STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics
Indian Institute of Technology Madras
Chennai 600036
School of Basic Sciences
Indian Institute of Technology Mandi
Mandi 175001
[email protected]
[email protected]
STiCM Lecture 29
Unit 9 : Fluid Flow, Bernoulli’s Principle
Curl of a vector, Fluid Mechanics / Electrodynamics, etc.
PCD_STiCM
1
Unit 9:
Fluid Flow, Bernoulli’s Principle
Definition of circulation,
curl
, vorticity,
irrotational flow.
Steady flow.
Bernoulli’s equation/principle, some illustrations.
PCD_STiCM
2
Unit 10: Fluid Flow, Bernoulli’s Principle.
Equation of motion for fluid flow. Definition of
curl, vorticity, Irrotational flow and circulation.
Steady flow. Bernoulli’s principle, some
illustrations. Introduction to applications of Gauss’
law and Stokes’ theorem in Electrodynamics.
Learning goals: Learn that both the divergence and
the curl of a vector field are involved (along with the
boundary conditions) in determining its properties.
Learn how a rigorous treatment of the velocity field is
necessary to explain quantitatively the observed
phenomena in fluid dynamics.
Get ready for a theory
of electrodynamics.
PCD_STiCM
3
Recall the discussion on directional derivative
d
 uˆ  
ds
 r dr
uˆ  lim

 s 0  s
ds
 s   r , tiny
increament
ds  dr ,
Gradient: direction in
which the function varies
fastest / most rapidly.
F  
Force: Negative gradient of
the potential
‘negative’ sign is the result
of our choice of natural
differential
motion as one occurring
increament from a point of ‘higher’
potential to one at a ‘lower’
PCD_STiCM
4
potential.
F  ma
Is it obvious that the ‘force’ defined by these F  
two equations is essentially the same?
Compatibility of the two expressions
Consistency in
these relations
exists only for
‘conservative’
forces.
 F  dr  0
holds if, and only if,
the potential  is defined in such way
that the work done by the force given by
 in displacing the object on which this
force acts, is independent of the path
along which the displacement occurred.
PCD_STiCM
b
 F  dr is
a
INDEPENDENT
of the path
a to b
PATH INTEGRAL
5
“CIRCULATION”
 F  dr  0
It is only when the line integral of the work done is
path independent that the force is conservative
b
 F  dr is
and accounts for the acceleration it generates
when it acts on a particle of mass m through the
a
INDEPENDENT
of the path
a to b
‘linear response’ mechanism expressed in the
principle of causality of
Newtonian mechanics:
F  ma
The path-independence of the above line integral is
completely equivalent to an alternative expression which
can be used to define a conservative force.
This alternative expression employs what is known as CURL of a
VECTOR FIELD
F
, denoted as
 F
PCD_STiCM
.
6
Definition of Curl of a vector:
 F is a vector point function
of the vector field F (r ) such that,
for an orthonormal basis
set of unit vectors uˆ (r ), i  1, 2,3

,
i
uˆi (r )    F (r )  lim
 F (r )  dr ,
Mean (average) ‘circulation’
per unit area taken at the
point when the elemental
area becomes infinitesimally
small.
s
where the path integral is taken over a closed path C, taken
over a tiny closed loop C which bounds an elemental vector
S 0
surface area S  S uˆi (r ).
The direction of the unit vector uˆi (r ) is such that a right-hand
screw would propagate forward along it when it is turned
along the sense in which thePCD_STiCM
path integral is determined.
7
uˆi (r )    F (r )  lim
 F (r )  dr ,
s
where the path integral is taken over
S  0
a closed path C, taken over a tiny
closed loop C which bounds an
elemental vector surface area
S  S uˆi (r ).
C traversed
one way
right-hand-screw
convention.
PCD_STiCM
C traversed
8
the other way
F (r )  dr

uˆ (r )   F (r )  lim
i
S 0
s
uˆ (r), i  1, 2,3
i
Cartesian unit vectors, of course,
do not change from point to point.
In general, the unit
vectors may depend
on the particular point
under discussion, and
hence written as
functions uˆi ( r ) of r.
They are constant vectors.
The above definition of CURL of a VECTOR is independent of
any coordinate frame of reference; it holds good for any
complete orthonormal set of basis set of unit vectors.
Mean (average) ‘circulation’ per unit area taken at the point
when the elemental area becomes infinitesimally small.
PCD_STiCM
9
circulation 
circulation and curl
 A(r ) dl
C
C
Consider an open surface S, bounded by a closed
curve C.
Circulation depends on the value of the vector at all
the points on C; it is not a scalar field even if it is a
scalar quantity. It is not a scalar point function.

lim
 S 0
C
A(r ).dl
S
=
 (  A) nˆ
‘Limiting’ circulation
per unit area
Shrink the closed path C; in the limit,
the circulation would vanish;
and so would the area S
bounded by C.
However, the ratio itself is finite in
the limit;
it is
a local quantity at that point. 10
PCD_STiCM
circulation and curl
circulation 
 A(r ) dl
C

lim
 S 0
C
A(r ).dl
S
 (  A) nˆ
C
‘Limiting’ circulation
per unit area
This limiting ratio defines a component of the curl
of the vector field; the curl itself is defined through
three orthonormal components in the basis
PCD_STiCM
nˆ1 , nˆ2 , nˆ3 
11

lim
 S 0
C
A(r ).dl
 (  A) nˆ
S
Curl measures how much
the vector
“
curls” around at a point
PCD_STiCM
12
curl of a vector field at a point represents the net
circulation of the field around that point.
the magnitude of curl at a given point represents
the maximum circulation at that point.
the direction of the curl vector is normal to the
surface on which the circulation (determined as
per the right-hand-rule) is the greatest.
If  F  0 in a region then there would be no
curliness/rotation, and the field is called irrotational.
PCD_STiCM
13
Remember!

lim
C
A(r ).dl
 (  A) nˆ
S
The criterion that a force field is conservative is that its
 S 0
path integral over a closed loop (i.e. “circulation”) is
zero. This is equivalent to the condition that
 F  0
If  F  0 in a region, then there would be no
curliness (rotation), and the field is called irrotational.
Conservative force fields: IRROTATIONAL
Examples for irrotational fields: electrostatic,
gravitational
PCD_STiCM
14
Curl in Cartesian Co-ordinate
Consider a point P( x0 , y0 , z0 ).
A(r ).dl
Consider a vector field A(r ) in some region of space.
Circulation over the peremeter of an elemental surface
to which the normal is eˆz is
 
y 
 y 

A
x
,
y

,
z

A
x
,
y

, z0    x 
x  0
0
 x 0 0 2 0
2



 
 
x
x



A
x

,
y
,
z

A
x

,
y
,
z
y 0
0
0   y
 y 0 2 0 0
2



 

C
Ay
Ax
A(r ).dl  
 y x 
 x y
y
x
3rd leg
x
Y
add up
4th y
leg
x
eˆz 1st leg
X
3rd leg
Z
4th
leg
2nd
leg
1st leg
Closed path C
which bounds
an elemental
surface
Ax Ay
 (curl A) eˆz  

 (curlA) z Make sure that you
y
x
PCD_STiCM
2nd
y
leg
understand the signs15
Ay
Ax
C A(r )  dl   y  y x  x  x y
Determining now the net circulation per unit area:
Ax Ay
(curl A)  eˆz  

 (  A) z
y
x
Color coded arrows are unit vectors
orthogonal to the three mutually
orthogonal surface elements bounded
by their perimeters.
Similarly if we get circulation per unit area along other two
orthogonal closed paths and add up, we get:
 Az Ay 
 Ay Ax 
 Ax Az 
Curl A  eˆx 



  eˆy 

  eˆz 
 z x 
 y z 
 x y 
PCD_STiCM
16
 Az Ay 
 Ax Az
curl A  eˆx 


  eˆy 
z 
x
 z
 y
 Ay Ax 



  eˆz 
y 

 x
The Cartesian expression for curl of a vector
field can be expressed as a determinant; but it
eˆx

curl A    A 
x
Can you interchange the 2nd and the
Ax
is, of course, not a determinant!
3rd row and change the sign of this
eˆy

y
Ay
eˆz

z
Az
‘determinant’? The curl is not a cross product of
two vectors; the gradientPCD_STiCM
is a vector operator!
17
examples of rotational fields, nonzero curl
z

V ( x, y)   yeˆx  xeˆ y
A(x,y)=(x-y)eˆ x  ( x  y)eˆy
 
  V  2eˆz
  A  2eˆ z
curl : along thePCD_STiCM
positive z-axis
18
rotational fields, nonzero curl
z
y
x
v  xeˆ y

  v  êz
PCD_STiCM
19
What is the DIVERGENCE and the CURL of the following
vector field?
As much flux
leaves a
volume
element as that
enters,
hence the
divergence is
zero
Y
X
F
 0 and Fx  0
y
 F 0
 F  0
Reference: Berkeley
Physics Course, Volume I
PCD_STiCM
20
What is the DIVERGENCE and the CURL of the following
vector field?
Clearly,
 F 0
 F  0
PCD_STiCM
Reference: Berkeley Physics Course, Volume I
21



  eˆx  eˆy  eˆz
x
y
z
curl of a gradient is zero
eˆx

   
x

x
eˆ y

y

y
eˆz

z

z
The final result will be
independent of the
coordinate system.
 
  
  
  
 eˆx 



  eˆy 
  eˆz 

 yz zy 
 zx xz 
 xy yx 
2
2

0
2
PCD_STiCM
2
2
2
22
Recall: from Unit 5
Motion in a rotating coordinate
system of reference.
PCD_STiCM
23
db  b(t  dt )  b(t ) | db | uˆ
nˆ  bˆ
where uˆ 
.
nˆ  bˆ
b sin 
  (nˆ, bˆ)
db  (b sin  )(d )
nˆ  bˆ
db  (b sin  )(d )
nˆ  bˆ
d
b
b sin 
d
db  d nˆ  b
d
since  
nˆ
dt

db

db
These two terms are equal and hence cancel.
db   dt   b
n̂
b(t  dt )
b(t )
d
  b   b
 dt  I
PCD_STiCM
24
24
d
  b   b
 dt  I
Remember! The vector
b itself did
not have any time-dependence in
the rotating frame.
If b has a time dependence in the rotating frame,
the following operator equivalence would follow:
d
d
  b   b    b
 dt  I
 dt  R
Operator
Equivalence:
d d
      
 dt  I PCD_STiCM
 dt  R
25
25
d d
      x
 dt  I  dt  R
Recall: from Unit 5
d
d
  r   r   x r
 dt  I
 dt  R
d
When   r =0,
 dt  R
d
  r  x r
 dt  I
d
  r  vI =  x r
 dt  I
PCD_STiCM
26
d
  r  vI =  x r
 dt  I
v r
eˆx
eˆy
eˆz
x
y
z
 v   (  r )    x  y z
  ( y z  z y)eˆx  (z x  x z )eˆy  (x y   y x)eˆz 
eˆx
eˆy
eˆz




x
y
z
( y z  z y ) (z x  x z ) (x y   y x)
 v  2(x eˆx   y eˆy  z eˆz )  2
PCD_STiCM
The ‘curl’ of the
linear velocity
gives a
measure of
(twice) the
angular
velocity; thus
justifying the
term ‘curl’. 27
Remember:
uˆi (r )   F (r )  lim
S 0
 F (r )  dr
s
The component of the curl of a vector field in the
direction uˆi (r ) is the circulation about the axis
of the vector field per unit area.
It measures the extent to which a particle being
carried by the vector field is being rotated about uˆi (r .)
PCD_STiCM
28
We shall see in the next class that we are now
automatically led to the STOKES THEOREM: William Thomson,
1st Baron Kelvin
(1824-1907)
 A(r )  dl   (  A)  dS
Note!
It is STOKES’
THEOREM
not
STOKE’S THEOREM
0
K temperature
George Gabriel
Stokes
(1819–1903)
This theorem is named after George Gabriel Stokes
(1819–1903), although the first known statement of
the theorem is by William Thomson (Lord Kelvin) and
appears in a letter of his to Stokes in July 1850.
PCD_STiCM
29
Reference: http://www.123exp-math.com/t/01704066342/
We shall take a break here…….
Questions ?
[email protected]
Comments ?
http://www.physics.iitm.ac.in/~labs/amp/
[email protected]
Next: L30
Unit 9 – Fluid Flow / Bernoulli’s principle
….. but which Bernoulli ?
PCD_STiCM
30
30
STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics
Indian Institute of Technology Madras
Chennai 600036
School of Basic Sciences
Indian Institute of Technology Mandi
Mandi 175001
[email protected]
[email protected]
STiCM Lecture 30
Unit 9 : Fluid Flow, Bernoulli’s Principle
Curl of a vector, Fluid Mechanics / Electrodynamics, etc.
PCD_STiCM
31
uˆi (r )   F (r )  lim
 F (r )  dr
S 0
s
The component of the curl of a vector field in the
direction uˆi (r ) is the circulation about the axis
of the vector field per unit area.
ABOVE RELATION: provides complete
DEFINITION
of CURL of a VECTOR.
uˆ (r); i  1, 2,3 orthonormal basis
i
PCD_STiCM
32
Proof of Stokes’ theorem follows from the very definition of the curl:
Definition : (curl A)  nˆ  lim
 A(r )  dl
C
 S 0
S
 (  A)  nˆ
For a tiny path δC, which binds a tiny area δS,
 A  dl   S  (curl A)  nˆ  curlA  S
C
We can split up a finite area S into infinitesimal
bits δSi bound by tiny curves δCi
n
 A(r )  dl   
C
i 1
n
A(r )  dl    curl A(r )  dS
Ci
i 1
 A(r )  dl   curl A(r )  dS
C
Stokes’ theorem
PCD_STiCM
33
consider a surface S enclosed by a curve C
Stokes’ theorem
ˆ
A
(
r
)

dl

curl
A
(
r
)

dSn


c
The Stokes theorem relates
the line integral of a vector
about a closed curve
to the surface integral of its
curl over the enclosed area
that the closed curve binds.
Any surface bound by the closed curve will work; you
can pinch the butterfly net and distort the shape of the
34
net any which way –PCD_STiCM
it won’t matter!
consider a surface S enclosed by a curve C
Stokes’ theorem
ˆ
A
(
r
)

dl

curl
A
(
r
)

dSn


c
The direction of the vector
surface element that appears in
the right hand side of the above
equation must be defined in a
C traversed one
manner that is consistent
way
with the sense in which the
closed path integral in the left
hand side is evaluated.
The right-hand-screw
convention must be followed.
PCD_STiCM
C traversed
35
the other way
Non-orientable surfaces
The surface under
consideration, however, better
be a ‘well-behaved’ surface!
A cylinder open at both ends is
not a ‘well-behaved’ surface!
A cylinder open at only one end
is ‘well-behaved’; isn’t it already
like the butterfly net?
PCD_STiCM
36
Consider a rectangular
strip of paper, spread flat at
first, and given two colors
on opposite sides.
The surface under
consideration, however, better
be a ‘well-behaved’ surface!
Now, flip it and paste the
short edges on each other
as shown.
Is the resulting object
three-dimensional?
How many ‘edges’ does
it have?
How many ‘sides’ does it have?
PCD_STiCM
37
Expression for ‘curl’ in cylindrical polar coordinate system
eˆ , eˆ , eˆ 


z
 A 
 
1 

ˆ
ˆ
ˆ
e    e    e z z   eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z) 


 A  eˆ     eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z )  






 1  
 eˆ      eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z )  


ˆ   ˆ
e z z   e  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z ) 
PCD_STiCM
38
 A 
 
1 

ˆ
ˆ
ˆ
e

e

e
z
     
  eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z) 

z


eˆ , eˆ , eˆ 


z
  
eˆ     eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z )  

 A  
 1  
 eˆ      eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z )  


ˆ   ˆ
e z z   e  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z ) 

eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z )  
  A  eˆ   

1  
ˆ
 e   
 eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z)  
   

 eˆ  A (  ,  , z )  eˆ z Az (  ,  , z )  39
eˆ z   eˆ  A (  ,  , z)PCD_STiCM
z
Expression for ‘curl’ in cylindrical polar coordinate system
eˆ , eˆ , eˆ 


z

ˆ


 eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z )  
  A  e   

1  
 eˆ    
 eˆ  A (  ,  , z )  eˆ  A (  ,  , z )  eˆ z Az (  ,  , z )  
   

ˆ
e z   eˆ  A (  ,  , z)  eˆ  A (  ,  , z)  eˆ z Az (  ,  , z) 
z
 1 Az A 
 A Az 
1     A  A 
= eˆ  




  eˆ  
  eˆ z 
z 
 
  
 
  
 z

PCD_STiCM
40
Expression for ‘curl’ in spherical polar coordinate system
eˆ , eˆ , eˆ 
r


  A=
 
1 
1
 
ˆ
ˆ
ˆ
=  er
 e
 e
   eˆ r Ar (r ,  ,  )  eˆ  A (r ,  ,  )  eˆ  A ( r,  ,  ) 
r 
r sin   
 r

A 
  sin  A  

 
 

1  1 Ar 
+eˆ  
  rA  
r  sin   r

1
  A=eˆ r
r sin 
Ar 
1 
 eˆ    rA  

r  r
 
PCD_STiCM
41
An important identity: divergence of a curl is zero
Gauss’ divergence theorem
   ( A)d    A dS
V
S
S2
Volume
Surface enclosing a volume
S1
c1
c2
Applying Stoke’s theorem

 
 
 
ˆ
ˆ
(


A
)

d
S

(


A
)

n
dS

(


A
)

n
1
1
2 dS 2



S
S1
 
 
  A  dl   A  dl  0
C1
C2
PCD_STiCM
S2
  
  (  A)  0
42
some definitions…..
To understand the term ‘ideal’ fluid, we first define
(i) ‘tension’, (ii) ‘compressions’ and (iii) ‘shear’.
Consider the force F on a tiny elemental area A passing
through point P in the liquid.
Stress at the point P is
S.
P
S  uˆ N  S
S: Tension
S  uˆ N  0
S: Shear
S  uˆ N   S
S: Compression
The unit normal
orientation.
uˆ N 
û N
A
A
can take any
An ideal fluid is one in which stress at any point is
43
essentially one of PCD_STiCM
COMPRESSION.
The curl of a vector is an important quantity.
A very important theorem in vector calculus is the
Helmholtz theorem which states that given the
divergence and the curl of a vector field, and
appropriate boundary conditions, the vector field is
completely specified. You will use this to study
Maxwell’s equations which provide the curl and the
divergence of the electromagnetic field.
Besides, the ‘curl’ finds direct application also in the
derivation of the Bernoulli’s principle, as shown below.
PCD_STiCM
44
2 sons of
Nicolaus
Bernoulli’s
Bernoulli
brothers
Bernoulli
Family
Math/Phys Tree
Johann’s work was assembled by the
Marquis de I;Hospital (1661-1704) under a
strange financial agreement with Johann in
1696 into the first calculus textbook. The
famous method of evaluating the
indeterminate form 0/0 got to be known as
I'Hospital's rule.
PCD_STiCM
Reference:
http://www.york.ac.uk/depts/maths/histstat/people/bernoulli_tree.htm
Posed the
brachistrocrone
problem
Bernoulli’s
Principle
45
References to read more about the Bernoulli Family:
http://www.york.ac.uk/depts/maths/histstat/people/bernoulli_tree.htm
http://library.thinkquest.org/22584/temh3007.htm
“...it would be better for the true physics
if there were no mathematicians on
earth”.
Quoted in The Mathematical
Intelligencer 13 (1991).
http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Bernoulli_Daniel.html
Daniel Bernoulli
1700 - 1782
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Bernoulli_Daniel.html
PCD_STiCM
46
dv  d 
d 
   v( r (t ) , t )    v  x(t ), y (t ), z (t ), t 
dt
 dt 
 dt 
 v dx  v dy  v dz  v




x dt y dt z dt t
d v dx  v dy  v dz  v  v




dt dt x dt y dt z
t


 v     v
t 

“CONVECTIVE DERIVATIVE OPERATOR” The term ‘convection’
d
  is a reminder of the fact that in the

convection process, the transport
i.e.
  v    PCD_STiCM
dt 
t  of a material particle is involved. 47
Result of the previous unit, Unit 8:

d
p

 v    t  v(r , t )  dt v(r , t )   (r )  
Use now the
following  A  B 
vector
identity: A   B  B   A  A    B  B    A



  
   
v  v  
v  v  v  v  v    v v    v
1
i.e. v  v   v   v  v    v 
2
 


1
v  p
 v  v  v  v 

 
2
t
 (r )
PCD_STiCM
48
 


1
v  p
 v  v  v  v 

 
2
t
 (r )

 p(r ) 

Now,
 


 (r )
  

 p(r )
 


 



1
v
 p(r ) 

 v v  v  v 
  
  
2
t

  

i.e.,


1
v
 p(r )

 v v  v  v 
  

2
t


 

PCD_STiCM
49
 




1
v
 p(r )

 v  v  v  v 
 

2
t


 

d 
d 
Recall that :   r    r   x r
 dt  I
 dt  R
v I  v R   x r R , where v I is just the
velocity that is employed in the equation
of motion for the fluid.

   vI    vR     x r R
To determine 
 x r 
R

we now use another vector
Identity, for the curl of cross-product of two vectors:

 
 

 A  B  B   A  APCD_STiCM
  B  A(  B)  B(  A)
50

 v I   v R    x r R

 

 



  A  B  B   A  A   B  A(  B)  B(  A)


    rR   rR        rR   (  rR )  rR (   )


    rR       rR   (  rR )  2
  v I    v R  2
In the rotating frame, v R  0,
hence , the VORTICITY,   v I    2
PCD_STiCM
51
 




1
v
 p(r )

 v  v  v  v 
 

2
t


 

2


1
v
 p(r )

 v  v  
  

2
t


 

2


v
 p(r )
 1
 v     
   v
t


 
 2
2


v
v
 p(r )

 v     
 

t
2 
 


For ‘STEADY STATE’
v
0
t
2


v
 p(r )

 v     
 

2 
 


2


v
 p(r )

Hence, 0  v  
 


2


PCD_STiCM


52
2


v
 p(r )

0  v  
 

2 
 


2


v
 p(r )

 
 
 must be ORTHOGONALto v,
2 
 


2


v
 p(r )

i.e.,  
 
 must be ORTHOGONALto STREAMLINES
2 
 


  must be ORTHOGONALto streamlines,
2
where  

p(r )

p(r )

 
 
v
2
v
.
2
2
 constant
PCD_STiCMalong a given streamline
Daniel Bernoulli’s Theorem
53
2

p(r )

 
v
2
 constant for a given streamline
We derived the above result for a ‘STEADY STATE’ and made use of the relation
2


v
 p(r )

 v     
 


2




v
0
t
If the fluid flow is both ‘steady state’ and ‘irrotational’,
 v    0
2

p(r )

 
v
2
Daniel Bernoulli’s Theorem
PCD_STiCM
54
is constant for the entire velocity field in the liquid.
WORK – ENERGY Theorem
Conservation of Energy
PCD_STiCM
55
Mass Current Density Vector J (r , t ) 
 ( r , t ) v( r , t )
Dimensions : ML 2T 1
G
For Steady State Flow,
vA  constant ,
A : cross - sectional area
since,
 d   J (r )  
volume
region
surface
enclosing
that
region
C
D
H
P
F
E
r
B
A

J (r )  dS  0, for STEADY STATE as
0
t
Work done on the fluid by the pressure
that the fluid exerts on Face 1 is:
W1  F1s  p1 A1s  p1 A1v1t
Work done by the fluid on Face 2 is:
Net work done on the fluid in the
parallelepiped by the pressure
that the fluid exerts on Faces 1
& 2 is:
W1  W2  p1 A1v1t  p2 A2 v2t
W2  F2s  p2 A2s  p2 A
PCD_STiCM
2 v 2t
56
Net work done on the fluid in the
parallelepiped by the pressure
that the fluid exerts at Faces 1 & 2 :
W1  W2  p1 A1v1t  p2 A2 v2t

Energy gained per unit mass by the
p1 A1v1  p2 A2 v 2
fluid as it traverses the x-axis of the E 2  E1 
m
parallelepiped across the Faces 1 & 2 :
 p1 A1v1  p2 A2 v2  t
m
t
 E 2  E1
1

1

  v2    U internal    v2    U internal 
2
2  2
1
 p1 A1v1  p2 A2 v2  t
  sA
1 2

1 2

  v    U internal    v    U internal 
2
2  2
1
PCD_STiCM
57
 p1 A1v1  p2 A2 v2  t
  sA
1 2

1 2

  v    U internal    v    U internal 
2
2  2
1
 p1 A1v1
p2 A2 v2 
1 2

1 2



  t   v    U internal    v    U internal 
2
2  2
1
   v1 t  A1   v 2 t  A2 
 p1 p2   1 2

1 2



v



U

v



U
internal 
internal 

 2


2
2 
1

 
1 2
1 2
p
p
0   v    U internal     v    U internal  
 2  2
 1
2
1 2
p
From slide
i.e.
v    U internal   constant
2

Daniel Bernoulli’s Theorem

PCD_STiCM
p(r )

53 :
v
 
2
2
58
is constant for the entire velocity field in the liquid.
2
p(r )

 
v
2
 constant
The swing of a ball is
governed by Bernoulli's
theorem.
A swing bowler rubs only
one side of the ball. The ball
is then more rough on one
side than on the other.
Ishant Sharma
Inswing / Outswing
bowler
A white ball has a thin lacquer that is applied to its surface to avoid discoloring
the ball. During play, the shiny surface of the white ball remains shinier than
that of a red ball, which has a rougher surface to begin with.
The difference between the rough and shiny surface of a white ball is
much more, and thus it swings more than the red ball.
PCD_STiCM
59
"Aerodynamics of sports balls", Rabindra D. Mehta,
in Annual Reviews of Fluid Mechanics, 1985. 17: pp. 151-189.
"On the swing of a cricket ball in flight", N. G. Barton,
Proc. R. Soc. of London. Ser. A, 1982. 379 pp. 109-31.
"An experimental study of cricket ball swing", Bentley, K., Varty,
P., Proudlove, M., Mehta, R. D., Aero. Tech. Note 82-106, Imperial College,
London, England, 1982.
"The effect of humidity on the swing of cricket balls",
Binnie, A. M., International Journal of Mechanical Sciences. 18: pp.497-499,
1976.
"The swing of a cricket ball", Horlock, J. H., in "Mechanics and
sport", ed. J. L. Bleustein, pp. 293-303. New York, ASME 1973.
"Factors affecting cricket ball swing", Mehta, R. D., Bentley, K.,
Proudlove, M., Varty, P., Nature, 303: pp. 787-788, 1983.
"Aerodynamics of a cricket ball", Sherwin, K., Sproston, J. L., Int. J.
Mech. Educ. 10: pp. 71-79. 1982.
http://web.archive.org/web/20071018203238/http://www.geocities.com/k_ac
PCD_STiCM
60
hutarao/MAGNUS/magnus.html#mehta
Olympic – HMS Hawke collision: 20 September
1911, off the Isle of Wight. Large displacement of
water by Olympic sucked in the Hawke into her
side. One crew member of the Olympic, Violet Jessop, survived the
collision with the Hawke, and also the later sinking of Titanic , and the
1916 sinking of Britannic, the third ship of the class.
"Popular Mechanics" Magazine December 1911
http://en.wikipedia.org/wiki/File:Hawke_-_Olympic_collision.JPG
PCD_STiCM
http://en.wikipedia.org/wiki/RMS_Olympic
61
Next: Unit 10
Classical Electrodynamics
Charles
Coulomb
1736-1806
Carl Freidrich
Gauss
1777-1855
Andre Marie
Ampere
PCD_STiCM
1775-1836
Michael
Faraday
1791-186762
Electrodynamics & STR
The special theory of relativity is intimately linked to the
general field of electrodynamics. Both of these topics belong
to ‘Classical Mechanics’.
James Clerk Maxwell
1831-1879
PCD_STiCM
Albert Einstein
1879 - 1955
63

E 
0
B  0
B
 E  
t
James Clerk Maxwell
1831-1879
Divergence
and Curl of
E, B
 J 1  E We shall take a break here.
  B  0
 2
t c t Questions ? Comments ?
1
c
 0 0
Helmholtz Theorem
Curl & Divergence;
+ boundary conditions
[email protected]
PCD_STiCM
[email protected]
64