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FLUID MECHANICS
FLUID – is either liquid or gas
NEWTONIAN FLUID - any shear stress applied to a fluid will result in motion of that fluid. Exhibits linear relationship
between the shear stress and the velocity gradient.
FLUID MECHANICS - is the study of fluids either in motion (fluid dynamics) or at rest (fluid statics).
PROPERTIES OF FLUID
PRESSURE (p) – it is the result of compressive forces acting on a unit area. Eq.: [p = F/A]
ATMOSPHERIC PRESSURE (patm) – is the pressure present in the atmosphere, pressure measured by a barometer.
Standard Atmospheric Pressure (40º latitude at sea level)
101.325 kPa
14.7 psi
29.92 in Hg
760 mm Hg
760 torr
1.013 bar
33.90 ft H2O
10.33 m H2O
1 atm
GAGE PRESSURE (pg) – is the pressure measured relative to atmospheric pressure. Gage pressure resulted by a
liquid column has the equation: Eq.: [pg = ρgh]
ABSOLUTE PRESSURE (pa) – is the pressure measure in the basis that it is zero in a volume that is void of molecules
(i.e. ideal vacuum). Eq.: [pa = pg + patm]
DENSITY (ρ) – mass per unit volume. Eq.: [ρ = m/V]
SPECIFIC VOLUME (υ) – volume per unit mass. Eq.: [υ = V/m] ; [υ =1/ρ]
SPECIFIC WEIGHT (γ) – weight per unit volume. Eq.: [γ = w/V] ; Specific weight relation to density Eq.: [γ = ρg]
SPECIFIC GRAVITY – is the ratio of density of a substance to the density of water and is often specified for a
liquid. Eq.: [SG = ρsubstance/ρwater]
VISCOSITY/DYNAMIC VISCOSITY/ABSOLUTE VISCOSITY (μ) - is a quantitative measure of a fluid’s resistance to
flow. More specifically, it determines the fluid shear rate(dv/dy, also, velocity gradient) that is generated by a
given applied shear stress(τ). Eq.: [μ = τ/(dv/dy)] [Note: 1 centi-Poise(cP) = 10−3 Pa-s]
KINEMATIC VISCOSITY (μk) – is the viscosity divided by density. Eq.: [μk = μ/ρ]
BULK MODULUS OF ELASTICITY (B) – is the measure of compressibility of the fluid. It is the ratio of change in unit
pressure to the change of volume per unit volume Eq.: [B = ∆p/(∆V/V)]. Bulk modulus of elasticity is also used to
determine the speed of sound (c) in liquid. Eq.: [c =
(B/ρ)1/2] B in Pa and ρ in kg/m3
Adhesion>Cohesion
Adhesion<Cohesion
SURFACE TENSION (σ) – is the result of attractive forces
between molecules. Eq.: [σdroplet=pr/2] [σbubble=pr/4]
CAPILLARITY – is the rise and fall of liquid in a capillary
tube (thin-bore tube). Height(h) of rise/fall of liquid can
be determine by the equation: Eq.: [h = 4σcosθ/(γd)]
VAPOR PRESSURE – pressure at which there is
equilibrium between the escaping and reentering
molecules. If the pressure is below the vapor pressure,
the molecules will escape the liquid; it is called boiling
when water is heated to the temperature at which the
FLUID MECHANICS
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vapor pressure equals the atmospheric pressure. If the local pressure is decreased to the vapor pressure,
vaporization also occurs.
THERMODYNAMIC PROPERTIES OF FLUID
IDEAL GAS – defined as the gas which has no attractive forces between particles and can be compressed
completely (i.e. particle has no volume of its own). Ideal gas behaves according to ideal gas law.
IDEAL GAS LAW - Eq.: [pV = mRT]
ENTHALPY(H,h) – is the sum of total internal energy and the product of the pressure and volume of the system.
Change of enthalpy is equal to the amount of heat transferred to a system at the condition of constant
pressure. Eq.: [H = U + pV] ; [in constant pressure: dH = dQ] ; [dh = mCpdT]
INTERNAL ENERGY(U,u) - is the energy contained within the system (i.e. excluding the kinetic energy of motion
of the system as a whole and the potential energy of the system as a whole due to external forces). Change in
internal energy is equal to the amount of heat transferred to a system at the condition of constant volume. Eq.:
[in constant volume: dU = dQ] ; [dU = mCvdT]
SPECIFIC HEAT RATIO(k) – ratio of Cp and Cv . Eq.: [k = Cp/Cv]
ISENTROPIC PROCESS – is an adiabatic(no heat transfer) and quasi-equilibrium(properties are constant
throughout the volume at an instant) process. For this process, the relationship Eq.: [T2/T1 = (p2/p1)(k-1/k)] can be
used for an ideal gas assuming specific heats are constant.
VISCOSITY-TORQUE(T) RELATION IN CONCENTRIC CYLINDER VISCOMETER
Eq.: [T = 2πRr2ωLμ/h] where: ω = angular velocity, R = radius of inner cylinder,
r=mean radius, L = length of cylinder, h = gap between inner and outer cylinder
SAMPLE PROBLEMS:
1) A vacuum of 25 kPa is measured at a location where the barometer measures 70.7 kPa. What is the absolute
pressure in millimeters of mercury? Answer: 343
2) A fluid mass occupies 2 m3. Calculate the density, specific weight and specific gravity if the fluid mass is 15
kg. Answer: 7.5 kg/m3 ; 73.6 N/m3 ; 0.0075
3) A viscometer, used to measure the viscosity of a liquid, is composed of two 12-cm-long concentric cylinders
with radii 4 and 3.8 cm. The outer cylinder is stationary and the inner one rotates. If a torque of 0.046 N-m is
measured at a rotational speed of 120 rpm, estimate the viscosity of the liquid. Neglect the contribution to the
torque from the cylinder ends and assume a linear velocity profile. Answer: 0.1628 N-s/m2.
4) Water at 30ºC is able to climb up a clean glass of 0.2 mm diameter tube due to surface tension. The waterglass angle is 0º with the vertical. How far up the tube does the water climb? At 30 ºC water surface tension =
0.0718 N/m and density = 996 kg/m3. Answer: 14.7 cm
5) A 0.5m x 2m flat plate is towed at 5 m/s on a 2-mm-thick layer of SAE-30 oil at 38ºC that separates it from a
flat surface. The velocity distribution between the plate and the surface is assumed to be linear. What force is
required if the plate and surface are horizontal? Answer: 250 N (Use: viscosity = 0.1 N-s/m2)
6) A machine creates small 0.5-mm-diameter bubbles of 20–C water. Estimate the pressure that exists inside the
bubbles. At 20 deg.C, surface tension of water is 0.0736 N/m. Answer: 589 Pa
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FLUID MECHANICS
7) Water is usually assumed to be incompressible. Determine the percentage volume change in 10 m3 of water
at 15ºC if it is subjected to a pressure of 12 MPa from atmospheric pressure. At 15 ºC water bulk modulus of
elasticity is 214x107 Pa. Answer: %change = -0.561%
8) How long would it take a small wave to travel under 22ºC water a distance of 800 m? At 22 ºC water bulk
modulus of elasticity is 220.6x107 Pa and density = 997.7 kg/m3. Answer: 0.539 s
9) If the viscosity of water at 68 deg.F is 0.01008 poise, compute its absolute viscosity in pound-seconds per
square foot. If the specific gravity at 68 deg.F is 0.998, compute its kinematic viscosity in square feet per second.
Answer: 2.11x10-5 lb-s/ft2 ; 1.09x10-5 ft2/s
10) A square block weighing 1.1 kN and 250 mm on an edge slides down an
incline(20º with horizontal) on a film of oil 6.0 micrometer thick. Assuming a linear
velocity profile in the oil, what is the terminal speed of the block? The viscosity of oil
7mPa-s. Answer: 5.16 m/s
is
11) A piston of weight 21 lb slides in a lubricated pipe, as shown. The clearance
between piston and pipe is 0.001in. If the piston decelerates at 2.1 ft/s2 when the
speed is 21ft/s, what is the viscosity of the oil? Answer: 1.36 x 10-4 lb-s/ft2
12) At a depth of 7 km in the ocean, the pressure is 71.6 MPa. Assume a specific weight at the surface of 10.05
kN/m3 and an average bulk modulus of elasticity of 2.34 GPa for that pressure range. Find (a) the change in
specific volume between the surface and 7 km; (b) the specific volume at 7 km; (c) the specific weight at 7km.
Answer: -0.0000299 m3/kg ; 0.000946 m3/kg ; 10 370 N/m3
13) Nitrogen gas occupies a volume of 4.0 ft3 at 2500 lb/ft2 absolute and 750 R. What are its (a) specific volume
and (b) specific weight? Answer: 16.56 ft3/lbm ; 0.0604 lb/ft3
14) Helium at 149 kPa absolute and 10ºC is isentropically compressed to one-fourth of its original volume. What
is its final pressure? Answer: 1488 kPaabs (Use: k = 1.66)
15) Estimate the height to which water at 70 deg.F will rise in a capillary tube of diameter 0.120in. Answer: 0.385
in (Use: surface tension = 0.00500 lb/ft, specific weight = 62.3 lb/ft, angle = 0)
16) A small drop of water at 80 deg.F is in contact with the air and has a diameter of 0.0200 in. If the pressure
within the droplet is 0.082 psi greater than the atmosphere, what is the value of the surface tension? Answer:
0.00492 lb/ft
17) A glass tube is inserter in mercury, as shown; the common
temperature is 20 deg.C. What is the upward force on the glass
as a result of surface effects? Answer: 0.0623N (Use: surface
tension = 0.514 N/m)
18) A 1in diameter soap bubble has an internal pressure 0.0045
lb/in2 greater than that of the outside atmosphere. Compute
the surface tension of the soap-air interface. Note that a soap
bubble has two interfaces with air, an inner and outer surface of
nearly the same radius. Answer: 0.00675 lb/ft
19) What force is required to lift a thin wire ring 6 cm in diameter
from a water surface at 20 deg.C? Answer: 0.0274N (Use: surface tension = 0.0728 N/m)
20) A farmer applies nitrogen to a crop from a tank pressurized to 1000 kPa absolute at a temperature of 25–C.
What minimum temperature can be expected in the nitrogen if it is released to the atmosphere? Answer: 119ºC (Use: k=1.4)