fluid - Physics

9.3 Density and Pressure
Density ! = M V
CHAPTER 9 – SOLID & FLUIDS
(9.3-9.10)
kg m"3 and can depend on temperature. Specific gravity
!
! water,4 ! C
.
Pressure in a fluid P = F A Nm!2
( 1 Nm !2 = 1 Pascal)
At any particular depth, the pressure is constant throughout a fluid.

Worked Example: Problem #16
A 70-kg man in a 5.0 kg chair tilts back so that all the weight is balanced on two legs of
the chair. Assume that each leg makes contact with the floor over a circular area with a
radius of 1.0 cm, and find the pressure exerted on the floor by each leg.
9.4 Variation of Pressure with Depth
In equilibrium, all points at the same depth
must be at the same pressure. Otherwise a
net force would be applied and the fluid
would accelerate.
Pick a volume of fluid a distance h below
the surface:
PA ! Mg ! P 0
A = 0
but w = Mg = "V
so
P = P 0
+ "gh
( ) g = ("Ah) g
P 0 =1.013x10 5 Pa at sea level
P increases with depth by an amount mgh.
NOTE that an increase in pressure applied to an enclosed fluid is transmitted
undiminished to every point in the fluid (including the walls of the container)- Pascal’s
Principle
Worked Example:
At what depth in the ocean is the pressure twice that of the atmosphere alone (the density
of seawater is about 1.02 kg m -3 )
What to wear at a depth of 200m

Application: Hydraulic Press
P = F 1
A 1
= F 2
A 2
! F 1
= F 2
A 1
A 2
Can you think of other applications of this principle
9.5 Pressure Measurements
Two devices for measuring pressure:
open-tube manometer
Manometer:
barometer
Note that P = absolute (true) pressure inside the bulb. P-P 0 is the gauge pressure, the
pressure that is added to the atmospheric pressure to equal P.
Barometer:
1 atm. ! pressure equal to a 0.76 m column of mercury at T=0°C and g=9.80665 m s -2 .

P = !gh = ( 13.595x10 3 kg m )( 3 9.80665 m s ) 2
( 0.760 m)
= 1.013x10 5 Pa = 1 atm
You can read Torecelli’s own description of his barometer here.
Blood pressure
Blood pressure is measured in terms of the
column of mercury (in millimeters) that
could be supported by the pressure inside
the arteries at two times: maximum thrust
by the heart, and when the heart is relaxed.
These are normally about 120 mm and 80
mm, respectively.
Recent medical guidelines suggest that the
familiar old “normal” 120/80 values are
too high, and that somewhat lower values
are desired!
Exercise:
A collapsible plastic bag contains a glucose
solution. If the average gauge pressure in
the artery is 1.33x10 4 Pa, what must be the
minimum height h of the bag in order to
infuse glucose into the artery Assume that
the specific gravity of the solution is 1.02.

9.6 Buoyant Forces and Archimedes’s Principle
Any object completely or partially submerged in a fluid is buoyed up by a force whose
magnitude is equal to the weight of fluid displaced by the object.
( ) A
buoyant force B = P 2
A ! P 1
A = P 2
! P 1
( ) = " fluid
gh
but P 2
! P 1
so B = " fluid
ghA = g" fluid
( hA)
= g" fluid
V displaced
= gM fluid displaced
Sink or Float Depends on B-w where w is the weight of the object.
For a completely submerged object, the volume of the object is the volume of displaced
fluid,
B ! w = " fluid
Vg ! " object
Vg = (" fluid
! " object )Vg
( ) < 0, B ! w < 0, object sinks
( ) > 0, B ! w > 0, object rises
( ) = 0, B ! w = 0, floats wherever
For " fluid
! " object
For " fluid
! " object
For " fluid
! " object
For an object floating partially submerged on the surface,
Worked Example: Problem 26
B ! w = 0 = " fluid
V fluid
g ! " object
V object
g
# " object
" fluid
= V fluid
V object
A frog in a hemispherical pod finds that he
just floats without sinking in a fluid of
density 1.35 g/cm 3 . If the pod has a radius
of 6.00 cm and negligible mass, what is the
mass of the frog
Worked Example: Problem 28
The density of ice is 920 kg/m 3 n and that of seawater is 1030 kg/m 3 . What fraction of the
total volume of an iceberg is exposed

9.7 Fluids in Motion
2 types of flow: laminar (streamline) & turbulent
Example: Numerical simulation of flow over a racing car. Here, the pressure is colorcoded,
with blue being low pressure and red being high pressure. The flow lines are
drawn in. Note that while the flow is laminar over much of the car, it breaks up into
turbulent eddies behind the car.
For more neat simulations go here.
Ideal Fluids:
1. nonviscous
2. incompressible
3. steady (does not depend on time)
4. not turbulent

Equation of Continuity
For an incompressible fluid, flowing with no added “sources” or “sinks”:
mass in = mass out
! 1
A 1
v 1
"t = ! 2
A 2
v 2
"t
! 1
A 1
v 1
= ! 2
A 2
v 2
but ! 1
= ! 2 ( incompressible)
# A 1
v 1
= A 2
v 2
Can you think of some examples of this principle
Bernoulli’s Equation
Here, we will look at how the pressure changes in a laminar fluid flow.
W 1
= F 1
!x 1
= P 1
A 1
!x 1
= P 1
V
W 2
= F 2
!x 2
= "P 2
A 2
!x 2
= "P 2
V
note sign as F 2
and v 2
opposite directions
and net work done by the fluid is
W fluid
= W 1
" W 2
= P 1
V " P 2
V
Now, part of the work goes into changing the KE of the fluid, and part goes into changing
the gravitational potential energy (mgh stuff).
!KE = 1 2 mv 2 " 1 2
2 mv 2
1
and !PE = mgy 2
" mgy 1
W fluid
= !KE + !PE
P 1
V " P 2
V = 1 2 mv 2 " 1 2
2 mv 2 + mgy
1 2
" mgy 1
divide by V : P 1
" P 2
= 1 2 #v 2 " 1 2
2 #v 2 + #gy
1 2
" #gy 1
P 1
+ #v 2 1
+ #gy 1
= P 2
+ 1 2 #v 2 + #gy
2 2
$ P + #v 2 + #gy = constant % Bernoulli's Law

Venturi Tube:
P 1
+ 1 2 !v 2 = P
1 2
+ 1 2 !v 2
2
("y = 0)
Since v 2
> v 1
# P 2
< P 1
The increase in velocity of the fluid is
accompanied by a drop in its pressure!
9.8 Other Applications
Aircraft Wing:
When air flows over the wing of an
aircraft, the flow is faster over the more
curved top than on the bottom, so that the
pressure is lower on top than on the
bottom. (Note: air is compressible, but the
effect is small in this case and can be
ignored). The tilt also aids lift. But
turbulence disrupts the flow, diminishing
the effect.
Atomizer:
A stream of air passing over a tube dipped
in a liquid causes the liquid to rise in the
tube. Used in perfume atomizer bottles and
paint sprayers.
Vascular Flutter:
The constriction in the blood
vessel speeds up going through
the constriction. The lower
pressure causes the vessel to
close, stopping the flow. Without
flow, there is no Bernoulli effect,
and blood pressure causes it to reopen.
The process repeats.

Applying Physics to the Home
Consider the portion of a home plumbing
system shown in the figure to the left. The
water trap in the pipe below the sink
captures a plug of water that prevents
sewer gas from finding its way from the
sewer pipe, up the sink drain, and into the
home. Suppose the dishwasher is draining,
so that water is moving to the left in the
sewer pipe. What is the purpose of the
vent, which is open to the air above the
roof of the house In which direction is air
moving at the opening of the vent, upward
or downward
Worked Example
What is the net upward force on an airplane wing of area 20.0 m 2 if the airflow is 300 m/s
across the top of the wing and 280 m/s across the bottom
Worked Example: Problem #43
A hypodermic syringe contains
a medicine with the density of
water. The barrel of the syringe
has a cross-sectional area of
2.50x10 -5 m 2 . In the absence of
a force on the plunger, the
pressure everywhere is 1.00
atm. A force F of magnitude
2.00 N is exerted on the
plunger, making medicine
squirt from the needle.
Determine the medicine’s flow
speed through the needle.
Assume that the pressure in the
needle remains equal to 1.00
atm and that the syringe is
horizontal.

9.9 Surface Tension, Capillary Action, and Viscous Fluid Flow
Surface Tension
The combined electrical attraction of molecules in a fluid gives rise to a force that tends
to minimize the surface area of the fluid. This makes raindrops spherical. If they weren’t
we would not see rainbows the way we do!
This surface tension " acts like a local force along the surface of the fluid:
! " F L
Nm #1 or Jm #2
Note that the units are the same as the spring constant.
The surface tension can support small objects placed on top of the surface (such as a
needle, which will float if placed carefully on the surface of still water), and hold back
others from leaving it (this impedes evaporation from a body of water, for example).
The surface tension of a fluid can be
measured with a device like that shown
here. If the force required to break free of
the water is F, then
! = F
2L = F
4"r
where r is the radius of the hoop. (Here we
need to factor of 2 because the surface
tension exerts forces on the inside and the
outside of the ring.
The following table lists the surface tension of some common fluids.
Note that " depends on temperature. At higher T, the molecules are not as tightly bound
together. You can also alter the surface tension of fluids using additives.

Surfaces of Liquids
When water sits
on a surface or in
a container, the
shape the water
takes depends on
whether it is
more strongly
attracted to itself
(cohesion) or to
the “other”
material
(adhesion).
Detergents – wet – allows water to penetrate clothes when washing and to spread over
glass surfaces better.
Repellants – water beads up & penetrates less.
Capillary Action
Wetting – pulls up
Examples – paper towels, sponges, mops,
finger-prick blood samples
F = ! L = ! 2"r
( ) so F vert.
= ! 2"r
w = Mg = $Vg = $g"r 2 h
Non-wetting – pushes down
( )cos#
! ( 2"r )cos# = $g"r 2 h %& h = 2!
$gr cos#
Worked Example: Problem#56
A staining solution used in a microbiology laboratory has a surface tension of 0.088 N/m
and a density 1.035 times that of water. What must be the diameter of a capillary tube so
that this solution will rise to a height of 5 cm (Assume a contact angle of zero).

Viscous Fluid Flow
Viscosity – the internal friction of a fluid. Resistance to shear stress.
F = ! Av Force needed to move plate
d
A = area in contact with fliud
! = "coefficient of viscosity"
Units N " s " m #2 SI
( ) poise (cgs)
where1 poise = 10 #1 N " s " m #2
Poiseuille’s Law
Rate of Flow !V
!t
= " ( R4 P 1
# P 2 )
8$L
Affects blood flow, squeezing Krazy Glue gel out of its tube, etc.
Reynolds Number
When is the onset of turbulence Fluid flow in a pipe of diameter d:
RN = !vd
"
v = average speed
d = diameter
" = viscosity
! = density
RN < 2000 laminar flow
2000 < RN < 3000 unstable…….
3000 < RN turbulent flow

9.10 Transport Phenomena
Diffusion
Net movement of a population across a “cross-section” by random walk from a region
where the concentration is higher to a region where it is lower.
(
= DA C " C 2 1)
L
!M
!t
D = "diffusion coefficient" m 2 s "1
( C 2
" C 1 )
= concentration gradient
L
Osmosis
Movement of water from a region where its concentration is high, across a selectively
permeable membrane, into a region where it is lower.
(READ THIS SECTION ON YOUR OWN)
Note use in artificial kidneys. Used in both hemodyalisis and paritoneal dialysis.
Motion through a Viscous Medium
Resistive force on spherical object of radius r: F r
= 6!"rv
Stokes's Law
Terminal Speed – net force goes to zero – velocity is constant v t
= 2r 2 g
(
9! " # " object fluid )
Sedimentation & Centrifugation - READ