in this problem we are looking at a very
academic example to try to see what the
continuity equation tells us and does
not tell us about a flow what we have is a
fluid flow with three components U V and
W and we're given U an V and we're
looking for W according to what the
continuity equation or the mass balance
equation tells us when the flow is
incompressible this is a completely
academic fluid flow there is no fluid
flow no real fluid flow that will look
like this and even if it did we wouldn't
know because fluid flow measurement
campaigns or fluid flow simulations using
computers typically produce arrays of
pixelized data and we're never able
to plot functions based on this so this
is very academic and you should not
focus on the meaning of those equations
it doesn't have any what instead we're
looking at is the mathematical nature
and structure of the physical laws that
we are playing [with]. so let's see what the
mass balance equation tells us about W
and this flow the mass balance equation
in incompressible flow tells us that the
divergence of velocity is equal to zero
the divergent is an operator it doesn’t
mean anything by itself and the
velocity field here has a whole field
of vectors and everywhere in space the
divergent of velocity has to be 0 so
this is not a vector this is a field of
scalar values that are all equal to zero
the divergent of velocity is a very funky
way of writing something that's perhaps
more easily understandable if you split
it out and this is the change with
respect to X of the X component of
velocity plus the change with respect to
Y of the Y component of velocity plus
the change with respect to Z of the Z
component of velocity note again this is
a scalar it's a sum of three values and
this is done everywhere in space yes so
it's not a three component
vector field it is really just a number
a scalar field okay in this equation
here we see that we have some term for W
appearing now what we want to do is to
isolate this W here as a function of all
the rest and try to figure out what we
can say about W and this velocity field
here so let's do that let's work it out
we're gonna say sorry we're gonna say
that the partial derivative of W with
respect to Z Y s equal to zero minus the
other two so that's minus partial u over
partial X minus partial v over let me
rewrite this V partial mm-hmm
let me grab the pen correctly partial V partial
V over partial Y like so and what we're
gonna do is to introduce in this instead
of this u here the components
that we have above so we have minus the
partial derivative with respect to X of
this whole expression over here which is
just I'm disciplined I'm just gonna
copy it over x squared minus y
squared plus Z and then here I have
minus the partial derivative with
respect to y of 3x y plus 3y z y like this okay so let's carry this
out this is not a very difficult
derivation so we have the partial
derivative with respect to X this whole
function here is just going to be 2x so
this is going to be minus 2x like this
and then over there the partial
derivative with respect to Y of this
whole expression is going to be 3x and
then 3z here so that's going to be here
3x plus 3z
so if you group this together it turns
out its sums up as minus 5x plus no
minus 3z like so yes and this is the
partial derivative of W with respect to Z
this is kind of cool but we are not
really looking for the partial
derivative of W we're looking for W and
so what do we do when we have a
derivative and we want the primitive we
have to integrate and so what I want to
do is now is integrate this function
so we write W W when I integrate this
with respect to Z look what I'm gonna
have I'm gonna have minus 5x Z minus 3
times 1/2 of Z squared PLUS
Plus and this is very important so let
me write it big here plus plus what well
usually when we do an integration we
have an integration constant which is
there but this time we didn't do quite
the usual integration we did an
integration based here on a partial
derivative which means W may be a
function of many different things it may
be a function of Z but it's also a
function of XY and time and so we don't
know what those functions are and so
instead of just an integration constant
at the end over here I'm gonna write a
function it's an unknown function which
I'm going to call F like this and this
function is a function of all the rest
that we didn't have in this derivative
here so there's going to be here a
function of X Y and T and look look what
happens if I try different things like
let me pretend here that this
function this thing here that
this was five x plus six million
times y cube plus three times the time
when you take the derivative of this
whole expression here like so
you will still land you take the
derivative of this with respect to Z you
will still land on this very same
expression here so this is a very
dangerous thing is that we don't we
don't know at all
what this function f is and the
continuity equation gives us no way out
of this problem so it gives us no hint
as to what this function is clearly
this is a very stupid fuess you cannot
have I mean if the velocity in Z
increases very strongly with Y with a
huge factor that's much larger than X
and T then it would be a good question
to answer what pushes this flow what
accelerates this flow what is the
physical cause of this huge acceleration
and the answer is we don't know to
answer this question we need an equation
that looks at the momentum of the flow
so we need to write something that says
the change of velocity of the flow now
it can only be due to certain forces and
this is of course the Navier-Stokes
equation but this is a story for another
chapter and another example in this
problem here the only thing we can tell
about W is this let me try to have
straight lines is that W is this
function of Z but then has unknown
components of X Y and T and this is
extremely frustrating about the
continuity equation the continuity
equation does not allow you to predict
the velocities it only allows you to
predict how wrong you are give it three
velocities and you take the derivative
of each velocity with one with respect
to one dimension you add up those three
numbers and that hopefully should be
zero but this is all it tells you it
doesn't tell you how to get to any of
those components
completely it just gives you hints
as to what it is and so this is a major
frustration for fluid dynamicists that
the continuity equation is not enough to
predict what's going to happen to flows
anyway this is how you calculate the
third component of a three component
velocity field based on the
incompressible continuity equation here
we go
