Fluid dynamics

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Fluid dynamics
Fluid is matter that flows under influence of
external forces (?)
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Pressure is a scalar quantity.
PF/A
The force exerted against any surface is always
normal to that surface.
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Hydrostatic equilibrium
Condition for hydrostatic equilibrium constant
pressure throughout the fluid volume (in the
absence of a gravitational field). Variation of
pressure creates net force in the direction of
decreasing pressure.
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Can pressure NOT be the same for different
directions?
No net force! A solid would be in
equilibrium. What about a fluid?
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Density measures the mass per unit volume.SI
units are kg/m3
Pressure measures the normal force per unit area
exerted by a fluid.SI units are N/m2 or Pa,
Pascal.
There are compressible and incompressible fluids
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Abrahams tank
mosquito
Mass 63 tons ( 63,000 kg)Caterpillar area 6
m2Ground pressure
Proboscis about 0.1 mm in diameter.Pressure of
105 Pa corresponds to
Equivalent mass is
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Abrahams tank
Equivalent mass is
A mosquito weighs about 2 mg, 0.002 g.An ant can
lift 50 its weight.
If the same is true for a mosquito, it can
produce a force equivalent to
and exert the same pressure as an Abrahams tank
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Do Newtons laws apply in fluid mechanics?
Of course they do, since they apply everywhere
unless the things get relativistic or
quantum.What is the problem with the fluid
mechanics then?Why do not we study it together
with the regular mechanics?
The trouble is we have to deal with a
continuumwhere the notions of a body is
difficult to define and it is not always
straightforward where the forces come from.
What do we do?For appropriate description of
continuum mechanics we have got to introduce new
notions and parameters.
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Newtons 2nd Law for a Fluid Element
Assume that the pressure only changes in the x
direction
For a fluid element with sides
Newtons 2nd law becomes
So we see that it is the negative pressure
gradient that results in the acceleration of a
fluid element
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Hydrostatic equilibrium with external forces
Gravity
Force from above
Force from below
Gravitational force (pulling down)
balance

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Hydrostatic equilibrium with Gravity
This is a differential equation.How do we
calculate the actual pressure?
We integrate the equation
And end up getting an integration constant,
P0.What is this constant about?
Mostly pain in the neck!..In order to find
pressure, P, in a given point, we need to know
the pressure P0 in some reference point, h0.Then
we compare the depth between the two points and
calculate P as
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In order to find pressure, P, in a given point,
we need to know the pressure P0 in some reference
point.
What might this reference point be?
Surface of water with is
the usual candidate.
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
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Drinking through a straw.
When I got my tooth extracted I was 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 large
pressure difference between the blood vessels in
your gums and your mouth, which may open the
wound.
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Pressure versus depth in water

• At what depth is the pressure twice
• atmospheric pressure?

(b) What is the pressure at the bottom of the
11.3km deep Marianas Trench?
Demonstration with column and plate.
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Force against a pool wall
Find the force of water on a pool wall that is
15m wide and 3m deep.
Substituting the values P01.0X105Pa and
r1000kg/m3 leads to F5.2X106 N.
What is the direction of this force? How will
the force on a dam depend on the size of the lake?
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The most remarkable thing about this expression
is what it does not include. The expression for
hydrostatic pressure is easy to see for the
straight, unobstructed column, but not obvious
for the cases of different geometry which are
shown.
Demonstration of equilibrium tubes.
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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 the cases of different geometry
which are shown.
The pressure force exerted on the bottom may be
strikingly different from the weight of the
liquid!
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The pressure force exerted on the bottom may be
strikingly different from the weight of the
liquid!
What about Newton? What supports the weight of
the liquid?
d1
The weight of the liquid is
h1
h2
However PA at the bottom PA at the lip is
d2
We see that
as it had to!!
What about the other containers on the previous
slide?
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The pressure on the sides of the containers
results in a force that has a vertical component
. It is this vertical component plus the force
from the pressure at the bottom that supports the
total weight of the liquid!
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Pascals Law
A couple of additional things that we should know
about fluids that are different than solid
objects
The first is Pascals law A pressure increase
anywhere in the fluid results in the same
pressure increase throughout the fluid.
As we shall see soon this is the basis for
hydraulic systems
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Hydraulic Lift
A multiplication of force can be achieved by the
application of fluid pressure according to
Pascal's principle, which for the two pistons
implies            P1P2 This allows the
lifting of a heavy load with a small force, as in
an auto hydraulic lift
but of course there can be no multiplication of
work, so in an ideal case with no frictional
loss Win Wout
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Hydraulic Lift
As an example, a small piston that exerts a force
of 450N P100Lbs and supports a car of mass
3200kg. The diameter of the piston that supports
the car is 120cm.
What is the diameter of the smaller piston in
order to maintain equilibrium?
From Pascals principle the pressure throughout
the is the same hence
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A barometer
Pressure of the air, Patmosphere,drives mercury
into the hollow tube.
There is no air and no pressure inside the
tube.Therefore Patmosphere is balanced by
hydrostatic pressure of the mercury.
Patmosphere rgh r Hg 13600 kg/m3
Patm 1.013X105Pa h Patm/rg .760m 760mm
What if we use wine rather than mercury?
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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 Patmosphere is now
balanced by hydrostatic pressure of the mercury.
Pres - Patmosphere rgh
A manometer is measuring gauge pressure
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Pressure and flow in a siphon.
h1
h2
Pressure in the static case.A flow causes
redistribution of pressures.