- Density (ρ)=
Pressure
mass (m)
volume (v)
Bubble rising up
at constant temperature
Normal force or thrust exerted by liquid
at rest per unit area
m
ρ v
P=
- unit= kg/m
- density of water=1000 kg / m3=1g/cc
ρgh = patm [n3-1]
3
- for same mass
ρ1v1=ρ2v2
Pressure depth relation
P= h ρ g
Patm
R= n r
h1
h1d1 = h2d2
h2
h1
h1d1 = h2d2 + h3d3
L1
d1
L2
d2
Conditions for equal forces on wall
and bottom in a cylinder
Hydrostatic paradox
if ρ1>ρ2
v1<v2
Patm
radius ‘r‛ becomes ‘R‛ when bubble rises
in liquid from bottom to the surface
F
A
4)
1) U-Tube manometer
Whatever the shape or width the pressure
at any particular depth is same
h2
h3
L3 d3
2) U-Tube type
PHYSICS
WALLAH
5)
h1d1 = h2 d2
Mixing of liquid
A
B
C
f2
D
PA=PB=PC=PD
Calculation of resultant/final density
1) Volumes are equal
d=
- Gauge pressure = P-Patm
= hρg
2) Masses are equal
2d1d2
3d1d2d3
Inclined barometer
d1d2+d2d3+d1d3
Total volume
m1+m2
= v +v
1
2
m =ρ1v1 & m =ρ2v2
2
1
v1 =
m1
ρ1
& v2 = ρ
2
1) Relative density of a body
ds
wa
=
(R.D)s=
dω wa-ww
2) Relative density of liquid
dL
(R.D)L=
dω
wa - w L
=
wa - w w
3) Relative density of a solid
to that of liquid
(R.D)L
=
wa
wa-wL
if θ=angle with horizontal
Sinθ =
m2
Relative density (R.D)/Specific gravity
(R.D)S
h2
L1
d2
h2
h1
Ll =
sp. gravity = (R.D) =
L2
Ll =
h
Ll
(hl)
h
(R.D) =
dL
dw
h
sinθ
L
h
Cosθ
L1
h1,d1
=
h3
hL
hw
L3 d3
hL
6)
L2
h
θ
Ll
ρm
h2,d2
L2
Special case: U - tube rotating
h
h2
L
tan
L
h2 d2
d3 h3
h = h1 - h2
h1
h1d1 + h3d3 = h2d2
L3
h
Px=Po+ hρmg
hρmg = Px- Po
L1
d1 h1
L3.
h
FLUID
MECHANICS 01
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h1d1 = h2d2 + h3d3
hw
U - tube accelerating
l
h2
L2
d2
h1d1 = h2d2
Px
Ll
h
h1 = h2 + h3
L1
d1
If one of the liquid H2O,
then, F1=F2
Patm
θ
if θ=angle with vertical
Cosθ =
h1
Note:
Manometer
d1 + d2
Total mass
d2
h1
L=R
=
3) The third liquid is in level with other
- Patm= 1.01325 x 105 Pa
3) Masses and volumes are different
d=
d1
R
2
For 3- liquid⟹ d=
f1
If
d1 + d2
For 2- liquid⟹ d=
d1
L
=
w
a
h =
g
=
h
L
=
w2l2
2g
a
g
l
h2
h1
x1
PHYSICS
WALLAH
W
x2
h1 =
w2x12
2g
h2 =
w2x22
2g
h = h2 - h1 =
w2 (x22 - x12)
2g
Archimedes principle
If gravity effect is neglected, the pressure at
every point of liquid in static equilibrium is same
Unit of Coefficient of viscosity
upthrust=weight of the liquid displaced= Vρ g
Apparent weight=Actual weight-upthrust
1) The CGS Unit of η is dyne s cm-2 and is called poise.
2) The SI unit of η is Nsm-2 or decapoise or poiseuille
Wapp=Wair-U
Fc
FB.Cos θ
Ab
Ac
FB
=Wair [1- _
ρ
θ
1 poiseuille = 10 poise
net force acting upward=V+ ρL+ g
)
)
θ
[
A
PHYSICS
WALLAH
Pascals Law
Poiseuille‛s formula
πPr4
Q= _
8 ηl
PHYSICS
WALLAH
FB.Sin θ
Law of floatation
W
AA
B
FA
C
W
Stoke’s law
F=6 ηπrv
W
Fnet=Apparent weight-viscous force
U
U
Application
U
Hydraulic Lift
(A) W>U
F
A
=
F
a
If the cylinders are connected
F1
R12
F1
R1
2
F1
D12
F1
D1
=
=
=
F2
R22
F2
R2
2
F2
D22
F2
D2
FLUID
MECHANICS 02
F >> f
f
a
A
ρb= ρL
ρb > ρL
As A>> a therefore
F
(B) W=U
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Terminal velocity
(C) W=U
2 r2 (ρ- ) g
Vt= _
9η
1) If ρ > ,the body will attain terminal velocity in the downward direction.
ρb < ρL
2) If ρ < the terminal velocity will be negative and the body will move in the
upward direction.
ρ
3) = ,the body remain suspended in the fluid.
W ⇒ Weight
U ⇒ Upthrust
Critical velocity
Fractional submerged volume
Displaced
volume(Vd) _
_
= ρ (submerged fraction)
Total volume(V)
Ve
Significances of Reynold number.
Exposed volume(Vl)
_ = 1- _ (Exposed fraction)
ρ
Total volume(V)
If Re lies between 0 and 2000 the folow is stream lined or laminar.
Vd
ρb
wa
weight of solid in air
_
_
Relative density of a solid= _ = w -w = ρW
a
w
Loss of weight in water
-wL _
ρL
Loss of weight in liquid w
_
a
Relative density of a liquid= _ = w -w = ρW
a
w
Loss of weight in water
Reynold number
ρvD
Re = _
η
If Re>3000,the liquid in turbulent.
σ
If Re lies between 2000 & 3000 the flow of liquid is unstable.It may
change from laminar to turbulent and vice versa.
P
Velocity gradient
dv
Velocity gradient= _
dx
F
d
dv
_
>F=- ηA _
A dv =
dx
x
_
F/A
F/A
_
F =
F/A
η= _
>coefficent of viscosity =
> η= _
dv/dx= v/l =d
Adv/dx
(x/l)
dt
shearing stress
=
>η= _
strain rate
Equation of continuity
V1A1 Δ t ρ1= V2A2 Δ t ρ2
v2
A2
v 2dt
since the liquid is incompressable ρ1= ρ2
V1A1=V2A2
Av=constant.
dv
=Q=
Av=
>Volume rate of flow
dt
B
A
A2
v1dt
v1
Energy of fluid in a study flow
SURFACE TENSION
VENTURIMETER
Device to measure the flow of speed of incompressible fluid
Surface tension T =
BERNOULLI’S PRINCIPLE
1
kinetic Energy =
mv2
2
v1 =
P₁V₁ - P₂V₂ = 1 m (v₂²-v₁²) +mg (h₂-h₁)
2
kinetic energy per unit mass = 1 v2
2
2hg
(A12/A22) -1
Unit in SI system =
1
P +
2
Potential energy per unit mass = gh
Potential energy per unit volume = ρgh
ρv + ρgh = constant
P
v
ρg + 2g + h = constant
2
Pressure energy = PV
P = pressure head
ρg
Pressure energy per unit mass = ρP
v2
2g = velocity head
Pressure energy per unit volume = P
h = Gravitational head
D
Soap film
T
Pi - Po =
Then
Pi - Po =
P
h
v= 2gh
H
B
C
Pa
T
p
T
x
p<p1
Pressure on concave side> pressure on convex side
But F = 2TL
2T
R
Pconcave - Pconvex =
FLUID 03
MECHANICS
4T
R
p>p1
p=p1
Pinside - Poutside =
2T
R
Pinside - Poutside =
4T
R
One surface
[ Liquid drop or air bubble ]
[ Soup bubble ]
Two surfaces
Capillarity
Meniscus
< 90
Meniscus
o
h<0
Water
)
R
Fa =
SL
SV
Glass
2(H-h)/g
LV
vx = Horizontal component of velocity
R is max. when h= H
2
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Mercury
> 90
o
Consider the equilibrium of at line of contact
O
O
l
Range R = vx xt
T
p
A
B1
Shape of liquid meniscus
Fa
R = 2 h (H-h)
A
T
F
Work done W = F x x
2T
R
SV
= 2gh x
A
p
h>0
(H-h)
Time of fall, t = 2(H-h)
g
P1
P1
T
Excess pressure inside a soap bubble
If tank is open, P = Pa
m
F
Excess pressure inside a liquid drop
Pa
R
F
l
= T (2LX)=T∆A
v= 2(P-Pa) + 2gh
ρ
Q
S
P1
A1
A
Energy of the additional surface = W = 2TLx
Torricelli‛s Law of Efflux
N
B
C
P
PRESSURE DIFFERENCE ACROSS A CURVED LIQUID SURFACE
W = 2TL x x
APPLICATIONS OF BERNOULLI’S PRINCIPLE
l
PHYSICS
WALLAH
SURFACE ENERGY
2
F
=
A
PHYSICS
WALLAH
Potential Energy = mgh
Length
Unit in CGS system = dyne / cm
(P₁ - P₂) V = 1 m (v22 - v1²) + mg (h₂-h₁)
2
mg
1 m
(P₁ - P₂) V =
(v22 - v1²) +
(h -h )
V ₂ ₁
2 V
kinetic energy per unit volume = 1 ρv2
2
Force
cos
LV
=
SL
=
Ascent formula:
sin
+
SV
LV
-
cos
SL
h =
2T
Rρg
LV
Water
= Angle of contact.
h =
2Tcos
rρg
h > 0
)
< 90
h < 0
)
> 90
o
)
o
)