
English: 
Hello welcome to my talk, All about Fluids. This talk is on an important topic on
how we can calculate a tensor divergence since this calculation would be used in
the derivation of navier-stokes equation, as well as in formulating the
transport equations for turbulence models. In this talk I will show how the
problem is caused and how we can get the correct formula for calculating
the tensor divergence.
In deriving Euler and Navier-Stokes equation, in the momentum equation we need
to include both the body force and the surface force acting on the fluid,

English: 
Hello welcome to my talk,All about the Fluids. This talk is on an important topic on
how we can calculate a tensor divergence since this calculation would be used in
the derivation of navier-stokes equation, as well as in formulating the
transport equations for turbulence models. In this talk I will show how the
problem is caused and the how we can get the correct formula for calculating
the tensor divergence.
In deriving Euler and and Navier-Stokes equation, in the momentum equation we need
to include both the body force and the surface force acting on the fluid,

English: 
especially, if the fluid viscous force or stress is involved. Based on
the Newton's 2nd law of motion, the momentum equation of the fluid dynamics
would be expressed as this. here scalar V is the fluid volume, with the
enclosing surface S; and the vector V is the flow velocity; vector f is the body
force per fluid volume; and the vector n is the normal vector on the surface S.
tensor capital T double arrow is the fluid total tensor, including pressure, viscous
stresses. Using the Gauss divergence theorem,
we can change the integral on the surface into the integral on the

English: 
especially, if the fluid viscous force or stress is involved. Based on
the Newton's second law of motion, the momentum equation of the fluid dynamics
would be expressed as this. here scalar V is the fluid volume, with the
enclosing surface S; and the vector V is the flow velocity; vector F is the body
force per fluid volume; and the vector n is the normal vector on the surface S.
tensor capital T double arrow is the fluid total tensor, including pressure, viscous
stresses. Using the Gauss divergence theorem,
we can change the integral on the surface into the integral on the

English: 
fluid volume V here, so we can write the  momentum equation in this
form, and we can employed Reynolds 
transport theorem for the left hand side
term, here F is an arbitrary function, can be either a scalar or a vector. here we can
take F equals Rho times vector V, and then we can obtain the differential
equation for the momentum as this. for incompressible flows we can have the
equation as this, here we can see double vector V and capital T double arrow
are both tensors. and the operations this and this, are the tensor divergences.

English: 
fluid volume V here, so we can write the  momentum equation in this
form, and we can employ Reynolds transport theorem for the left hand side
term, here F is an arbitrary function, can be either a scalar or a vector. here we can
take F equals (to) Rho * vector V, and then we can obtain the differential
equation for the momentum as this. for incompressible flows we can have the
equation as this, here we can see double vector V and capital T double arrow
are both tensors. and the operations this and this, are the tensor divergences.

English: 
Based on the stoke's assumption for the Newtonian fluid, we could have them total
tensor, T double arrow expressed as in this form. Here the first term on the
right hand side is from the pressure, and the negative sign here means the
pressure is always opposite with the surface normal vector.
and this term is the tensor for the fluid compressibility effect, and
LAMBDA is called the second viscosity coefficient.
the last term is the fluid viscous stress tensor.
so on the specific surfaces, for instance, on the small cube, on the surface

English: 
Based on the stoke's assumption for the Newtonian fluid, we could have them total
tensor, T double arrow expressed as in this form. Here the first term on the
right hand side is from the pressure, and the negative sign here means the
pressure is always opposite with the surface normal vector.
and this term is the tensor for the fluid compressibility effect, and
LAMBDA is called the second viscosity coefficient.
the last term is the fluid viscous stress tensor.
so on the specific surfaces, for instance, on the small cube, on the surface

English: 
S1, S2 and S3, we have the tensor components expressed in this form.
here based on the book, 'Marine Hydrodynamics', TAU_21 denotes the stress acting in the
y direction, this is indicated by the first index '2', on a surface of
constant x, which is indicated by the second index '1'.
Accordingly, TAU_12 denotes a stress acting in the x direction on a surface
of constant y. Similarly we have TAU_11, denotes a

English: 
S1, S2 and S3, we have the tensor components expressed in this form.
here based on the book, 'Marine Hydrodynamics', TAU_21 denotes the stress acting in the
y direction, this is indicated by the first index '2', on a surface of
constant x, which is indicated by the second index '1'.
Accordingly, TAU_12 denotes a stress acting in the x direction on a surface
of constant y. Similarly we have TAU_11, denotes a

English: 
stress acting in the x direction on a surface of constant x.
For a closed surface, S, with a volume of V. The surface force acting on the fluid
due to the viscous stress tensor, TAU double arrow would be calculated as this
so if we apply the Gauss divergence theorem, we can change the integral on a
surface S into the volume integral on V, we have this expression.
here the stress tensor divergence can be
regarded as a force of a unit volume.

English: 
stress acting in the x direction on a surface of constant x.
For a closed surface, S, with a volume of V. The surface force acting on the fluid
due to the viscous stress tensor, TAU double arrow would be calculated as this
so if we apply the Gauss divergence theorem, we can change the integral on a
surface S into the volume integral on V, we have this expression.
here the stress tensor divergence can be regarded as a force of a unit volume.

English: 
now the question is how we can calculate the divergence of a tensor in Einstein
summation convention.
Based on the relevant references, we have two different expressions for
the divergence of a tensor in Einstein notation,
based on their famous book, 'Marine Hydrodynamics', we have the expression for
the divergence of tensor, given by this.
and that we can also find from the book, 'Turbulence modelling for CFD', the divergence
of a tensor is given by this. so we can see here the difference is the
derivation with the xj or with xi. Why we have such two different expressions?

English: 
now the question is how we can calculate the divergence of a tensor in Einstein
summation convention.
Based on the relevant references, we have two different expressions for
the divergence of a tensor in Einstein notation,
based on their famous book, 'Marine Hydrodynamics', we have the expression for
the divergence of tensor, given by this.
and that we can also find from the book, 'Turbulence modelling for CFD', the divergence
of a tensor is given by this. so we can see here the difference is the
derivation with the xj or with xi. Why we have such two different expressions?

English: 
basically they came from different expressions for a vector, for example, we
can express a vector using a column, given as this. for the normal vector, the components,
n1, n2 and n3 are put into a column,
so the Gauss divergence theorem is given as this, and here the final result
would be the vector force, given in a form
of a column. Otherwise we may express the
vector in a row, given as this. and if we use the same Gauss divergence
theorem, we have this, and the vector force would be given in a form of a

English: 
basically they came from different expressions for a vector, for example, we
can express a vector using a column, given as this. for the normal vector, the components,
n1, n2 and n3 are put into a column,
so the Gauss divergence theorem is given as this, and here the final result
would be the vector force, given in a form of a column. Otherwise we may express the
vector in a row, given as this. and if we use the same Gauss divergence
theorem, we have this, and the vector force would be given in a form of a

English: 
row, so if we look at this expression, the summation is on j and the vector is
indicated with i. and here the summation is on i and the vector is indicated
with the index j.
if we write the divergence of the tensor in details for different
expressions, we have the first expression as this, so the first row we can take
i = 1, and the summation for j from 1 to 3.
and the second row taken i = 2, summation on j and 3rd row, i  equals to

English: 
row, so if we look at this expression, the summation is on j and the vector is
indicated with i. and here the summation is on i and the vector is indicated
with the index j.
if we write the divergence of the tensor in details for different
expressions, we have the first expression as this, so the first row we can take
i = 1, and the summation for j from 1 to 3.
and the second row taken i = 2, summation on j and 3rd row, i  equals to

English: 
3 and the summation on j. So for the second expression, we have with the
vector expressed in a row, so the first column would be given by j = 1 and
the summation on i from 1 to 3; the second column would be on j = 2,
and the summation on i; and the 3rd column j = 3,
and the summation on i. so if we compare the component Fx,
for the first expression, we have the expression as this and the second
expression the force component given by
this. Obviously we can see the difference

English: 
3 and the summation on j. So for the second expression, we have with the
vector expressed in a row, so the first column would be given by j = 1 and
the summation on i from 1 to 3; the second column would be on j = 2,
and the summation on i; and the 3rd column j = 3,
and the summation on i. so if we compare the component Fx,
for the first expression, we have the expression as this and the second
expression the force component given by this. Obviously we can see the difference

English: 
is this: in here we have TAU_11, TAU_12 and TAU_13, but here we have TAU_11
TAU_21 and TAU_31.
now we look at the expression for the fluid viscous stress. for a Newtonian fluid,
the conventional flow viscous stress is
assumed as a symmetric stress tensor,
given by this TAU_ij. this is the Cauchy's symmetrical viscous stress tensor.
so if we substitute this expression into the expression in this, so we have this
and we have this. So we can write this by exchanging the

English: 
is this: in here we have TAU_11, TAU_12 and TAU_13, but here we have TAU_11
TAU_21 and TAU_31.
now we look at the expression for the fluid viscous stress. for a Newtonian fluid,
the conventional flow viscous stress is assumed as a symmetric stress tensor,
given by this TAU_ij. this is the Cauchy's symmetrical viscous stress tensor.
so if we substitute this expression into the expression in this, so we have this
and we have this. So we can write this by exchanging the

English: 
order of the derivation with regard to xi, and with regard to xj, we
have this. And for the second expression we have
this and we can obtain the expression as this, and further we
can write the expression as this. for this term, we exchange the order of the
derivation with regard to xi and with regard to xj.
so we can see these terms actually are same, because both are vectors,
with the remaining index here is i and here is j, but the summation here and the

English: 
order of the derivation with regard to xi, and with regard to xj, we
have this. And for the second expression we have
this and we can obtain the expression as this, and further we
can write the expression as this. for this term, we exchange the order of the
derivation with regard to xi and with
regard to xj.
so we can see these terms actually are same, because both are vectors,
with the remaining index here is i and here is j, but the summation here and the

English: 
summation here, physically these two terms are same. and these terms are same as well.
So we can conclude these two expressions for the divergence of a tensor would be same,
if the stress tensor is symmetric. So here if we apply the continuity equation
for the incompressible flows, and we can see this term would be 0. therefore we
have the expression this and final expression as this, same for the second
expression, this term would be 0, and the expression would be this and write
it into the vector form, we have this. So we can see both expressions are same for
the incompressible flows.

English: 
summation here, physically these two terms are same. and these terms are same as well.
So we can conclude these two expressions for the divergence of a tensor would be same,
if the stress tensor is symmetric. So here if we apply the continuity equation
for the incompressible flows, and we can see this term would be 0. therefore we
have the expression this and final expression as this, same for the second
expression, this term would be 0, and the expression would be this and write
it into the vector form, we have this. So we can see both expressions are same for
the incompressible flows.

English: 
However, if the fluid viscous stress tensor
is not symmetric, as recently shown by the author, see the reference below.
The fluid viscous stress tensor component should be given as this, here I would
like to say this expression it from the direct physical definition of the
fluid viscous stress, so for the first expression of the
tensor divergence, we have this and we have this.
So this is equivalent to this. For the 2nd expression of the tensor
divergence, so we have this, and we have this and then if we consider

English: 
However, if the fluid viscous stress tensor
is not symmetric, as recently shown by the author, see the reference below.
The fluid viscous stress tensor component should be given as this, here I would
like to say this expression it from the direct physical definition of the
fluid viscous stress, so for the first expression of the
tensor divergence, we have this and we have this.
So this is equivalent to this. For the 2nd expression of the tensor
divergence, so we have this, and we have this and then if we consider

English: 
the continuity equation for the incompressible flow, so this term would
be zero, and obviously these two expressions are not same.
now we have the question: which one is correct formula for the Einstein
notation of the tensor divergence? or where both of them are incorrect?
From the result of the asymmetric tensor, the 2nd expression would lead
to a zero viscous force in the incompressible flows. Hence it is reasonably
deduced the second formula is incorrect, but is there any proof for
this conclusion?

English: 
the continuity equation for the incompressible flow, so this term would
be zero, and obviously these two expressions are not same.
now we have the question: which one is correct formula for the Einstein
notation of the tensor divergence? or where both of them are incorrect?
From the result of the asymmetric tensor, the 2nd expression would lead
to a zero viscous force in the incompressible flows. Hence it is reasonably
deduced the second formula is incorrect, but is there any proof for
this conclusion?

English: 
so we go back to the fluid element, here in the figure. So on the surfaces of the
cube we can have the force due to the stress component, TAU_22, so we
can see on both sides the black arrows
the net force in y direction would be given as this, here Delta_V is given
by this, and if we consider the tensor component, TAU_23, we can see on
the bottom and upper surfaces, we can calculate the net force due to the
tensor component TAU_23, given by this.
and similarly if we consider the tensor component TAU_21, this would

English: 
so we go back to the fluid element, here
in the figure. So on the surfaces of the
cube we can have the force due to the stress component, TAU_22, so we
can see on both sides the black arrows
the net force in y direction would be given as this, here Delta_V is given
by this, and if we consider the tensor component, TAU_23, we can see on
the bottom and upper surfaces, we can calculate the net force due to the
tensor component TAU_23, given by this.
and similarly if we consider the tensor component TAU_21, this would

English: 
be given as this, the net force in y direction would be given in this.
so if we put all these together and the net force because of the
viscous stress tensor in y-direction would be given by this.
so based on the calculation in the previous slide, the total force in y
direction due to the fluid viscous stresses would be given as this,
therefore the total force acting in y- direction per volume would be given by
this. so the expression in the ellipse is the
correct expression for the net force in y direction due to the stress tensor,

English: 
be given as this, the net force in y direction would be given in this.
so if we put all these together and the net force because of the
viscous stress tensor in y-direction would be given by this.
so based on the calculation in the previous slide, the total force in y
direction due to the fluid viscous stresses would be given as this,
therefore the total force acting in y- direction per volume would be given by
this. so the expression in the ellipse is the
correct expression for the net force in y direction due to the stress tensor,

English: 
because this formula is from direct physical analysis.
if we compare these two tensor divergence expressions, this and this, and for
the component in y-direction, we have this and this. so if we compare this
to this result from the direct physics, we can see the first
expression is correct and the second expression is not correct.
However, in this slide, it should be noted that this talk proves the
correct form of tensor divergence, but this development should not be taken for
excluding the use of the vector in rows if only a slightly different

English: 
because this formula is from direct physical analysis.
if we compare these two tensor divergence expressions, this and this, and for
the component in y-direction, we have this and this. so if we compare this
to this result from the direct physics, we can see the first
expression is correct and the second expression is not correct.
However, in this slide, it should be noted that this talk proves the
correct form of tensor divergence, but this development should not be taken for
excluding the use of the vector in rows if only a slightly different

English: 
form of the surface force expression is employed, given by this. here TAU'
double arrow is the transpose of the viscous stress tensor TAU_double arrow
and the vector here is given in a form as this, and the corresponding Gauss divergence
theorem should be given in this. And as such, we can still deduce the
correct form of the tensor divergence in Einstein notation as this.

English: 
form of the surface force expression is employed, given by this. here TAU'
double arrow is the transpose of the viscous stress tensor TAU_double arrow
and the vector here is given in a form as this, and the corresponding Gauss divergence
theorem should be given in this. And as such, we can still deduce the
correct form of the tensor divergence in Einstein notation as this.
