So this is a quick little video to show you
how to take two velocity components and see
whether the flow flow is irrotational and
whether it satisfies conservation of mass.
Also shows that it is incompressible.
So we have the x-component of velocity, which
is u, and y-component of velocity, which is
v, and the first question is the flow irrotational?
You probably understand what that means physically,
but what that means mathematically is that
this omega z, which is the rotation, which
equals one half dv/dx minus du/dy.
If this equals 0, then the flow is irrotational,
and notice that what this is rotation in the
z-plane, and what we are looking at are what
are called cross derivatives.
dv/dx minus du/dy.
So lets find dv/dx, and if we look at dv/dx
we find that the derivative is 2y.
Lets look at du/dy, and we find that this
derivative is 2y, and we can see that 2y-2y
equals 0.
So the answer to that question is yes.
It is irrotational.
Second to satisfy the conservation of mass.
Now we look derivatives of velocity with respect
to the direction that the velocity is in,
and instead of subtracting we add.
To satisfy the conservation of mass, du/dx
plus dv/dy has to equal 0.
So now lets find du/dx, and if we look at
our equation for u.
We find that du/dx is minus 1 minus 2x.
Now we look at dv/dy, and that's 2x plus 1,
and again this is an x.
When we add the two of them together we get
fortunately 0, which means that conservation
of mass is satisfied.
Why does it also mean that it is incompressible?
The answer to that is this is also the volume
dilatation rate.
So if the volume dilatation rate equals zero.
What it means is again since the mass does
not change then the volume does not change,
and therefore the density does not change,
and it is incompressible.
Hope this helps
