introduction and basic fluid properties · 2015-06-08 · in fluid mechanics we must describe...
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Introduction and Basic Fluid Properties
fluid mechanics is defined as the science that deals with the behavior of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interaction of fluids with solids or other fluids at the boundaries.
Fluid mechanics is also referred to as fluid
dynamics by considering fluids at rest as a special case of motion with zero velocity
Fluid mechanics itself is also divided into several categories:-
The study of the motion of fluids that are practically incompressible (such as liquids, especially water, and gases at low speeds) is usually referred to as hydrodynamics.
A subcategory of hydrodynamics is hydraulics, which deals with liquid flows in pipes and open channels.
Gas dynamics deals with the flow of fluids that undergo significant density changes, such as the flow of gases through nozzles at high speeds.
The category Aerodynamics deals with the
flow of gases (especially air) over bodies such as aircraft, rockets.
• What Is a Fluid?
You will recall from physics that a substance exists in three primary phases: solid, liquid, and gas. A substance in the liquid or gas phase is referred to as a fluid.
The main distinction between solids and fluids:
• A solid can resist an applied shear stress by deforming while a fluid deforms continuously under the influence of shear stress, no matter how small.
stress is defined as force per unit area and is determined by dividing the force by the area upon which it acts.
The normal component of the force acting on a surface per unit area is called the normal stress, and the tangential component of a force acting on a surface per unit area is called shear stress
In a fluid at rest, the normal stress is called pressure.
Application Areas of Fluid Mechanics
Dimensions and Units In fluid mechanics we must describe various fluid characteristics in
terms of certain basic quantities such as length, time and mass
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, θ
We can derive any secondary quantity from the primary quantities
i.e. Force = (mass) x (acceleration) : F = M L t-2
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
Dimensions and Units
Primary
Dimension SI Unit
British
Gravitational
(BG) Unit
English
Engineering
(EE) Unit
Mass [M] Kilogram (kg) Slug Pound-mass
(lbm)
Length [L] Meter (m) Foot (ft) Foot (ft)
Time [t] Second (s) Second (s) Second (s)
Temperature [θ] Kelvin (K) Rankine (°R) Rankine (°R)
Force [F] Newton
(1N=1 kg.m/s2) Pound (lbf) Pound-force (lbf)
Units of Force: Newton’s Law: F=m.a • SI system: Base dimensions are Length, Time, Mass, Force
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)
• BG system: Base dimensions are Length, Time, Mass, Force
A pound-force (lbf) is the force which when applied to a mass of 1 slug
produces an acceleration of 1 ft/s2,(ie 1lbf = 1slug.1. ft/s2)
1lbf≈ 4.4482 N and thus 1slug=32.174 lbm
• EE system: Base dimensions are Length, Time, Mass, Force
The pound-force (lbf) is defined as the force which accelerates
1pound-mass (lbm): 1lbf = 1lbm.1. ft/s2)
Dr Mustafa Nasser 12
PROPERTIES OF FLUIDS
• Some familiar properties are pressure P, temperature T, volume V, and mass m. The list can be extended to include less familiar ones such as viscosity, thermal conductivity, modulus of elasticity, thermal expansion coefficient,…etc.
Temperature
• Temperature, T, in units of degrees celcius, oC, is a measure of “hotness” relative to the freezing and boiling point of water.
• A thermometer is based on the thermal expansion of mercury
Temperature • Microscopic point of view:
Temperature is a measure of the internal molecular motion, e.g., average molecule kinetic energy
• At a temperature of –273.15oC molecular motion start.
• Temperature in units of degrees Kelvin, K, is measured relative to this absolute zero temperature, so 0 K = -273oC
In general, T in K = T in oC + 273
T in oF= 1.8 oC + 32
Density and Specific Volume
The density of a fluid, designated by the Greek symbol (rho), is
defined as its mass per unit volume
=m/V
In SI the unit of is kg/m3. In BG system, has units of
slug/ft3 and in EE, has units of lbm/ft3
Density is used to characterize the mass of a fluid system.
The value of density can vary widely between different fluids, but
for liquids, variations in pressure and temperature generally have
only a small effect on the value of density.
The specific volume, v, is the volume per unit mass
is the inverse of density: v=1/ρ
and thus, density = 1/specific vol.
Example
An ethyl alcohol substance has a mass of 18.5 g and occupies a
volume of 23.4 ml. (milliliter).
The density can be calculated as
ρ = [(18.5 g) / (1000 g/kg)] / [(23.4 ml) / (1000 ml/l) (1000 l/m3) ]
= (18.5 10-3 kg) / (23.4 10-6 m3)
= 770 kg/m3
Example The density of titanium is 4507 kg/m3. Calculate the mass of 0.17 m3 titanium. The mass of titanium is m = (0.17 m3) (4507 kg/m3) = 766.2 kg
Specific Weight
The specific weight of a fluid, designated by the Greek
symbol (gamma), is defined as its weight per unit
volume (with unit N/m3).
g
In fluid mechanics, specific weight represents the force exerted by gravity on a unit volume of a fluid. For this reason, units are expressed as force per unit volume (e.g., lb/ft3 or N/m3). Specific weight can be used as a characteristic property of a fluid.
Specific Weight
Under conditions of standard gravity
g= 9.807 m/ s2 , for simplicity use g= 9.81 m/ s2
Example: water at 70ºF has a specific weight of 9.80
kN/m3.
The density of water is 998.9 kg/m3.
g
Specific Weight
• Weight per unit volume (e.g., @ 20 oC, 1 atm)
water = (1000 kg/m3)(9.81 m/s2)
= 9810 N/m3
[= 62.4 lbf/ft3]
air = (1.205 kg/m3)(9.81 m/s2)
= 11.82 N/m3
[= 0.0752 lbf/ft3]
]/[]/[ 33 ftlbformNg
Specific Gravity
The specific gravity of a fluid, designated as SG, is
defined as the ratio of the density of the fluid to the
density of WATER (for liquids) or to the density of AIR
(for gases) at some specified temperature.
Substances with a specific gravity of 1 are neutrally buoyant in water, those with SG greater than one are denser than water, and so (ignoring surface tension effects) will sink in it, and those with an SG of less than one are less dense than water, and so will float
Specific Gravity • Ratio of fluid density to water or air density at STP
(e.g., @ 20 oC, 1 atm)
3/998 mKgSG
liquid
water
liquid
liquid
3/205.1 mkgSG
gas
air
gasgas
•Water SGwater = 1 •Mercury SGHg = 13.6 •Air SGair = 1
H2O , 4oC= 999kg/m3
≈1000kg/m3 .
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Example: 5.6m3 of oil weighs 46 800 N. Find its density, and relative density (SG).
Solution 1: Weight 46 800 = mg
Mass m = 46 800 / 9.81 = 4770.6 kg
Density ρ = Mass / volume
= 4770.6 / 5.6 = 852 kg/m3
Relative density (SG) is
852.0/1000
8523
mKgSG
water
liquid
liquid
Simple Flows • Flow between a fixed and a moving plate
• Fluid in contact with the plate has the same
velocity as the plate
u = velocity in x-direction= (U/b)y
u=U Moving plate
Fixed plate
y
x
U
u=0
b yb
Uyu )(
Fluid
• Flow through a long, straight pipe, the fluid in contact
with the pipe wall has the same velocity as the wall
r
x R
2
1)(R
rUru
U Fluid
Simple Flows
Fluid Deformation • Flow between a fixed and a moving plate • Force causes plate to move with velocity U and the fluid deforms
continuously.
u=U Moving plate
Fixed plate
y
x
u=0
Fluid
t0 t1 t2
Fluidity of Fluid 1/3
How to describe the “fluidity” of the fluid in Simple Flows?
The bottom plate is rigid fixed, but the upper plate is free
to move.
If a solid, such as steel, were placed between the two
plates and loaded with the force F, the top plate would
be displaced through some small distance, a.
The vertical line AB would be rotated through the small
angle, , to the new position AB'.
F =τA
where τ is shearing
stress Acting
opposite to F on the
area A
F F
Fluidity of Fluid 2/3
What happens if the solid is replaced with a fluid such as water?
When the force F is applied to the upper plate, it will
move continuously with a velocity U.
The fluid “sticks” to the solid boundaries and is referred
to as the NON-SLIP conditions.
The fluid between the two
plates moves with velocity
u=f(y) that would be
assumed to vary linearly,
u=(U/b)y.
In such case, the velocity
gradient is
slope= du/dy = U/b
F
Fluidity of Fluid 3/3
In a small time increment, δt, an imaginary vertical line AB
would rotate through an angle, δβ , so that
tan (δβ) = δa/b =δβ
Since δa = U δt it follows that
δβ= U δt / b
Defining the rate of shearing strain, as
If the shearing stress is increased by Force, the rate of
shearing strain is increased in direct proportion,
dy
du
b
U
tt
0lim
dydu /The common fluids such as water, oil, gasoline,
and air, the shearing stress and rate of shearing
strain can be related with a relationship.
µ is the viscosity coefficient
F
dy
du
A
F
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. In everyday terms (and for fluids only), viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity. Simply, the less viscous the fluid is, the greater its ease of movement (fluidity). Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries causes the fluid to shear. The force required for this action is a measure of the fluid's viscosity.
Viscosity
Viscosity coefficients µ
• Newton’s Law of Viscosity
• Viscosity
• Units
• Water (@ 20oC)
– = 1x10-3 N.s/m2
• Air (@ 20oC)
– = 1.8x10-5 N.s/m2
dydU /
dy
dU
ms
kg
ms
smkgsPa
m
sN
msm
mN
.
...
//
/222
2
In ASTM standards, as centipoise (cP). Water at 20 °C has a viscosity of 1.0 cP. 1P=100 cP 1 P = 0.1 Pa·s, 1 cP = 0.001 Pa·s.
Viscosity coefficients µ Viscosity coefficients can be defined in two ways: Dynamic viscosity, also absolute viscosity, the more usual one (typical units Pa·s, Poise, cP); Kinematic viscosity is the dynamic viscosity divided by the density (typical units cm2/s, Stokes, St).
Kinematic Viscosity
Defining kinematic viscosity
The dimensions of kinematic viscosity are L2/t.
The units of kinematic viscosity in BG system are ft2/s and SI system are m2/s.
In the CGS system, the kinematic viscosity has the units is called a stoke, abbreviated St.
• It is sometimes expressed in terms of centiStokes (cSt).
• 1 St = 1 cm2·s−1 = 10−4 m2·s−1.
• 1 cSt = 1 mm2·s−1 = 10−6m2·s−1.
• Water at 20 °C has a kinematic viscosity of about 1 cSt.
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Example: Shear stress
The space between two plates, as shown in the figure, is filled with water. Find
the shear stress and the force necessary to move the upper plate at a constant velocity of 10 m /s. The gap width is yo=0.1 mm and the area A is 0.2 m2. The viscosity of water is 0.001 Pa.s
Vo F
A
yo Water
= F/A
= (v/ y) thus = 10x0.001/0.0001= 100 N/m2
And F= x A = 100 x 0.2 =2 0 N
38
Example
39
Fig.2
Example: A Newtonian fluid having a specific gravity of 0.92 and a kinematic viscosity of ( ) flows past a fixed surface. The velocity profile near the surface is shown in Fig. 2. Determine the magnitude of the shear stress developed on the surface of the plate. (Hint the flow is Laminar)
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Example Newtonian Fluid Shear Stress 1/3
Example Newtonian Fluid Shear Stress 2/3
Example Newtonian Fluid Shear Stress 3/3
Viscosity and Temperature 1/3
Liquid viscosity decreases with
an increase in temperature.
For gases, an increase in
temperature causes an increase
in viscosity.
Viscosity and Temperature 2/3
The liquid molecules are closely spaced, with strong
cohesive forces between molecules, and the resistance
to relative motion between adjacent layers is related to
these intermolecular force. As the temperature
increases, these cohesive force are reduced with a
corresponding reduction in resistance to motion. Since
viscosity is an index of this resistance, it follows that
viscosity is reduced by an increase in temperature.
The Andrade’s equation μ= DeB/T
where D and B are empirical constants
Viscosity and Temperature 3/3
In gases, the molecules are widely spaced and intermolecular force negligible. The resistance to relative motion mainly arises due to the exchange of momentum of gas molecules between adjacent layers. As the temperature increases, the random molecular activity increases with a corresponding increase in viscosity.
The Sutherland equation μ= CT3/2 / (T+S)
where C and S are empirical constants
Dimension and Viscosity The dimension of μ : Ft/L2 or M/Lt.
The unit of μ:
In EE & BG : lbf . s/ft2 and slug/(ft.s)
In SI : kg/(m . s) or N . s/m2 or Pa . s
– In the Absolute Metric: poise=1 g/(cm . s)
The primary parameter correlating the viscous
behavior of all Newtonian fluids is the dimensionless Reynolds number (Re)
ρuD uD
Re=--------- = --------
μ ν
u is the average velocity, D is the diameter and ν is the Kinematic viscosity
Generally, the first thing a fluids engineer should do is estimate the Reynolds number range of the flow under study.
Very low Re indicates viscous creeping motion, where inertia effects are negligible.
Moderate Re implies a smoothly varying laminar flow.
High Re probably spells turbulent flow, which is slowly varying in the time-mean but has superimposed strong random high-frequency fluctuations.. For a given value of u and D in a flow, these fluids exhibit a spread of four orders of magnitude in the Reynolds number
Example Viscosity and Dimensionless Quantities
Example Viscosity and Dimensionless Quantities
Type of Viscosity
Newton's law of viscosity, given by: Viscosity, is the slope of each line, varies among materials. This is a constitutive equation (like Hooke's law, Fick's law) and is linear quation. Non-Newtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient and is non-linear equation
Non-Newtonian Fluids
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.
non-Newtonian Fluids 1/3
Shear thinning fluids
(Pseudoplastic) : The
viscosity decreases with
increasing shear rate
The harder the fluid is
sheared, the less viscous it
becomes.
Many polymer solutions are
shear thinning, clay solutions
are examples.
non-Newtonian Fluids 2/3
Shear thickening fluids : The viscosity increases
with increasing shear rate.
The harder the fluid is sheared, the more viscous
it becomes.
water-sand mixture are examples.
non-Newtonian Fluids 2/2
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, paint, blood, ketchup are common
examples.
Non-Newtonian Fluids Non-Newtonian Fluid
dr
duzrz
Newtonian Fluid
dr
duzrz
η is the apparent viscosity and is not constant for non-Newtonian fluids.