Re: [Phys-L] Dirigible Flight Question

[John Denker <jsd@av8n.com>]

On 11/05/2012 10:49 AM, Paul Nord wrote:
> If I've got a blimp inflated to neutral buoyancy and I hang a small 
> mass from it, what will the acceleration of the blimp be immediately 
> after I attach the mass?

Egads.  This is a nightmarishly complicated question.  Much depends
on non-idealities ... and on what you mean by "immediately".

In particular, in the real world, the dirigible is not 100% rigid,
so in the first femtosecond after you attach the weight, it will
be bending and flopping in response to the new stress, such that
there will be no such thing as "the" position, velocity, or acceleration
of "the" aircraft.

To make progress, you might imagine a slightly longer timescale, and
pretend that the aircraft is super-rigid.   Then in the first microsecond
after the weight is attached, you can ignore the motion of the surrounding
air.  On short timescales, the air acts mostly like a spring ... and in the
first microsecond you haven't even compressed the spring very much.  The 
air hasn't yet had time to set up a flow-field in response to what the 
dirigible is  doing.  The timescale for setting up such a flow-field depends 
on the speed of sound in air.

If you want to ask questions about viscous drag, you need to look on a
very much longer timescale.  You probably want to cross out the word
"immediately" and ask about the long-term steady state.

> Question 1: Of course Pascal's Principle says that air pressure will
>  distribute itself equally on all sides.

That's for statics only.  This is a dynamical situation.  There will be
all sorts of lift and drag pressures, not equally distributed.

> Question 2: Flow through a vicious fluid is typically modeled with a 
> term which rises as a function of the square of the velocity.

Sometimes. It depends on the Reynolds number.  For very small velocities,
such as we might get from attaching a very small mass, the force is only
linear in velocity.  Also FWIW in a heavier-than-air aircraft the drag
includes a contribution from /induced/ drag, which is a /decreasing/
function of velocity!

> If it use the mass of the displaced fluid in the calculation above,
> am I already accounting for some of the "drag" force which is
> normally accounted for in this velocity squared term?

That's a smart question.  In this business, there are very often multiple 
ways of explaining the same thing, such that when you try to quantify it 
you run the risk of counting something and its synonym as two things when 
really there is only one thing.

In this case it comes down to a question of terminology:  By convention the
unadorned term "drag" includes everything that acts like drag.

Also note that everything is /indirectly/ dependent on everything else.  For
example, even if you think the drag does not directly depend on the displaced
mass, it indirectly depends, because it is related to the overall size of the
aircraft, and drag certainly depends on size.  At the end of the day the most
you can achieve is consistency, i.e. a consistent set of numbers for the size, 
mass, acceleration, velocity-field, pressure-field, et cetera.  You might wish 
for some nice simple separation of variables, but you're not going to get it.

============

Beware that fluid dynamics is verrrry complicated.  People live surrounded
by fluids, and they think they understand what's going on, but they don't
... with the rarest of exceptions.  I spent a couple of years trying to
figure this stuff out, and I'm still not an expert.  That's a couple of
years /on top of/ a PhD in physics.  It's a couple of orders of magnitude
more complicated than the topics ordinarily discussed in this forum.

If you are interested in /unsteady/ flow, such as what happens "immediately" 
after the weight is added, that adds yet another order of magnitude of 
complexity.  I would seriously not recommend this as a starting point.