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Mechanical Engineering: Module 8
Welcome back. You will recall that in a previous module we applied the
Reynolds Transport Theorem to the principle of the conservation of mass in
order to derive a local form of that principle that is going to work for us in
spatial coordinates. In this module, what we would like to do is start the
same process for the balance of linear momentum.
That is ultimately going to lead us to a set of equations of motion for viscous
fluids as equations of motion which are known as Navier-Stokes equations,
and along the way we shall derive the set of equations governing the motion
of an inlet viscid fluid as well as a viscous fluid.
Again, the balance of linear momentum is nothing more than a statement of
Newton’s Second Law of Motion,
and so we will be applying the Reynolds Transport Theorem to that,
expressed in material coordinates,
to give us the spatial coordinate form.
Now, Newton’s Second Law of Motion applied to a particular piece of the
continuum tells us the following in words. It says that the time rate of
change, or increase, of the linear momentum within a fluid material volume
that we will indicate by capital V, just as we did with conservation of mass,
is equal to the resultant external force acting on the fluid in the volume,
capital V. So that is a very simple statement of Newton’s Second Law.
We need to examine both sides of this equation. We need to look at the time
rate of increase of linear momentum within the fluid material volume (it
turns out that is the easy part),
and we also need to look at the types of forces that are going to be acting on
the fluid in that volume. We will start with the forces to begin our process.
There are two types of forces that we are going to be dealing with. The first
type of force is called a body force. Body forces are very long-range forces,
which permeate the matter.
The examples, the easy examples to think of, are gravity and
electromagnetic forces. In addition, body forces depend upon the mass of the
body on which they are acting.
Gravitational force, my weight, depends upon my mass, not just my volume
or my surface area. So body forces depend upon that mass.
The other type of force is a surface force. Surface forces are forces that are
exerted by material on one side of a surface element, an imaginary surface
element,
on the material, on the opposite side of that surface element. The length
scale over which these surface forces act, unlike the body forces,
which are long-range forces, is intermolecular distance. So surface forces act
– the material on this side of this surface is exerting a force on the material
on the other side of the surface. Now, this concept of surface forces seems
like a simple one to us,
but it comes from something called the Cauchy Stress Principle, which we
will examine a statement of right now.
The Cauchy Stress Principle dates back to about 1827. The statement that we
are going to be looking at is due to Clifford Truesdell,
and the statement is the following. Upon any imagined closed surface S,
closed surface, there exists a distribution of stress vectors
(we are going to call those stress vectors t. We will be using t for something
else later, but for right now it is the stress vector),
whose resultant and moment are equivalent to the actual forces of material
continuity exerted by the material outside the surface S
(it is a closed surface) upon the material inside the surface capital S.
Now this seems like a very simple concept, but as Truesdell points out, a
whole generation of mathematicians and hydrodynamicists were trying to
come up with an expression for the forces of material continuity.
That means the forces that keep the material from tearing itself apart. Now, a
whole generation of people had tried to do this, in mostly an ad hoc fashion,
without any success, until Cauchy postulated this very simple statement.
Okay, so this is how we are going to be modeling these intermolecular
forces that exist between pieces of the fluid continuum. That is just a
distribution of the stress vectors acting on surfaces.
Now we are going to be designating the two types of forces differently.
Body forces we will designate with vector lowercase f.
That is going to indicate the body force per unit mass, so that if we want the
total body force acting on a particular body of mass,
we are going to need to multiply f by m to get the total force. Surface forces,
on the other hand, are just a force per unit area,
and we are going to indicate surface forces by the vector t. As I said, later
we will be using t to indicate something else,
but for right now just think of that vector t as the surface force exerted on
some element. That force does not have to be normal to the element,
and it does not have to be tangential to the element. In fact, in general it will
contain both components.
Now, what we are going to do is create a mathematical statement of
Newton’s Second Law of motion,
as applied to an arbitrary fluid material volume, capital V of lowercase t,
bounded by a surface capital S of lowercase t.
And we can do that in a very simple fashion, now that we know how to
represent the forces.
The left-hand side just represents the time rate of change following the
motion (we are using the material derivative notation capital D by capital D
lowercase t) of the total linear momentum of the fluid in the material volume
capital V of lowercase t.
So lowercase v is the velocity vector, rho is the density function, as I used
with the continuity equation derivation, lowercase d capital V is the
differential volume element, so rho times lowercase d capital V represents
the differential mass multiplied by the velocity,
integrated over the entire volume. That is the time rate of change of linear
momentum following the motion, is equal to the sum of the forces.
And for the body forces, remember f is the body force per unit mass, rho
times lowercase d capital V is the mass of a differentially small element,
so if we integrate that over the entire volume, we get the total body force.
Lowercase t, on the other hand, is the surface force per unit area.
We multiply that by the differentially small area lowercase d capital S, and
integrate over the entire area S of t, which bounds the material volume
capital V of lowercase t. So that is our global statement now of Newton’s
Second Law of Motion, in terms of material formulation.
And again, we are just going through the various changes, so the red terms
indicate the time rate of change of linear momentum in capital V.
The blue terms, the middle terms indicate the total body force acting on the
fluid material in V of t.
And finally, the magenta terms are the total surface forces acting on the fluid
material in V of t. So that is our integral form of Newton’s Second Law.
Using the corollary to the Reynolds Transport Theorem – you will remember
that corollary stated if conservation of mass held,
then we had capital D by capital D lowercase t of the integral of rho times
something. We could write that just as the integral over capital V of
lowercase t of rho times the material derivative of what is left. So therefore,
we have the integral over capital V of lowercase t of rho times the material
derivative of the velocity field, integrated over the volume. This is not a
working form, however.
We want a local form of this equation. We want form of this equation valid
at every point in the fluid, and in order to pursue that local form of the
equation,
we are going to have to prove a little theorem called local equilibrium. The
theory of local equilibrium tells us that the stress forces are in local
equilibrium at each point
(that means that balance themselves at each point) in the flow field.
So in order to prove that, it is a pretty simple proof, the first thing we are
going to have to do is begin by recognizing that volumes are of the order of
a length scale cubed. ‘l’ is a length scale representative of our volume capital
V of lowercase t. So capital V is of order lowercase l cubed.
This big-O notation indicates the way that capital V behaves as lowercase l
changes. In particular, we are going to be concerned with how it behaves as l
goes to zero.
So the volume goes to zero very quickly as the length scale cubed, whereas
the surface, on the other hand, goes to zero as the length squared.
Alright, so S is of order l squared. We can then rewrite our global form that
we have in the following manner.
That is our form, now we are going to rewrite it.
We are going to divide by lowercase l squared on both sides of the equation,
and we are going to lump the two volume integral terms together on the lefthand side of the equation.
So all we have done is taken the body force integral to the left hand side of
the equation, and combined it with the time rate of change of the linear,
and we have divided by lowercase l squared. And now, we take the limit.
If we take the limit as lowercase l goes to zero, we recognize that nothing in
here varies with length, but the volume goes as l cubed,
and so something that goes as l cubed divided by something that goes as
lowercase l squared in the limit as lowercase l goes to zero, will vanish.
That means, therefore, that the limit of the right-hand side as lowercase l
goes to zero must also vanish.
And, as we shrink down to a point, that is going to tell us that our
distribution of stress vectors is going to be at equilibrium at that particular
point.
Okay, so that is our principle of local equilibrium. And now what we wish to
do is to apply that principle of local equilibrium to a very particular fluid
volume.
And in undergraduate courses, this is usually done to a rectangular piped.
Here, we are going to be a little bit more sophisticated with it,
and apply it to a trapezoidal fluid volume. I am sorry, not trapezoidal, but
tetrahedral. Pardon the mistake.
So this is the tetrahedron bound by the three coordinate axes x subscript one,
x subscript two, and x subscript three.
And I see my figure has overlapped there, which is a bit of a mistake. We
are going to indicate the unit outward vector at each face.
So lowercase n here, indicated in red, represents the unit outward normal, on
the slanted face which is colored blue,
and it has components lowercase n subscript one times lowercase i, where
lowercase i is the unit vector along the lowercase x subscript one axis,
plus lowercase n subscript two times lowercase j, where lowercase j is the
unit vector along the lowercase x subscript two axis,
plus n subscript three times k, and n subscript one, n subscript two, and n
subscript three are the three direction cosines.
The unit vector on this back face, marked by the x subscript one/x subscript
two plane, is minus lowercase k, it’s the unit outward normal,
and the area of that back face is n subscript three times sigma, the area of the
slanted face, where n subscript three is again, that direction cosine that
appears right here.
The unit vector here is minus i. The associated area is n subscript one times
sigma, and the unit vector here is minus j, with an area n subscript two times
sigma.
I will clean this up before you see the printed version of it. Okay, so we have
indicated now the unit vector and the area of each one of the faces of that
stress tetrahedron.
If we apply local equilibrium to that, that says the integral of the stress
vector, integrated over the surface has to vanish when divided by the area in
the limit,
as the area goes to zero. We had lowercase ‘l’ before, and that was
lowercase l squared, but it is perfectly valid to use sigma for the area of the
slanted face in this because as
we shrink the volume, the shape of the volume maintains the same. So we
have the stress vector times the area.
So for the slanted face, that is just lowercase t times sigma. For the first face,
it is lowercase bold t, evaluated at lowercase x and time, t, for minus
lowercase i. That is the unit normal for that face. The area of that face is
lowercase n subscript one sigma.
For the second face it is lowercase t, associated with the unit vector minus j,
times area lowercase n subscript two, sigma, and for the third face, it is t,
associated with the unit vector minus k times the area n subscript three,
sigma.
And so when we divide by the area, and take the limit of that quantity as the
area goes to zero, we get the following.
First of all, we recognize that sigma just divides out to one, and so when we
take the limit, this is our result. This has to vanish.
Now everything has been evaluated at the point x, to which the tetrahedron
has shrunk.
That is the origin of the coordinate system that we have indicated the
tetrahedron to have, associated with it.
The stress vector lowercase bold t, however, has to be an odd function of the
outward normal vector lowercase n. Why is that?
If I think of a surface element, and if I talk about the stress vector on this
particular setter, or the stress vector on this side exerting a force on that side,
the unit outward normal would be in this direction. Visa-versa, if I wanted
the force for the fluid on this side, being exerted on the fluid on the opposite
side, the outward normal would be this way. Why does the stress vector have
to be an odd function of the unit outward normal vector?
Because Newton’s Third Law of motion, action/reaction, tells us that this
fluid is an equal and opposite force on that fluid, and that fluid exerts back
on this fluid.
Consequently, if I have lowercase t associated with a unit vector minus
lowercase i, that is going to be minus lowercase t associated with the unit
vector plus lowercase i.
And that has been taken into account here, because we have taken these
three terms to the right-hand side of the equation, and replaced the zero with
those three terms.
Well, that equation simply tells us that it is possible to represent the stress
vector t as a linear combination of the components of its unit outward
normal.
Remember, the unit outward normal vector, n, is n subscript one, i, plus n
subscript two, j, plus n subscript three, k.
So this represents such a linear combination of the outward normal
components, and that may be written in tensor notation in the following way.
Lowercase t subscript lowercase i is now the stress vector. Lowercase n is
the unit outward normal vector, and it tells us that lowercase t subscript
lowercase i is given by the coefficient matrix here, or tensor, capital T
subscript lowercase i lowercase j times n subscript lowercase j.
The summation convention applies, because we have a repeated index
lowercase j.
In general, capital T subscript lowercase i lowercase j is a function of
position, x, and time, t.
And that quantity capital T subscript lowercase i lowercase j is what we call
the stress tensor.
It represents the i-component of force per unit area exerted across a surface
element normal to the j-direction at position x at time t.
So again, it is the i-component of force exerted per unit area, exerted on a
surface who’s unit outward vector is in the j-direction at position x at time t.
Having that at our disposal, we are now going to be able to substitute that
back into our equations of motion that we have derived thus far,
and complete the process of coming up with a working form for the local,
spatial equations of motion.
We have discussed the nature of surface body forces in this module, and we
have developed the principle of local equilibrium,
and we have seen that when we apply that principle of a local equilibrium to
a tetrahedral element,
we arrive at the existence of the stress tensor. So we will continue this in the
next module.