let us discuss an important concept, directly
related to bernawlli’s theorem that is,
pressure velocity trade off. for this we can
write that, for a fluid, flowing at a horizontal
level, according to bernawlli’s theorem,
we use, pressure plus, half ro v square, is
a constant at all points.
because, at horizontal level, there is no
difference in gravitational potential energy
per unit volume, so we can always take up,
the sum of pressure energy and ki-netic energy
per unit volume as constant.
that means at any point if, velocity of fluid
increases, it has to sacrifice for the pressure
value at that point, as their sum is a constant.
and if at a point, pressure is increased automatically
the flow velocity will decrease.
this phenomena is called pressure velocity
trade off because, pressure as well as velocity
both have to trade off, at any point in the
flowing fluid. and based on this there are
many applications we are going to see, like
we can see this is a picture of an aerofoil,
this is an airplane wing, which is moving
in forward direction say with a velocity v
p. and due to the shape and motion of this
wing, we can say, the stream lines will become
denser in the upward zone, or we can say the
flow velocity of air relative to plane, will
be more at a point above the wing compared
to a point which is below the wing.
so if we consider there is a point ay and
another point b, ay is below the airplane
wing and b is above the airplane wing.
here we can state directly that, velocity
at point ay is less than velocity at point
b. so in this situation if at point ay velocity
is less according to the concept of pressure
velocity trade off which we have discussed,
here we can state, pressure at point ay will
be more than pressure at point b. if at point
ay pressure is more than that at b, then due
to this pressure difference, this airplane
will experience, an upward lift force, due
to this pressure difference, and this force
will balance the whole weight of the airplane.
and such wings are on both the sides of airplane,
so total lift force on both the wings, has
to balance the weight of airplane.
this is the concept how airplane fly. and
here we have just considered that, points
ay and b are very close to each other and
almost at the same level, from the ground,
and respective heights of both of these points
we consider to be equal as the figure we have
drawn is quite a magnified one.
similarly we can discuss one more application
for this pressure velocity trade off here.
here we can see this is an atomizer, using
which we can spray, any liquid filled in it.
say a liquid is filled in it and from this
balloon if we push it and release it, then
what happen is the air, which is just above
the nozzle of, this tube connected with the
balloon, it’ll move in forward direction.
and as air starts blowing, the pressure at
this point say this point is x, and here there
is a point y, we can simply state, the air
particles which are at point y are at rest,
and those particles which are at point x just
above this vertical tube which is dipped in
the liquid, will start moving.
so we can say pressure at point x in this
situation, this pressure at point x can be
taken as, lesser than pressure at point y.
and if pressure at point y is more we can
directly state that liquid will be pushed
in downward direction and it’ll start moving
in upward direction.
and from the topmost point, due to the container’s
flow of air particles as the liquid is reaching
the top, it’ll split into small number of
particles, or the tiny droplets of the liquid,
will move away, or these are blown away.
so this is the way how, liquid jets or sprays
are developed, and this is the concept which
is based on again the pressure velocity trade
off because just at this point, as well as
the point which is below the nozzle, we can
simply state, the pressure difference is developed
due to which, the concept of atomizer can
be understood.
similar to these applications, another example
can be seen.
if, through a vertical tube, air is blown
in upward direction, and a plastic ball is
kept, in the air blow.
then it’ll keep on floating in a stable
iquilibrium because, say if the ball is floating
and is pushed towards right, then we can simply
state, more stream lines will pass from the
region which is left of the ball, and the
flow velocity of air in the left side will
increase and on right it’ll be less.
or the air which is outside the blow, is almost
at rest.
so, in the region of no flow, pressure is
high, and the region where more air is passing,
the pressure will be low.
so if pressure is high in this region low
in this region, the ball will be pushed back
by the atmospheric pressure.
similarly if the ball is pushed towards left,
this’ll be the region of no flow, and on
the right side most of the air particles will
be moving, so pressure will be decreased here,
and it’ll be more in this direction, so
again atmospheric pressure will push the ball
in this direction.
so we can say, if a light weight ball is kept
floating above, a continuously blowing air
nozzle, then it’ll be, maintaining its floating
position in stable iquilibrium. even if it
is slightly displaced it’ll have a tendency
to automatically maintain its position.
this is again because of pressure velocity
trade off or the concept of bernawlli’s
theorem by which we can explain the phenomena.
