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Properties of Fluids for Fluid Mechanics

- P M V Subbarao
- Associate Professor
- Mechanical Engineering Department
- IIT Delhi

Basic Steps to Design.

Continuum Hypothesis

- In this course, the assumption is made that the

fluid behaves as a continuum, i.e., the number of

molecules within the smallest region of interest

(a point) are sufficient that all fluid

properties are point functions (single valued at

a point). - For example
- Consider definition of density ? of a fluid

- dV limiting volume below which molecular

variations may be important and above which

macroscopic variations may be important.

Static Fluid

For a static fluid

Shear Stress should be zero.

For A generalized Three dimensional fluid

Element, Many forms of shear stress is possible.

One dimensional Fluid Element

Y

uU

u0

X

?

Fluid Statics

- Pressure For a static fluid, the only stress is

the normal stress since by definition a fluid

subjected to a shear stress must deform and

undergo motion.

Y

X

Z

- What is the significance of Diagonal Elements?
- Vectorial significance Normal stresses.
- Physical Significance ?
- For the general case, the stress on a fluid

element or at a point is a tensor

Stress Tensor

First Law of Pascal

Proof ?

Simple Non-trivial Shape of A Fluid Element

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Fluid Statics for Power Generation

- P M V Subbarao
- Associate Professor
- Mechanical Engineering Department
- IIT Delhi

Steps for Design of Flow Devices.

Pressure Variation with Elevation

- For a static fluid, pressure varies only with

elevation within the fluid. - This can be shown by consideration of equilibrium

of forces on a fluid element

- Basic Differential Equation
- Newton's law (momentum principle) applied to a

static fluid - SF ma 0 for a static fluid
- i.e., SFx SFy SFz 0

1st order Taylor series estimate for pressure

variation over dz

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- For a static fluid, the pressure only varies with

elevation z and is constant in horizontal xy

planes. - The basic equation for pressure variation with

elevation can be integrated depending on - whether ? constant i.e., the fluid is

incompressible (liquid or low-speed gas) - or ? ?(z), or compressible (high-speed gas)

since g is constant.

Pressure Variation for a Uniform-Density Fluid

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Reading Material

- Fluid Mechanics Frank M White, McGraw Hill

International Editions. - Introduction to Fluid Mechanics Fox McDOnald,

John Wiley Sons, Inc. - Fluid Mechanics V L Streeter, E Benjamin Wylie

K W Bedfore, WCB McGraw Hill. - Fluid Mechanics P K Kundu I M Cohen, Elsevier

Inc.

Pressure Variation for Compressible Fluids

Basic equation for pressure variation with

elevation

Pressure variation equation can be integrated for

?(p,z) known. For example, here we solve for the

pressure in the atmosphere assuming ?(p,T) given

from ideal gas law, T(z) known, and g ? g(z).

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Draft Required to Establish Air Flow

Flue gas out

Air in

Natural Draft

Zref,,pref

pA pref Dp

Hchimney

Tgas

Tatm

B

A

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Zref,,pref

pA pref Dp

Hchimney

Tgas

Tatm

B

A

Pressure variations in Troposphere

Linear increase towards earth surface

Tref pref are known at Zref.

a Adiabatic Lapse rate 6.5 K/km

Reference condition At Zref TTref p pref

Pressure at A

Pressure variation inside chimney differs from

atmospheric pressure. The variation of chimney

pressure depends on temperature variation

along Chimney. Temperature variation along

chimney depends on rate of cooling of hot gas Due

to natural convection. Using principles of Heat

transfer, one can calculate, Tgas(Z). If this is

also linear T Tref,gas agas(Zref-Z). Lapse

rate of gas, agas is obtained from heat transfer

analysis.

Natural Draft

- Natural Draft across the furnace,
- Dpnat pA pB

- The difference in pressure will drive the

exhaust. - Natural draft establishes the furnace breathing

by - Continuous exhalation of flue gas
- Continuous inhalation of fresh air.
- The amount of flow is limited by the strength of

the draft.

Pressure Measurement

- Another Application of Fluid Statics

Pressure Measurement

Pressure is an important variable in fluid

mechanics and many instruments have been devised

for its measurement. Many devices are based on

hydrostatics such as barometers and manometers,

i.e., determine pressure through measurement of a

column (or columns) of a liquid using the

pressure variation with elevation equation for an

incompressible fluid.

PRESSURE

- Force exerted on a unit area Measured in kPa
- Atmospheric pressure at sea level is 1 atm, 76.0

mm Hg, 101 kPa - In outer space the pressure is essentially zero.

The pressure in a vacuum is called absolute zero. - All pressures referenced with respect to this

zero pressure are termed absolute pressures.

- Many pressure-measuring devices measure not

absolute pressure but only difference in

pressure. This type of pressure reading is called

gage pressure. - Whenever atmospheric pressure is used as a

reference, the possibility exists that the

pressure thus measured can be either positive or

negative. - Negative gage pressure are also termed as vacuum

pressures.

Manometers

U-tube or differential manometer

Right Limb fluid statics

Left Limb fluid statics

Point 3 and 2 are at the same elevation and same

fluid

Gauge Pressure

Absolute, Gauge Vacuum Pressures

System Pressure

Gauge Pressure

Absolute Pressure

Atmospheric Pressure

Absolute zero pressure

Absolute, Gauge Vacuum Pressures

Atmospheric Pressure

Vacuum Pressure

System Pressure

Absolute Pressure

Absolute zero pressure

Stress Tensor for A Static Fluid

0

0

0

0

0

0

An important Property of A Fluid under Motion

Shear stress(t) Tangential force on per unit

area of contact between solid fluid

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Elasticity (Compressibility)

- Increasing/decreasing pressure corresponds to

contraction/expansion of a fluid. - The amount of deformation is called elasticity.

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Surface Tension

- Two non-mixing fluids (e.g., a liquid and a gas)

will form an interface. - The molecules below the interface act on each

other with forces equal in all directions,

whereas the molecules near the surface act on

each other with increased forces due to the

absence of neighbors. - That is, the interface acts like a stretched

membrane, e.g.

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Vapour Pressure

- When the pressure of a liquid falls below the

vapor pressure it evaporates, i.e., changes to a

gas. - If the pressure drop is due to temperature

effects alone, the process is called boiling. - If the pressure drop is due to fluid velocity,

the process is called cavitation. - Cavitation is common in regions of high velocity,

i.e., low p such as on turbine blades and marine

propellers.

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