Fluid dynamics

1
Pressure is a scalar quantity.
Fluid exerts the same pressure in all
directions means that the force transmitted
through an interface near a given point does not
depend on the orientation of the interface.
Positive pressure means that adjacent fluid
elements are pushing each other
The pushing force acts through the interface and
is proportional to the area of the interface. 2x
area gt 2x force.
point of interest
2
Hydrostatic equilibrium
Condition for hydrostatic equilibrium constant
pressure throughout the fluid volume. Variation
of pressure creates net force in the direction of
decreasing pressure.
3
Hydrostatic equilibrium with external forces
Gravity
Force from above (pushing down)
Force from below (pushing up)
Gravitational force (pulling down)
Balance (equilibrium) equation
4
Hydrostatic equilibrium with external forces
Gravity
We have got a differential equation Where do
we go from here?
We need to integrate it. Can we?
If both r and g are constant, certainly yes.
What is P0? And BTW, what is h?
P0 is the constant of integration the value of
pressure at h 0.
It is natural to count h downwards from the fluid
surface (h depth). Then P0 is the pressure at
the surface the atmospheric pressure.
If the atmospheric pressure changes, does the
pressure at a given depth change?
Yep!
5
If the atmospheric pressure changes, the pressure
at all depths changes accordingly.
Pascals lawa pressure change anywhere in a
fluid is felt throughout the fluid.
How can we find the pressure inside a closed
vessel?
We need to know the pressure P0 at some point.
Then to calculate the pressure everywhere else we
use
Only applicable in the hydrostatic equilibrium.
6
Hydrostatic equilibrium with external forces
Gravity
What is the depth, h, where P 2P0?
The density, r, is only constant in liquids.
Gasses are compressible, and one must assume
variable r(h).
Earth atmosphere
7
Pressure at the surface of water or at 0 altitude
is the usual good candidate for P0
Ocean (or cup).
Atmosphere
Here h is the depth.
Here h is the altitude.
Pressure grows by about 1 atm (105 Pa) every 10 m
Pressure drops by about 120 Pa every 10 m
8
Drinking through a straw.
When you get a tooth extracted you are warned not
to drink through a straw. Why so?
The pressure of liquid at the top of the straw is
For you to drink, the pressure inside your mouth
should drop below Ptop.This implies a
substantial negative pressure difference, rgh,
between the blood vessels in your gums and your
mouth, which may open the wound.
9
The total force acting on the bottom of the
vessel is
The most remarkable thing about the expression
for pressure is what it does not include. The
expression for hydrostatic pressure is easy to
see for the straight, unobstructed column, but
not obvious for more contorted geometries.
The force exerted on the bottom may be strikingly
different from the weight of the liquid!
10
A multiplication of force can be achieved by the
application of fluid pressure according to
Pascal's principle, which for the two pistons
implies            P1 P2 This allows the
lifting of a heavy load with a small force, as in
an auto hydraulic lift
Hydraulic lift
but of course there can be no multiplication of
work, so in an ideal case with no frictional
loss Win Wout
11
A vacuum cleaner
Does the vacuum suck the dust?
Strictly speaking the pump of the cleaner creates
lower pressure inside it and the air and dust are
driven into the hose by the pressure of the
atmosphere.
12
A barometer
evacuated volume
Pressure of the air, Patm,drives mercury into
the hollow tube.
There is no air and no pressure inside the
tube.Therefore Patm is balanced by
hydrostatic pressure of the mercury.
Patm rgh
r 13600 kg/m3 - density of mercury
13
A manometer pressure gauge
Now there is a fluid (gas) under pressure in the
reservoir.
The difference between the pressure inside the
reservoir, Pres, and Patm is now balanced by
hydrostatic pressure of the mercury.
gauge pressure
absolute pressure
A manometer is measuring gauge pressure
14
Buoyant force
Archimedes principle The buoyant force on an
object is equal to the weight of the fluid
displaced by the object.
The buoyant force is applied to the center of
gravity of the fluid (center of the submerged
volume of the body).