1. HW No. 5
given KW

2. Introduction to Fluid Mechanics
Fluid Mechanics is concerned with the behavior of fluids at rest and in motion
Distinction between solids and fluids:
According to our experience: A solid is “hard” and not easily deformed. A fluid is “soft” and deforms easily.
Fluid is a substance that alters its shape in response to any force however small, that tends to flow or to conform to the outline of its container, and that includes gases and liquids and mixtures of solids and liquids capable of flow.
A fluid is defined as a substance that deforms continuously when acted on by a shearing stress of any magnitude.

5. Units of Force: Newton’s Law F=m.g
SI system: Base dimensions are Length, Time, Mass, Temperature
A Newton is the force which when applied to a mass of 1 kg produces an acceleration of 1 m/s2.
Newton is a derived unit: 1N = (1Kg).(1m/s2)
EE system: Base dimensions are Length, Time, Mass, Force and Temperature
The pound-force (lbf) is defined as the force which accelerates 1pound-mass (lbm), ft/s2.

6. Properties of Fluids
Fundamental approach: Study the behavior of individual molecules when trying to describe the behavior of fluids
Engineering approach: Characterization of the behavior by considering the average, or macroscopic, value of the quantity of interest, where the average is evaluated over a small volume containing a large number of molecules
Treat the fluid as a CONTINUUM: Assume that all the fluid characteristics vary continuously throughout the fluid

9. Measures of Fluid Mass and Weight
Density of a fluid, r (rho), is the amount of mass per unit volume of a substance: r = m / V
For liquids, weak function of temperature and pressure
For gases: strong function of T and P
from ideal gas law: r = P n/Ru T
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)

10. Measures of Fluid Mass and Weight
Specific volume: u = 1 / r
Specific weight is the amount of weight per unit volume of a substance: g = w / V = r g
Specific Gravity (independent of system of units)

11. Fluid Statics
The word “statics” is derived from Greek word “statikos”= motionless
For a fluid at rest or moving in such a manner that there is no relative motion between particles there are no shearing forces present: Rigid body approximation

12. Definition of Pressure
Pressure is defined as the amount of force exerted on a unit area of a substance:
P = F / A
Fluid statics
Chee 223

13. 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.

14. 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

15. 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)

16. Hydrostatic forces on plane surfaces
Case 1: Horizontal surface exposed to a gas F
P=constant everywhere
F = P . A
Case 2: Horizontal surface exposed to a liquid F
P=rgh+Patm h
P = Patm
P=constant along the horizontal surface
F = P . A

17. Hydrostatic forces on plane surfaces
Case 3: Vertical surface exposed to air F
Pressure varies linearly with height (see also equation 2.4): P=rgh
However, because r of gases is very low, the dependence is very week
Therefore we can assume that P=constant everywhere
F = P . A

18. Hydrostatic forces on plane surfaces
Case 4: Vertical surface exposed to liquid
Example: The lock gate of a canal is rectangular, 20 m wide and 10 m high. One side is exposed to the atmosphere and the other side to the water. What is the net force on the lock gate?
Vertical rectangular wall (wall width = W) F H h P
Here the pressure varies linearly with depth (see also equation 2.4): P=rgh

19. Buoyancy H 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

20. 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
From (2.8):
FB = r g H (DxDy)
Thus the buoyant force is:
FB = r g V
(2.8)
(2.9)
where r = the fluid density

22. Newton’s law of Viscosity
(6.6)
m [=N/m2 . s=Pa . s]: Viscosity
n = m/r : Kinematic viscosity [=m2/s]
Newtonian fluids: Fluids which obey Newton’s law: Shearing stress is linearly related to the rate of shearing strain.
The viscosity of a fluid measures its resistance to flow under an applied shear stress.

23. Effect of temperature on viscosity
Viscosity is very sensitive to temperature
The viscosity of gases increases with temperature:
Power-law
Sutherland equation
The viscosity of liquids decreases with temperature:

24. Non-Newtonian fluids
Non-Newtonian fluids: Fluids which do not obey Newton’s law: Shearing stress is not linearly related to the rate of shearing strain.
Bingham plastics
Shear thinning
Shear thickening
The study of these materials is the subject of rheology

26. The Balance Equation / Mass Balances – Continuity Equation

27. The Mass Balance - Rate of mass flow into CV
Surroundings, S
Control Volume
(CV)
Accumulation
or depletion
Loss
through outlet
Addition
through inlet
Boundary, B (Control Surface, CS)
The law of conservation of mass states that mass may be neither created nor destroyed. With respect to a control volume (CV):
Rate of mass flow into CV
Rate of mass efflux from CV
Rate of accumulation of mass into CV - =
(3.1)

28. Conservation of Mass- Continuity Equation
The mass balance (equation 3.1) can be written:
(3.2)
where:
Rate of addition of mass into the system
Rate of removal of mass from the system
Rate of accumulation of mass in the system
denotes mass flow rate [mass/time=kg/s]

29. Steady and Unsteady Flows
If there is no accumulation of mass (dm/dt = 0) then the system is at steady-state conditions
The fluid flow in general is a function of location and time f(x, y, z, t). If the flow at every point in the fluid is independent of time, the flow is at steady state conditions.
For a steady-state flow: dX/dt = 0, where X is a physical quantity (for example mass, temperature or velocity)

30. Mass and Volumetric Flow Rates
Recall from definition of density:
(3.3)
In terms of “rates”:
where
= The mass flow rate: The mass of fluid flowing past a section per unit time [=kg/s]
Q = The volumetric flow rate :The volume of fluid flowing past a section of a pipe or channel per unit time [=m3/s]
(3.4)
(3.5)

31. Conservation of Mass- Continuity Equation
For steady-state conditions equation 3.2 combined with 3.3, 3.5 becomes: or
(3.7)
(3.6)
Aout, rout, Vout
Inlet
Outlet
Ain, rin, Vin

32. Conservation of Mass- Continuity Equation
For incompressible fluids (r=const) from (3.6) and (3.7):
(3.8)

33. Conservation of Mass- Continuity Equation
For multiple inlets – outlets, starting from (3.2):
If there is no accumulation of mass (dm/dt = 0) then the system is at steady-state conditions
(3.7)

34. Conservation of Mass- Continuity Equation
For incompressible flow:
(3.8)

35. Summary
The first of the fundamental laws of fluid flow, the conservation of mass, has been considered. It is also called the continuity equation.
The continuity equation has been derived using the “Control Volume”, or “integral method of analysis”, by applying the general balance equation to a finite control volume.
Significant simplification of the continuity equation can be made by using the steady state and incompressibility assumptions.

36. Bernoulli’s Equation- Mechanical Energy Balance

37. Simplifications (5.3) In terms of “heads” (units of length) (5.4)
Assumptions:
Steady, incompressible, frictionless (inviscid) flow
No heat transfer or change in internal energy, no shaft work
Single input-output
Bernoulli’s equation (5.1) reduces to:
(5.3)
In terms of “heads” (units of length)
(5.4)

39. The Bernoulli equation can be expressed in three different ways as follows:
energies:
pressures:
heads:

41. at the exit of the tube so that P1 = P2 = Patm
Analysis We take point 1 to be at the free surface of water in the bottle and point 2
at the exit of the tube so that P1 = P2 = Patm
(the bottle is open to the atmosphere and water discharges into the atmosphere),
V1  0 (the bottle is large relative to the tube diameter),
and z2 = 0 (we take point 2 as the reference level).
Then the Bernoulli Equation simplifies to
0.25 in
1.5 ft
2 ft 1 2

42. Substituting, the discharge velocity of water and the filling time are determined as follows:
(a) Full bottle (z1 = 3.5 ft):
(b) Empty bottle (z1 = 2 ft):

43. Flow in Pipes – Fluid Friction

44. The Pressure-Drop ExperimentQ L

45. Laminar vs. Turbulent Flow
Laminar flow: Fluid flows in smooth layers (lamina) and the shear stress is the result of microscopic action of the molecules.
Turbulent flow is characterized by large scale, observable fluctuations in the fluid and flow properties are the result of these fluctuations.

46. Reynolds Number
The Reynolds number can be used as a criterion to distinguish between laminar and turbulent flow:
(6.1)
For flow in a pipe
Laminar flow if Re < 2100
Turbulent flow if Re > 4000
Transitional flow if 2100< Re < 4000
For very high Reynolds numbers, viscous forces are negligible: inviscid flow
For very low Reynolds numbers (Re<<1) viscous forces are dominant: creeping flow

47. Case 1: Laminar Flow

48. Laminar flow: Velocity and Shear stress profiles

49. Fully Developed Flow Flow in the entrance region of a pipe is complex.
Once the velocity profile no longer changes, we have reached fully developed flow. Mathematically dV/dx = 0
Typical entrance length, 20 D < Le < 30 D

50. Case 2: Turbulent Flow

51. Turbulent Flow
When the flow is turbulent the velocity and pressure fluctuate very rapidly. The velocity components at a point in a turbulent flow field fluctuate about a mean value.

52. Turbulent Flow – Velocity Profile
For turbulent flow in tubes the time-averaged velocity profile can be expressed in terms of the power law equation. n =7 is usually a good approximation.
where Vc is the velocity at
The centerline

53. Friction factor: The Moody Chart
The Moody Chart (Figure 6.10 textbook) provides a convenient representation of the functional dependence f = f(Re, e/D)
For laminar flow:
f = 16 / Re
For turbulent flow:
(6.18)
Colebrook formula
(6.19a )
(6.19b )
For turbulent flow, with Re<105 and for hydraulically smooth surfaces:
(6.20)
Blasius formula

54. To find f:
Calculate Re
If Re<2100 (laminar flow) find f from eq. (6.18)
If Re>2100 (turbulent flow)
Estimate roughness, e
Find f from Moody chart

55. Calculation of Friction Factor

56. Other considerations: Non circular conduits
Define hydraulic diameter, Dh:
Then use Reynolds number based on hydraulic diameter, Dh:

57. Non circular conduits Pipe of circular cross-section
Annulus (inside diameter D1, outside D2)
Rectangular conduit (area ab)