Biofluid properties introduction

1. CHAPTER 2
Fluid Mechanics
Session 1: Fluid statics

2. Learning outcome
At the end of this chapter and having completing the
essential readings and activities, the students should be
able to:
• Understand the concept of fluid statics
• Differentiate absolute and gage pressure
• Explain the characteristics of pressure in fluid
• Understand how to measure atmospheric pressure
• Explain different mechanism of pressure measurement
• Understand the concept of Buoyancy

4. Fluid Statics
• Fluid statics is the part of fluid mechanics that deals
with fluids when there is no relative motion between the
fluid particles
• A fluid at rest has no shear stress. Any force developed
is only due to normal stresses which is pressure
• This Session will cover pressure and its variation
throughout a fluid

5. Pressure
• Pressure is defined as the amount of force exerted on a
unit area of a substance.
• In algebraic form this definition may be stated as:
• Pressure acts uniformly in all directions
• Units for Pressure
1 pascal (Pa) = 1 kg m-1 s-2
1 bar = 1 x 105 Pa
1 atmosphere (atm) =101,325 Pa
1 torr = 1 / 760 atm
1 atm =760 mm Hg

6. Absolute and Gage pressure
• Atmospheric pressure: It is the pressure in the
surrounding atmosphere
• Gage pressure: pressure measured with measuring
instruments. It is above atmospheric pressure.
• Vacuum pressure: It is the negative pressure
measured below atmospheric pressure.
• Absolute pressure: The actual pressure at a given
position. It is measured relative to absolute vacuum
(i.e., absolute zero pressure).

7. Cont.

8. Sample problem 1
• The pressure inside a tank is at 4.2 atm at sea
level. What is the gauge pressure inside the
thank.
• In the above question If the pressure inside the
tank is 0.9 atm what will be the gauge pressure

9. 1
Why do you think nose
bleeding and shortness of
breath happens at high
elevation

10. Characteristics of liquid pressure
Pressure increases with depth
• Pressure in a static liquid increases linearly with
depth(vertical depth)
• pressure at the base = ρgh ….how? Hint pbase=F/A
• The fluid pressure is directly proportional to the depth of
the fluid

11. Cont.
Shape, size and Area
• At any depth h below the surface of the water in any
column, the pressure P is the same. The shape and area
are not factors

12. Cont.
All points at the same level have same
pressure
• The pressure is the same at all points on a given
horizontal plane in the same fluid

13. Cont.
Density and Gravity
• Liquid with different density have different pressure
• Liquid pressure increase with its density
• Liquid with larger force of gravity exhibits higher
pressure

14. Measurement of Atmospheric
Pressure
Barometer: It is device used to measure the atmospheric
pressure at any point on the earth
There are two types of barometer
• Liquid barometer: it measures the pressure with help
of column of liquid
• Aneroid barometer: It measures atmospheric
pressure by its action on an elastic lid of evacuated box.

15. Cont.
Aneroid barometer has aneroid cell which is made of
alloy of beryllium and copper. This alloy under goes
expansion and contraction on the change in pressure.
Thus, the mechanical levers of aneroid cell expand and
contract and display the reading on the aneroid
barometer by the movement of the pointer.

16. Cont.
Liquid Barometer
• It consists of a transparent tube which is open from one
end only.
• The tube is filled with liquid and is inserted in a jar
also containing same liquid.
• The liquid initially drops in tube due to gravity but
stabilizes at certain level under the action of
atmospheric forces.
• The atmospheric pressure is then measured as height of
liquid at which it stabilizes.

17. Cont.
• Three forces acting on fluid are
Patm(A)=Force of atmospheric Pressure
W=Weight of liquid
Pvap(A)= Force of vapour pressure
A=Cross-sectional area of tube

18. Sample Problem 2
• A diver is located 20 m below the surface of a lake (ρ =
1000 kg/m3). What is the Absolute pressure and gage
pressure( In pascal and bar) due to the water at a sea
level?

19. Sample problem 3
In the figure below, the height of the water and oil are
15m and 8m respectively. The container is open to the
atmosphere. What is the gauge and absolute pressure
A. At the oil water interface? Density of oil 750 kg/m3
B. At the bottom of the container.

20. Sample problem 4
A mercury barometer is exposed to air at sea
level. What is the height of the mercury column? (
the density of the mercury is 13,600 kg/m3)

21. Sample problem 5
The mercury column in a barometer is 72cm tall
at a certain elevation(density of mercury 13600
kg/m3)
A. What is the pressure of the air at this elevation
B. If the barometer was filled with water instead
of mercury , what height of water can this air
pressure support

22. Sample problem 6
• The height of water (p=1000kg/m3) and
oil(p=720 kg/m3) are 6m and 5m inside an
open barometer. What is the atmospheric
pressure in Kpa and atm

23. Pressure at a point(Pascal’s Law)
• States that
“An external pressure applied to a confined fluid is
transmitted uniformly throughout the volume of the
liquid”
• Pressure at any point in fluid is same in all directions
when the fluid is at rest
• This can be demonstrated by considering a small wedge
shaped fluid element that was obtained by removing a
small triangular wedge of fluid from some arbitrary
location within a fluid mass.

24. Cont.
• Since there are no shearing stresses at rest, the only
external forces acting on the wedge are due to the
pressure and the gravity(Weight).

25. Cont.
• For simplicity the forces in the x direction are not
shown, and the z axis is taken as the vertical axis so
the weight acts in the negative z direction
• From Newton’s second law, a force balance in the y-
and z directions gives
• where ps, py and pz are the average pressures on the
faces, γ and ρ are the fluid specific weight and density

26. Cont.
• The last term in Eq. b drops out as δx ,δy and δz→0
and the wedge becomes infinitesimal, and thus the fluid
element shrinks to a point.
• Thus substituting and simplifying results
• Thus we conclude that the pressure at a point in a fluid
has the same magnitude in all directions.
• This important result is known as Pascal’s law

27. Sample problem 7
A downward force of 100N is applied to the small piston
with a diameter of 50cm in the hydraulic lift.
A. What is the upward force exerted by the large piston
with a diameter of 2m?
B. What is the mechanical advantage of this hydraulic
lift?

28. Pressure Measurement
The devices below are used for pressure
measurement
1. Piezometer
2. Manometer
3. Mechanical Pressure Transducer
(bourdon gauge)( Reading Assignment)

29. Cont.
Piezometer
• It is used to measure pressure in pipes or vessels via
the rise of a fluid column.
• It consists of a transparent tube open from other ends
• The diameter of tube should > 13mm to avoid
capillarity action
• A piezometer tube is used only for measuring moderate
pressure

30. Cont.
Limitations:
• It must only be used for liquids
• Liquid is exposed to the atmosphere
• It cannot measure vacuum pressure

31. Manometer
A. Simple Manometer
• It consists of a U shaped tube, part of which is filled
with manometric fluid.
• One end of tube is connected with the pipe whose
pressure is required to be determined.
• Due to pressure, level of manometric fluid rises on one
side while it falls on other side.
• The difference in levels is measured to estimate the
pressure

32. Sample problem 8
The open tube manometer shown below contains liquid
mercury(p=13,600kg/m3)
A. What is the pressure of the gas in the bulb
B. What is the gauge pressure
C. Calculate A and B using distance of 40cm(instead of
60cm).

33. Sample problem 9
The u-tube manometer shown below is filled with fluid of
unknown density. The height difference between the two
columns is 1.3m. If the pressure of the gas in the bulb is
92,000 pa and the other side is open to the air at sea
level, what is the density of the unknown fluid.

34. Sample problem 10
A manometer is used to measure the pressure in a tank.
The fluid used has a specific gravity of 0.85, and the
manometer column height is 55 cm, as shown in the Fig.
If the local atmospheric pressure is 96 kPa, determine the
absolute pressure within the tank.

35. Cont.
B. Multifluid Manometer
• Many engineering problems and some manometers
involve multiple fluids of different densities stacked on
top of each other.
• Such systems can be analysed easily by remembering
that
1. The pressure change across a fluid column of height h
is ∆P = ρgh,
2. Two points at the same elevation in a continuous fluid
at rest are at the same pressure.

36. Cont.
• The last principle, which is a result of Pascal’s law,
allows us to “jump” from one fluid column to the next in
manometers without worrying about pressure change
as long as we don’t jump over a different fluid, and the
fluid is at rest.
• Then the pressure at any point can be determined by
starting with a point of known pressure and adding or
subtracting ρgh terms as we advance toward the point
of interest.

37. Sample problem 11
• The water in a tank is pressurized by
air, and the pressure is measured by
a multifluid manometer as shown in
Fig. below. The tank is located on a
mountain at an altitude of 1400 m
where the atmospheric pressure is
85.6kPa. Determine the air pressure
in the tank if h1 =0.1m, h2 =0.2m,
and h3 = 0.35m. Take the densities of
water, oil, and mercury to be 1000
kg/m3, 850 kg/m3, and 13,600
kg/m3, respectively.

38. Cont.
Differential Manometer
• Manometers are particularly well suited to measure
pressure drops across a horizontal flow section
between two specified points .
• This is done by connecting the two legs of the manometer
to these two points, as shown in the Fig.
• Standard manometer are used to measure pressure in
container by comparing it to atmospheric pressure

39. Cont.
• But, Differential manometer are used to compare the
pressure of two different container
• They reveal both which container has greater pressure
and how large the difference between the two is.

40. Cont.
• A relation for the pressure difference P1-P2 can be
obtained by starting at point 1 with P1, moving along
the tube by adding or subtracting the ρgh terms until
we reach point 2, and setting the result equal to P2:
• Note that we jumped from point A horizontally to point
B and ignored the part underneath since the pressure at
both points is the same. Simplifying

41. Sample Problem 12
The U tube contains mercury. Calculate the difference in
pressure if h =1.5 m, h2 = 0.75 m and h1 = 0.5 m. The
liquid at A and B is water and the specific gravity of
mercury is 13.6.

42. Assignment 2.1
• Explain the advantages and limitation of
manometers
Reading Assignment
• Pressure measurement using bourdon
gauge

43. Buoyancy and Floatation
• When a body is immersed wholly or partially in a fluid,
it is subjected to an upward force which tends to lift
(buoy)it up.
• The tendency of immersed body to be lifted up in the
fluid due to an upward force opposite to action of
gravity is known as buoyancy.
• The force tending to lift up the body under such
conditions is known as buoyant force or force of
buoyancy or up-thrust.

44. Cont.
• The magnitude of the buoyant force can be determined
by Archimedes’ principle which states
“ When a body is immersed in a fluid either wholly or
partially, it is buoyed or lifted up by a force which is
equal to the weight of fluid displaced by the body”
• The buoyant force is caused by the increase of pressure
in a fluid with depth.

45. Cont.
• Consider, for example, a flat
plate of thickness h
submerged in a liquid of
density ρf parallel to the free
surface, as shown in the Fig.
• The area of the top (and also
bottom) surface of the plate
is A, and its distance to the
free surface is s.
• The pressures at the top and
bottom surfaces of the plate
are ρf gs and ρf g(s + h),
respectively.

46. Cont.
• Then the hydrostatic force Ftop = ρf gsA acts downward
on the top surface, and the larger force
Fbottom = ρf g(s + h)A acts upward on the bottom surface
of the plate.
• The difference between these two forces is a net
upward force, which is the buoyant force,

47. Sample Problem 13
A 10 kg block of aluminum(p=2700 kg/m3) is attached to
a rope. What is the tension force if the block is in
water(p=1000kg/m3)?

48. Sample Problem 14
An iceberg(p=917kg/m3) float in a seawater(p=1025kg/m3)
A. What fraction of the iceberg is beneath the surface of
the water
B. What fraction of the iceberg is above the surface of the
water

49. Sample Problem 15
• A cube having 15cm side length is suspended between
the oil(p=800kg/m3) and water(p=1000kg/m3) layers
as shown in the figure below . What is the density of
the cube block.