1. CE319F: Elementary Mechanics of Fluids
FLUID PROPERTIES
Chapter 2
CE319F: Elementary Mechanics of Fluids

2. Fluid Properties Define “characteristics” of a specific fluid
Properties expressed by basic “dimensions”
length, mass (or force), time, temperature
Dimensions quantified by basic “units”
We will consider systems of units, important fluid properties (not all), and the dimensions associated with those properties.

3. Systeme International (SI)
Length = meters (m)
Mass = kilograms (kg)
Time = second (s)
Force = Newton (N)
Force required to accelerate 1 1 m/s2
Acceleration due to gravity (g) = 9.81 m/s2
Weight of 1 kg at earth’s surface = W = mg = 1 kg (9.81 m/s2) = 9.81 kg-m/s2 = 9.81 N
Temperature = Kelvin (oK)
oK = freezing point of water
oK = oC

4. Système International (SI)
Work and energy = Joule (J)
J = N*m = kg-m/s2 * m = kg-m2/s2
Power = watt (W) = J/s
SI prefixes:
G = giga = 109 c = centi = 10-2
M = mega = 106 m = milli = 10-3
k = kilo = 103 m = micro = 10-6

5. English (American) System
Length = foot (ft) = m
Mass = slug or lbm (1 slug = 32.2 lbm = kg)
Time = second (s)
Force = pound-force (lbf)
Force required to accelerate 1 1 ft/s2
Temperature = (oF or oR)
oRankine = oR = oF
Work or energy = ft-lbf
Power = ft-lbf/s
1 horsepower = 1 hp = 550 ft-lbf/s = 746 W
Banana Slug
Mascot of UC Santa Cruz

6. Density Mass per unit volume (e.g., @ 20 oC, 1 atm)
Water rwater = 1,000 kg/m3 (62.4 lbm/ft3)
Mercury rHg = 13,500 kg/m3
Air rair = kg/m3
Densities of gases = strong f (T,p) = compressible
Densities of liquids are nearly constant (incompressible) for constant temperature
Specific volume = 1/density = volume/mass

7. Example: Textbook Problem 2.8
Estimate the mass of 1 mi3 of air in slugs and kgs. Assume rair = slugs/ft3, the value at sea level for standard conditions

8. Example
A 5-L bottle of carbon tetrachloride is accidentally spilled onto a laboratory floor. What is the mass of carbon tetrachloride that was spilled in lbm?

9. Specific Weight Weight per unit volume (e.g., @ 20 oC, 1 atm)
gwater = (998 kg/m3)(9.807 m2/s)
= 9,790 N/m3
[= 62.4 lbf/ft3]
gair = (1.205 kg/m3)(9.807 m2/s)
= 11.8 N/m3
[= lbf/ft3]

10. Note: SG is dimensionless and independent of system of units
Specific Gravity
Ratio of fluid density to density of 4oC
Water SGwater = 1
Mercury SGHg = 13.55
Note: SG is dimensionless and independent of system of units

11. Example
The specific gravity of a fresh gasoline is If the gasoline fills an 8 m3 tank on a transport truck, what is the weight of the gasoline in the tank?

12. Ideal Gas Law (equation of state)
P = absolute (actual) pressure (Pa = N/m2)
V = volume (m3)
n = # moles
Ru = universal gas constant = 8.31 J/oK-mol
T = temperature (oK)
R = gas-specific constant
R(air) = 287 J/kg-oK (show)

13. Example
Calculate the volume occupied by 1 mol of any ideal gas at a pressure of 1 atm (101,000 Pa) and temperature of 20 oC.

14. Example
The molecular weight of air is approximately 29 g/mol. Use this information to calculate the density of air near the earth’s surface (pressure = 1 atm = 101,000 Pa) at 20 oC.

15. Example: Textbook Problem 2.4
Given: Natural gas stored in a spherical tank
Time 1: T1=10oC, p1=100 kPa
Time 2: T2=10oC, p2=200 kPa
Find: Ratio of mass at time 2 to that at time 1
Note: Ideal gas law (p is absolute pressure)

16. Viscosity

17. Some Simple Flows Flow between a fixed and a moving plate
Fluid in contact with plate has same velocity as plate (no slip condition)
u = x-direction component of velocity
u=V
Moving plate
Fixed plate y x V
u=0 B
Fluid

18. Some Simple Flows Flow through a long, straight pipe
Fluid in contact with pipe wall has same velocity as wall (no slip condition)
u = x-direction component of velocity r x R V
Fluid

19. Fluid Deformation Flow between a fixed and a moving plate
Force causes plate to move with velocity V and the fluid deforms continuously.
u=V
Moving plate
Fixed plate y x
u=0
Fluid t0 t1 t2

20. Fluid Deformation For viscous fluid, shear stress is proportional
to deformation rate of the fluid (rate of strain)
u=V+dV
Moving plate
Fixed plate y x
u=V
Fluid t
t+dt dx dy da dL

21. Viscosity V+dv V Air (@ 20oC): m = 1.8x10-5 N-s/m2 Kinematic viscosity
Proportionality constant = dynamic (absolute) viscosity
Newton’s Law of Viscosity
Viscosity
Units
Water 20oC): m = 1x10-3 N-s/m2
Air 20oC): m = 1.8x10-5 N-s/m2
Kinematic viscosity
Kinematic viscosity: m2/s
1 poise = 0.1 N-s/m2
1 centipoise = 10-2 poise = 10-3 N-s/m2

22. Shear in Different Fluids
Shear-stress relations for different fluids
Newtonian fluids: linear relationship
Slope of line = coefficient of proportionality) = “viscosity”
Shear thinning fluids (ex): toothpaste, architectural coatings; Shear thickening fluids = water w/ a lot of particles, e.g., sewage sludge; Bingham fluid = like solid at small shear, then liquid at greater shear, e.g., flexible plastics

23. Effect of Temperature Gases:
greater T = greater interaction between molecules = greater viscosity.
Liquids:
greater T = lower cohesive forces between molecules = viscosity down.

25. Typical Viscosity Equations
T = Kelvin
S = Sutherland’s constant
Air = 111 oK
+/- 2% for T = 170 – 1900 oK
Gas:
Liquid:
C and b = empirical constants

26. Force acting ON the plate
Flow between 2 plates
Force is same on top and bottom
Thus, slope of velocity profile is constant and velocity profile is a st. line
u=V
Moving plate
Fixed plate y x V
u=0 B
Fluid
Force acting ON the plate

27. Flow between 2 plates Shear stress anywhere between plates y
Moving plate
u=V V t
Shear on fluid B t x
Fixed plate
u=0

28. Flow between 2 plates
2 different coordinate systems r x B V y x

29. Example: Textbook Problem 2.33
Suppose that glycerin is flowing (T = 20 oC) and that the pressure gradient dp/dx = -1.6 kN/m3. What are the velocity and shear stress at a distance of 12 mm from the wall if the space B between the walls is 5.0 cm? What are the shear stress and velocity at the wall? The velocity distribution for viscous flow between stationary plates is

31. Example: Textbook Problem 2.34
A laminar flow occurs between two horizontal parallel plates under a pressure gradient dp/ds (p decreases in the positive s direction). The upper plate moves left (negative) at velocity ut. The expression for local velocity is shown below. Is the magnitude of the shear stress greater at the moving plate (y = H) of at the stationary plate (y = 0)?

33. Elasticity (Compressibility)
If pressure acting on mass of fluid increases: fluid contracts
If pressure acting on mass of fluid decreases: fluid expands
Elasticity relates to amount of deformation for a given change in pressure
Ev = bulk modulus of elasticity
How does second part of equation come about?
Small dV/V = large modulus of elasticity

34. Example: Textbook Problem 2.45
Given: Pressure of 2 MPa is applied to a mass of water that initially filled 1000-cm3 (1 liter) volume.
Find: Volume after the pressure is applied.
Ev = 2.2x109 Pa (Table A.5)

35. Example
Based on the definition of Ev and the equation of state, derive an equation for the modulus of elasticity of an ideal gas.

36. Surface Tension air water
Below surface, forces act equal in all directions
At surface, some forces are missing, pulls molecules down and together, like membrane exerting tension on the surface
Pressure increase is balanced by surface tension, s
surface tension = magnitude of tension/length
s = N/m 20oC)
water
air
No net force
Net force inward
Interface

37. Surface Tension
Liquids have cohesion and adhesion, both involving molecular interactions
Cohesion: enables liquid to resist tensile stress
Adhesion: enables liquid to adhere to other bodies
Capillarity = property of exerting forces on fluids by fine tubes or porous media
due to cohesion and adhesion
If adhesion > cohesion, liquid wets solid surfaces at rises
If adhesion < cohesion, liquid surface depresses at pt of contact
water rises in glass tube (angle = 0o)
mercury depresses in glass tube (angle = o)
See attached information

38. Example: Capillary Rise
Given: 20oC, d = 1.6 mm
Find: Height of water W

39. Example: Textbook Problem 2.51
Find: Maximum capillary rise between two vertical glass plates 1 mm apart. t s q h

40. Examples of Surface Tension

41. Example: Textbook Problem 2.48
Given: Spherical soap bubble, inside radius r, film thickness t, and surface tension s.
Find: Formula for pressure in the bubble relative to that outside. Pressure for a bubble with a 4-mm radius?
Should be soap bubble

42. Vapor Pressure (Pvp)
Vapor pressure of a pure liquid = equilibrium partial pressure of the gas molecules of that species above a flat surface of the pure liquid
Concept on board
Very strong function of temperature (Pvp up as T up)
Very important parameter of liquids (highly variable – see attached page)
When vapor pressure exceeds total air pressure applied at surface, the liquid will boil.
Pressure at which a liquid will boil for a given temperature
At 10 oC, vapor pressure of water = atm = 1200 Pa
If reduce pressure to this value can get boiling of water (can lead to “cavitation”)
If Pvp > 1 atm compound = gas
If Pvp < 1 atm compound = liquid or solid

43. Example
The vapor pressure of naphthalene at 25 oC is 10.6 Pa. What is the corresponding mass concentration of naphthalene in mg/m3? (Hint: you can treat naphthalene vapor as an ideal gas).

44. Vapor Pressure (Pvp) - continued
Vapor pressure of water (and other liquids) is a strong function of temperature.

45. Vapor Pressure (Pvp) - continued
Pvp,H2O = Pexp( a – a2 – a3 – a4)
P = 101,325 Pa a = 1 – (373.15/T) T = oK
valid to +/- 0.1% accuracy for T in range of -50 to 140 oC
Equation for relative humidity of air = percentage to which air is “saturated” with water vapor.
What is affect of RH on drying of building materials, and why? Implications?

46. Example: Relative Humidity
The relative humidity of air in a room is 80% at 25 oC.
What is the concentration of water vapor in air on a volume percent basis?
If the air contacts a cold surface, water may condense (see effects on attached page). What temperature is required to cause water condensation?

48. Saturation Vapor Pressure