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 .

23

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)

40

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.