(Since this was partially written a couple days ago, I decided to
complete it and post it anyway even though I'm sure some were hoping the
the discussion had died. :-))

Alan Baker wrote:
Jim Logajan wrote:
To that question I say: "Nothing in the physics rules it out." The
math appears to allow a solution wherein the fluid moves such that
the *forces*, *momentum*, and *mass flows* are *all* conserved _long
before_ any downwash strikes the earth.

Sorry, but you're wrong.

Tsk. Here's the math that shows there are solutions wherein momentum is
conserved yet yields zero average mass flow. It uses two conservation
laws (one of them being the first derivative of conservation of
momentum):

A toy helicopter is hanging from a line in a sealed room initially at
rest, and is released from the line and engine started at the same time
so it remains stationary. The law of conservation of mass requires that
the net flow of mass through the surface of any volume in an
incompressible fluid must be zero. For this problem the volume of the
room bounded by the walls above the plane made by helicopter blades and
said plane is used for one of the needed constraints.

The vertical forces through the plane made by the blades is described by
two areas: the area made by the blades (call it A1, in m^2) and the
remaining area of that same plane (call it A2, in m^2; the area from the
blade disk to the walls):

(1) F1 = M1d*V1
(2) F2 = M2d*V2

Whe
F1: Force of downwash through A1, in N. Positive number.
F2: Force of any upward flow through A2, in N. Positive number.
M1d: Average mass flow, in kg/s, of through A1. Positive number.
M2d: Average mass flow, in kg/s, of through A2. Positive number.
V1: Average velocity, in m/s, of flow through A1. Positive number.
V2: Average velocity, in m/s, of flow through A2. Positive number.

Since the chosen volume remains completely filled with fluid at all times
due to the constraints on it, its center of mass never moves, indicating
zero net force. The helicopter never moves either, so the force the
helicopter exerts on that volume, the force any upflow exerts on it, and
the force of gravity on the helicopter must all sum to zero:

(3) Fz - F1 + F2 = 0
Whe
Fz: Gravitational force, in N, on helicopter. Positive number.

Substituting (1) and (2) into (3):

(4) Fz - M1d*V1 + M2d*V2 = 0

(5) M1d - M2d = 0
{Conservation of mass; that is, the net flow of mass through the
surface of any volume in an incompressible fluid must be zero. So
the net flow into our chosen volume must be zero.}

Solving for V2 in terms of M1d, V1, and Fz (exercise left for the reader;
see also note [1])) yields:

(6) V2 = V1 - Fz/M1d {for F1 Fz, i.e. V1 Fz/M1d}

Now an example:

Given a downward gravity force of
Fz = 10 N
with a downwash of
V1 = 6 m/s
M1d = 2 kg/s
the equations yield the following average upward flow values:
V2 = 1 m/s
M2d = 2 kg/s

Plugging those back into equations (4) and (5) indicates they are valid
solutions. Forces balance and momentum is conserved. Mass flow is
conserved (no sinks, no sources.) So the SOLID PART part of the earth
DOES NOT have to move up to insure conservation of momentum because part
of the FLUID PART of the earth has already moved up and done that job.

[1] The M1d and M2d variables actually contain V1 and V2 respectively;
e.g. M1d = rho*A1*V1 where rho = density of the fluid in kg/m^3 and A1 =
area in m^2 that V1 is measured through.

The only way to get momentum is to have mass in motion. That it is
greater and greater mass as the system evolves means it is moving more
and more slowly, but it is still moving until it can transfer its
momentum to something else.

You've contradicted yourself. You are now saying there is a reduction in
velocity of the downwash. That can only happen if a force is acting
upward on it. That upward force, that you inadvertently overlook, would
be due to the air itself. It has inertia and transmits the pressure
exerted on it to the ground much as a solid object would - at the speed
of sound in that material.

Here's what really happens: the instant the blades started to move the
first tiny bit of fluid downward, conservation of mass in a fluid
required an equal tiny bit of mass of the fluid to flow upward. The fluid
flow starts as a tiny closed circular-like flow and grows into a large
circular-like flow. Whatever their shape and size, the flows form closed
circuits.

Meanwhile, the fluid masses that are forced to move cause pressure waves
to move out at the speed of sound (which is infinite in a perfectly
incompressible flow) and it is those pressure changes that will appear
elsewhere, such as the earth's surface. The earth accelerates upward
until the downward pressure balances the upward gravitational pull
yielding net zero force: in the idealized case of a perfectly
incompressible fluid, the instantaneous rise in pressure at the surface
yields a net force just balancing the pull of the aircraft's mass and the
earth doesn't move upward at all.

I can't adjust my thinking to your way of thinking until you explain
which directions the fluid is flowing in my hypothetical case to my
satisfaction.

Your trying to obfuscate and I'm not buying it.

You obfuscate - I clarify: I've just presented a simplified mathematical
model (more than you've presented) and a better conceptual model for
incompressible flow as it relates to flight than yours.

You've already admitted ignorance of fluid mechanics when you said the
law of conservation of masses was only relevant to chemistry - so I
should really be charging you tuition.

Net force up on plane: net force down on air.

That's correct.

Net force down on air: net momentum down.

Transmitted at the speed of sound, not at the speed of the downwash.

Net momentum down: net velocity down.

Transmitted at the speed of sound, not at the speed of the downwash.