1.3 Analysis of Fluid Behavior

1.3 Analysis of Fluid Behavior

 Fluid Statics : When the fluid is at rest.

 Fluid Dynamics : When the fluid is moving.

 Governing equations :

 

 



mics) themodyna

of law (First

energy of

on Conservati

law)

second

(Newtons momentum

of on Conservati

mass of

on

Conservati

1.4 Measures of Fluid Mass and Weight

 Density,  : Mass of a fluid per unit volume [slug/ft3, kg/m3]

For water at 5^oC,

_water = 1.940 slugs/ft³ = 1000 kg/m³ For air at standard pressure and at 20^oC,

_air = 2.377 10^-3 slugs/ft³ = 1.225 kg/m³

V lim m

V

V



 



 Specific volume,  : Volume per unit mass.

 

 1

 Specific weight,  : Weight per unit volume

For water at 5^oC,



For air at standard pressure and at 20^oC,



 g



 ^[lb/ft

^{, N/m}

^]

 Specific gravity, SG : The ratio of the density of the given fluid to the density of water at some specified temperature, usually at 4^oC(39.2^oF).

Specific gravity of gases is usually based on dry air as the reference fluid.

3 3

C 4 at O

H

1 . 94 slugs / ft , or 1000 kg / m SG

o 2

 



 

 Pressure : the normal compressive force per unit area acting on a real or imaginary surface in the fluid.

 Microscopically, pressure represents molecular momentum and intermolecular forces within the fluid.

A lim F

p

A

A 

 







 



pressure zero

to relative defined)

(or measured

Pressure :

pressure Absolute

pressure atmosphere

local to

relative measured

Pressure :

pressure

Gage

 Absolute pressure

= Gage pressure + Atmosphere pressure in vicinity of gage

 Vacuum pressure : pressure below local atmosphere pressure Vacuum pressure = Atmosphere pressure – Absolute pressure

= - Gage pressure

 The subscripts “g” and “a” indicates whether the pressure is gage or absolute.

(ex. 10 psig = 10 pounds per square inch, gage ; 10 psia = 10 pounds per square inch, absolute)

 Standard value of

atmospheric pressure is 101.333 kPa (14.496 psia, 29.92 in. Hg, abs)

[Pascal ; Pa=N/m2]

 Temperature : defined as a measure of (not equal to) the energy contained in the molecular motion of the fluid

T (Rankine) = T (Fahrenheit) + 459.67 T (Kelvin) = T (Celsius) + 273.15

T (Rankine) = 1.80 T (Kelvin)

 Internal energy (U) : Energy contained in random molecular motions and intermolecular forces, U = U(T)

 Specific internal energy ( ) : Internal energy per unit mass

u

 Specific heat at constant volume,

 Specific heat at cont. pressure,

 For an incompressible fluid, all processes are constant specific volume and

So c_p = c_v ( for incompressible fluid)

v

v

T

c u 







 

p

p

T

c h 







 

pv p u

u h

; enthalpy

specific  

 



  0

T p v T

pv

p p

 







 

 







1.5 Ideal (Perfect) Gas Law

(=Equation of State for an ideal gas)



 R is the specific gas constant and is equal to the universal gas constant (R₀) devided by the molecular weight (MW) of the gas :

v RT RT

p   

R . lbm /

lb . MW ft K 1545

. kg / m . MW N 8314 MW

R  R

 

 Liquids exhibit slight variation of density with temperature and pressure.

 No simple, exact equations are available for properties of liquids. For most practical purposes, liquids are treated as incompressible fluids.

1.6 Viscosity

Newton’s law of viscosity



where the constant of proportionality, , is called the absolute viscosity, dynamic viscosity, and simply viscosity of the fluid.

Dimension = [lb.s/ft2], [N.s/m2]

dy

 du





viscosity [Pa·s]

viscosity [cP]

liquid nitrogen@ 77K 1.58 × 10⁻⁴ 0.158

acetone* 3.06 × 10⁻⁴ 0.306

methanol* 5.44 × 10⁻⁴ 0.544

benzene* 6.04 × 10⁻⁴ 0.604

water 8.94 × 10⁻⁴ 0.894

ethanol* 1.074 × 10⁻³ 1.074

mercury* 1.526 × 10⁻³ 1.526

nitrobenzene* 1.863 × 10⁻³ 1.863

propanol* 1.945 × 10⁻³ 1.945

Ethylene glycol 1.61 × 10⁻² 16.1

sulfuric acid* 2.42 × 10⁻² 24.2

olive oil .081 81

glycerol* .934 934

castor oil* .985 985

corn syrup* 1.3806 1380.6

HFO-380 2.022 2022

pitch 2.3 × 10⁸ 2.3 × 10¹¹

viscosity [cP]

honey 2,000–10,000

molasses 5,000–10,000 molten glass 10,000–1,000,000 chocolate syrup 10,000–25,000 molten

chocolate^* 45,000–130,000 ^[19]

ketchup^* 50,000–100,000 peanut butter ~250,000 shortening^* ~250,000

Viscosity of Liquids at 25^oC

 Sutherland equation : (for gases)

where C and S are empirical constants and T is absolute temperature.

 Andrade’s equation : (for liquids)

where D and B are constants. T is absolute temperature.

 Kinematic viscosity : [ft2/s, m2/s]

 In CGS (centimeter-gram-second) unit, the dynamic viscosity has the unit of dyne.s/cm² (=poise, abbreviated as P).

The kinematic viscosity has the unit of cm²/s (=stoke, St)

** 1 dyne = (1g) x (1cm/s²)

S T CT

 



T /

De







 



Newtonian and Non-Newtonian Fluid

 Newtonian fluid :

Fluids that obey the Newton’s law of viscosity.

(The shearing stress is linearly related to the rate of shearing strain)

Most common fluids, both liquids and gases, are Newtonian.

 Non-Newtonian fluid :

Fluids for which the shearing stress is not linearly related to the rate of shearing strain

Shear-thinning fluid

 The coefficient of resistance decreases with increasing strain rate.

 Ex. Ketchup (It all comes out of the bottle at once)

Colloidal suspensions Polymer solutions,

Latex paint (It does not drip from the brush because the shear rate is small and the shear stress is large. However, it flows smoothly onto the wall because the thin layer of paint between the wall and brush causes a large shear rate (large du/dy) and a small shear stress.)

Shear-thickening fluid

 Fluids having the characteristics that the shear stress increases with increasing the shear strain. The harder the fluid is sheared, the more viscous it becomes.

 Ex. Water-corn starch mixture

Water-sand mixture (quicksand):

The difficulty in removing an object from quicksand increases dramatically as the speed of removal increases.

Bingham plastic

 This is neither a fluid nor a solid.

 This material can withstand a finite shear stress without motion ( hence, not a fluid), but once the yield stress is exceeded it flows like a fluid (i.e., not a solid).

 Ex. Toothpaste, Mayonnaise

 No-slip condition : Whenever a fluid is in contact with a solid surface, the velocity of the fluid at the surface is equal to the velocity of the surface; that is, the fluid “sticks” to the surface and does not “slip” relative to it.

This condition is true regardless of the type of the fluid, type of surface, or surface roughness, so long as the continuum hypothesis is valid.

 Inviscid fluid : the fluid with zero viscosity, i.e.,  = 0.

Consequently,  = 0.

The assumption of an inviscid fluid is often useful for analyzing flow remote from the solid boundaries.

 Ideal fluid :  = 0 and  = constant(incompressible)

1.7 Compressibility of Fluids

 Bulk modulus, E_v : measure of the compressibility of fluid [psi, Pa]

 Large values of the bulk modulus indicate that the fluid is relatively incompressible-that is, it takes a large pressure change to create a small change in volume.

 Common liquids have large value of E_v, For example, at atmospheric pressure and a temperature of 60^oF it would require a pressure of 3120 psi to compress a unit volume of water 1%. i.e., E_v=3.12x10⁵ psi (=2.15x10⁹ Pa) for water.

 For most practical engineering problems, we consider the liquids are incompressible.



 







d dp V

dV E dp

V m v

 For isothermal process, =constant,

 For isentropic process, constant

where (for air k=1.4) and R=c_p-c_v p  RT



p / RT

d RTd /

d E dp

RT d

v dp   





 



 





^k  p

p kp k k

) const / (

d

d k

) const (

/ d

E dp

1 k d

k ) const (

v dp _{k 1}  

 



 





 



  ^





 ^

v

p

c

c

k 

 For air under standard atmospheric conditions with p=14.7 psi and k=1.4, the isentropic bulk modulus is 20.6 psi.

Comparing this value with that of water (E_v,water=312,000 psi), the air is approximately 15,000 times as compressible as water.

Speed of Sound

 Speed of sound : defined as

Since the disturbance is small, there is negligible heat transfer and the process is assumed to be isentropic.

Thus, for ideal gases the speed of sound is proportional to the square root of the absolute temperature.

  d c dp

kp kRT E

d c dp

process isentropic

undergoing gas

for

v

gas ideal

 











 For example, for air at 60^oF with k=1.4 and R=1716 ft.lb/slug.^oR, c=1117 ft/s(340 m/s).

 For water at 20^oC, E_v=2.19 gN/m² and =998.2 kg/m³ so that c=1481 m/s or 4860 ft/s.

 The speed of sound in water is much higher than in air.

 If a fluid is truly incompressible (E_v=), the speed of sound would be infinite.

Speed of sound as a function of depth at north Hawaii

Sound during the Day Sound in the Evening

Sound during the Day Sound in the Evening Warm

Cold Warm

Cold

Mach Number

 Mach number, Ma : defined as Ma=

 Subsonic flow regime : Ma < 1.0

 Sonic flow : Ma = 1.0

 Supersonic flow regime : Ma > 1.0

 Transonic flow regime : 0.7~0.8 < Ma <1.2~1.5 (depends on the configuration of flying object)

c

V

1.8 Vapor Pressure

 Evaporation takes place because some liquid molecules at the surface have sufficient momentum to overcome the intermolecular cohesive forces and escape into the atmosphere.

 When the saturation is reached, the pressure that exerts on the liquid surface is termed the Vapor Pressure.

 Since the development of a vapor pressure is closely associated with molecular activity, the value of vapor pressure for a particular liquid depends on temperature (because the molecular activity (internal energy) depends on temperature).

 Generally, as the temperature increases, the vapor pressure of a fluid also increases.

 Boiling is initiated when the absolute pressure in the fluid reached the vapor pressure.

 Water at standard atmospheric pressure will boil when the temperature reaches 212^oF (100^oC)-that is, the vapor pressure of water at 212^oF is 14.7 psi abs.

 However, if at a higher elevation, say 10,000 ft above sea level, where the atmospheric pressure is 10.1 psi abs, the boiling will start at about 193^oF. At this temperature the vapor pressure is 10.1 psi abs.

 Thus, boiling occurs at a given pressure acting on the fluid by raising the temperature, or at a given fluid temperature by lowering the pressure.

Cavitation phenomena in the pump, valve, marine propeller, etc.

(mmHg)

760 mmHg 14.7 psia

Cavitation

Erosion by Cavitation Bubble

Erosion by Cavitation Bubble

Supercavitating Torpedo

VA-111 Shkval Torpedo

 Length: 8.2 m (27 feet)

 Diameter: 533 mm

 Weight: 2700 kg (5940 pounds)

 Warhead weight: 210 kg

 Speed

Launch Speed: 50 kt (93 km/h)

Maximum Speed: 200+ kt (370 km/h)

 Range: Around 7000 m to 13000 m (New version)

• Research is on going by PNU CFD lab.

German “Barracuda”

 Western countries are not far behind though, with Germany currently developing the "Barracuda", which is guided and has been offically stated as

being capable of 360Km/h, but has been rumoured to travel at up to 800km/h.

 It looks like the Russians have been them to the

punch again though, with the Shkval-II already

deployed and rumoured of being cable of at least

720km/h whilst also being guided.

Iran

Underwater Express

 DARPA (Defense Advanced Research Projects Agency) / ATO (Advanced Technology Office)

 Period : April 2006 ~ August 2009

 Technology development and demonstration program (Model scale=1/4~1/2)

 Demonstrate stable and controllable high-speed underwater transport through supercavitation for future littoral missions

 Speed ~100 knots

 Size : 8 ft diameter, 60 tones for super-fast submerged transport (SST)

- comparable in size to current special purpose craft such as the MK V Special Operations Craft and the Advanced Seal Delivery Vehicle

- Mark V: 82 feet long aluminum monohull surface craft, 40 knots

• Research is on going by PNU CFD lab.

RAMICS (RAPID AIRBORNE MINE CLEARANCE SYSTEM)

AHSUM (Adaptable High-Speed Munitions)

1.9 Surface Tension

 The surface tension is due to the unbalanced cohesive forces acting on the liquid molecules at the fluid surface.

 Molecules in the interior of the fluid mass are surrounded by the molecules that are attracted to each other equally.

However, molecules along the surface are subjected to a net force toward the interior.

This unbalanced force along the surface creates the membrane. The tensile force along the surface is called the surface tension.

 Unit = [lf/ft], [N/m]

2  R   p  R

p R p

p

2 









 Capillary : In fig (a), the attractive(adhesive) force between the wall of the tube and liquid molecules is strong enough to overcome the mutual attractive (cohesive) force of the molecules.  The liquid is said to “wet” the solid surface.







 R

h  2 R cos

h h





 cos

 2

 Note that the height in a capillary tube is inversely proportional to the tube radius, and thus the rise of a liquid becomes increasingly pronounced as the tube radius is decreased.

 If adhesion of molecules to the solid surface is weak compared to the cohesion between molecules, the liquid will not wet the surface and the level in a tube placed in a nonwetting liquid will be depressed as shown in fig.1.8(c).

Mercury is nonwetting liquid when it is contact with the glass,

  130^o.

Surface Tension

 Application:

- Detergent

- Washing in hot water