in this problem we want to calculate the
acceleration field given the properties
of a fluid flow with this velocity field
right here so we want to know what the
acceleration is of every particle
everywhere all the time a shortcut
answer to this that is wrong would be to
say "well since the velocity here does
not depend on time acceleration which is
the change in time of velocity should
also be zero" and so the wrong answer to
this problem would be to say that the
acceleration field is zero vector is
zero everywhere all the time
this is not true and I will show you why
to see why it's not so you have to try
to plot this vector field here if you
plot this vector field you should get
something that looks a little bit like
this
you could try plotting the V X component
so all the horizontal components of the
velocity so you would put u as a
function of x and y so you can see that
U is only a function of X does not
change with Y so if you plot u here and
you increase you progressively with with
X it increases with increasing X like
this let me try to increase this one
perhaps also draw straight lines
something like this it gets it gets
longer all the time like this
it does not change with Y so you have
the same identical vectors as a function
of Y and so if you plot this whole thing
like this and you go further in the Y
direction then you will see those
vectors the X components of those
vectors does not change but now if you
plot the y component you see that the Y
component is a constant minus something
that increases with Y and so if you plot
the y component of this you may have a
long vector here and this vector becomes
a bit smaller here and ultimately it
becomes smaller and smaller like this
and the Y component does not depend on X
ok so we have something like this as a
vector field
and something like that now if you draw
some of those two components you'll get
the velocity vector view and you get
something that would look like this
something like this and looks something
like that on the top and from this I
think you can see you can picture that
the velocity is the same as a function
of time but certainly it's not the same
as a function of space and so if you
drop a particle in there so let's have a
let's have a pink red particle that you
would drop in here and you would follow
this trajectory as it follows the
velocity field and you would be here the
next time and then perhaps here the next
time step and then here and then here
and then here and so and so forth this
particle here would have a velocity that
changes with time the change in time of
the velocity of a particle is its
acceleration and this is what we want to
calculate we want to not only calculate it
for one particle but we want to
calculate it for all of the particles so
what we want to calculate overall if the
acceleration here is the total time
derivative of velocity so what we want
to calculate let me switch to something
a bit smaller yes this the total time
derivative of the velocity field and the
total time velocity total time
derivative of the velocity field is by
definition written like this it is the
partial derivative of the velocity field
with respect to time like this plus an
operator which we call the advective
operator sometimes called also the
convective operator and this operator is
the velocity dot the nabla operator this
is the convective operator and this
convective over here is applied to the
velocity field this whole thing here is
a shortcut it's a shortcut for
expressing the velocity in a slightly
more tedious way but perhaps some
that's a bit clear and so this lets
reproduce this is this guy here so
partial Z over partial T but then
overall over here we have then here we
have U the X component of velocity
multiplied by the change of the velocity
field with respect to X plus V times the
change with respect to Y of the same
velocity field
let me perhaps rewrite this one so we're
not confusing it with the nabla
operator and then plus at the back here
we have w times the change over z of the
velocity vector like this so this
expression here at the bottom is what we
want to calculate this is the
acceleration field for our flow so let
me move this perhaps up to here like so
let's try to calculate this in our case
in our specific case we have the
conditions that U the X component of
the velocity field is 2 plus 3x yeah we
have V is 4 minus 3y like this this is
completely arbitrary by the way there's
no physical flow that would follow those
rules but in this case we're interested
in the mathematical side of the physics
to understand what this acceleration
field is so we're not too worried about
having an unrealistic flow field so
let's calculate this
overall let's take here the total
time derivative this component here this
expression here is actually three
expressions it is a vector expression so
when we write it out we need to write
three components perhaps let's split
them out and then let's fill them in we
have over here let me pick a better
color something like this we have in
green the total time derivative of
velocity it is three components in X is
going to be the change in time of X
component
so partial partial T of U yes plus
this component the X component of this
part here and the X component of this
will be here U multiplied by the
change over X of the X component of V
which is U and then here we have V
multiplied by the change in Y so V let
me rewrite this plus here your like this
V multiplied by the change in Y of the X
component of the velocity which is u and
then we have W here multiplied by the
change in Z of the X component of V
which is U minus so this is the X
component of that expression here and
now we add to this the Y and the Z
component like so so this is a change in
time plus U times the change in X oops
that's too fast plus V times the change
in Y plus W times the change in Z and
the same thing again you know for the Z
components so change over time plus u
times the change in X plus V times the
change in Y plus W times the change in
Z like this and now I can fill in the
terms and on the second line here we
have all the Y terms so all the
components here will be V the Y
component of the velocity
like this and on the bottom it will be w
w w w and w here so these are the three
components that we want to calculate
when we say we want the acceleration
field we want all those three
equations so let me move those down a
little bit like so yeah let's try to
calculate them with the specific
conditions that are highlighted here
which is that U is 2 plus 3 X and V is 4
minus 3 y ok let's work it out
let me move
this a little bit higher I know this is
moving behind the thumbnail but I need a
little bit more space so let's let's
move through the page so I have a bit
more space here we have the total time
derivative oops wrong color
total time derivative of the velocity or
if you want the acceleration field this
is equal to the following the change in
time of U is the change in time of this
component doesn't depend on time so it's
zero plus u multiplied by the change in
X of that component so let's take U
which is 2 plus 3x here let's multiply
it by the partial derivative of this
with respect to X which is just going to
be 3 and then we add here V times the
partial derivative of U with respect to
Y Z is here and the partial derivative
of U with respect to Y is going to be 0
so I have here 0 and then I am here W
which is going to be 0 in our case
multiplied by the partial derivative of
U with respect to Z which is also 0 so
we have 0 over here alright that's not
too hard we see a lot of terms are
going away let's take the second term
now partial derivative with respect to
time of V is 0 u multiplied by the
partial derivative of V with respect to
X U is here but the partial derivative
of V with respect to X is 0 so this is
going to be 0 here plus V times the
partial derivative of V with respect to
Y so V is here and the partial
derivative of V with respect to Y is
going to be minus 3 so let's write it
out V is four minus 3 Y and I
multiply this by minus 3 the partial
derivative of V with respect to Y and
then to this I have to add W multiplied
by partial derivative of V with respect
to Z and both of those terms are 0
so that's another zero and then we move
on to the W lines with the Z component
and the change of zero with respect to
time is zero here we have the change of
zero with respect to X it's also zero
here we have the change of zero with
respect to Y is also 0 and you have you
guessed already that the change of zero
with respect to Z is also zero like this
yes it's making sense now so let's bring
this down to a simpler expression we
have on the X component we have
there's going to be 6 plus 9 X here and
on the middle here we're
gonna have minus 12 plus 9 Y and on
the bottom of course we have zero here
like so and this my friends is the
acceleration field so this is the
acceleration everywhere in space and in
time
inside this strange drawing that we did
over there which is which is above so
inside this field here that we have here
everywhere in space and time you take one
particle you drop it here that was
coordinates X and those coordinates Y
here when you bring those in into the
acceleration field you put values of x
and y in here and you're gonna land on
the acceleration of that particle at
that moment in time so here I this is
how you calculate the acceleration field
our fluid flow given its velocity field.
