ESE-CIVIL ENGINEERING.

PAPER-1 CUT OFF

Year UR OBC SC ST OH HH EWS
2011 125 121 107 107
2012 149 146 124 105 87 87
2013 165 153 110 102 113 101
2014 202 186 156 100 90 101
2015 262 228 180 187 91 114
2016 229 214 187 183 83 76
2017 202 177 148 151 102 68
2018 207 194 169 188 114 87
2019 188 185 143 159 88 52
2020 238 238 202 227 160 62 238
2021 249 243 196 213 159 59 246
2022 233 233 197 198 127 77 233
ESE-CIVIL ENGG. PAPERS MAINS CUT OFF
Year UR OBC SC ST OH HH EWS
2011 283 239 224 221 125 159
2012 342 296 280 255 126 126
2013 352 301 267 252 159
2014 391 373 315 293 158 158
2015 516 470 376 411 197 225
2016 464 418 387 395 223 143
2017 520 459 415 439 315 145
2018 546 502 467 513 308 161
2019 541 500 453 482 252 145
2020 651 576 486 575 365 180 600
2021 617 582 484 519 264 156 581
2022 551 525 474 476 345 263 534

ESE-CIVIL ENGG. PAPER FINAL CUT OFF

Year UR OBC SC ST OH HH EWS
2011 451 427 369 387 216 251
2012 512 484 422 422 274 393
2013 500 475 412 419 299
2014 572 540 467 453 329 263
2015 674 630 539 552 380 323
2016 623 588 538 539 414 217
2017 691 650 564 578 437 311
2018 710 679 609 671 476 423
2019 702 668 596 640 466 241
2020 807 762 688 725 567 371 759
2021 804 762 642 667 553 272 772
2022 756 729 646 698 506 519 711

A
 SSC JE PAPER 1 CUT OFF (OUT OF 200)
Year UR OBC SC ST OH HH EWS
2012 62.25 52.5 47.75 43.5 30 30
2013 78 70.5 66.25 63.5 60 40
2014 93.75 82 75.75 70 69 40
2015 103.75 91.25 88 87.75 78 30
2016 100 92.5 84.5 58.5 72.5 40
2017 117 110.75 101.75 105 91.5 61.75
2018 127.4 122.91 107.61 107.01 97.45 61.61 118.99
2019 123.52 115.93 101.70 102.61 92.24 55.73 112.28
2020 120.02 114.21 99.15 99.15 78.83 48.86 108.14
2022 110.57 107.99 86.36 86.32 80.28 40 89.08
SSC JE PAPER 2 CUT OFF

Year UR OBC SC ST OH HH EWS

2012 84 54 62 40 40 45
2013 83 69 62 56 59 32
2014 136 100 80 79 79 30
2015 131 62 50 50 50 40
2016 220.5 186 164 163.75 139.5 87.5
2017 244.75 244.75 220.75 228 231.25 152
2018 250.49 209.38 193.68 201.54 162.01 132.68
2019 315.55 265.07 235.51 243 169.02 122.79 270.44
2020 257.84 234.28 201.59 188.11 150.23 137.52 229.58
2022 323.40 310.53 290.01 296.86 244.69 186.65 314.32

SSC JE FINAL CUT OFF RANGE

Year UR OBC SC ST EWS

2012 184  276.75 190  258.5 168  238.75 144.25  237.25
2013 199.75  280.75 226  291.5 213.25  245.5 205  232.25
2014 285  353 287.75  334.5 265.25  307.75 250  392.75
2015 235.5  284.75 236.25  276.25 213.5  289.75 236.25  267.75
2016 238.25  264.75 236  252.25 205.25  225 228.25  245
2017 289.75  301.75 283.5  292.0 262.75  272.25 279.0  291.0
2018 295  275 280  250 270  235 270  245 295  265
2019 362.02  319.14 349.50  305.23 320.17  272.13 318.56  267.61 348.99  304.64
2020 306.76  264.13 280.83  261.3 258.75  227.34 248.38  216.32 297.14  252.26
2022 347.37  323.40 345.43  310.53 322.01  290.01 328.88  296.86 339.41  314.32

B
GATE-CIVIL ENGG. CUT OFF

Year UR OBC SC / ST
2011 25 22.5 16.67
2012 33.03 29.73 22.02
2013 27.13 24.42 18.09
2014 26.57 23.91 17.71
2015 27.52 24.77 18.34
2016 25 22.5 16.6
2017 28.7 25.8 19.1
2018 26.9 24.2 17.9
2019 28.2 25.4 18.8
2020 32.9 29.6 21.9
2021 29.20 26.20 19.4
2022 30.4 27.3 20.2
2023 26.6 23.9 17.7

C
 SSC JE APPEARED CANDIDATES DETAILS

GATEAPPEARED CANDIDATES (CIVILENGINEERING)

SSC JE NO. OF FORM

ESE CIVILENGG. VACANCY DETAILS

SSC JE Civil Engg. Vacancy Details

F
G
H
 How To Get Free Books On Each Data Error

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 Overall Exam Strategy

• Gather study materials: Collect the best study materials, including textbooks,
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Duao Me Yaad Rakhna..........

 Fluid Mechanics 335

Properties of Fluid 1
Basic Conversion:
Density (  )
1kg/m3 = 0.0624 lb/ft3; 1 lb/ft3 = 16.018 kg/m3
Unit weight, (  or    g )
Units: N/m3, kg/m3 (kg (wt)/m3);
1kg/m3 (kg(wt)/m3) 1N/m3 = 0.102kg/m3
= 9.807N/m 3

(Force)
1kg wt = 9.807 N 1N = 1 kgm/sec2
(Newton); 1N = 0.102 kg wt
1gm wt = 980.7 dynes 1lb wt = 32.2 poundals
1kg(wt) = 9.807N = 2.204 lb(wt) = 9.807 × 105 dynes
1N = 0.102kg(wt) = 0.2247 lb(wt) = 105 dynes
1 poundal = 13825.5 dynes = 14.102 gm(wt).
(Pressure)
Units: N/m = Pascal (Pa); kg/m2
2

1kg/cm2 = 104 kg/m2 100kPa = 10,200kg/m2

1kg/m = 9.807 N/m ;
2 2
1Pa = 1N/m2 = 0.102kg/m2
= 9.807 Pa 1kPa = 0.01 kg/cm2
1 Standard Atmospheric = 1.02 kg/cm2
Pressure = 1.013 bar
= 760 mm of Hg
= 101.325 kPa
= 10.33 m water head
1kg/cm = 98.07kPa 100kPa = 1.02kg/cm2
2
336 Civil Booster (Civil Ki Goli Publication 9255624029)

 Any substance in gaseous/liquid phase is called as fluid.

 Fluid is a substance that deforms (flow) continually under an applied
shear stress.
 Fluid does not have tensile strength, but it has compressive strength only
when it is kept in closed container.
 Fluid Static – Study of fluid in rest condition.
 Fluid dynamic – Branch of mechanics dealing with forces & torque in
moving situation and look for the reasons behind the motion and forces.
(a) Kinetics – study of fluid in motion considering forces.
(b) Kinematics – study of fluid in motion without considering forces.
Special Point: In solids, stress is proportional to strain but in fluid stress
is proportional to strain rate.

Types of fluid -

(a) Real fluid - Fluids which are not ideal.

Ideal fluid do not exist, so practically all fluid are real.
(b) Ideal fluid -
Bulk modulus is infinite.
They are incompressible and do not have viscosity & surface tension.
For an ideal fluid, no shear stresses exist and pressure is same in all
directions.
Some Basic fluid properties:
mass
1. Density = measured in kg/m3.
volume
2. If relative density < 1, then fluid is lighter than water.
3. The density of water is maximum at 4oC
4. Fathometer is used to measure ocean depth
5. Density of liquid & gas is directly proportional to pressure and inversely
to temperature
Density of liquid
6. Specific gravity/relative density =
Density of water at 4ºC
Weight of substance
7. Specific weight = , (  = g in N/m3 )
Volume of substance
 Fluid Mechanics 337

8. Some Important Relation

1 milibar = 10–3 bar =100 N/m2
1 mm of Hg = 10–3 m of Hg = 10–3 × 13.6 m of water = 10–3 × 13.6 ×
9810 N/m2 = 133.42 N/m2
1 N/mm2 = 106 N/m2
9.81 N
1 Kgf/cm2 = 4 2 = 98.1 × 103 N/m2
10 m
N KN
9. water = 9810 3  9.81 3
m m
10. mercury = 13.6 w
1
11. Specific volume = Density

Special points:
1. Higher temperature, more chances of cavitation.
2. At 100ºC, vapour pressure of water = Atmospheric pressure.
3. Air cavitation is less damaging than vapour cavitation.
dp dp

Bulk modulus: Bulk modulus of elasticity (k) =     d  
dv
 v    
P

1 d P
Compressibility (  ) = 
K dp
P
Note:
  
If density does not change with pressure,   0  then fluid is incompressible
 dp 
Vapour Pressure
Liquid molecules escaping from the free surface to air is known as
vapourisation.
Vapourisation increase with temperature like as for water, vapour pressure
at 0oC is 0.063 m while its value is 10.336 m of column of water when
temperature is 100oC.
Viscosity: It is the measure of resistance of fluid to its deformation. It is
due to the internal frictional forces, which is developed among different
layers of fluid when they are forced (external/internal) to move relative to
each other.
338 Civil Booster (Civil Ki Goli Publication 9255624029)

du.dt
u + du

dy
d
u
du . dt
From above diagram, Shear strain (d) =
dy
d  du
Rate of change of shear strain dt  dy ( velocity gradient)
For Newtonian fluids,
  Rate of change of shear strain
d  du du
   =  dy
dt dy
Where = dynamic viscosity, absolute viscosity or coefficient of viscosity
Dynamic viscosity is the property of a fluid in motion.
NS kg
Unit of  = 2
or or pascal sec (SI)
m m.s
Dyne  sec
(CGS unit) or Poise
cm 2
NS
1 (SI) = 10 poise
m2
dynamic viscosity  
Kinematic viscosity =  v
density  
cm 2
CGS unit = or stoke
sec
m2
SI unit =
sec
10 m /s (SI) = 1 stoke
–4 2

Special point: water = 55 air, air = 15.2 water at 20ºC, water = 1 centipoise
at 20ºC.
Viscosity of liquids is due to cohesion but for gases it is due to molecular
momentum transfer among the gas molecules.
 Fluid Mechanics 339

Effect of temperature and pressure of fluid’s viscosity:

Viscosity
L iqu
id s

es
Ga s

Temperature
For liquids,  does not depends on pressure except at high pressure.
For gases also, gas does not depends on pressure but as  is inversely
proportional to pressure
1
Therefore ,  gas 
Pressure
Newtonian and Non-Newtonian fluids
du
 If  =  dy then Newtonian fluids otherwise non-Newtonian
pic ic pas
te
o tro a st m tic )
0

i x P l s u
p s g
Th Gy pla in
1, B

m
ha tic udo thinn
0 ing e c
n<

B B p Pse ar
1,
n = B  0 Rh
eo e
n >1 , ( Sh
Newtonian
1
<
n

 =1
0,
B=

,n Dilatant
=0
B 1 (Shear Thickening)
0, n>
B= Ideal Fluid
du/dy
n
 du 
  = A    B (General shear equation)
 dy 
 Slope of the curve gives apparent viscosity.
 Pseduo plastic are shear thinning but Dilatants are shear thickening fluids.
 Study of Non-Newtonian fluid is called Rheology.
 Ex.
(a) Thixotropic Ink, ketchup, Enamels etc.
(b) Bingham plastic Sewage, Sludge, Drilling mud, Gel, Toothpaste,
Cream
(c) Rheopectic Gypsum in water & Bentonite slurry.
(d) Pseudo Plastic Paint, Paper, Pulp, Blood, syrup, Polymer, Lipstick,
Nailpaint, Milk.
(e) Dilatant Quick sand, Sugar in water, Butter, rice starch solution.
340 Civil Booster (Civil Ki Goli Publication 9255624029)

Civil Ki Goli:
Tu Bhi Ruhani Pakki Naughty Dramebaj Hai
     
Thixotropic Bingham Rheopectic Pseudo Newtonian Dilatant

Special Points:
1. Wetting property is due to surface tension.
2. Ideal fluids  No-viscosity  No “No slip” condition
3. No slip condition is due to fluid viscosity.
Surface tension and capillary effect: It occurs at the liquid-gas interface
or at the interface of two immiscible liquids while a thin film is apparently
formed due to attraction of liquid in the surface which is similar to tension
force
in stretched membrane known as surface tension measured as
length
N
(Unit )
m
 Surface tension is caused by force of cohesion between liquid molecules.
Net down force is shown Here
Tension Tension
C
B

Liquid Surface
Net Cohesive Force
Work done
Surface tension =
Change in area to work done
water/air 0.073 N/m, At critical point it becomes zero
Then
2
(a) Pressure inside jet P =
d
d


P = Gauge pressure

8
(b) Pressure inside soap bubble P =
d
 Fluid Mechanics 341




d = diameter. 

d = diameter
4
(c) Pressure inside water droplet P =
d

 = Surface tension d

Capillary Effect :
Rise or fall in the surface of liquid when a small diameter (less than 6
mm) tube is inserted into the liquid is called capillary rise or capillary depression
respectively.

 

Water Mercury

 < 90º Cohesion < Adhesion Wetting of surface Concave top surface Rise in capillary tube
 > 90º Adhesion < cohesion Does not wets the Convex top surface Drop in capillary tube
surface

4 cos 
h =
d
 for water glass = 0º, mercury glass = 128º, Kerosene glass = 26º
Special points
1. Capillary effect is due to adhesion and surface tension both
2. Water in soil is able to rise a considerable distance above ground
water table due to capillary action.
3. When a liquid like Hg is spilled on a smooth horizontal surface, It
gathers into droplets because the force of cohesion is more than
force of Adhesion.
4. Force of attraction between molecules of different types is called
adhesion but in molecules of same type, it is called cohesion.
342 Civil Booster (Civil Ki Goli Publication 9255624029)

Pressure and its

Measurement 2
 Normal force exerted by a fluid per unit area is called pressure. It is a
scalar Quantity (it has magnitude but no direction).
 Atmospheric Pressure: Pressure exerted by atmosphere. It is
measured by Barometer. At MSL, atmospheric pressure is 1.01 × 10 5
Pascal or 1 Bar or 10.3 m of Height of water or 76 cm height of mercury.
Special Point: If head of water is ‘h’ meter, then equivalent pressure is
wh and if head of mercury is ‘h’ m then equivalent pressure will be Hgh
Gauge Pressure: It is the pressure w.r.t. atmospheric pressure as datum.
It is measured using Manometer or Bourdon gauge.
 It can be +ve, –ve or zero.
Absolute Pressure: It is the pressure wrt absolute zero or complete
vacuum.It is the actual pressure & measured by Aneroid Barometer.
Patm P local
Pvaccum
P gauge Pabsolute
P local
Patm

Pabs Patm

Absolute Absolute
Vacuum Vacuum
Pabsolute = P atm - P Vacuum Pabsolute
= P + Pa tm gauge

Special point: ‘h’ m of water vacuum means pressure of –hw

Facts about pressure

1. Longer runway is needed at higher altitude due to reduced drag & lift.
2. Nose bleeding starts at higher altitude because of difference in body’s
blood pressure and atmosphere pressure.
 Fluid Mechanics 343

3. Motor capacity reduces at higher altitude because oxygen available

for burning of fuel is less in unit volume of gas.
4. Cooking takes longer time at higher altitudes.
Special Points: As per Pascal’s law, pressure applied at the surface a
confined fluid increases the pressure throughout by the same amount.
1 atm > 1 bar > 1 kgf/cm2
Hydrostatic law : Pressure at a point in a fluid at rest is independents of
shape & cross-section of container in which it is kept. It varies in vertical
direction & remains constant in horizontal direction.

Measurement of fluid pressure

Manometer Mechanical gauges
Based on principle of balancing Mechanical pressure measuring
a column of fluid by the same instruments with a deflecting
or other column needle (used in filling air in tyres)

Simple Differential
manometer manometer
To measure pressure at a point To measure the pressure difference
U-Tube manometer Inverted differential
Single column manometer manometer
Piezometer Micro manometer

No. Type of Manometer Fluid Types Pressure measurement

1. Piezometer Liquid Positive

(Gauge pressure)

2. U-tube Manometer Both liquid & Both positive &

gases Negative Pressure

3. Inclined Tube Gases Both (+ve & -ve)

Manometer ( for very low pressure) (mostly +ve)

4. Differential &
Inverted Differential Both liquid & Pressure difference
gases Between 2 points

5. Bourdon Pressure Both liquid & It measures pressure

gauge gases at a point
344 Civil Booster (Civil Ki Goli Publication 9255624029)

1. Piezometer
 Use for small & +ve pressure
 Very long column of piezometer is required if pressure is large.
 Generally Diameter of Tube > 10 mm

Pat m
h

PA = h
2. U-Tube manometer - Measure absolute pressure at a point.
 For large pressure measurement, gas pressure & –ve pressure
 Simple manometer/U-tube manometer can measure both +ve and –
ve pressure.
Pressure at A = Pressure at H = Patm + G2wy + G1wh
G1
G2 (Sp. gravity)
(Sp. gravity)
B
y
C D A
G h h
A H E Air
F
G
Pipe
PA = G h
Special Points:
 Liquid in U-tube manometer, should have specific gravity more than
the liquid whose pressure is to be measured.
 Manometric liquid should be completely immiscible (oil & water does
not mix) with the liquid whose pressure is to be measured.
 Liquid should have small thermal coefficient & vapour pressure.
 Mercury is used in manometer & barometer because of high density,
Immiscible & low Vapour pressure.
Special Case: To increase the sensitivity, one leg is inclined.
PA = PB = PC = Gw h = Gw (l sin ) measured reading of tube = ‘l’
 Fluid Mechanics 345

A
l h
B C


Concept of differential manometer:
PA – PB = (G2 – G1)h
PA – PB = (head loss) G1
PA  PB G 
=  2  1 h
G1    G1 

A B

G1 h

G2

2. Micromanometer: It measures very small pressure difference or

for measuring the pressure difference with high precision.
PA – PB = (G1 – G2)x

A B
G3
y1
Area=A
y y
y2 G2 G2
Area=a x/2 Original level
x/2 x/2

G1

Special points: Multi fluid manometer is used in measuring pressure in a

pressurised water tank.
• Mechanical gauges are used for measuring high pressure values which
does not requires high precision.
346 Civil Booster (Civil Ki Goli Publication 9255624029)

Hydrostatic–Forces 3
Forces at every point on the plane surface can be added algebrically to
obtain the magnitude of resultant force on the plane surface.
F.L.S

yp yc

Centroid
C.P
C.P – centre of pressure
When the surface is Curved, then at every point, the direction of force
due to stationary fluid is Normal to the surface
IG sin 2 
y P = yC 
Ayc
y P = Centre of pressure from liquid surface
I G = MOI about the centroidal axis
y C = Centroid from the liquid surface
Special point: Magnitude of the resultant force acting on a plane surface
of a completely submerged plate in a homogenous fluid is equal to the
product of pressure at centroid of surface & Area ‘A’ of the surface
F = PCA and this force acts at yP. Also P =  g yc
 As we go deeper, difference of yP & yC will reduce.

Concept of Pressure
Horizontal Plane Vertical Plane Inclined Plane
Surface Surface Surface


x x
xp
x xp
C.G.
Area A C.P.
C.G.
C.G. C.P
 Fluid Mechanics 347

F = Ax F = Ax F = Ax

Ig I g sin 2 
xp  x xp = x  xp = x 
Ax Ax

x & x p are same horizontal plane surface from liquid surface

Special point: In case of vertical surface, when depth of Immersion is

very large. Then centre of pressure = Centre of Gravity. As the depth of
Immersion increases, distance between centre of pressure and centre of
gravity decreases.

Surface C.G. x C.P. x p

h h 2h
1.
b
2 3

h 2h 3h
2.
3 4
b
b
h h
h
3 2

r 5
3. r r
4
4r 3r
4. r
3 16

h 3h 5h
5.
b 5 7
b
2h 4h
h
5 7
a
h  b  2a   h   a  3b  h
6.     
 a  b  3   a  2b  2
b
348 Civil Booster (Civil Ki Goli Publication 9255624029)

Hydrostatic forces on the curved surface:

Horizontal Force (FH): It is the resultant hydrostatic force ‘fx’ of curved
surface, may be calculated by projecting the surface upon a vertical plane
& multiplying the projected area by the pressure at its own centre of area.
Vertical Force (FV): It is the weight of the liquid contained in the zone
bounded by two verticals drawn from the two ends of the curved surface,
the curved surface & the free surface (which is applying pressure on the
curved surface)

Resultant force (F): F= FV2  FH2

Fy FV
tan  = =
Fx FH
  Angle b/w line of action & Horizontal axis
Hydrostatic Paradox:

H H
F F F

The hydrostatic force 'F' is the same on the bottom of all 3 containers if the
bottom cross-section area & the fluid are the same even though the weight
of liquid above are quite different.It is known as hydrostatic paradox.
For the same force, it is shows that the pressure at a certain horizontal level
in a static fluid is proportional to the vertical distance to the surface of fluid.
Clear Your Doubt

 Centre of Gravity-It is defined as a point through which the whole

weight of the body is assumed to be act
 Centre of mass-It is the point in a body at which the entire mass may
be assumed to be concentrated.
 Centre of Percussion-It is the point on an extended massive object
attached to a pivot where a perpendicular impact will produce no reactive
shock at the pivot.
 Fluid Mechanics 349

Buoyancy and
Floatation 4
Archimedes Principle: When a body is wholly or partially submerged in a
liquid, then the vertical upward force acting on the body (known as Buoyant
force) is equal to the weight of the liquid displaced by the immersed part of
the body.
Buoyant force = Net upward force = weight of liquid displaced

Special Points: Point of application of buoyant force is the C.G. of the

displaced liquid & it is called centre of buoyancy.
 Buoyant force is independent of distance of body from free surface of
liquid and also the density of solid body.
 When a ship moving on sea water and it enters in a river, it is ex-
pected to sink a little

Principle of Floatation: A body will float in a liquid, if weight of body is

equal to weight of liquid displaced by its immersed part.

Gm

G
H
B h
h = GmH

B = Centre of buoyancy at a distance of h/2 from base of cylinder.

G = Centre of gravity, at a distance of H/2 from base of cylinder,
Gm = Specific gravity of material wrt liquid, which should be < 1
Linear Stability: When a small linear displacement of body sets up a
restoring force, then the body is in linear stability
350 Civil Booster (Civil Ki Goli Publication 9255624029)

A B
a b

C D
c d
Submerged body Floating body
It remains in neutral It remains in stable equilibrium
equilibrium against linear against vertical displacement
displacement & in neutral equilibrium
against horizontal displacement
Rotational Stability: When a small angular displacement sets up a
restoring couple, then stability is known as rotational stability.

FB = Buoyant Force

B Couple (Restoring)

G G

W
Submerged body Floating body
Stable equilibrium G below B M above G
BM > BG
GM = MB – BG = +Ve
Unstable equilibrium G above B M below G
BM < BG
GM = MB – BG = –Ve
Neutral equilibrium G & B coincide M & G coincide
GM = 0
Metacentre (M) is the point of intersection of lines of action of buoyant
force before and after rotation.
GM = metacentric height
GM = BM – BG
 Fluid Mechanics 351

G
B

I
Where BM =
V
I = MOI of top view of the immersed part of the body about longitudinal
axis.
 Larger the metacentric height, greater is stability & comfort will decrease.
Time period of oscillation: If a floating body oscillates, then its time
period of transverse oscillation wrt metacentre is given by

I MK 2G K G2
T = 2 W.GM  2 = 2
W  GM  GM.g
Where KG Radius of Gyration about centre, W = weight of floating body.
 Larger the time period, more will be the comfort of passenger,
 For cargo / merchant ships, GM is 0.5-1m, comfort & stability both considered.
 For passenger ship, GM is less(0.5-1m), so more comfortable.
 For battle ship, GM is 1-1.5m, Stability is prime consideration.

Movements of a ship:
If a ship is safe in rolling, it must be safe in pitching.

Z
Yawning y (Longitudinal axis)

Rolling Pitching
X
Transverse
axis
352 Civil Booster (Civil Ki Goli Publication 9255624029)

Liquid in Relative
Equilibrium 5
When a liquid is contained in a moving container, then it behaves like as a
rigid body (liquid is moving but not flowing)
From Newton’s law of motion

P 
= g ax
x
P 
= ay
y g

P 
= g (az  g )  Euler’s equation
z
 p dz 
 P    dx.dy
 z 2 
dx

P
dz z
W
x y

dy
 p dz 
 P    dx.dy
 z 2 
Adding above equations,
 P P ˆ P ˆ  
  iˆ  j k = (ax iˆ  a y ˆj  (az  g )kˆ)
  x y  z  g
Following are the various conditions:
1. When fluid at rest ax = ay = az = 0
 Fluid Mechanics 353

P P P
then x  y  0 ,  g  P  g z
z
2. When fluid moves in downward direction with constant acceleration
(-az). Then
p p p
ax  a y  0   0,  ( g  az )
x y z

  az 
P  Pa   g  a z  h   h 1  
g  g
3. When fluid moves in upward direction with constant acceleration
(+az). Then
p p p
ax = ay = 0    0,  ( g  az )
x y z

 az 
P – Pa =    g  a z  h   h 1  
 g 
4. With constant aceleration ax in x-direction
P A =  gh
ax
Slope (tan  = g
eff

 a x .x 
Z = H  g   Equation of free surface
 eff 

Constant pressure
at free surface

h z
H
z A
x
x
Rotation in Cylindrical Container

P V 2
= ...(a)
r r
P
= –(az + g) ...(b)
z
Combining (a) and (b)
354 Civil Booster (Civil Ki Goli Publication 9255624029)

V 2
dP = dr  (az  g )dz
r
Free Vortex motion : In it, angular momentum remains conserved as
external torque is zero, so mvr = constant. In it, Bernoulli's equation can
be applied.
C
So V 
r
 So as radius increases, velocity decreases, pressure Increases. Ex:
whirlpool in rivers, whirling mass of liquid in wash basin.
 A free Vortex motion is that in which the fluid may rotate without Any
external force applied on it.
Forced Vortex motion : In it, fluid is rotated about a vertical axis at constant
speed in such a way that every particle has the same angular velocity.
 The surface profile of forced Vortex flow is paraboloid.
 Ex. Rotational Vortex is forced Vortex motion.
Rotating Cylinder and flow inside Centrifugal pump.
 A force Vortex motion is that in which the fluid mass is made to rotate by
means of some external source of power.
R

2 R 2  w2R 2 
h=
2g P  gh  g  
H  2g 
V = r w 2 R 2
P
 2

Hence as radius increases, velocity increases, pressure decreases. Ex:

Flow inside centrifugal pump.
Special point: Rankine Vortex motion is a combination of free and forced
vortex motion. For no spilling case, rise above original water level = Fall
below original water level.
 Amount of water spilled out = Original volume – Remaining volume
 Remaining volume = Volume of cylinder – Volume of shaded paraboloid.
 Volume of cylinder = R2H

1  2R 2 
 Volume of paraboloid = ( R 2 )  
2  2g
 eff 
 Fluid Mechanics 355

Fluid Kinematics 6
Fluid Kinematics: It deals with the motion of the fluids without necessarily
considering the forces & moments which cause the motion.
Generally these two approach are used:
 In Lagrangian concept, study of motion of single fluid particle
 In Eulerian concept, study of motion of fluid through a particular section
or a control volume
Special point: In F.M., We generally follow Eulerian concept, because it
is difficult to keep the track of a single fluid particle.

Types of fluid:
1. Steady and Unsteady Flow: At any given location, the flow and fluid
properties do not change with time, then its steady flow otherwise
unsteady.

V P 
= 0,  0,  0  Steady flow
t t t
2. Uniform and Non-Uniform Flow: A flow is said to be uniform flow
in which velocity & flow both in magnitude and direction do not change
along the direction of flow for given instant of time.
3. Rotational and Irrotational Flow: When fluid particles rotate about
their mass centre during movement. Flow is said to be rotational
otherwise irrotational.
 Flow above the drain having a wash basin is a free vortex motion
(Irrotational flow).
 Rotational Flow  Forced Vortex, Flow inside boundary layer.
 Irrotational Flow  Free Vortex, Flow outside boundary layer.
 In a straight tube of uniform diameter & uniform roughness, the
356 Civil Booster (Civil Ki Goli Publication 9255624029)

flow properties does not vary across the length of the pipe. Hence,
Uniform flow.
4. Laminar and Turbulent Flow: Turbulent flow particles have the
random & erratic movement, intermixing in the adjacent layers. Which
causes continuous momentum transfer between different layers.
A water supply pipe carries water at high speed leading to rapid
mixing which causes highly turbulent conditions.
 In laminar flow, the particles moves in layers sliding smoothly over
the adjacent layers
 Flow of blood in veins and arteries occurs as a viscous flow. Hence,
Laminar flow.
5. One, two or three Dimensional Flow: If flow parameters varies in
one dimension wrt space only, then its 1 D otherwise its 2 or 3 dimension
respectively.
V = V(x, t)  one dimensional
V = V(x, y, t)  two dimensional
V = V(x, y, z, t)  three dimensional
6. Compressible and Incompressible Flow: In compressible flow
density of fluid changes from time to time while in Incompressible
flow it remains constant.
Flow lines

Stream lines Streak line Path line

 Stream Line: There are a set of concentric circle with origin at centre.
 Stream lines neither touch nor cross each other. Line tangent to it give
direction of Instantaneous velocity.
 Tracing of motion of different fluid particle.
dx dy dz
 = Equation of stream line
u v w
 Streak Line: It is line traced by series of fluid particles passing through
a fixed point. It is formed by continuous introduction of dye or smoke
from a point in the flow.
 Path Lines: It is actual path traced by a fluid particle over a period of time. It is
based on lagrangian concept. Two path lines can intersect each other.
Continuity Equation: It is based on principle of conservation of mass.
Fluid mass can neither be created nor can be destroyed hence mass of fluid
 Fluid Mechanics 357

entering a fixed region should be equal to mass of fluid leaving that fixed
region in a particular time.
Various forms of continuity Equation:
Cartesian co-ordinate System:
(i) Steady Flow in 1-D,  AV = Constant
 1A 1V 1 =  2A 2V 2
(ii) Steady Incompressible in 1-D, A 1 V 1 = A2 V 2

Acceleration of fluid:
 ˆ ˆ ˆ
V = u ( x, y, z, t )i  v( x, y, z , t ) j  w( x, y, z , t ) k
u u u u
ax = u x  v y  w z 
t

v v v v
ay = u x  v y  w z 
t

w w w w
az = u x  v y  w z 
t
 Total Acceleration = Convective acceleration with respect to space +
local acceleration with respect to time.
Convective Temporal
Type of flow
Acceleration Acceleration
Steady & uniform 0 0
Steady & non-uniform Exists 0
Unsteady & uniform 0 Exists
Unsteady & non-uniform Exists Exists
Acceleration on a stream line

Tangential Acceleration Vn (s,n,t) Vs (s,n,t) Normal Acceleration

It occurs due to change in It occurs due to the
magnitude of velocity. If change in the
spacing b/w stream line direction of fluid
changes, then tangential acceleration moving on a curved
n
exists path
s
358 Civil Booster (Civil Ki Goli Publication 9255624029)

vs vs vn vn

as  Vs + an  Vs +
s t s t
convective local tangential convective local
tangential acceleration normal normal
acceleration acceleration acceleration

No Acceleration Tangential Convective

Acceleration

Both Normal and

Tangential Convective
Normal Convective Accelation
Acceleration

Angular Velocity: It is the average of rotation rate of two initially

perpendicular lines that intersect at that point.

 = x iˆ   y ˆj  z kˆ
1  w v 
x =   
2  y z 
1  u w 
y =   
2  z x 
1  v u 
z =   
2  x y 

 iˆ ˆj kˆ 
 
1    
 =
2  x y z 
 
u v w
 Fluid Mechanics 359

Velocity Potential or Potential Function (): It is the scalar function

of space & time in such a way that its negative derivative wrt any direction
gives velocity of flow in that direction. It must satisfy Laplace eq.
In Cartesian co-ordinate System,  =f(x, y, z, t)
  
= u,  v, w
x y z
Special Points: If angular velocity is zero, flow will be irrotational.
Vorticity () = Twice of Angular Velocity
Circulation = Vorticity × Area of loop
Circulation () = line integral of tangential component of velocity
vector along a closed curve.
 Velocity potential exists only for ideal & irrotational flow.
 Equipotential line is the line joining points having same potential function.
 Velocity of flow is in direction of decreasing potential function.
Stream Function (): It is a scalar function of space & time in such a
way that its partial derivative wrt any direction gives the velocity component
at right angles (in anti clock wise direction) to this direction.
Cartesian co-ordinate system
 
= v,  u
x y
 1   2  Discharge per unit width.

Special Points:
(a) If two points lie on same stream line, then  will be constant.
(b) If Stream function () satisfies the Laplace equation, then flow is
irrotational otherwise rotational.

 2  2 
Laplace equation ,  = 0
x2 y2
Cauchy-Riemann Equation: For incompressible irrotational flow

    

u =  , v= 
x y  y x
360 Civil Booster (Civil Ki Goli Publication 9255624029)

Fluid Dynamics 7
• It is the study of motion of fluid along with the forces causing the motion.
Dynamic behaviour of fluid flow is analysed by Newton's 2nd law of
motion F = ma
(a) Newton’s equation of motion
      
Fg  FP  FV  Ft  Fc  F  ma
(b) Reynold’s equation of motion
    
Fg  FP  FV  Ft  ma
(c) Navier-stokes equation of motion - Use for viscous flow.
   
Fg  FP  FV  ma
(d) Euler’s equation of motion
  
Fg  FP  ma
where, Fg = Gravity force
FV = Viscous force
Ft = Turbulence force
FP = Pressure force
Fc = Compressibility force
F = Surface tension force
Special Points:
 Energy equation can be used to find the pressure at a point in a pipeline
using Bernoulli’s eq.
 Continuity equation is used to find out the flow rate/velocity betweeen
two sections of tapering pipes.
 Euler equation is based on momentum conservation while Bernoulli is
based on energy conservation.
 Fluid Mechanics 361

 Impulse momentum principle is used to find out the force on a moving

vane.
 Concept of moment of momentum (Angular momentum principle) is
used in lawn sprinkler problems.
Bernoulli’s Equation: It is the integration of Euler’s equation of motion
along a stream line under steady incompressible flow conditions.It represent
total energy per unit weight.
Assumptions:
(i) Along Stream line
(ii) Effect of friction is negligible (Ideal flow)
(iii) Steady, Incompressible & ir-rotational .
 p   V2 
Total head H =  g    2g   Z = Constant
   
 p   v2 
Piezometric Head =   z  , Dynamic Head =  
 g   2g 

 p   v2 
Stagnation Head =  g    2g 
   
Static pressure
head Dynamic Hydrostatic pressure
pressure head head

Stagnation pressure head

Piezometric pressure head

P V2
+ + gz = Constant
 2
Pressure Energy Kinetic Energy Potential Energy
+ +
Mass Mass Mass
Special Points: When normal acceleration is zero , (when particles move
on a straight line), then the piezometric head is a constant.
362 Civil Booster (Civil Ki Goli Publication 9255624029)

 Free vortex equation is based on “Principal of Conservation of angular

momentum”.
Kinetic Energy Correction Factor ()
Actual K.E.
 =
K.E. Calculated from Average Velocity
A v dA
3

v dA
 = 3
Vavg  
AVavg A A
Momentum Correction Factor ()
Actual linear momentum/sec
 =
Linear momentum/sec calculated from Average Velocity

 v dA
2

v dA
 = A
2
Vavg  
AV avg
A A
 
Laminar flow between circular pipes 2 4/3
Laminar flow b/w parallel plates 1.543 1.2
Turbulent flow in pipes 1.03 - 1.06 1.015

Applications of Energy Equation:

1. Orificemeter - An orifice is called a large orifice if water head is
less than the five times the diameter of the orifice.
(1) (2)
Plate h

Flow a0 a2 stream lines

a1
(1) (2)
Vena-contracta
 It is cheaper & less accurate instrument, measures discharge but
has more head losses. Hence cd = 0.61 – 0.65.
 Region of minimum flow area is called Vena-contracta, in it stream
lines are assumed to be nearly parallel.
 Fluid Mechanics 363

a2 Area of Vena contractra

cc = 
a0 Area of opening
a1a0
 Qactual = cd 2 gh
a12  a02
Special Point: If the discharge is changed, then the position of Vena-
contracta will also change & then stream lineas will not be parallel at sec
(2)-(2).
2. Venturimeter
 To find discharge from a large diameter pipe
 Accuracy is quite good It has much smaller head loss.

(2) h
P1 P2
 
22° 5°–7°
z1
z2 (2) Throat
(1)
Datum
 Angle of convergence = 20° - 30° (Generally 22°)
 Angle of divergence = 5° - 7°
1 1
 d   to  D , commonly d = D/2, where d = dia of throat
3 2
D = dia of pipe
 The divergent cone angle in a ventruimeter is generally kept lesser
than the convergent cone angle to avoid separation of flow.
 Principle : Reduction in Area leads to increase in velocity & decrease
in pressure, this pressure reduction is noted & used in Bernoulli to
calculate discharge.
V22 V12
Piezometric head difference h = 
2g 2g
a1a2
 Qactual = cd 2 gh
a12  a22
Where a1, a2 cross-sectional areas at section 1 and 2
364 Civil Booster (Civil Ki Goli Publication 9255624029)

a1
a2 = area ratio
cd  discharge coefficient
a1a2 2 g
, because this depends only on dimensions of venturimeter,,
a12  a22
it is called venturi-constant.
h  hL Q actual
 cd =  0.98 =
h Q theoretical

Special Point: If in venturimeter, the pipe is not contracted such that

cc = 1 , then it is termed as Nozzle meter and also used for calculating
discharge.
3. Pitot tube:
 Used to measure fluid flow velocity, water speed of a boat
 To measure liquid, air & gas flow velocity in certain industrial
applications.
 Used to measure ship’s speed relative to be water. They are used on
both surface ships & submarine.

h
pc pa  pA VA2 
   2g 
   

C
Zc A
ZA
Datum

 It measures the velocity of fluid at any point by measuring stagnation

pressure.
VA2
 hmeasured  , VA = 2gh
2g

 VA actual = C V 2gh , CV = 0.98 (coefficient of velocity)

 At stagnation point, velocity is already zero. There is no need to measure
velocity at stagnation point.
 Fluid Mechanics 365

Special Points: Anemometer measures gas and air velocity.

Preston Tube is used for Boundary shear stress measurement.
Type of Accuracy Cost Loss of Typical
flow meter total head value of Cd
Venturimeter High High Low 0.95 to 0.98
Orificemeter Low Low High 0.60 to 0.65
Flow nozzle Intermediate b/w 0.7 to 0.8
venturimeter & Orificemeter
4. Pitot Static tube (Prandtl tube) - measure Dynamic pressure
 Pitot Tube is based on principle of Conversion of Kinetic Head into
pressure Head. The point at which velocity reduces to zero is called
stagnation Point.
 It measures the piezometric head at the same point where velocity is
to be measured.
 Velocity head is indicated by the difference in liquid level between
the pitot tube and the piezometer.

Rise due to
stagnation

h
Rise only due to
pressure only.
PA velocity has no.
component
2
VA
 PA
2g
Prandtl tube
VA  C V 2gh, CV = 0.99

 It can also used on rough boundaries.

 Velocity head is found out from difference of toal head and piezometric
head.
5. Elbow meter or Bend meter Measures discharge
  Rotameter is used to measure discharge but current meter is used to
measure velocity in open channel.
  Hot Wire Anemometer:It is used for measurement of Instantaneous
velocity & temperature at a point in flow.
Special Point :The pirani gauge is a robust thermal conductivity gauge used
for measurement of pressure in vaccum system.
366 Civil Booster (Civil Ki Goli Publication 9255624029)

Momentum Equation
and Application 8
Rate of change of linear momentum in any direction of a body wrt a fixed
frame of reference is equal to external forces acting on the body in that
direction.

 Net rate of flow 

Fx external   of linear 
   
 on control  =  momentum out 
 volume   
   of control volume 
in x-direction 
Rate of change of Angular momentum in any direction of a body wrt a
fixed frame of reference is equal to torque applied on the body in that direction.

M z external 
on control   Net torque oncontrol 
  =  
 volume   volumein that direction 
Special case:
Q1




Q2
Q(1  cos )
Q1 =
2
Q(1  cos )
Q2 =
2
 Fluid Mechanics 367

Force acting on a pipe bend:

y

V2 x
P2, A2,

P1A 1
V1
1 W

In y-direction: Fy = Q(v2 sin 2 – v1 sin 1)

P1A1 sin 1 – P2A2 sin 2 + Ry - W = Q(- v2 sin 2 – v1 sin 1)
In x-direction:
Fx = Q(v2 cos 2 – v1 cos 1)
P1A1 cos 1 – P2A2 cos 2 - Rx = Q(v2 cos 2 – v1 cos 1)
Resultant force R = R 2x  R 2y
Magnus effect: It is an observable phenomenon in a real fluid flow, in
which local circulation can be produced through surface drag by rotating the
cylinder itself. The sudden deviation of a ball which has been chopped (as in
Table Tennis or volley ball ) or sliced (as in lawn tennis) by player, from its
normal trajectory is simple illustration of the Magnus effect.
Venna-Contracta - point in fluid stream where diameter of stream is least
and fluid velocity is maximum.
368 Civil Booster (Civil Ki Goli Publication 9255624029)

Weir and Notches 9

Weir Notch
1. Constructed in an 1. Used for measuring
open channel to the discharge through
measure its discharge. a small channel or a tank.
2. It is bigger in size 2. It is smaller in size
3. It is concrete or masonary 3. It is generally metallic plate
structure. 4. Useful in model analysis

H Nappe

Crest or sill

Weir or notch
Crest/Sill: The bottom edge of a notch/Top of a weir over which water
flows is known as crest/sill.
Classification of Weirs Based on
Shape of Effect of sides on Shape of crest Nature of
opening emerging nappe discharge

Rectangular With end Sharp edge crested

Ordinary weir
Trapezoidal contraction Narrow crested
Submerged weir
Triangular Without end Broad crested
Cipolletti contraction Ogee-shaped
1. Rectangular sharp-crested Suppressed weir:
 Suppressed – without end contraction.
 Fluid Mechanics 369
Nappe

H
Crest
H Outside
air supplied

2
 Qactual = cd L 2g H3/ 2 , cd  0.62
3
H  depth of water above crest level
 If velocity of approach (Va) is also considered , then
Q Va2
,
Va = (H + H')L ah  , Q = 2 c d 2 g L [ (H  ha )3/ 2  ha 3/ 2 ]
2g 3
 Effect of end contraction, if not suppressed L is replaced by Left
1 2 3 4

Leff = L – 0.1 nH
n = Number of end contractions (It is 4th in the above diagram)
2 3/ 2
Q = cd 2 g L eff H
3
2. Trapezoidal Notch or weir:
2 3/2 8  5/ 2
Q = cd1 2g L H + cd2 2g tan H
3 15 

 H 
2 2
L

3. Flow over V-Notch or triangular weir:

 End contraction is not consider in this case.
h

H dh


8 
 Q= cd 2 g tan H5 / 2 cd = 0.52
15 
370 Civil Booster (Civil Ki Goli Publication 9255624029)

 If we consider velocity of approach then

8 
Q= cd 2 g tan [(H + ha )5 / 2  ha 5 / 2 ]
15 
Advantages
(a) cd nearly constant with depth.
(b) Only one dimension is to be measured, therefore more accurate
(c) Even for small discharge, high head is obtained. Therefore, no effect.
or viscosity and surface tension.
(d)A triangular notch gives much more accurate results in low discharge
conditions as compared to conventional rectangular notch.
4. Cipolletti-Weir:It is a trapezoidal weir whose slopes are adjusted in
such a way that:
• Reduction in discharge due to end contraction in rectangular weir =
Increase in discharge due to triangular portion.(Side slope 1 H: 4V)
 4V
 H 2
2 1H
L

 1
tan = ,   28º
2 4
2
Q = cd 2g L H3/2
3
c d = 0.63
5. Broad Crested weir
 Consider a Nappe in such a way that stream lines become straight
& pressure variation become hydrostatic over the weir.
 Q = cd Lh 2g(H - h)
 In this, flow adjusts itself to give max. discharge at available head H.
H
h

 For maximum , discharge

2
h = H Qmax = 1.7 cd LH3/2
3
c d = 0.85 – 1,
 Fluid Mechanics 371

h = critical depth as discharge is maximum

 If Velocity of approach is also considered
Q = 1.7 cd L [(H + ha)3/2 – ha3/2]

Type of Weir Discharge Coefficient

Submerged Broad Crested Weir 0.83 – 0.85
Free Broad Crested Weir 0.85 – 1.0
6. Proportional weir:
 Q  H
 a
Q = K H  
 3
c d = 0.6 – 0.65, K = cd L 2ga
 If there are fluctuations in discharge, then there will be less fluctuations
in ‘H’ as compared to rectangular & triangular weirs.
7. Flow through orifice
 Orifice is small opening in tank
 Q = cd a 2 gh
h

a x ac = Area of
vena
contracta
y
ac
cc = , c = cc × cv
a d

8. Flow through mouth piece

 Mouth piece is short length of tube with length < (2 –3) diameter of
orifice.
 Q = cd a 2 gh , cd = 0.82

h
Area 'a'
372 Civil Booster (Civil Ki Goli Publication 9255624029)

9. Ogee spillway
 Profile of the crest is made in such a way that it matches with the
shape of water profile over sharp crested weir to avoid development
of –ve pressure below nappe (or Adhering Nappe).

0.115 H

2
Q= cd L 2 g H3/ 2 , cd = 0.62
3
10. Borda’s weir
Q = cd a 2 gh

h
Area = a

11. Submerged weir

 When downstream water level is above the crest of the weir, then it
is said to be submerged
 Sharp crested weir is more susceptible to submergence than a broad
crested weir
2
 Q= Cd L 2g (H – H')3/2 + cd2 LH' 2 g (H  H')
3 1
H – H
H
H

12. Discharge through sluice gate

v = 2gh

Q = cd a L 2gh
h = Water depth from ground
L = Inside length
 Fluid Mechanics 373

v v
h
a

Free flow Drowned flow

Effect on discharge due to error in head measurement

For infinite small errors in head measurement
Q = KH n
dQ = Kn Hn–1dH
dQ  dH 
 100 = n   100 
Q  H 
% error in discharge = n × % error in head measurement
here n = 1 Proportional weir, sutro weir
n = 1.5 rectangular weir,
n = 2.5 triangular weir.

Actual Velocity
Special Point - Cv 
Theoretical Velocity
Actual velocity is always less than theoretical velocity because in ac-
tual fluid are real & in real fluid head losses are takes place, hence the
value of Cv is always less than 1.
For Pitot tube Cv = 0.97 - 0.99, sharp edge orifice Cv = 0.98
For Orifice meter Cd = 0.64 - 0.67, sharp edge orifice Cd = 0.611
For Venturimeter Cd = 0.94 - 0.98, Cc = 1
The Relationship between Cd, Cv, Cc for orifice is given by Cd = CV × Cc

CIVIL Ki Goli :- Cd  Cc  Cv

CV:- Curriculum Vitae

(Resume for job)
374 Civil Booster (Civil Ki Goli Publication 9255624029)

Laminar Flow 10
In Laminar flow fluid particles move along the straight parallel paths in layers.
It occurs at a very low velocity, & Viscous force predominates the inertial
forces. (Couette flow: When one plate is moving and other is at rest)
Nature of flow according to Reynold's number (Re)

Laminar Transition Turbulent

Flow in pipe Re < 2000 2000 < Re < 4000 Re > 4000
Flow between Re < 1000 1000 < Re < 2000 Re > 2000
parallel plate
Flow in open channel Re < 500 500 < Re < 2000 Re > 2000
Flow through soil Re < 1 1 < Re < 2 Re > 2

Flow through flat plate in circular pipe (steady uniform flow)

dp d  r dp
1.  , 
dx dy 2 dx
x is the direction of flow
y is perpendicular to x

 y
P.dA x
 p 
 P+ dx  dA
 x 
dx

 r2 
2. V = Vmax 1  2 
 R 
1  dp  2
3. Vmax =  R
4  dx 
 Fluid Mechanics 375

r  dp  r
Variation of shear
4. =    stress  linear
2  dx 
2 Vmax  P2 - P1  R
5. max = = 
R  L 2

Shear Power input Velocity

stress per unit variation
variation volume
   dp  4
6. Hagen Poiseuille Formula , Q =  D
  dx 
Q 1  p  2
7. Vavg =     D , Vmax = 2Vavg
 R 32  x 
2

R
8. V = Vavg at r = = 0.707 R
2
flV 2 (4 f )lV 2 64
9. hL =  , ( f = friction factor = , f' = coefficient of
2 gD 2 gD Rc
friction)
32 VL 128 QL
10. hL = =
D 2
D 4

Flow between two fixed parallel plates

1  -dp  B dy
 (By  y )
2
1. u =  y
2  dx  dx x

1  dp  3
2. Q =  B
12  dx 

du 1  dp 
3.      (B  2y) 
dy 2  dx 
376 Civil Booster (Civil Ki Goli Publication 9255624029)

Q 1  dp  2
4. Vavg =   B
A 12  dx 
1  dp   3
5. Vmax =    , Vmax = Vavg
8  dx  2
B 3B
6. V = Vavg at y = 
2 6
3B
6
3B
6

12Vavg.L
7. hL =

For couette flow (one plate moving other at rest)
V

B y

Velocity Shear
distribution stress variation
Vy 1  dp 
u =    (By – y2)
B 2  dx 
V  dp  B 
 =    y 
B  dx  2 
Entrance length: Entrance length in a pipe is the length where boundary
layer increases and flow becomes fully developed.
 For Laminar Flow L = 0.07 Re D
 For Turbulent FlowLe = 50 D
Exam Points:
Hele Show flow: Laminar flow between parallel plates
Stoke’s Law: Settling of fine particles.
Hagen Poiseuille flow: Laminar flow in Tubes/pipes.
Measurement of viscosity - (a) Rotating cylinder method, (b) Capillary
tube method, (c) Orifice type viscometer (Eagler viscometer or Bolt red
wood)
 Fluid Mechanics 377

Turbulent Flow 11
Turbulent flow results from the instability of laminar flow & due to continuous
mixing among different layers. Then momentum transfer occurs which gives
rise to addition shear called Turbulent shear.
 For Turbulent flow, the velocity profile will be flatter than that in Laminar
flow.

More Reynold's
Number (Turbulent
flow)

Less Reynold's Number

(laminar flow)

 Shear stress at boundary (w) is much less in Laminar flow as compared

to turbulent flow because velocity gradient near the boundary is large in
turbulent flow (due to additional shear stress).

Shear stress in turbulent flow

For Turbulent flow

(1) Boussineq's (2) Reynold's (3) Prandtl
 =  du  = l2 du
2
 =  (u v )
dy dy
 = eddy viscosity or u, v are fluctuating l = mixing length, it is
turbulent mixing coefficient components of velocity the distance covered by
(which depends on flow property) in x & y direction a particle before striking
 respectively any other particle
= eddy kinematic viscosity (or before its momentum

is changed)
 Total = Laminar+  Turbulent
378 Civil Booster (Civil Ki Goli Publication 9255624029)

Hydrodynamically smooth and rough boundary

 If average height of roughness (K) is more than laminar sub layer, it is
hydrodynamically rough boundary.
 In hydrodynamically smooth boundaries, average height of roughness
(K) is much less than the laminar sub layer ().
v
 = 11.6 u
*

w
u * = Shear velocity = ; w= Boundary shear stress

vKinematic Viscosity
Velocity distribution for turbulent flow in smooth as well as Rough
pipe
y
u  uavg  y
1.  5.75log10    3.75
u* R R y

Here, y is measured from the boundary surface, not from centre.

U* f  U max 
2. As U = ,    1  1.33 f
avg . 8  U avg 
 Umax  Uavg 
3. U  Umax at y  R, So 
u*
  3.75
 
1
4. th power law of velocity distribution for smooth pipes
7
1
u  y 7
=   (As per Nikuradse)
umax R
dp  R  R  P  R
5. In pipe flow, w   ,    ( h L )
dx  2  2  L  2L

w R R  flV 2  D  flV 2 
= 2L ( ghL )  g  g 
 2L  2 gD  4L  2 gD 

w f 2 w f
= V ,  u*  Vavg
 8  8
 Fluid Mechanics 379

Friction factor ‘f’ for Turbulent flow (Artificial Roughness)

1. For smooth pipes
0.316
(a) f  , 4000  R e  105
(R e )1/ 4

64
Special Point: For laminar flow f  R circular pipe
e

0.221
(b) f  0.0032  (R )0.237 , 5 10  Re  4 10
4 7

(c)
1
f
 
 2 log10 Re f  0.8, 5×104 < Re < 4 × 107 (Nikuradse)

2. For Smooth Commercial pipes

1 R  R/K 
 2log10    1.74  2log10 1  18.7 
f K
   Re f 
3. For Rough pipes
1 R
 2log10    1.74 , R  Radius of pipe
f K
R
   Relative Smoothness
K

Dependence of friction factor

Transition flow Laminar flow Turbulent flow
f = g(Re, k) f = g(Re) Smooth pipe Rough pipe
Reynold's no & Only on Reynold's f = g(Re) f = g(k)
average height of Number Only on Only on
Roughness Reynold's average height of
number roughness

Laminar flow Turbulent flow

In it fluid layer moves in straight line. Fluid flo w in zig-zag manner
Occurs when fluid flow with low Occurs when fluid flow with high
velocity & in small diameter pipes velocity & in large diameter pipes
Shear stress depend only on viscosity Shear stress depends on density
Fluid flo w is very orderly There is mixing of different layers
380 Civil Booster (Civil Ki Goli Publication 9255624029)

Boundary Layer
Thickness 12
It is the region in the immediate vicinity of the boundary surface in which the
velocity of flowing fluid increases gradually from zero at the boundary surface
to the velocity of the main stream.
 Flow outside the boundary layer has Ir-rotational characteristic but that
within the boundary layer is rotational characteristic.
 It was developed by Prandtl in 1904
 Valid for infinitely large medium of real fluid & not for ideal fluid.

V0
y Laminar
sub layer
11.6v
Flat x   u
Plate *

Laminar Turbulent
region region
Leading edge Transition
(stagnation point) region

Essential boundary conditions:

1. x = 0,  = 0 2. y = , u = 0.99V0
du
3. y = 0, u = 0 4. y =  , dy  0

du d 2u
Desirable boundary conditions: At y  ,  0, 2  0
dy dy
Salient points regarding boundary layer:
1. As the roughness of plate increases, length of laminar region decreases
2. With increase in velocity, boundary layer thickness decreases but with
increase in viscosity boundary layer thickness increases.
3. +ve pressure gradient increases boundary layer thickness but reduces
the length of laminar region.
 Fluid Mechanics 381

4. Rex = 5 × 105 is called critical Reynold’s number.

If Re < Rex then laminar boundary layer region in case of flat plates
Re > Rex then Turbulent boundary layer region, in case of flat plates
5. On a smooth plate, in turbulent layer region, there is very thin layer
adjacent to the boundary where the flow remains laminar. This region
is called laminar sub layer.
Velocity profiles

Laminar region Laminar sublayer Turbulent Region

2 1/ 7
u  y u  u*  y  u  y
     
u0    u0  v  u0   
Boundary layer Thickness (): It is the distance form the boundary
surface in which velocity reaches 99% of the free stream velocity.
At y = , V= 0.99 V0
Displacement Thickness () : It is the distance by which boundary
should be shifted in order to compensate for the reduction in mass flow rate
on account of boundary layer formation.

 V
* =  1  V  dy
0 0 

V 0 = Free stream velocity

V = Velocity at any distance y from the boundary
Reduction in mass flow rate per unit width = V0
Momentum Thickness (): It is the distance by which boundary should
be shifted in order to compensate for the loss of momentum due to formation
of Boundary layer.

V V
 = V
0
1   dy
0  V0 

Energy thickness (  E )
It is the distance by which boundary should be shifted in order to compensate
loss of energy due to boundary layer formation.

V  V2 
E   1  2  dy
V
0 0 
V0 
382 Civil Booster (Civil Ki Goli Publication 9255624029)

V03
Loss of energy due to boundary layer formation = E
2
Special Points:

*
 Shape factor =

 * >E > 
FRICTION COEFFICIENT

Local skin Average drag coefficeint

friction coefficient
0 Wall shear stress (Drag Force on plate)
C fx   C favg  Avg. wall shear stress
 V02  Dynamic pressure
, V02 =
   area of plate Dynamic pressure
 2  2

Special Point :
If Boundary layer is laminar through out

II I
F Drag force on I half  1  2,  1
L L Drag force on II half
2 2

Blassius experiments results/when velocity profile is not given

Laminar Turbulent Transition region

R0< 5 × 105 5 × 105 < Re < 107 107 < Re < 109
5x 0.376 x 0.22 x
 R ex (R ex )1/ 5 (R ex )1/ 6

0.664 0.059 0.37

C fx (R ex )1/ 2 (R ex )1/ 5 (log 10 R ex ) 2.58

1.328 0.074 0.455

C favg R eL (R eL )1/ 5 (log10 R eL )2.58
 Fluid Mechanics 383

Special Points :
1 1
 In laminar region Cfx  , but in turbulent region C fx  1/5 So, 0
x x
decreases more rapidly in laminar region than in turbulent region.
 In Laminar region   x , while in turbulent region   x4/5. So 
increases more rapidly in turbulent region than in laminar region

Drag forces on a plate which has both Laminar and Turbulent

regions
V 20
Drag force = C D   Area
2
Separation of Boundary layer
 du 
  = 0 is called separation point
 dy  y  0
 du 
  > 0, Attached flow
 dy  y  0
 du 
  < 0, Already separated flow
 dy  y  0
Separation of flow occurs in turbines, pumps, aerofoils, open channel
transitions etc:
Boundary U
layer
U U U
Separated
stream line
C
D
B Separation
point Solid body
A  u   u  E
  0  u    0
 y y0    0  y y 0
 y  y 0 Pressure
dp dp distribution
0 pmin 0
dx dx
B C D
Effect of pressure gradient on boundary layer separation.
 In the region ABC of curved surface, the area of flow decrease & so
the velocity increases. Hence, the flow get accelerated in the region &
the pressure decreases in the direction of flow.
384 Civil Booster (Civil Ki Goli Publication 9255624029)

dp
 Therefore,  0 , and the entire boundary layer moves forward.
dx
 Along the region CDE of curved surface, the area of flow increases &
so velocity of flow decreases in the fluid.
 The pressure is minimum at point C.
 To delay the point of separation, a trip wire is mounted near the leading
edge of body.
dp
 Due to decrease of velocity,  0 . Therefore, in the region CDE, the
dx
velocity of flow goes on decreasing because the kinetic energy of the
layer is used to overcome the frictional resistance of the surface. The
combined effect of +ve pressure gradient & surface resistance decrease
the momentum of the fluid.
 A condition comes, when the momentum of the fluid is unable to over
come the surface resistance & the boundary layer starts separating from
the surface at point D.
 D/s of the point D, where the flow takes place in the reverse direction &
the velocity gradient becomes –ve.
 So, the +ve pressure gradient helps in boundary layer separation.
 Large turbulent eddies are formed at D/s of the point of separation. The
region is called the turbulent wake.
Consequences of boundary layer separation
(a) Separation of boundary layer increases flow losses in case of internal
flow like pipes.
(b) There is increase in pressure drag if there is boundary layer separation
in case of external flow

Methods to control separation

1. Rotating boundary in flow direction
2. Stream lining of the body.
3. Suction of fluid from boundary layer.
4. Supplying additional energy from blower
5. Providing a bypass in the slotted wing
6. Accelerating the fluid in boundary layer by injecting fluid.
7. Providing guide blades on bends.
 Fluid Mechanics 385

Dimensional Analysis
and Model Studies 13
Dimensional homogeneity: It states that every term in an equation when
reduced to its primary (fundamental)dimensions must contain identical powers
of each dimension.
Dimensions of Few Physical Quantities
(a) Kinematic Quantities:
1. Angular velocity T –1
2. Vorticity T –1
3. Angular acceleration T –2
4. Kinematic viscosity L 2 T –1
5. Stream function L 2 T –1
6. Circulation L 2 T –1
(b) Dynamic Quantities:
1. Specific weight ML –2T –2
2. Surface tension MT –2
3. Modulus of elasticity ML –1T –2
4. Dynamic viscosity ML –1T –1
5. Bulk modulus ML –1T –2
6. Angular momentum ML 2T –1

Methods of Dimensional Analysis

(a) Rayleigh's Method:
 It does not provides any information regarding the number of
dimensionless groups to be obtained as a result of dimensional
analysis.
 It is used for determining the expression for a variable which
depends upon maximum of 3 to 4 variables.
(b) Buckingham's -theorem: If there are n no. of variables in a
dimensionally homogenous equation & these variables contain m
386 Civil Booster (Civil Ki Goli Publication 9255624029)

fundamental dimensions, then the no. of dimensionless groups which

can be formed shall be (n-m).These dimensionless groups are called
-terms.
Similitude : To achieve similarity between the flow in the model & its
prototype, every dimensionless parameter referring to that conditions in the
model must have the same numerical value as the corresponding parameter
referring to the prototype

Geometric Kinematic Dynamic

(similarity of shape) (similarity of motion) (similarity of forces)

Special points: For kinematics similarity, Geometric similarity must exist

& for dynamic similarity both Geometric & kinematic similarity must exist.
These are necessary conditions but not sufficient conditions (if kinematic
similarity exists, then geometry similarity will be definitely there while
geometric similarity exists then kinematic similarity may or may not exist).
(also similar for dynamic similarity)
 A model can be larger or smaller than prototype.
 Normally larger models are made when:
(a) Flow field is very small.
(b) Flow velocity is very large.

Forces acting of Fluid mass

1. Inertia Force (Fi) =  L2V2
2. Gravity Force (Fg) =  L3g
3. Pressure Force (Fp) = PL2
4. Viscous Force (Fv) = VL
5. Surface tension Force (F) = × L
6. Elasticity force (Fe) = KL2
Dimensionless-Parameters
Compressibility force are predominant when mach no 0.3.
Mach number is used for water hammer pressure.
0.3 < Low subsonic, 0.3-0.8  High subsonic, 0.8-1.2  Transonic,
1.2-5.0  Supersonic, 5-10  Hypersonic, 10 -25  High Hypersonic
 Fluid Mechanics 387

Number Equation Uses

Fi VL
Reynolds No.  Aeroplanes, submarines, pipe flow
Fv 

Fi V

Eulers No. Fp p Cavitation problem, high pressure

flow in pipe

Fi V
Mach No.  Aerodynamic testing, rocket, missile
Fe C

Fi V
Froude No.  OCF, spillway, weir , Harbour model
Fg gL

Fi v
Weber No.  Blood in arteries and veins, rising
F  / L
bubble, seepage through soil capillary
rise, study of droplet, flow over weir
for small head
Reynold’s law Fraude’s law
 r L2r Lr
Time Ratio (Tr)
r gr
r
Velocity Ratio (Vr) L r r Lr  g r
 2r
Acceleration Ratio (ar) gr
 2r L3r
3r
Power Ratio (Pr)
2r Lr r L3.5
r gr
1.5

 2r
Force Ratio (Fr)
r r L3r g r

r Lr
Discharge Ratio (QR) r r L2.5
r gr
0.5
388 Civil Booster (Civil Ki Goli Publication 9255624029)

River model law (Distorted model law)

Hp
(a) Vertical scale ratio (Lrv )=
Hm
Lp Bp
(b) Horizontal scale ratio (LrH )= 
Lm Bm

(c) Velocity ratio (Vr )= L rv

(d) Area ratio (Ar )= Lrv × LrH
(e) Discharge ratio (Qr )= LrH .L3/rv2

Model / Prototype Laws for

(a) Head ratio (Hr )= N 2r D r2 , ( where Dr = diameter ratio )

(b) Discharge ratio (Qr )= N r D3r , ( where Nr= rotational speed ratio)

(c) Power ratio ( P )= r N3r D5r

 Fluid Mechanics 389

Pipe Flow 14
Practically, all the flow in the pipes is turbulent in nature.
Head loss

Frictional loss Losses in pipe fittings

Major loss  (80 – 90)% Minor loss 10 – 20)%
 Major loss in pipe due to friction is given by Darcy–Weisbach equation.

Major Losses
(a) Darcy’s weisbach equation

f LV 2
hf = , f  4f  , (f = friction factor, f' = coefficient of friction)
2gD
(b) Chezy’s formula
V = C RS
A  D2 D
R =  R=
P  D 4
hf
 Slope ( S ) =
L
8g
By equating both the above equations , we can get C =
f

Minor losses
(a) Due to sudden expansion

P1 V2 A2
A1 V1
Eddies
390 Civil Booster (Civil Ki Goli Publication 9255624029)

2
(V1  V2 ) 2 V12  A1 
hL =  1  
2g 2g  A2 
A1 = Area of smaller diameter pipe
A2 = Area of bigger diameter pipe
V1 = Velocity of smaller diameter pipe
2
V2  A1 
hL = K 1 Where K =  1  
2g  A2 
(b) Losses due to sudden contraction
vena contractra
(1)
(2) V2
AC

(Vc  V2 ) 2 KV2 2 0.5V22

hL =  =
2g 2g 2g
2
 1  Ac
where K =   1 , Cc 
 Cc  A2
Special Points:
 Momentum equation and Bernouilli’s equation are uses in derivation of
losses
 Loss in expansion is much higher than loss in contraction
 Losses are always expressed in terms of velocity of smaller diameter
pipe.
(c) Exit loss (due to impact)

hL
KV 2
hL =
V 2g

Special Point: In exit loss due to impact, K is the kinetic energy correction
factor. For Laminar , its K = 1 & forTurbulent, its K = 1.
(d) Entry loss
hL
0.5V 2
hL =
V entry in pipe 2g
 Fluid Mechanics 391

(e) Loss due to pipe fittings and bends

KV 2
hL =
2g

Type of fitting K
Standard Tee 1.8
Standard Elbow 0.9
45º Elbow 0.4
90º Bend T (Sharp) 1.2
Gate valve (half open) 5.6
Angle valve 5.0
Foot valve of pump 1.5

Hydraulic gradient line and Total energy line

P 
Line joining the points of piezometric head   z  at various points in a
 
flow is called HGL.
2
V1
2g 2
V2
TEL 2g
2
HGL V2 Hydraulic grade line
2
2g joining top pipe surface
V1
2g
exit
V2

TEL HGL

datum

 P V2 
Line joining the points of total energy    2 g  z  at various points in
 
a flow is called TEL.
Special Points:
 HGL may rise or fall in the flow direction, depending upon the velocity
head (which varies with the area of cross section)
 TEL always fall down. But if there is a pump or turbine placed in the
flow, then there will be sudden rise or fall repectively
 TEL is horizontal in case of idealised Bernoulli's flow as losses are
zero.
392 Civil Booster (Civil Ki Goli Publication 9255624029)

Pipe connections
Parallel connection Series connection
1
2 A 1 2 3 B
Q Q
l1, d1 l3 ,d 3
l2,d 2
3
Same head loss Same discharge
Q  Q1  Q2  Q3 hL  hL  hL  hL
AB 1 2 3

hL  hL  hL  hL Q = Q1  Q2  Q3
AB 1 2 3

Equivalent pipe: A pipe which can replace existing compound pipe

while carrying same discharge under same losses. For series connection
equivalent pipe of length ‘L’ and diameter ‘D’ will be
L l l l
5
 15  25  35
D d1 d 2 d 3
Special Point: Increase in discharge by adding a pipe of same diameter
in mid way of a pipe but keeping that head constant is 26.53%
l
d

d New
pipe
l/2
In pipe flow of municipal water supply, a parallel pipe is Installed mainly for
increasing the discharge .
Flow through syphon:
 A pipe which rise above its hydraulic grade line has –ve pressure & is
known as syphon.
(2)

Patm
l

(1) hS

H
 Fluid Mechanics 393

Special point: For No vapourisation Ps > Pvaporization, otherwise vapours

will form & flow will be stopped.
Power transmitted through pipe
1
H/2 2 head available = H – h f
H 'D'
H/2 3

H  hf
Efficiency () =
H
Power (P) = Q (H–hf)
dP H
For max power  0, h f 
dQ 3
max= 66.67,% , min. power lost = 33.33%
H T
dh 2g a
 Time required to empty the reservoir 
0 h

0
 dt , here K is
K A
head loss constant.
 Time required to empty the top half of tank form 1 to 2 be t1 & for
bottom half from 2 to 3 be t2 , then t1= 0.414 t2
Special point: It Nozzle of area ‘a’ attached at exit, then for maximum
fla 2 1
efficiency  where A corresponds to area of diameter D.
DA 2 2
Special case of head loss
Loss of head due to friction in tapering pipe

D1 D2

x
L
L
fQ 2 dx
hf = 
0 12.1(D1  kx )
5
394 Civil Booster (Civil Ki Goli Publication 9255624029)

 D1  D 2 
K =  
 L 
Water hammer Pressure: Sudden/rapid closure of valve in a pipe
carrying flowing liquid destroys the momentum of flowing liquid & sets up a
high pressure wave. This pressure wave travels with the speed of sound &
causes hammering action in pipe called Knocking/water hammer.
 Surge tanks are used to absorb the Increase in the pressure due to water
hammer phenomenon.
Velocity of Pressure Wave (c)

In Rigid Pipe In Elastic Pipe

1
 2
Value
K  1 
C  
   D  K  
 1   t  E  
L     

K
C=

D = diameter of pipe, K = Bulk modulus of Liquid,t = thickness of pipe
 = mass density of liquid, E = modulus of elasticity of material,
Water hammer pressure =  VC
4L
 Time period for complete cycle of water hammer pressure =
C
2L
Critical time equations T0 =
C
Water hammer pressure

Rapid closure Slow/gradual/closure

T  T0

T0 << T T0 < T  1.5T0

Special Point : Equations used in solving branching of pipes connecting

reservoirs of different levels are (a) Darcy weisbach equation, (b)
Bernoulli’s equation (c) Continuity equation.
 Fluid Mechanics 395

Hardy-Cross method of solving closed loop pipe networks.

1. It is a trial & error solution
2. Flow rate at the entry of any junction must be equal to flow out of
each junction.
3. Loss of head due to flow in clockwise direction must be equal to loss
of head due to flow in anti clockwise direction.

rQn
Modification in discharge Q =
 rnQn1

where hf = rQn & Q is algebrically added.

Various Instruments Used in Measurement

Device Measurement
Venturimeter Discharge or rate of flow
Flow nozzle Discharge or rate of flow
Orifice & mouthpiece Discharge or rate of flow
Rotameter Discharge or rate of flow
Bendmeter Discharge or rate of flow
Hydrometer Density or specific gravity
Hygrometer Moisture
Pyrometer Solar radiation
Pycnometer Water content & specific gravity
Hot wire anemometer Air & gas velocity
Current meter Velocity in open channel flow
Barometer Local atmospheric pressure
Pitot tube Fluid velocity
Notches & weir Discharge or rate of flow
through small channels