  P.G. Lemarie Rieusset further explained the concept: one

English: 
for compressible flows in his papers in 2005 titled 'Navier-Stokes revisited'
and in 2012 titled 'Beyond Navier-Stokes', he gave the definition
of the substantial derivative as this. Here the vector Vm is the Eulerian mass
transport velocity and for the fluid motion we should have two different
velocity vector Vm, the mass velocity and the vector Vv the volume velocity.
According to the book 'the Navier-Stokes
Problem in the 21st century', P.G. Lemarie Rieusset further explained the concept: one

English: 
must distinguish the Euler mass transportation velocity Vm and the
Lagrangian fluid particle velocity vector, Vv in the compressible flows.
In the definition the momentum equation should be given as this. Here we can see
the fluid volume velocity, vector Vv is used as the representation of the fluid
velocity, but in that substantial derivative the velocity here is
different. It is the mass velocity.
And a formula is given the difference between
these two velocities as this. Obviously if the flow is incompressible and these two
velocities would be same but they are different in compressible flows.

English: 
must distinguish the Euler mass transportation velocity Vm and the
Lagrangian fluid particle velocity vector, Vv in the compressible flows.
In the definition the momentum equation should be given as this. Here we can see
the fluid volume velocity, vector Vv is used as the representation of the fluid
velocity, but in that substantial derivative the velocity here is
different. It is the mass velocity.
And a formula is given the difference between
these two velocities as this. Obviously if the flow is incompressible and these two
velocities would be same but they are different in compressible flows.
