1. Fluid Mechanics Course Code: CHPE207 and CIVL213
Lecturer: Dr Mustafa Saleh Nasser
Office Number: 5D-44
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9. Dimensions and Units
In fluid mechanics we must describe various fluid characteristics in terms of certain basic quantities such as length, time and mass
A dimension is the measure by which a physical variable is expressed qualitatively, i.e. length is a dimension associated with distance, width, height, displacement.
Basic dimensions: Length, L
(or primary quantities) Time, T
Mass, M
Temperature, Q
We can derive any secondary quantity from the primary quantities i.e. Force = (mass) x (acceleration) : F = M L T-2
A unit is a particular way of attaching a number to the qualitative dimension: Systems of units can vary from country to country, but dimensions do not
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11. Units of Force – E system
To make Newton’s law dimensionally consistent we must include a dimensional proportionality constant:
where
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12. 1 2
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14. 3
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15. Density for Gasses For gases: strong function of T and P
from ideal gas law: r = P M/R T (can you drive this)
where R = universal gas constant, M=mol. weight
R= J/(g-mole K)= (liter bar)/(g-mole K)=
(liter atm)/(g-mole K)=1.987 (cal)/(g-mole K)=
10.73 (psia ft3)/(lb-mole °R)= (atm ft3)/(lb-mole °R)
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16. Pressure Pascal’s Laws Pascals’ laws:
Pressure is defined as the amount of force exerted on a unit area of a substance:
P = F / A
Pascal’s Laws
Pascals’ laws:
Pressure acts uniformly in all directions on a small volume (point) of a fluid
In a fluid confined by solid boundaries, pressure acts perpendicular to the boundary – it is a normal force.
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17. Direction of fluid pressure on boundaries
Furnace duct
Pipe or tube
Heat exchanger
Pressure is due to a Normal Force
(acting perpendicular to
the surface)
It is also called a Surface Force
Dam
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18. Absolute and Gauge Pressure
Gauge pressure: Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere.
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19. Units for Pressure Unit Definition or Relationship 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
760 mm Hg
1 atm
pounds per sq. in. (psi)
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20. Pressure distribution for a fluid at rest
We will determine the pressure distribution in a fluid at rest in which the only body force acting is due to gravity
The sum of the forces acting on the fluid must equal zero
Consider an infinitesimal rectangular fluid element of dimensions Dx, Dy, Dz z y x
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21. Pressure distribution for a fluid at rest
Let Pz and Pz+Dz denote the pressures at the base and top of the cube, where the elevations are z and z+Dz respectively.
Force at base of cube: Pz A=Pz (Dx Dy)
Force at top of cube: Pz+Dz A= Pz+Dz (Dx Dy)
Force due to gravity: m g=r V g = r (Dx Dy Dz) g
A force balance in the z direction gives: Fg
For an infinitesimal element (Dz0) 
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22. Variation of pressure with elevation

23. General variation of pressure in a static fluid due to gravity

24. General variation of pressure in a static fluid due to gravity
for vertical direction: q = 0

25. Equality of pressure at the same level in a static fluid

26. Incompressible fluid By using gauge pressures we can simply write:
Liquids are incompressible i.e. their density is assumed to be constant.
When we have a liquid with a free surface the pressure P at any depth below the free surface is:
where Po is the pressure at the free surface (Po=Patm) and h = zfree surface - z
By using gauge pressures we can simply write:
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27. Equality of pressure at the same level in a continuous fluid

28. Examples
SG= 13.6
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29. Solution:
1.1
1.2
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30. 1.3
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31. Calculate the weight of a reservoir of oil if it has a mass of 825 kg.
We have

32. If the reservoir from previous Example has a volume of 0
If the reservoir from previous Example has a volume of 0.917m3, compute the density, the specific weight, and the specific gravity of the oil.

33. 3.7 Pressure expressed as the Height of a Column of Liquid
Then we can use this as a conversion factor,

34. Pressure Gages and Transducers
For those situations where only a visual indication is needed at the site where the pressure is being measured, a pressure gage is used.
In other cases there is a need to measure pressure at one point and display the value at another.
The general term for such a device is pressure transducer, meaning that the sensed pressure causes an electrical signal to be generated that can be transmitted to a remote location such as a central control station where it is displayed digitally.

35. Below Figs shows the Bourdon tube pressure gage.
Pressure Gages
Below Figs shows the Bourdon tube pressure gage.

36. Pressure Gages
The pressure to be measured is applied to the inside of a flattened tube, which is normally shaped as a segment of a circle or a spiral.
The increased pressure inside the tube causes it to be straightened somewhat.
The movement of the end of the tube is transmitted through a linkage that causes a pointer to rotate.

37. Pressure Gages
Figure below shows a pressure gage using an actuation means called Magnehelic pressure gage.

38. 3.8.2 Strain Gage Pressure Transducer
Fig 3.17 shows the strain gage pressure transducer and indicator.
The pressure to be measured is introduced through the pressure port and acts on a diaphragm to which foil strain gages are bonded.
As the strain gages sense the deformation of the diaphragm, their resistance changes.
The readout device is typically a digital voltmeter, calibrated in pressure units.

39. Strain Gage Pressure Transducer

40. LVST-Type Pressure Transducer
A linear variable differential transformer (LVDT) is composed of a cylindrical electric coil with a movable rod-shaped core.
As the core moves along the axis of the coil, a voltage change occurs in relation to the position of the core.
This type of transducer is applied to pressure measurement by attaching the core rod to a flexible diaphragm.
Fig shows the Linear variable differential transformer (LVDT)-type pressure transducer.

41. LVST-Type Pressure Transducer

42. Piezoelectric Pressure Transducer
Certain crystals, such as quartz and barium titanate, exhibit a piezoelectric effect, in which the electrical charge across the crystal varies with stress in the crystal.
Causing a pressure to exert a force, either directly or indirectly, on the crystal leads to a voltage change related to the pressure change.
Fig shows the digital pressure gage.

43. Piezoelectric Pressure Transducer

44. Buoyancy Archimedes Principle
Laws of buoyancy discovered by Archimedes:
A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces
A floating body displaces its own weight in the fluid in which it floats
Free liquid surface F1 h1
The upper surface of the body is subjected to a smaller force than the lower surface
 A net force is acting upwards H h2 F2
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45. Buoyancy where r = the fluid density
The net force due to pressure in the vertical direction is:
FB = F2- F1 = (Pbottom - Ptop) (DxDy)
The pressure difference is:
Pbottom – Ptop = r g (h2-h1) = r g H
FB = r g H (DxDy)
Thus the buoyant force is:
FB = r g V
where r = the fluid density
If the fluid density is greater than the average density of the object, the object floats. If less, the object sinks
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46. Example
Consider a solid cube of dimensions 1ft x 1ft x 1ft (=0.305m x 0.305m x 0.305m). Its top surface is 10 ft (=3.05 m) below the surface of the water. The density of water is rf=1000 kg/m3.
Consider two cases:
The cube is made of cork (rB=160.2 kg/m3)
b) The cube is made of steel (rB=7849 kg/m3)
In what direction does the body tend to move?
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48. Shear in Different Fluids
Shear-stress relations for different types of fluids
Newtonian fluids: linear relationship
Slope of line (coefficient of proportionality) is “viscosity”

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50. Viscosity
The constant of proportionality is designated by the Greek symbol  (mu) and is called the absolute viscosity, dynamic viscosity, or simply the viscosity of the fluid.
The viscosity depends on the particular fluid, and for a particular fluid the viscosity is also dependent on temperature.

51. Viscosity and Temperature 1/3
For fluids, the viscosity decreases with an increase in temperature.
For gases, an increase in temperature causes an increase in viscosity.
WHY? molecular structure.

52. Fluid Properties Examples
Explain why the viscosity of a liquid decreases while that of a gas increases with a temperature rise. The following is a table of measurement for a fluid at constant temperature. Determine the dynamic viscosity of the fluid.
du/dy (rad s-1)
0.00
0.20
0.40
0.60
0.80
t (N m-2)
0.01
1.90
3.10
4.00
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53. Dr Mustafa Nasser 53 Using Newton's law of viscocity
where m is the viscosity. So viscosity is the gradient of a graph of shear stress against vellocity gradient of the above data, or
Plot the data as a graph:
Calculate the gradient for each section of the line
du/dy (s-1)
0.0
0.2
0.4
0.6
0.8
t (N m-2)
1.0
1.9
3.1
4.0
Gradient -
5.0
4.75
5.17
Thus the mean gradient = viscosity = 4.98 N s / m2
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54. Example 2: 5. 6m3 of oil weighs 46 800 N
Example 2: 5.6m3 of oil weighs N. Find its mass density, and relative density.
Solution 2: Weight = mg
Mass m = / 9.81 = kg
Mass density r = Mass / volume = / 5.6 = 852 kg/m3
Relative density 
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55. Example 3
The velocity distribution of a viscous liquid (dynamic viscosity m = 0.9 Ns/m2) flowing over a fixed plate is given by u = 0.68y - y2 (u is velocity in m/s and y is the distance from the plate in m). What are the shear stresses at the plate surface and at y=0.34m?
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56. Calculate the shear stress at the plate face
At the plate face y = 0m,
Calculate the shear stress at the plate face
At y = 0.34m,
As the velocity gradient is zero at y=0.34 then the shear stress must also be zero.
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57. Example 4
In a fluid the velocity measured at a distance of 75mm from the boundary is 1.125m/s. The fluid has absolute viscosity Pa s and relative density What is the velocity gradient and shear stress at the boundary assuming a linear velocity distribution.
m = Pa s
g = 0.913
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58. Newtonian and Non-Newtonian Fluid
Fluids for which the shearing stress is linearly related to the rate of shearing strain are designated as Newtonian fluids
Most common fluids such as water, air, and gasoline are Newtonian fluid under normal conditions.
Fluids for which the shearing stress is not linearly related to the rate of shearing strain are designated as non-Newtonian fluids.

59. Non-Newtonian Fluids

60. Non-Newtonian Fluids Newtonian Fluid Non-Newtonian Fluid
η is the apparent viscosity and is not constant for non-Newtonian fluids.

61. η - Apparent Viscosity
The shear rate dependence of η categorizes non-Newtonian fluids into several types.
Power Law Fluids:
Pseudoplastic – η (viscosity) decreases as shear rate
increases (shear rate thinning)
Dilatant – η (viscosity) increases as shear rate increases
(shear rate thickening)
Bingham Plastics:
η depends on a critical shear stress (t0) and then becomes constant

62. Non-Newtonian Fluids Bingham Plastic: sludge, paint, blood, ketchup
Pseudoplastic:
latex, paper pulp, clay solns.
Newtonian
Dilatant:
quicksand

63. non-Newtonian Fluids 1/2
Shear thinning fluids: The viscosity decreases with increasing shear rate – the harder the fluid is sheared, the less viscous it becomes. Many colloidal suspensions and polymer solutions are shear thinning. Latex paint is example.

64. non-Newtonian Fluids 2/2
Shear thickening fluids: The viscosity increases with increasing shear rate – the harder the fluid is sheared, the more viscous it becomes. Water-corn starch mixture water-sand mixture are examples. Bingham plastic: neither a fluid nor a solid. Such material can withstand a finite shear stress without motion, but once the yield stress is exceeded it flows like a fluid. Toothpaste and mayonnaise are common examples.