Okay, so let's continue.
In the previous segment, we began looking at the strong form of the problem for
linear elliptic PDEs in three dimensions with scalar, with a scalar unknown, right?
We wrote out the, the strong form, we wrote it out in coordinate notation,
learned a little bit about the quantities that show up.
And the very last thing we did was make a connection with the,
the physical problem of heat conduction at steady state.
Let's carry on with this and
what I want to do first in this segment is for completeness,
also make the connection with mass diffusion in three dimensions, okay?
So just as we looked at heat conduction,
let us now consider the problem of mass diffusion.
Okay? Of course, the, the,
the strong form is the same so we're not going to rewrite it.
Let's just say what the variables would translate to here,
u in this case would be a concentration, right?
Concentration of some field.
All right, now it could be a concentration either in terms of mass or
unit volume or maybe number of particles of a certain kind per unit volume.
Al right, so this could be a concentration in
terms of mass over unit volume or number.
Volume unit for u or it may even be,
normalized by some reference concentration to be rewritten
into something to call a composition, right?
So I will just write this here.
Composition is a essentially a re-parametization
of the composition that is often used in physics, okay?
So that's what you would be right?
What about the j?
Now that we know direct notation it's safe to use still, to just do that, right?
J in this case would be the mass flux, right?
The mass flux or the number flux.
Right?
Which would be,
mass flow perpendicular,
perpendicular to a unit
area or unit time.
Okay, or alternatively,
if you were doing number flux it would be the same sort of thing, right?
We would be talking here of the number of some particle flowing
perpendicular to a unit area by unit time, right?
So number flow Number of particles
Flowing Perpendicular to unit area and so on.
Right? You can complete that statement,
all right?
So that's what j would be.
Our constitutive relation j
equals minus kappa.
Now, now if actually, I realize that this is probably the very time I'm writing
the entire constitutive relation in direct notation, okay?
So, the way we would write a temperature gradient would be that, right?
Being the spatial gradient operator, okay?
Could be this or perhaps you are familiar also with writing that
gradient as partial of u with respect to the vector x, okay?
So if we had this constitutive relation
kappa then would be the diffusivity tensor.
Okay? And as we did from the case of heat
conduction, we would make the observation kappa equals kappa transposed.
It's symmetric and it's positive semi-definite, right?
So, if c
not equal to zero and
c being a vector, okay?
If, if c is a non zero vector,
then c dot kappa over c is greater than or equal to zero.
okay?
To the same, the same mathematical properties as
the heat conductivity tensor.
All right, and then the condition that u equals u,
g, would simply be a condition of stating that we are controlling
the concentration or the composition, right?
On Dirichlet boundary, all right?
On the specified sub set of the boundary.
And finally saying
that j, sorry,
minus j dot n equals j
sub n on omega is sub j
would be the mass in flux
boundary condition.
Okay?
So, you know, everything else is well, everything else is really the same
between the heat conduction and the mass diffusion problem in 3D, okay?
At steady state it turns out, actually, when we, when we take away the steady
state assumption also there is this analogy between the two.
And of course the equations would be the same,
we will study those equations as well and of the finite element methods for them.
So, okay.
So, at this point we've looked at the strong form and, and coordinate notation.
We've also looked at some direct notation just for completeness let me write
out the PDE of the strong form in fully indirect notation, okay?
Okay, so this is,
find u given u sub g,
j sub n, f, and
it's constitutive relation
J equals minus kappa,
u, okay?
Find u, given all this stuff such that
Now, j, I, comma, I, is essentially the diversions of j, 'kay?
So we get minus del
dot j, right?
And del dot is the diversions operator in direct notation.
Equals f in omega, 'kay?
The boundary conditions, the [INAUDIBLE] boundary condition is
straightforward because it's only on the scalar unknown.
U equals u sub j on partial omega u and
the boundary condition is
minus j dot n equals j,
j sub n on partial omega sub j, okay?
All right, and this actually is it, right?
For as far as our strong form is concerned in direct notation, okay?
And one, one useful thing to look at here is what happens to this
equation when we make the substitution of the constitutive relation, right?
So when we make the substitution, we so substituting.
J equals minus kappa grad
u in the PDE, right?
We get minus
del dot minus
kappa grad u equals
f, right?
In omega, okay?
And then something that's commonly done,
is that if if kappa is specially uniform, right?
So that means kappa is not a function of position, right?
What does gives us then is the following equation, right?
It gives us minus del, sorry.
It gives us, if kappa is uniform,
it tells us that we get kappa contracted
with the hessian of u, okay?
This equals f, all right?
Now it's useful to write this coordinate notation to see exactly what is
meant here.
That cont, the the double dot there which I've referred
to as the contraction essentially is an extension to
two tenses of the idea for that product, all right?
And what this thing is doing for
us is, it is doing kappa, i, j,
multiplying u, i, comma j, okay?
In this context, what we have here,
that is the hessian operator.
Okay?
We could also take the special case, which is often done.
Which is that which is to assume or
to consider cases where kappa i j is
can be written as a scalar kappa multiplying the chronicle delta, okay?
Sometimes called the Kronecker delta tensor.
Okay, when we do this, what it implies for us, is that now if we go back to
the problem, where that we wrote just about, the form that we wrote just about.
And here we're noting that not only is the tensor kappa
representable as a scalar multiplying the chronic of delta.
And that furthermore here too we're seeing that kappa is the,
the scalar kappa is uniform, right?
If this thing is uniform then what we arrive at is kappa delta ij, u comma ij.
All right?
Equals f, right?
And here if you observe what
the action of the Kronecker delta
is it reduces that relation,
that PDE to kappa u comma ii equals f, okay?
In omega.
Okay? All right, so
this from of the equation is often called the Poisson equation.
Okay, so this is good.
Okay, in direct notation the same thing will be.
It would be kappa.
Now, there's square u but note that I do not have an under bottom on the square,
implying that it's not a tensile this is just a Laplace operator, right?
It has very a few [INAUDIBLE] Laplace.
Okay, all we've done is write the,
the Laplacian using that operated [INAUDIBLE], okay?
So and of course, in this sort of case if we have indeed that
kappa is that the tensor kappa can be represented as a scalar kappa, right?.
If we say that we have that representation then for the Neumann boundary condition.
Right, so the Neumann boundary condition
which is minus j dot n equals j sub n.
This is actually reduces to
the requirement that minus kappa
grad u dotted with n equals j sub n, okay?
It reduces in this case to a requirement on the normal
radiant of temperature, okay?
So this thing is this is the normal, normal gradient
Of the temperature u, okay?
And these are simplifications that are commonly used.
In fact, it is very often assumed that kappa does indeed have this
representation by the scale of kappa and the chronicle delta, right?
And what this translates to,
is that the physical the physical sort of situation that this represents.
So we have kappa equals
that, sorry.
If kappa ij equals just the scalar kappa,
multiplying the chronicle delta, what we have is,
what is called isotropic heat conduction
Okay?
We're specifying here that this body is such that when heat is flowing in here,
heat flows only in the direction of the temperature gradient, right?
So if we in, int, introduce a temperature gradient in one direction,
it does not induce heat flow in any other direction, okay?
That's, that's what it reduces to and furthermore,
it also says that the amount of heat flow we get by introducing a temperature
gradient in one direction is exactly the same heat flow that we get.
By introducing a temperature preheating in another direction.
In both cases,
the heat flow back would align with the direction of the temperate preheat itself.
And the amount of heat flow would work to be the same,
if we specify that the same temperature created in different directions.
Okay?
So, those are the sort of simplifications that are often used in representing heat
conduction, all of those equivalently mass diffusion as well.
Okay, we're actually going to end the segment here when we return and
pick up the next segment.
We are going to find ourselves ready to look at the weak form of the problem.