
English: 
Hello welcome to my talk, All about Fluids. This talk is on Reynolds transport theorem
which is a theorem for calculating the time rate of change of
physical quality in the control volume in the continuum mechanics, especially
in fluid mechanics. This theorem has been widely used for deriving the
Navier-Stokes equation on the basis of the conservation laws. A simplest
example is the fluid continuity equation which can be derived using the
Reynolds transport theorem based on the mass conservation.
in deriving Navier-Stokes equation, different methods can be used. The most

English: 
Hello welcome to my talk ,All about Fluids. this talk is on Reynolds transport theorem
which is a theorem for calculating the time rate of change of
physical quality in the control volume in the continuum mechanics, especially
in fluid mechanics. This theorem has been widely used for deriving the
Navier-Stokes equation on the basis of the conservation laws. A simplest
example is the fluid continuity equation  which can be derived using the
Reynolds transport theorem based on the mass conservation.
in deriving Navier-Stokes equation, different methods can be used. The most

English: 
used methods include element method and the method using the Reynolds
transport theorem.
The former uses the intuitive fluid element for formulating the fluid dynamic equation
For the fluid element we can easily calculate the quality of a fluid
parameter flowing in from the surface at the left-hand side and flowing out from the
surface at the right-hand side of the element
thus we can build the dynamic equation in a very direct way, based on the
principle of conservation laws. However, in such a case we must use  a special

English: 
used methods include element method and the method using the Reynolds transport theorem.
The former uses the intuitive fluid element for formulating the fluid dynamic equation
For the fluid element we can easily calculate the quality of a fluid
parameter flowing in from the surface at the left-hand side and flowing out from the
surface at the right-hand side of the element
thus we can build the dynamic equation in a very direct way, based on the
principle of conservation laws. However, in such a case we must use a special

English: 
derivative named substantial derivative or material derivative.
but it would be a problem, especially for compressible flows, in which we must use
the substantial derivative carefully, while using the Reynolds transport theorem
to derive Navier-Stokes equation is easy, but the principle
behind the Reynolds transport theorem is not straightforward. This is
why we need to derive the theorem. so we can understand the principle
behind the theorem, this is the purpose of this talk.
What is the Reynolds transport theorem?

English: 
derivative named substantial derivative or material derivative.
but it would be a problem, especially for compressible flows, in which we must use
the substantial derivative carefully, while using the Reynolds transport theorem
to derive Navier-Stokes equation is easy, but the principle
behind the Reynolds transport theorem is not straightforward. This is
why we need to derive the theorem. so we can understand the principle
behind the theorem, this is the purpose of this talk.
What is the Reynolds transport theorem?

English: 
Reynolds transport theorem is the theorem can be used for calculating
the time rate of change of a quality of a physical parameter, especially in fluid
mechanics and the large deformation solid mechanics
Principally, Reynolds transport theorem works on the whole fluid domain of  interest,
or so called the fluid control volume for the
whole fluid domain as seen in this figure. The broken black line CV is the
control volume, which would be in a manner that we can avoid to use the
substantial derivative directly, hence the choice of the control volume must

English: 
Reynolds transport theorem is the theorem can be used for calculating
the time rate of change of a quality of a physical parameter, especially in fluid
mechanics and the large deformation solid mechanics
Principally, Reynolds transport theorem works on the whole fluid domain of interest,
or so called the fluid control volume for the
whole fluid domain as seen in this figure. The broken black line CV is the
control volume, which would be in a manner that we can avoid to use the
substantial derivative directly, hence the choice of the control volume must

English: 
be different from the fluid element.  Reynolds transport theorem calculates the
transport of the quality of a physical parameter using the control volume.
This includes the change of the quality within the control volume and
the quality flowing in and flowing out of the control volume due to the fluid flowing
motion, such as dilatation, rotation or shear deformation.
The Navier-Stokes equation can be directly derived using the Reynolds transport
theorem, together with the basic conservation laws of continuum
mechanics, such as law of conservation of mass,
the law of conservation of linear momentum or the law of the conservation of energy

English: 
be different from the fluid element. Reynolds transport theorem calculates the
transport of the quality of a physical parameter using the control volume.
This includes the change of the quality within the control volume and
the quality flowing in and flowing out of the control volume due to the fluid flowing
motion, such as dilatation, rotation or shear deformation.
The Navier-Stokes equation can be directly derived using the Reynolds transport
theorem, together with the basic conservation laws of continuum
mechanics, such as law of conservation of mass,
the law of conservation of linear momentum or the law of the conservation of energy

English: 
To derive Reynolds transport theorem, we can consider the fluid domain or the
control volume. At time t, the fluid volume is
represented by V(t) and its closed surface S(t),
we can imagine at the time t plus delta-t, the volume would become
V(t+delta t) and the enclosed surface would be S(t+delta t)
Here we must note the fluid volume change with time is not the motion of
fluid domain or the control volume. Principally the control volume would be

English: 
To derive Reynolds transport theorem we
can consider the fluid domain or the
control volume. At time t, the fluid volume is
represented by V(t) and its closed surface S(t),
we can imagine at the time t plus delta-t, the volume would become
V(t+delta t) and the enclosed surface would be S(t+delta t)
Here we must note the fluid volume change with time is not the motion of
fluid domain or the control volume. Principally the control volume would be

English: 
fixed in the space, but due to the fluid deformation, the fluid volume can be
changed, so the volume V would be a function of the time only.
the volume integral of a physical parameter or the quality of the physical
parameter or simply the quality would be a function of time only as this,
even though the physical parameter is the function of spatial parameters r
and the time t, but the volume integral is the function of time only.
Here we use scalar V to represent the fluid volume
so the time rate of change of the

English: 
fixed in the space, but due to the fluid deformation, the fluid volume can be
changed, so the volume V would be a function of the time only.
the volume integral of a physical parameter or the quality of the physical
parameter or simply the quality would be a function of time only as this,
even though the physical parameter is the function of spatial parameters r
and the time t,  but the volume integral is the
function of time only.
Here we use scalar V to represent the the fluid volume
so the time rate of change of the

English: 
volume integral would be same, regardless the derivative is the substantial
as this or the normal as this. The reason for this is because the integral is the
function of time only, it's independent of the spatial parameters x, y and z.
So this is why we see both expressions in different textbooks.
And we can understand this: the system of the whole fluid is very similar
to the system of a solid in principle, even though the fluid
inside the control volume is very complicated, but the overall behaviour of
the whole fluids would be similar to the solid dynamics

English: 
volume integral would be same, regardless the derivative is the substantial
as this or the normal as this. The reason for this is because the integral is the
function of time only, it's independent of the spatial parameters x, y and z.
So this is why we see both expressions in different textbooks.
And we can understand this: the system of the whole fluid is very similar
to the system of a solid in principle, even though the fluid
inside the control volume is very complicated, but the overall behavior of
the whole fluids would be similar to the solid dynamics

English: 
For a convenience we can define a quality I as the volume integral of a
physical parameter F. This is a defined quality.
Here both the volume V and the volume integral I are functions of time only.
For example, we can define the following qualities: the fluid mass if we take F
equals to the fluid density Rho, we have the mass of the total fluid.
and we can define the fluid momentum as this if we take F equals to

English: 
For a convenience we can define a quality I as the volume integral of a
physical parameter F. This is a defined quality.
Here both the volume V and the volume integral I are functions of time only.
For example, we can define the following qualities: the fluid mass if we take F
equals to the fluid density Rho, we have the mass of the total fluid.
and we can define the fluid momentum as  this if we take F equals to

English: 
Rho times vector V. Here vector V is the flow velocity.
The Reynolds transport theorem is for calculating the time rate of
change of the quality of F as this, the question is what is the result of
with the time rate of change of the quality?
So this is the Reynolds transport theorem for giving the formulation to calculate the
time rate of change of the volume integral of a physical parameter F
Here the increment of the defined quality I over the time interval delta t

English: 
Rho times vector V. Here vector V is the flow velocity.
The Reynolds transport theorem is for calculating the time rate of
change of the quality of F as this, the question is what is the result of
with the time rate of change of the quality?
So this is the Reynolds transport theorem for giving the formulation to calculate the
time rate of change of the volume integral of a physical parameter F
Here the increment of the defined quality I over the time interval delta t

English: 
given by this. The increment for the defined quality
comes from two different changes: the change of the physical parameter F
and the change of the fluid volume, V.
So we can calculate this using the Taylor series and keep the first-order term
and drop all the higher-order terms
we have this for the volume, and for the physical parameter F
and we can also obtain the result for this change in the physical parameter in
the time interval delta-t.

English: 
given by this. The increment for the defined quality
comes from two different changes: the change of the physical parameter F
and the change of the fluid volume, V.
So we can calculate this using the Taylor series and keep the first-order term
and drop all the higher-order terms
we have this for the volume, and for the physical parameter F
and we can also obtain the result for this change in the physical parameter in
the time interval delta-t.

English: 
So for calculating the increment of defined quality I over the time interval delta t,
we can separate the integral on volume, so we have this expression.
if we re-order the formulation we have this,
And we can further write the formulation as this. using the
formulation increment of the physical parameter F, so we have this equation,
the increment of the defined quality I is made up by two parts:
the first part is due to the change of the physical parameter F within the

English: 
So for calculating the increment of defined quality I over the time interval delta t,
we can separate the integral on volume, so we have this expression.
if we re-order the formulation we have this,
And we can further write the formulation as this. using the
formulation increment of the physical parameter F, so we have this equation,
the increment of the defined quality I is made up by two parts:
the first part is due to the change of the physical parameter F within the

English: 
control volume and the second part is the change due to the volume
change delta V
Look at the figure we can see the change of the volume delta V can be calculated
at the small surface dS moving in its normal direction n at the velocity Un
in the time interval delta t, thus dV is given as this.
Therefore, we can change the integral on delta V this into the integral over the
surface S as this

English: 
control volume and the second part is the change due to the volume
change delta V
Look at the figure we can see the change of the volume delta V can be calculated
at the small surface dS moving in its normal direction n at the velocity Un
in the time interval delta t,  thus dV is given as this
therefore we can change the integral on delta V this into the integral over the
surface S as this.

English: 
Now we can obtain the time rate of change of the defined quality I as this
this is known as the Reynolds transport theorem.
Basically the surface integral can be
simply understood for representing the transport of the quality of F out of
the volume V as a result of the change of the fluid boundary S,
here because of the fluid velocity
Now we can do some more derivations on Reynolds transport theorem.
We can use the Gauss divergence theorem so we can change the surface integral

English: 
Now we can obtain the time rate of change of the defined quality I as this
this is known as the Reynolds transport theorem.
Basically the surface integral can be
simply understood for representing the transport of the quality of F out of
the volume V as a result of the change of the fluid boundary S,
here because of the fluid velocity
Now we can do some more derivations on Reynolds transport theorem.
We can use the Gauss divergence theorem so we can change the surface integral

English: 
into the volume integral, so we have this Reynolds transport theorem.
This is the form of Reynolds transport theorem which is used
for deriving the Navier-Stokes equation including the continuity equation based
on the mass conservation, the momentum equation based on the Newton's second
law of motion or the momentum conservation, and the energy equation
based on the energy conservation
if we take the physical parameter F as the fluid density, then the defined
quality is actually the fluid mass, therefore within the control volume,

English: 
into the volume integral, so we have this Reynolds transport theorem.
This is the form of Reynolds transport theorem which is used
for deriving the Navier-Stokes equation including the continuity equation based
on the mass conservation, the momentum equation based on the Newton's second
law of motion or the momentum conservation, and the energy equation
based on the energy conservation
if we take the physical parameter F as the fluid density, then the defined
quality is actually the fluid mass, therefore within the control volume,

English: 
the fluid mass would be a constant, thus the time rate is zero
by employing the Reynolds transport theorem, we can easily obtain the Integral
continuity equation as this. This is the integral continuity equation.
And from this equation we can easily derive the differential continuity equation as this
this is the form we can see in many textbooks.
And this is the universal continuity
equation for both compressible and incompressible flows
if the flow is incompressible, its density would be a constant. Then we
could have a simpler continuity equation as this for incompressible flows

English: 
the fluid mass would be a constant, thus the time rate is zero
by employing the Reynolds transport theorem, we can easily obtain the Integral
continuity equation as this. This is the integral continuity equation.
And from this equation we can easily derive the differential continuity equation as this
this is the form we can see in many textbooks.
And this is the universal continuity
equation for both compressible and incompressible flows
if the flow is incompressible, its density would be a constant. Then we
could have a simpler continuity equation as this for incompressible flows

English: 
For the flow momentum equation we can take the physical parameter F as rho times
the fluid velocity vector V. As such the defined equality would be the total
momentum of the fluid. Based on the Newton's second law of
motion, the time rate of the change of the fluid momentum would equal to the
total force acting on the fluid, including the body force as this and the
surface force as this. Here vector f is the body force density and the tensor T
double arrow is the total tensor of the fluid.

English: 
For the flow momentum equation we can take the physical parameter F as rho times
the fluid velocity vector V. As such the defined equality would be the total
momentum of the fluid. Based on the Newton's second law of
motion, the time rate of the change of the fluid momentum would equal to the
total force acting on the fluid, including the body force as this and the
surface force as this. Here vector f is the body force density and the tensor T
double arrow is the total tensor of the fluid.

English: 
Applying the Reynolds transport theorem to this term,
we can have the momentum equation as this
We can apply the Gauss divergence theorem for this term, so we can change
the integral of the surface into the volume integral of the tensor divergence
so we have the integral equation for the momentum. From this we can easily derive
the differential momentum equation as this
Here double vector V is actually a tensor
In this slide, we can carry out some mathematical manipulations for the

English: 
Applying the Reynolds transport theorem to this term,
we can have the momentum equation as this
We can apply the Gauss divergence theorem for this term, so we can change
the integral of the surface into the volume integral of the tensor divergence
so we have the integral equation for the momentum. From this we can easily derive
the differential momentum equation as this
Here double vector V is actually a tensor
In this slide, we can carry out some mathematical manipulations for the

English: 
momentum equation, together with the continued equation. The continuity
equation in vector form is as this, and we can write this continuity equation in
Einstein summation as this. Here we have two subscripts j which
means the summation as this. So for the momentum equation, this is the
momentum equation in a vector form, its corresponding Einstein summation is as this
now we can use the rule of calculus, we can
obtain the equation as this. and the first term becomes two terms
in the square bracket and the second term in this momentum equation, we have two

English: 
momentum equation, together with the continued equation. The continuity
equation in vector form is as this, and we can write this continuity equation in
Einstein summation as this. Here we have two subscripts j which
means the summation as this. So for the momentum equation, this is the
momentum equation in a vector form,  its corresponding Einstein summation is as this
now we can use the rule of calculus, we can
obtain the equation as this. and the first term becomes two terms
in the square bracket and the second term in this momentum equation, we have two

English: 
terms as these. so if we re-order the equation as this,
so we can see inside the square brackets
is actually the continuity equation, so it is zero
so drop the first term in the left hand side we can have this equation.
if we write this momentum equation back into the vector form, we can have the
equation as this. Here we can see the fluid acceleration
term given by the substantial derivative and we can also see this momentum

English: 
terms as these. so if we re-order the equation as this,
so we can see inside the square brackets
is actually the continuity equation, so it is zero
so drop the first term in the left hand side we can have this equation.
if we write this momentum equation back into the vector form, we can have the
equation as this. Here we can see the fluid acceleration
term given by the substantial derivative and we can also see this momentum

English: 
equation would be same for both incompressible and compressible flows.
The difference for incompressible and compressible flows is whether
the density here is a constant or not. For incompressible flow, the density is a
constant, and for the compressible flow the density is not a constant.
It would be the function of the spatial variables x, y and z or the time t.

English: 
equation would be same for both incompressible and compressible flows.
The difference for incompressible and compressible flows is whether
the density here is a constant or not. For incompressible flow, the density is a
constant, and for the compressible flow the density is not a constant.
It would be the function of the spatial variables x, y and z or the time t.
