
English: 
Hello welcome to my talk, All about Fluids.
this talk is the second part of the talk on whether the Navier-Stokes equation is
correct. And this part would be for a solution on how we can better
formulate the fluid viscous stress tensor based on the solid physics,
(on) how we can derive the correct Navier-Stokes equation using the newly
defined fluid viscous stress tensor and how we can solve those inconsistencies
I have talked about in the first part.
Let's first have a look at the inconsistencies behind the Navier-Stokes

English: 
Hello welcome to my talk All about Fluids
this talk is the second part of the talk on whether the Navier-Stokes equation is
correct. And this part would be for a solution on how we can better
formulate the fluid  viscous stress tensor based on the solid physics,
(on) how we can derive the correct Navier-Stokes equation using the newly
defined fluid viscous stress tensor and how we can solve those inconsistencies
I have talked about in the first part.
Let's first have a look at the inconsistencies behind the Navier-Stokes

English: 
equation and then I will show you how we can solve those inconsistencies via a
new fluid viscosity tensor and how we can better derive the Navier-Stokes
equation. As we have shown in the first part of
the talk on whether Navier-Stokes equation is correct. There are 4
inconsistencies identified in that talk. These inconsistencies include
the classical Couette flow has shown that the shear stresses are asymmetric for fluid.
there exist viscous stresses in the pure rotation motion of fluids.
Therefore, the ignorance of the viscous stress due to the rotation motion

English: 
equation and then I will show you how we can solve those inconsistencies via a
new fluid viscosity tensor and how we can better derive the Navier-Stokes
equation. As we have shown in the first part of
the talk on whether Navier-Stokes equation is correct. There are 4
inconsistencies identified in that talk. These inconsistencies include
the classical Couette flow has shown that the shear stresses are asymmetric for fluid.
there exist viscous stresses in the pure rotation motion of fluids
therefore the ignorance of the viscous stress due to the rotation motion

English: 
would not be physically consistent with the facts
The original form of the symmetrical stress tensor has an artificial factor 2
and the explanations to this factor in some famous reference books: it is merely
used for a purpose of convenience. But there is no physical basis to support it.
In real flows, there have been found exceptions for the Cauchy symmetric stress tensor
the question is why do we have exceptions for the universally correct laws?
The key is the formulation of the fluid viscous stress tensor.

English: 
would not be physically consistent with the facts
The original form of the symmetrical stress tensor has an artificial factor 2
and the explanations to this factor in some famous reference books: it is merely
used for a purpose of convenience. But there is no physical basis to
support it
In real flows, there have been found exceptions for the Cauchy symmetric  stress tensor
the question is why do we have exceptions for the universally correct laws?
The key is the formulation of the fluid viscous stress tensor.

English: 
it is asymmetric, not necessary limited to be symmetric.
As for whether the asymmetric stress tensor could cause a spinning motion in fluid,
the answer is NO
for instance, if we look at this fluid element and apply a shear stress
TAU-23, and this shear stress can simply cause
a flow as seen in this figure. This is fundamentally different from the solids.
As we know from the Newton's definition of fluid friction or stress,
the fluid viscous force or stress is caused due to the different fluid speeds,

English: 
it is asymmetric, not necessarily limited to be symmetric.
As for whether the asymmetric stress tensor could cause a spinning motion in fluid,
the answer is NO.
for instance, if we look at this fluid element and apply a shear stress
TAU-23, and this shear stress can simply cause
a flow as seen in this figure. This is fundamentally different from the solids.
As we know from the Newton's definition of fluid friction or stress,
the fluid viscous force or stress is caused due to the different fluid speeds,

English: 
at two joint spatial locations. Therefore, we can examine two neighbouring
points in the fluid, A and B. And the relative fluid velocity of B
with regard to A at a certain time is studied here.
Point A the position and the velocity are given these and B the location
and the velocity are given as this. So the increments are given as delta x, delta
y and delta z in the position and for the velocity, it is delta u, delta v
and delta w. Based on the chain rule of differentiation we can
calculate delta u as this, here the velocity u is the function of

English: 
at two joint spatial locations. Therefore we can examine two neighboring
points in the fluid, A and B. And the relative fluid velocity of B
with regard to A at a certain time is studied here.
Point A the position and the velocity are given these and B the location
and the velocity are given as this. So the increments are given as delta x, delta
y and delta z in the position and for the velocity, it is delta u, delta v
and delta w. Based on the chain rule of differentiation we can
calculate delta u as this, here the velocity u is the function of

English: 
the independent spatial variables x, y and z
Similarly we can calculate the velocity increments delta v and delta w in y and
z directions. Put all these together we can calculate
the relative velocity of point B with regard to the point A as this,
and this can be written as a matrix form as this. Here the matrix is actually a tensor
and from this formulation we can define the tensor A double arrows as the strain
rate tensor, and this tensor is correct for
all fluids, including the Newtonians and non-Newtonians

English: 
the independent spatial variables x, y and z
Similarly we can calculate the velocity increments delta v and delta w in y and
z directions. Put all these together we can calculate
the relative velocity of point B with regard to the point A as this,
and this can be written as a matrix form as this. Here the matrix is actually a tensor
and from this formulation we can define the tensor A double arrows as the strain
rate tensor, and this tensor is correct for
all fluids, including the Newtonians and non-Newtonians

English: 
because this formulation is simply from the direct physical analysis, without
assuming the fluid types in the analysis
by analogue to the definition of Newtonian fluid
viscous stress,  it is proportional to the rate of strain,
like, TAU_xy equaling to the fluid viscosity timing the gradient of the
velocity u with regard to y.
Therefore, based on this principle we can construct
the general fluid viscous just tensor as this for the Newtonian fluid.
This formula is correct for the isotropic flows because we assume MU is isotropic

English: 
because this formulation is simply from the direct physical analysis, without
assuming the fluid types in the analysis
by analogue to the definition of Newtonian fluid
viscous stress,  it is proportional to the rate of strain,
like, TAU_xy equaling to the fluid viscosity timing the gradient of the
velocity u with regard to y.
Therefore, based on this principle we can construct
the general fluid viscous just tensor as this for the Newtonian fluid.
This formula is correct for the isotropic flows because we assume MU is isotropic

English: 
and constant for representing the fluid viscosity
For a more general expression for the fluid viscosity stress tensor, it can be simply
expressed as this. Here the stress tensor can be asymmetric,
therefore with the newly defined asymmetrical stress tensor, we can see
the relative velocity between two neighbouring points is from the
direct physical analysis, thus it is universally correct
the symmetry for the stress tensor is not required for the real stress tensor
for fluids. it is because the shear stress will simply cause a flowing motion,
rather than a spinning motion in fluids.  The shear stresses directly follow the

English: 
and constant for representing the fluid viscosity
For a more general expression for the fluid viscosity stress tensor, it can be simply
expressed as this. Here the stress tensor can be asymmetric
therefore with the newly defined  asymmetrical stress tensor, we can see
the relative velocity between two neighboring points is from the
direct physical analysis, thus it is universally correct
the symmetry for the stress tensor is not required for the real stress tensor
for fluids. it is because the shear stress will simply cause a flowing motion,
rather than a spinning motion in fluids.  The shear stresses directly follow the

English: 
definition of Newtonian viscous stress which is physically correct for
Newtonian fluids. And this can be easily adopted for the non-Newtonian fluids
if the fluid stress is calculated for the actual fluids.
the momentum equation for fluid motion is based on the Newton's second law
of motion, which can be simply expressed as this for the fluids
here scalar V is volume and vector V is the fluid velocity. This part is the
time rate of change of fluid momentum,  this part is the body force and this is

English: 
definition of Newtonian viscous stress which is physically correct for
Newtonian fluids. And this can be easily adopted for the non-Newtonian fluids
if the fluid stress is calculated for the actual fluids.
the momentum equation for fluid motion is based on the Newton's second law
of motion, which can be simply expressed as this for the fluids
here scalar V is volume and vector V is the fluid velocity. This part is the
time rate of change of fluid momentum, this part is the body force and this is

English: 
from the surface force.  So basically this equation is the Newton's second law
of motion: the time rate of change of fluid
momentum equals to the total force acting on the fluid.
by applying the Gauss divergence theorem, we can change the surface force
of the integral on the fluid surface into the integral on fluid volume
as this, so we have the equation for the flow momentum as this
Based on this momentum equation you can see here the capital D in the
differentiation means the substantial derivative

English: 
from the surface force.  So basically this equation is the Newton's second law
of motion: the time rate of change of fluid
momentum equals to the total force acting on the fluid.
by applying the Gauss divergence theorem, we can change the surface force
of the integral on the fluid surface into the integral on fluid volume
as this, so we have the equation for the flow momentum as this
Based on this momentum equation you can see here the capital D in the
differentiation means the substantial derivative

English: 
however the derivative of the volumetric integral on the whole fluid with
regard to time would make no difference whether this is substantial
derivative or the normal derivative, since the volumetric integral on the
whole fluid domain is a function of time and the volume, but it is
independent of the spatial variables x, y and z. this is why in this
expression the substantial derivative is the same as the normal derivative.
however for calculating this term, we need to employ Reynolds transport theorem as

English: 
however the derivative of the volumetric integral on the whole fluid with
regard to time would make no difference whether this is substantial
derivative or the normal derivative, since the volumetric integral on the
whole fluid domain is a function of time and the volume, but it is
independent of the spatial variables x, y and z. this is why in this
expression the substantial derivative is the same as the normal derivative.
however for calculating this term, we need to employ Reynolds transport theorem as

English: 
this. Here we can assume f0 equals to rho times vector V, applying the Reynolds
transport theorem, the left hand side of the equation would become to this and
then we can obtain the integral momentum equation for fluids as this
so from this integral momentum equation we can easily obtain the differential
momentum equation for fluid, expressed as this
here rho double vector V and T double arrow are both tensors.
so following Stokes, the total tensor is given as the combination of the tensors

English: 
this. Here we can assume f0 equals to rho times vector V, applying the Reynolds
transport theorem, the left hand side of the equation would become to this and
then we can obtain the integral momentum equation for fluids as this
so from this integral momentum equation we can easily obtain the differential
momentum equation for fluid, expressed as this
here rho double vector V and T double arrow are both tensor.
so following Stokes, the total tensor is given as the combination of the tensors

English: 
due to the pressure by this term; due to the fluid compressibility by
this term, here LAMBDA is the second viscosity coefficient; and due to the
viscous stress tensor. The only difference from the
conventional derivation of Navier-Stokes equation is the shear stress tensor.
Here the shear stress tensor is defined by this component. it can be seen the stress
tensor is an asymmetric tensor. So if we apply this to the momentum
equation for this total tensor, the divergence of the total tensor is
calculated this, it's a vector.

English: 
due to the pressure by this term; due to the fluid compressibility by
this term, here LAMBDA is the second viscosity coefficient; and due to the
viscous stress tensor. The only difference from the
conventional derivation of Navier-Stokes equation is the shear stress tensor.
Here the shear stress tensor is defined by this component. it can be seen the stress
tensor is an asymmetric tensor. So if we apply this to the momentum
equation for this total tensor, the divergence of the total tensor is
calculated this, it's a vector.

English: 
here we are going to examine the last 3 terms in the first row of the total
tensor divergence.  and we use the definition for the asymmetric stress
tensor, so we have this and it can be written as this and this. Similarly we can
have the equation for the last 3 terms in the second row and the last 3 terms
for the third row. We have all these. and then the divergence of the total tensor
can be written as this
For deriving Navier-Stokes equation we considered the genetic form of the
Navier-Stokes equation here. First the continuity equation is given as this in

English: 
here we are going to examine the last 3 terms in the first row of the total
tensor divergence.  and we use the definition for the asymmetric stress
tensor, so we have this and it can be written as this and this. Similarly we can
have the equation for the last 3 terms in the second row and the last 3 terms
for the third row. We have all these. and then the divergence of the total tensor
can be written as this
For deriving Navier-Stokes equation we considered the genetic form of the
Navier-Stokes equation here. First the continuity equation is given as this in

English: 
vectors and it can be written as this, use the Einstein summation form,
the momentum equation from the Reynolds transport theorem given as this
using the Einstein summation we can write the momentum equation as this
so for the divergence of double velocity vector, it is calculated as this
and combine this with the continuity equation we can obtain the generic
Navier-Stokes equation as this
we can see the substantial derivative in the square brackets

English: 
vectors and it can be written as this, use the Einstein summation form,
the momentum equation from the Reynolds transport theorem given as this
using the Einstein summation we can write the momentum equation as this
so for the divergence of double velocity vector, it is calculated as this
and combine this with the continuity equation we can obtain the generic
Navier-Stokes equation as this
we can see the substantial derivative in the square brackets

English: 
and this universal Navier-Stokes is equation is same for both compressible
and incompressible flows
in this slide we are going to look at the incompressible flows, where the flow
density is constant and the continuity equation is given by this
now look at the divergence of the total tensor, so we can drop the fluid
compressibility term,  thus we have with the moment the equation the x direction
and in y direction , z-direction as these
so if we write this back in the vector form, we have the navier-stokes equation
for incompressible flows as this

English: 
and this universal Navier-Stokes is equation is same for both compressible
and incompressible flows.
in this slide we are going to look at the incompressible flows, where the flow
density is constant and the continuity equation is given by this
now look at the divergence of the total tensor, so we can drop the fluid
compressibility term, thus we have with the moment the equation the x direction
and in y direction, z-direction as these
so if we write this back in the vector form, we have the Navier-Stokes equation
for incompressible flows as this

English: 
Now for compressible flows, the continuity equation is given as this
again we look at the divergence of the total tensor. Here we have to keep
with the compressibility term, however, the second viscosity can be
assumed reasonably as LAMBDA equals to 1/3 of the fluid viscosity
This can be understood that in the fluid dynamic equation, we have used the fluid
compressibility three times as we can see in this total tensor.
Therefore, it is reasonable to evenly distribute the contribution so that's why we have

English: 
Now for compressible flows, the continuity equation is given as this.
Again we look at the divergence of the total tensor. Here we have to keep
with the compressibility term, however, the second viscosity can be
assumed reasonably as LAMBDA equals to 1/3 of the fluid viscosity
This can be understood that in the fluid dynamic equation, we have used the fluid
compressibility three times as we can see in this total tensor.
Therefore, it is reasonable to evenly distribute the contribution so that's why we have

English: 
the one-third of the viscosity. By comparison to the Stokes assumption
given the second viscosity coefficient LAMBDA equals to -2/3 MU. However,
this is still very controversial, this can be seen from the book
'Aerodynamics for Engineering Students'
so we apply the assumption, LAMBDA equals to 1/3 MU, we can get
the momentum equation for the compressible flow as this
here it's the term for the fluid compressibility
we can write back the momentum equation into the vector form, we have this

English: 
the one-third of the viscosity. By comparison to the Stokes assumption
given the second viscosity coefficient LAMBDA equals to -2/3 MU. However,
this is still very controversial, this can be seen from the book
'Aerodynamics for Engineering Students'
so we apply the assumption, LAMBDA equals to 1/3 MU, we can get
the momentum equation for the compressible flow as this.
here it's the term for the fluid compressibility
we can write back the momentum equation into the vector form, we have this

English: 
this is exactly same as the momentum equation in the conventional Navier-Stokes
equation for compressible flows.
We are going to solve all these inconsistencies we found for the
symmetric stress tensor for fluids. The first solution would be for the shear stress
in Couette flow. For such a laminar flow the velocity distribution is given as
this. And based on the newly defined
stress tensor, we can calculate the horizontal shear stress as this,
TAU_12 is nonzero, this is understandable since there is a
horizontal flow in Couette flow. And we can also calculate the

English: 
this is exactly same as the momentum equation in the conventional Navier-Stokes
equation for compressible flows.
We are going to solve all these inconsistencies we found for the
symmetric stress tensor for fluids. The first solution would be for the shear stress
in Couette flow. For such a laminar flow the velocity distribution is given as
this. And based on the newly defined
stress tensor, we can calculate the horizontal shear stress as this,
TAU_12 is nonzero, this is understandable since there is a
horizontal flow in Couette flow. And we can also calculate the

English: 
vertical shear stress TAU_21 equals to this. It is zero.  This is also correct,
since there is no vertical flow in the Couette flow.
As such the first inconsistency is solved.
For the second inconsistency we will look at how the rotation motion is
included for contributing to the fluid viscosity stress.
The asymmetric stress tensor can be separated in the two parts
the symmetric part and the antisymmetric part as this. In the conventional

English: 
vertical shear stress TAU_21 equals to this. It is zero.  This is also correct,
since there is no vertical flow in the Couette flow.
As such the first inconsistency is solved.
For the second inconsistency we will look at how the rotation motion is
included for contributing to the fluid viscosity stress.
The asymmetric stress tensor can be separated in the two parts
the symmetric part and the antisymmetric part as this. In the conventional

English: 
derivation of Navier-Stokes equation, the symmetric part, which represents
the shear deformation of fluid, was (modified and) used in deriving the Navier-Stokes equation
and these are the symmetric stresses
but the anti-symmetric strain rate tensor which corresponds to the rotation of fluid
with OMEGA defined as this, but this had been dropped in the conventional
derivation of Navier-Stokes equation, because the rotational motion was
believed to make no contributions to the viscous stress in fluid. But this is not

English: 
derivation of Navier-Stokes equation, the symmetric part, which represents
the shear deformation of fluid, was (modified and) used in deriving the Navier-Stokes equation
and these are the symmetric stresses
but the anti-symmetric strain rate tensor which corresponds to the rotation of fluid
with OMEGA defined as this, but this had been dropped in the conventional
derivation of Navier-Stokes equation, because the rotational motion was
believed to make no contributions to the viscous stress in fluid. But this is not

English: 
correct. For fluids, the pure rotational motion will cause the stress, so if it is
asymmetric stress tensor is used, the contributions from both fluid
shear deformation and the fluid rotation would be included automatically
Therefore, the second inconsistency is solved by
using the newly defined asymmetric stress tensor.
For the 3rd inconsistency, if the newly defined asymmetric stress tensor
is employed, we can derive the Navier-Stokes equation directly as we
have shown in the previous slides in this talk
the asymmetric stress tensor is given by this,  here A_ij is the asymmetrical rate

English: 
correct. For fluids, the pure rotational motion will cause the stress, so if it is
asymmetric stress tensor is used, the contributions from both fluid
shear deformation and the fluid rotation would be included automatically
Therefore, the second inconsistency is solved by
using the newly defined asymmetric stress tensor.
For the 3rd inconsistency, if the newly defined asymmetric stress tensor
is employed, we can derive the Navier-Stokes equation directly as we
have shown in the previous slides in this talk
the asymmetric stress tensor is given by this, here A_ij is the asymmetrical rate

English: 
strain for the fluid, defined as this. Unlike the original derivation of
Navier-Stokes equation, in formulating the symmetric stress tensor for
Navier-Stokes equation, an artificial factor 2 must be employed
as this. Here S_ij is the symmetrical strain rate component
which represents the shear deformation for the fluid element.
Based on the famous textbooks, the explanation of the artificial
factor 2 is merely used for a convenience. More specifically,

English: 
strain for the fluid, defined as this. Unlike the original derivation of
Navier-Stokes equation, in formulating the symmetric stress tensor for
Navier-Stokes equation, an artificial factor 2 must be employed
as this. Here S_ij is the symmetrical strain rate component
which represents the shear deformation for the fluid element.
Based on the famous textbooks, the explanation of the artificial
factor 2 is merely used for a convenience. More specifically,

English: 
it's used to cancel out the factor 1/2 in the symmetric strain rate expression
otherwise the derivation of Navier-Stokes equation would not be correct.
Therefore by employing the newly defined asymmetrical stress tensor, the artificial
factor would not be necessary, hence it would be more consistent with the real
physics in the formulation, and thus the 3rd inconsistency is solved.
The fourth inconsistency is for some exceptions for the symmetric stress tensor
In the book of Kundu, Cohen and Dowling, 'Fluid Mechanics',
It mentioned one exception:

English: 
it's used to cancel out the factor 1/2 in the symmetric strain rate expression
otherwise the derivation of Navier-Stokes equation would not be correct.
Therefore by employing the newly defined asymmetrical stress tensor, the artificial
factor would not be necessary, hence it would be more consistent with the real
physics in the formulation, and thus the 3rd inconsistency is solved.
The fourth inconsistency is for some exceptions for the symmetric stress tensor
In the book of Kundu, Cohen and Dowling, 'Fluid Mechanics',
It mentioned one exception:

English: 
for the electrolytes fluid, the symmetry of the stress tensor is
violated and for the analysis the anti-symmetric part must be included.
So if we look at this asymmetrical stress tensor, we can see it includes the
symmetric part and the anti-symmetric part automatically
therefore this asymmetrical stress tensor automatically includes the anti-
symmetric stress tensor
In the Wilcox's book, 'Turbulence Modeling for CFD', it mentioned the
symmetric stress tensor is not for the anisotropic flows. This requirement can

English: 
for the electrolytes fluid, the symmetry of the stress tensor is
violated and for the analysis the anti-symmetric part must be included.
So if we look at this asymmetrical stress tensor, we can see it includes the
symmetric part and the anti-symmetric part automatically.
Therefore, this asymmetrical stress tensor automatically includes the anti-
symmetric stress tensor
In the Wilcox's book, 'Turbulence Modeling for CFD', it mentioned the
symmetric stress tensor is not for the anisotropic flows. This requirement can

English: 
be achieved by using the asymmetric stress tensor, either for Newtonian flows or
for generic fluids, used generic stress tensor, even for the anisotropic flow.
because the asymmetric stress tensor has no requirement on the
symmetry on the stress tensor, thus this asymmetrical stress tensor
could provide more flexibility for constructing the fluid viscous stress tensor
for both Newtonian and the generic flows, including those anisotropic flows
so in this slide, the summary is given for the correctness of Navier-Stokes equation.

English: 
be achieved by using the asymmetric stress tensor, either for Newtonian flows or
for generic fluids, used generic stress tensor, even for the anisotropic flow.
because the asymmetric stress tensor has no requirement on the
symmetry on the stress tensor, thus this asymmetrical stress tensor
could provide more flexibility for constructing the fluid viscous stress tensor
for both Newtonian and the generic flows, including those
anisotropic flows
so in this slide, the summary is given for the correctness of Navier-Stokes equation.

English: 
The talk is for better understand the fundamentals and the issues with the Navier-Stocks equation, so to provide
better physics for understanding the equation, rather than to prove the
incorrectness of Navier-Stokes equation. To distinguish the difference of
the motions for fluids and solids, the flowing motion in fluids causes the
fundamental difference between the fluid motion and solid motion
To solve the inconsistencies behind the Navier-Stokes equation, there are
conflicts between the requirement of the symmetrical stress tensor and the
definition of fluids, the newly defined  asymmetric stress tensor is the key
To formulate the mathematical equation and the viscous stress tensor are all

English: 
The talk is for better understand the fundamentals and the issues with the Navier-Stocks equation, so to provide
better physics for understanding the equation, rather than to prove the
incorrectness of Navier-Stokes equation. To distinguish the difference of
the motions for fluids and solids, the flowing motion in fluids causes the
fundamental difference between the fluid motion and solid motion
To solve the inconsistencies behind the Navier-Stokes equation, there are
conflicts between the requirement of the symmetrical stress tensor and the
definition of fluids, the newly defined asymmetric stress tensor is the key
To formulate the mathematical equation and the viscous stress tensor are all

English: 
based on the solid physics. There is no artificial factor or parameters in
the formulation and all formulations have sound physical foundations
To provide the flexibility for flow motion there is no requirement or
limit for the fluid stress tensor which can be either asymmetric or symmetric
totally depend on the actual flows.
It is hoped that such a flexibility could
provide the new prosperities in turbulence modeling since we are no more limited on the symmetrical stress tensor

English: 
based on the solid physics. There is no artificial factor or parameters in
the formulation and all formulations have sound physical foundations
To provide the flexibility for flow motion there is no requirement or
limit for the fluid stress tensor which can be either asymmetric or symmetric
totally depend on the actual flows.
It is hoped that such a flexibility could
provide the new prosperities in turbulence modeling since we are no more limited on the symmetrical stress tensor.
