Download FLUID MECHANICS PART II(1)

FLUID MECHANICS PART (II)
Objective:
1. To study the Archimede’s principle
2. To derive Euler’s equation of motion for a moving fluid.
3. To understand Toricelli’s theorem.
4. To get an idea about irrotational contiuous flow of inviscid fluids.
Module 1
Archimede’s principle:
This principle states that a body wholly or partially immersed in fluid is buoyed
up with a force which is equal to the weight of the volume of fluid the body
displaces and which acts vertically upwards through the centre of gravity of the
immersed part of the body. The buoyancy force need not act through the centre of
gravity of the immersed body.
Let us consider that a fluid is kept in the gravitational field. Now, the total body
force acting on volume V of the fluid is,
F

b
o
d
y
fpdA
V
(1)
The total force due to pressure on the surface area A of volume V of the fluid is,
F
p 
âpdA
A
(2)
where â is the outward normal unit vector on any point on the surface. Now,
since the fluid is in the equilibrium, we have
Fp F
body 0
or
Fp Fbody
(3)
We may not that Fp is the total force due to pressure on the surface of the
volume V , whether volume V is occuppied by the fluid or not. This clearly
reveals that a body immersed in a fluid experiences a force Fp due to pressure,
equal and oppositte to the body force Fbody which would be exerted on the volume
V if it were occupied by the fluid in equilibrium. This is the Archemede’s
principle.
Module 2
Euler’s Equation of Motion for a Moving Fluid:One can describe the motion of a fluid by following the motion of each individual
fluid particle. However, that is a difficult one. There is a more convenient method
for many purposes. In this process, one has to specify the density and velocity of
the fluid at each point in space as a function of time. Here, we will follow this
treatment. We specify the motion of fluid by density
x, y, z,tand velocity v(x, y, z,t).
We may note that any quantity which is used to describe the state of a fluid is
function of space co-ordinate
 x, y, z 
and time t . This will have a definite
volume at each point in space, and at each instant of time. We may note that there
are two time rate of change for such a quantity. For example, one can consider
pressure. The rate of change of pressure with time at a fixed point in space will be
p
. i.e., the total time derivative.
t
Thus we have
d
ppp


xp

yp

z
  
(4)
d
t 
t x

t y

t z

t
Here
z
x y
and
are the component of the fluid velocity V
t
t , t
Rewriting equation (4) as
d
p
p 
p 
p 
p

V
V
V
x 
y 
z
d
t 
t 
x 
y 
z
Or
d
p 
p
 V
.
p(5)
d
t 
t
One finds that a similar relation holds between partial and total derivative of any
other quantity which describes the state of the fluid. One can express equation (4)
symbolically as
d 
 V.
dt t
(6)
This shows that the total time rate of increase can be resolved into a purely
temporal part and a purely spatial part. However, for a stationary state or a steady
state,

 0 .Let us consider a small volume element dV of a non viscous fluid
t
when its equilibrium, then the body force f dV equals p dV . Now, for the
motion of this volume element, we have
2
d
r
f
d
V


p
d
V


d
V2 (7)
d
t
d 2r
Here
is the acceleration of the element. From equation (7) one obtains
dt 2
V
p
 x 
x
t
(9)
Vy
 Vx
Vz 
 Vy
 Vz
Vx

y
z 
 x
fx 
fy 
Vy
p


y
t
Vy
 Vy
V 
 Vy
 Vz z 
Vx
y
z 
 x
(a)
(9) (b)
fy 
V
p
 z 
y
t
V
 Vy
V 
 Vy y  Vz z 
Vx
y
z 
 x
(9)(c)
ˆ and z
ˆ respectively,
Now, multiplying equations (9)(a), (9)(b) and (9)(c) by xˆ, y
and adding, one obtains
f  p  
V

t
 1 V  V    V 
 2

(10)
2
Equation (10) is called the Euler’s equation of motion for a moving fluid.
Module 3
Toricelli’s Theorem: Speed of Efflux of a Fluid From a Large
Vessel:
Using eqution (9) one can calculate the equilibrium velocity of the liquid coming
out from a hole in the bottom of the vessel filled upto a height
say h . Let us assume that the density of the liquid to be a constant.
Let the cross section of the vessel is large and the opening, i.e., the hole is small
and equipped with nozzle directing the outflowing jet. During the outflow, the
whole volume of the fluid in the vessel is in motion and can be divided into
stream tubes. One can consider flow of the liquid as stationary. Let us consider
the z- axis vertically down, and the surface of the liquid in the vessel as z  0 .
Obviously, the velocity of the liquid coming out will be along the z- axis only.
There fore, using equation (9) (c) one obtains
dV
dp
 0  Vz z
dz
dz
since f =  g
g  
equation by dz and
Now multiplying both the sides of the above
integrating, one obtains
1 2

V

g
z
pc (11)
z 
2
where c is a constant. We may note that what has been said applies to any motion
in which all the moving particle of the ideal fluid are in identical initial
conditions. Under the condition mentioned above, the velocity of the liquid is zero
in the vessel itself. We may assume that p is equal to the hydrostatic pressure,
 gz . Then the constant c  0 . If the hole for the efflux is at z  h , then the
2
2
g
ho
rV
2
g
h
pressure p vanishes, so that V
. This result, observed
z
z
originally by Toricelli in 1643, states that the speed of efflux is the same as that
acquired by fluid falling freely through the height h; it is commonly known as
Toricelli’s theorem. We may note that the velocity V is completely independent
of the direction of the jet relative to the horizontal line; it is the same for all the
values of the angle of inclination of the jet.
Module 4
Irrotational Continous Flow of Inviscid Fluids:
Let us consider that the body force density is derivable say u which is usually the
case, one finds
f  u
(12)
In this case, equation (10) taken the form
u  p  
dV
dt
(13)
   1 p 2    V 
 2

where V
The vector field
(14)
V is called velocity. When the motion of the fluid is
such that velocity is zero every where, the flow is said to be irrotational or
potential flow. A flow initially irrotational, will remain so if not disturbed
externally. To make it more clear, let us find out the integral of the normal
component of N  V across a surface S bounded by a curve C . Using Stoke’s
theorem, we obtain

Vn
s

V
.d
s
.ˆd


(15)
s
c
The line interal in (15) is taken around the curve C . Now, if the curve
C surrounds a vortex in the fluid (figure 1 a), the r.h.s of equation (15) has non-
zero, i.e finite value which is a measure of the speed at which the fluid whirls
around the vortex. This reveals that   V is a sort of measure of the rotation of
the fluid per unit area. We may note that   V may also be non-zero in regions
where there is no vortex, provided there is a transverse velocity gradient (figure 1
b) Figure 1 a figure 1 b A vortex A transverse velocity gradient From equation
(15),
we may note that a point P in the fluid, the vorticity V is clearly the
circulation of V around a unit area pependicular to the direction of  . Now,
cosider that a little piece of dirt is part in the neighborhood of point P. One finds
that the piece of dirt will rotate in the angular velocity

2 . In order to prove
this, let us consider an area element inside the fluid at P having an extremely
small radius r . In accordance with equation (15), the dirt particle will be carried
by the rotating fluid element along with it. From equation (15), we have
or
or
r2 V2r
=2V =2
r

 2
However, in the above crude approximation, we have assumed that the velocity
vector V and displacement vector ds are always parallel and  and ds are also
parallel. One can also derive the above result rigorously. For this purpose, let us
consider a co-ordinate system rotating with angular velocity  . Let V ' be the
velocity of the fluid at P relative to this system, then we have VV'r
where r
is a vector form of the axis of rotation to point P, and
V

r



V
'

r


Since  is a constant vector for all point in the fluid, we can express it as
ˆ
ˆ
ˆ

xx


yy
zz
One obtains
  r   y z   x y  xˆ
  z x   x z  yˆ   x y   y x  zˆ
Now, taking the curl of both sides of
the above and keeping in mind that  x ,  y and  z are not functions of
x, y and z , we have
   r   2x xˆ  2 y yˆ
2z zˆ  2



V
=V

' 2

But, at a point P in the fluid, we have
2V
Then at this point V'  0
Thus in a co-ordinate system rotating with angular velocity  V2, the
fluid flow is irrotational at the point P. interpreting V=2, i.e the angular
velocity of the fluid near P. Now, if   V is constant, then it is possible to select
a rotating co-ordinating system in which the flow is irrotational everywhere.
If at any time t,   0 everywhere,

also vanishes. This means  is still zero
t
everywhere at time t  t . This shows that flow is permanently irrotational or
potential (i.e  V=0 at each instant of time throughout the volume) i.e if a flow
was started with zero rotation, it would always have zero rotation. Let us consider
that at any point V =0 . This means that an element of the fluid at that point
will have no net angular velocity about that point, although its shape and size may
be changing. Let us imagine a small paddle wheel immersed in the moving fluid
as shown in figure 2. Now if the paddle wheel moves without rotating, the motion
is irrotational, otherwise the motion is rotational or vortex, i.e, in this region
V  0 . Obviously, rotational flow of a fluid includes vortex motion, such as
whirlpools.
Figure 2 Irrotational fluid motion
Equation (10) takes the following form for irrotational motion

V1
2
f


 


V
(16)
2

t
In case if we dealing with a type of fluid flow for which   V vanishes in the
entire region, i.e space, the integration of equation (16) become relatively simple.
In this case, one may represent V as the gradient of the velocity potential   
Conclusion:
Archimede’s principle states that abody wholly or partially immersed in a
fluid is buoyant up with a force which is equal to the weight of the volume
of the fluid the body displaces and which acts vertically upwards through
the centre of gravity of the immersed part of the body.


V
2
1

is called the Euler’s equation of
f


p


V

V



V

2


t
motion for a moving fluid. Toricelli’s theorem states that the speed of
efflux is same as that acquired by the fluid falling freely from the height h .
Assignments
1. State Archimede’s principle.
2. Give the Euler’s equation of motion for moving fluid.
3. State Toricelli’s theorem.
4. When will the flow of a fluid is irrotational?
Bibiliography:
1. A text book of fluid mechanics, by R.K Rajput
2. Fluid Mechanics, A Course Reader by Jermy. M
3. An introduction to fluid dynamics by Batchelor G.T
4. Elementary Fluid Dynamics by Acheson D.J
FAQ’s:
1. State Archimede’s principle.
Archemedi’s principle states that abody wholly or partially immersed in a
fluid is buoyant up with a force which is equal to the weight of the volume of
the fluid the body displaces and which acts vertically upwards through the
centre of gravity of the immersed part of the body.
2. Give the Euler’s equation of motion for moving fluid.
The Euler’s equation of motion for moving fluid is given by


V
2
1


f


p


V

V



V

2


t
3. State Toricelli’s theorem.
Toricelli’s theorem states that the speed of efflux is same as that acquired
by the fluid falling freely from the height h .
4. When the flow of a fluid is said to be irrotational?
When the motion of the fluid is such that the vorticity is zero everywhere,
the flow is said to be irrotational or potential flow. The vector field
V is called the vorticity
Quiz:
1. When a fluid is in equilibrium
a. Fp  Fbody b. Fp Fbody
c. Fp  Fbody
2. Toricelli’s theorem was postulated in the year
a.
1543
b. 1643
c. 1653
3. The vector --------------------- is called vorticity
a. V
b. V
c. V
Quiz Answers:
1.b
2.b
3.c
Glossary:
Buoyant force:
The upward force exerted by the fluid when an object is placed in the fluid.
Nozzle:
A nozzle is a device dessigned to controll the direction or charecteristics
of a fluid flow as it exits or enter an enclosed chamber or pipe via an on fice.