Gv 3 Newtonian Compressible Flows

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MEE206Fluid Mechanics

G. Vinayagamurthy Dr. Engg.School of Mechanical and Building Sciences

VIT University Chennai

Lecture - 5

Newtonian and Non-Newtonian Fluids

Compressibility

Incompressible and Compressible flows

VINAYAGAMURTHY, SMBS, VIT CHENNAI


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Viscosity

where is called absolute or dynamic viscosity. Dimensionsand units for are and , respectively. [In

the absolute metric system basic unit of co-efficient of viscosity

is called poise. 1 poise =

The viscosity is measure of the fluidity of the fluid which is not captured simply by

density or specific weight. A fluid can not resist a shear and under shear begins to flow.

The shearing stress and shearing strain can be related with a relationship of the followingform for common fluids such as water, air, oil, and gasoline:

dy

du

No SlipCondition

VINAY

AGAMURTHY, SMBS, VIT CHENNAI
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For Non-Newtonian fluid: . Examples are sugar solution andpolymers. Therefore,General expression for shear stress

where, m is the flow consistency and n is the flow behaviour index.

Also known as powerLaw model

also known as apparent viscosityVINAYAGAMURTHY, SMBS, VIT CHENNAI


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Viscosity: Measurements

A Capillary Tube Viscosimeter is one method of measuring the viscosity of the fluid.

Viscosity Varies from Fluid to Fluid and is dependent on temperature, thus

temperature is measured as well.

Units of Viscosity are Ns/m2or lbs/ft2

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Viscosity



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Newtonian Fluids



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Non-Newtonian Fluids



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Non-Newtonian Fluids



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No slip condition

No-slip conditionfor viscous fluids states that ata solid boundary, the fluid will have zero velocity

relative to the boundary.

No Slip

Condition

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Compressibility

Compressibility of any substance is the measure of its change in

volume under the action of external forces. The normal compressive stress on any fluid element at rest is

known as hydrostatic pressure p and arises as a result of

innumerable molecular collisions in the entire fluid.

The degree of compressibility of a substance is characterized by the

bulk modulus of elasticity E defined as



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E for liquids are very high as compared with those of gases (except at very high

pressures). Therefore, liquids are usually termed as incompressible fluids though, in

fact, no substance is theoretically incompressible with a value ofE as

For example, the bulk modulus of elasticity for water and air at atmospheric

pressure are approximately 2 x 106kN/m 2and 101 kN/m 2respectively. It indicates

that air is about 20,000 times more compressible than water. Hence water can be

treated as incompressible.VINAYAGAMURTHY, SMBS, VIT CHENNAI


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For gases another characteristic parameter, known as compressibility

K, is usually defined , it is the reciprocal of E

K is often expressed in terms of specific volume



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Incompressible and a Compressible

Flow

In order to know, if it is necessary to take into account the compressibility ofgases in fluid flow problems, we need to consider whether the change in pressure

brought about by the fluid motion causes large change in volume or density.

By Bernoulli's equation

p + (1/2)V2= constant (V being the velocity of flow), change in pressure, p, in

a flow field, is of the order of (1/2)V2 (dynamic head).

Invoking this relationship into

So if / is very small, the flow of gases can be

treated as incompressible with a good degree of

approximation.VINAYAGAMURTHY, SMBS, VIT CHENNAI


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where, Ma is the ratio of the velocity of flow to the acoustic velocity in theflowing medium at the condition and is known as Mach number.

So we can conclude that the compressibility of gas in a flow can be neglected

if /is considerably smaller than unity, i.e. (1/2)Ma2


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Compressibility of Fluids: Speed of Sound

vE

cord

dp

c

kpc

A consequence of the compressibility of fluids is that small disturbances introduced ata point propagate at a finite velocity. Pressure disturbances in the fluid propagate as

sound, and their velocity is known as the speed of sound or the acoustic velocity, c.

Isentropic Process (frictionless, no heat exchange because):

Ideal Gas and Isentropic Process:

kRTc



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Compressibility of Fluids: Compression of Gases

RTp Ideal Gas Law:

P is pressure, is the density, R is the gas constant, and T is Temperature

COMPRESSIBILITYOFFLUIDS: SPEEDOFSOUNDSpeed of Sound in Air at 60 F 1117 ft/s or 300 m/s

Speed of Sound in Water at 60 F 4860 ft/s or 1450 m/s

If a fluid is truly incompressible, the speed of sound is infinite, however, all

fluids compress slightly.

Ideal Gas and Isentropic Process:

smc

KkgKJc

kRTc

/6.296

219*)/9.286(*40.1

Example: A jet aircraft flies at a speed of 250 m/s at an altitude of 10,700 m,

where the temperature is -54 C. Determine the ratio of the speed of the aircraft,

V, to the speed of sound, c at the specified altitude. Assume k = 1.40



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84.0

/6.296

/250

Ratio

sm

smRatio

c

VRatio

Compressibility of Fluids: Speed of Sound

Example(Continued):

The above ratio is known as the Mach Number, Ma

For Ma < 1 Subsonic Flow

For Ma > 1 Supersonic Flow

For Ma > 1 we see shock waves and sonic booms:

1) Wind Tunnel Visualization known as Schlieren method

2) Condensation instigated from jet speed allowing us to see a

shock wave

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Gv 3 Newtonian Compressible Flows

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