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METU Civil Engineering Department CE 272 FLUID MECHANICS

1.01

INTRODUCTION

DEFINITION OF FLUID

plate

F

at t = 0 t > 0

solid

= F/A

F plate Up

fluid

FLUID IS A SUBSTANCE THAT CAN NOT

SUPPORT SHEAR FORCES OF ANY MAGNITUDE

WITHOUT CONTINUOUS DEFORMATION

THE PROCESS OF CONTINUOUS DEFORMATION

IS KNOWN AS FLOW OF FLUIDS

t0 t1 t2 t3

Fluid

Fluids cannot support tension either


1.02

SCOPE OF FLUID MECHANICS

Civil Engineering Applications

Utilitarian

Water supply

Energy production

Transportation

of fluids,

of material,

as waterways

Mechanical

Pipelines

Hydraulic structures

Fluvial hydraulics

Coastal hydraulics

Groundwater flow

Wind forces on structures

Ships, Cars, Fast Trains, Aeroplanes…

Machines, Industrial Plants…

The circulatory system of human body…


1.03

CONCEPT OF CONTINUUM

Actual molecular structure A hypothetical medium

mlim

d

d

Continuum assumption is the continuous distribution of

matter in the flow field without any discontinuity.

A fluid particle is defined as the mass contained in the smallest

fluid volume for which the continuum assumption is not

violated.


1.04

PR

IMA

RY

DIM

EN

SIO

NS

DESCRIBING PHYSICAL ENTITIES

Quantity MLT FLT SI units

Mass M FL-1T2 kg

Force MLT-2 F N

Length L L m

Time T T s

Temperature C

Area L2 L2 m2

Velocity LT-1 LT-1 m/s

Acceleration LT-2 LT-2 m/s2

Force MLT-2 F N

Pressure ML-1T-2 FL-2 Pa

Energy ML2T-2 FL Joule

Power ML2T-3 FLT-1 Watt

Angle 1 1 radian

DE

RIV

ED

Qualitative description

DIMENSIONS

Quantitative description

UNITS

PR

IMA

RY

amF

or


1.05

PHYSICAL PROPERTIES OF FLUIDS

Density, : Mass per unit volume, = m/

[]=ML-3

Specific Weight, : Weight per unit volume, = W/

[]=FL-3

Specific Gravity, SG: The ratio of the density of the fluid to the

density of water (or air) at standard conditions.

Density and Specific Weights of some fluids (g=9.81m/s2)

Fluid Temperature C

Density

kg/m3

Specific Weight

N/m3

Water 4.0 1000. 9810.

Mercury 20.0 13600. 133416.

Gasoline 15.6 680. 6671.

Alcohol 20.0 789. 7740.

Air 15.0 1.23 12.0

Oxygen 20.0 1.33 13.0

Hydrogen 20.0 0.0838 0.822

Methane 20.0 0.667 6.54

Liq

uid

s G

ases

w

liquid)SG(

air

gas)SG(

Note that = g


1.06

Viscosity:

Deformation of fluid for a short time interval t

or

F Up

S

h A

B B´

y u(y)

hA

hFUp

dt

d

tlim

h

t

hlim

h

t

Slim

h

U

0t

0t0tp

h

Up

dt

d

Thus

Shear stress is proportional to the rate of angular deformation

y

h

B B’

A Δθ


1.07

Newton’s

Law of

viscosity dy

du

dy

du

Therefore

Viscosity can be made independent of

fluid density; kinematic viscosity is

defined as the ratio

112

1

2

TMLTFL

L

LT

FL

dy

du

12

3

11

TLML

TML

For the linear velocity profile

y

)y(u

h

Up

yh

U)y(u

p

dt

d

h

U

dy

du p

or

The proportionality constant is known as dynamic viscosity

of the fluid.

Fluid Temperature

(C)

(Ns/m2)

(m2/s)

Water 20 1.00E-03 1.01E-06

Air 20 1.80E-05 1.51E-05


1.08

In general

n

dy

duK

n<1

n>1

1

= 0 ideal fluid

n=1 Newtonian fluids

n1 Non-Newtonian fluids

n>1 Shear thickening

n<1 Shear thinning

A typical variation of shear stress

0y

wdy

du

Frictional drag force

y

dy

du

Umax

u(y)

(y)

w

Wall shear stress

u(y)

w


1.09

Dynamic (absolute) viscosity of some common

fluids as a function of temperature


1.10

Surface tension ,

Intermolecular Attraction Forces

The intensity of the molecular

attraction per unit length along

any line on an interface is called

the surface tension.

Solid

Gas A>C

Liquid

Capillary

effects

Capillary rise

(wetting fluid)

Capillary drop

(non-wetting fluid)

R

cos2h

1FL

Cohesive Forces (C)

Liquid to liquid

Gas to gas

Adhesive Forces (A)

Liquid to solid

Gas to liquid

h

Patm

Patm

2R

z

A>C

h

h

C>A


1.11

Vapor pressure, pv

Water

Vapor p

Heat

Boiling occurs when

ppv

Vapor pressure for water

Temperature C pv (kPa)

0 0.61

10 1.23

25 3.17

60 19.92

100 101.33=patm

2 3

Vapor pockets

p3

p1>pv p2pv p3>pv

1

Cavitation


1.12

Compressibility of Fluids

A

0 V

F

ρ/ρd

dp

/d

dpE

0v

Bulk Modulus

of Elasticity

dp

ρ/ρd

dp

/dK 0

Compressibility

dp

p=F/A

d/0

1

(Ev)water=2.15x109Pa (STP)

(Ev)air=1.42x105Pa (STP)

Esteel=2.00x1011Pa


1.E01

EXAMPLES

Example 1.1 Calculate the velocity gradient and the shear stress for y=0, 0.1, and 0.5 m if the

velocity profile of the flow is a parabola given by

u = 50 (2y-y2) , m1y0

where u is in (m/s) and y is in (m). Draw the shear stress distribution. Also calculate the

frictional drag force of the fluid on the bottom boundary on an area of 10 m2. Use dynamic

viscosity =0.001 Pas.


1.E02

Example 1.2 A space h=25 mm wide between two plane surfaces is filled with crude oil at

20C for which oil =7.18x10-3 Pas. What force is required to drag a very thin plate of 0.5 m2

area between the surfaces at a speed of v=0.15 m/s. Assume linear velocity profile.

a) If the plate remains equidistant from the two surfaces?

b) If it is at a distance of 10 mm from one of the surfaces.

Lower stationary plate

Upper stationary plate

F, V h


1.E03

Example 1.3 When a torque T is applied to the shaft, the disk A rotates with a constant angular

velocity The fluid in between transmits this torque T to the disk B. What will be the angular

velocity 2 for the disk B?

ω1

ω2

h

R0

Disk A

Disk B


1.E04

Example 1.4 Two capillary tubes of different diameter are submerged into water as seen in the

figure. Find the elevation difference of water between the two tubes.

σ

h1

h2

x

D1

σ θ

θ

D2


1.H01

HOMEWORK PROBLEMS

1.1 If F=QU/g, where Q is discharge, is specific weight, U is velocity and g is the

gravitational acceleration, what are the dimensions of F?

1.2 An expression for the volume rate of flow, Q flowing over a dam of length, B, is given

by the equation Q=3.09 BH3/2 where H is the depth of the water above the top of the dam

(called as head). This formula gives Q in ft3/s when B and H are in feet. Is the constant,

3.09, dimensionless? Would this equation be valid if units other than feet and seconds

were used?

1.3 A liquid when poured into a graduated cylinder is found to weigh 6 N when occupying a

volume of 500 ml (milliliters). Determine its specific weight, density and specific gravity.

1.4 A gas is compressed. The measured volume and absolute pressure before compression

are 0.30 m3 and 50.7 kPa, respectively. After compression the volume and the pressure

becomes 0.111 m3 and 202.8 kPa, respectively. What is the compressibility and bulk

modulus of elasticity of this gas?

1.5 Develop an expression for the pressure variation in a liquid in which the specific weight

increases with depth, h, as =Kh+o, where K is constant, o is the specific weight at the

free surface.

1.6 An 8-kg flat block of metal slides down a = 20 inclined plane while lubricated by a thin

film of oil. The contact area, A, is 0.2 m2. What is the terminal velocity of the block?

oil=0.29 Pa.s, t=2 mm.

Contact area, A

t


1.H02

1.7 Calculate the shear stress for y= 0, 3

and 6mm. If the velocity profile of the

flow in an open channel is given as,

)2

y(SinUu max

where u is in (m/s) and y in (mm).

Draw the shear stress distribution.

=1.8*10-5 kg/m.s, δ=6 mm, Umax=10

m/s.

1.8 A triangular shaft is pulled in a

triangular bearing housing (see figure)

at a constant velocity of 0.3 m/s. Find

the force required to pull the shaft,

if the length of the shaft is 2 m.

The viscosity of the lubricating oil

filling the clearing between the shaft

and the housing is =1x10-1 Ns/m2.

t1=t2=t3=1 mm, l =10 cm.

1.9 A 25 mm-diameter shaft is pulled

through a cylindrical bearing as shown

in the figure. The lubricant that fills the

0.3 mm gap between the shaft and

bearing is an oil having a kinematic

viscosity of 8x10-4 m2/s and a specific

gravity of 0.91. Determine the

force P required to pull the shaft

at a velocity of 3 m/s. Assume the

velocity distribution in the gap is

linear. L=0.5 m.

y δ

Umax

u=UmaxSin( )

P

L

Lubricant

Shaft

Bearing

t3 t2

t1

Shaft

60 60

l

oil


1.H03

1.10 A torque of T=4 Nm is required to

rotate the intermediate cylinder at

=30 rad/min. Calculate the viscosity

of the oil. All cylinders are 450 mm

long. Neglect the end effects.

R=0.15 m, t=0.003 m.

1.11. The device shown consists of a disk that is rotated by a shaft. The disk is positioned very

close to a solid boundary. Between the disk and boundary there is viscous oil.

a) If the disk is rotated at a rate of 1 rad/s, what will be the ratio of the shear stress in

the oil at r=2 cm to the shear stress at r =3 cm?

b) If the rate of rotation is 2 rad/s, what is the tangential velocity of the oil in contact

with the disk at r=3 cm?

c) If the oil viscosity is 0.01 N.s/m2 and the spacing y is 2 mm, what is the shear stress

for the conditions noted in (b)?

1.12. A conical body is made to rotate at a

constant speed of =42 rad/sec. A film

of oil having a viscosity of 0.5 poise

(gr/cm.s) separates the cone from the

container. The film thickness, t, is

0.025 cm. What torque is required to

maintain this motion? The cone radius

at the base, R, is 10 cm and cone has a

length of h=30 cm.

t t

R

Oil

Disk

D

y

r

R

t t

t

h oil


1.H04

1.13. Compute the torque T required to rotate a conical object at a constant angular speed .

The clearance between the object and the casing is constant in thickness (h) and filled

with oil of . ( =30)

1.14. Small droplets of carbon tetrachloride at 68F are formed with spray nozzle. If the average

diameter of the droplets is 200 m what is the difference in pressure between the inside

and outside of the droplets? (=2.69x10-2N/m for carbon tetrachloride at 68F)

1.15 Water is filled between two parallel plates of infinite length, a distance d apart. Find the

capillary rise between these two plates, where

surface tension

angle of contact

ddistance between plates

unit weight of water

h : capillary rise

W

d

d

h

oil

,

D

h

h

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