Unit 14:
Fluid Flow, Bernoulli’s Principle
Definition of circulation, curl, vorticity,
irrotational flow.
Steady flow.
Bernoulli’s equation/principle, some illustrations.
PCD-08
Recall the discussion on directional derivative
r
dψ
= uˆ • ∇ψ
ds
uur uur
δ r dr
=
uˆ = lim
δ s →0 δ s
ds
uur
δ s = δ r , tiny
Gradient: direction in
which the function varies
fastest / most rapidly.
ur
r
F = −∇ψ
Force: Negative gradient of
the potential
increament
uur
ds = dr , differential
increament
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‘negative’ sign is the result
of our choice of natural
motion as one occurring
from a point of ‘higher’
potential to one at a ‘lower’
potential.
ur
r
F = ma
ur
r
Is it obvious that the ‘force’ defined by these F = −∇ψ
two equations is essentially the same?
Is it obvious that the ‘force’ defined by
these two equations is essentially the
same?
The compatibility of the two expressions
emerges if, and only if,
the potential ψ is defined in such way
that the work done by the force given by
r
−∇ψ in displacing the object on which this
force acts, is independent of the path
along which the displacement occurred.
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Consistency in
these relations
exists only for
‘conservative’
forces.
ur uur
∫ F • dr = 0
ur uur
∫ F • dr is
b
a
INDEPENDENT
of the path
a to b
PATH INTEGRAL
“CIRCULATION”
ur uur
∫ F • dr = 0
It is only when the line integral of the work done is
path independent that the force is conservative
ur uur
∫ F • dr is
b
and accounts for the acceleration it generates
when it acts on a particle of mass m through the
a
INDEPENDENT
of the path
causality relation of
Newtonian mechanics:
ur
r
F = ma
a to b
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
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ur
F , denoted as
ur ur
∇ × F.
ur ur
∇ × F is a vector point function
Definition of Curl of a vector:
ur r
of the vector field F ( r ) such that for an orthonormal basis set
r
of unit vectors
Mean (average) ‘circulation’
{uˆ (r ), i = 1, 2,3} ,
i
r ur ur r
uˆi (r ) • ∇× F (r ) = lim
ur r uur
∫ F (r ) • dr
per unit area taken at the
point when the elemental
,
area becomes infinitesimally
Δs
small.
where the path integral is taken over a closed path C, taken
over a tiny closed loop C which bounds an elemental vector
uuur
r
surface area ΔS = ΔS uˆi (r ), the direction of the unit vector
ΔS →0
being that in which a right-hand screw would propogate
forward when turned along the sense in which the path
integral is determined.
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r ur ur r
uˆi (r ) • ∇× F (r ) = lim
ur r uur
∫ F (r ) • dr
ΔS →0
r
uˆi (r ), i = 1, 2,3
{
}
Cartesian unit vectors, of course,
do not change from point to point.
They are constant vectors.
Δs
In general, the unit
vectors may depend
on the particular point
under discussion, and
hence written as
r
r
functions uˆi (r ) of r.
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.
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circulation and curl
circulation =
∫
r r r
A(r ).dl
C
Consider an open surface S, bounded by a closed curve C.
Circulation depends on the value of the vector at all points on C and
hence is not a scalar field; it is however a scalar quantity even if not a
scalar field which must be described/defined through a scalar point
function.
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.
∫
lim
δ S→0
C
r r r
A ( r ).dl
δS
r r
= ( ∇ × A ).nˆ
limiting circulation per unit area
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C
This limiting ratio defines the
component of the curl of the
vector field; the curl itself is
defined through three
orthonormal components in
the basis nˆ1 , nˆ2 , nˆ3
{
}
r
( curl A ).nˆ = lim
δ S→0
∫
C
r r r
A ( r ).dl
δS
r r
= ( ∇ × A ).nˆ
limiting circulation per unit area
Curl measures how much the vector
“curls” around at a point
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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.
ur r r
∇×F=0
If
in a region then there would be no
curliness/rotation, and the field is called irrotational.
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r
( curl A ).nˆ = lim
∫
C
δ S→0
r r r
A ( r ).dl
δS
r r
= ( ∇ × A ).nˆ
Remember!
The criterion that a force field is conservative is that its path
integral over a closed loop (i.e. “circulation”) is zero. This is
ur r r
equivalent to the condition that ∇×F=0
ur r r
If ∇×F=0 in a region, then there would be no curliness
(rotation), and the field is called irrotational.
Conservative force fields are IRROTATIONAL
Examples for irrotational fields: electrostatic, gravitational
PCD-08
Y
3rd leg
Curl in Cartesian Co-ordinate
δx
Consider a point P( x0 , y0 , z0 ).
2nd
4thδ y
δy
ur r
leg
leg
Consider a vector field A(r ) in some region of space.
δx
eˆz 1st leg
Circulation over the peremeter of an elemental surface
to which the normal is eˆz is
⎡ ⎛
δy ⎞
δ y ⎞⎤
⎛
x
y
z
x
y
−
−
+
A
,
,
A
,
, z0 ⎟ ⎥ δ x +
x ⎜ 0
0
⎢ x⎜ 0 0 2 0⎟
2
⎠
⎝
⎠⎦
⎣ ⎝
δx
δx
⎡ ⎛
⎞
⎛
⎞⎤
x
y
z
x
y
z
+
−
−
A
,
,
A
,
,
y⎜ 0
0
0 ⎟⎥ δ y
⎢ y⎜ 0 2 0 0⎟
2
⎠
⎝
⎠⎦
⎣ ⎝
∫
C
r r r
∂Ay
∂Ax
A(r ).dl = −
δ yδ x +
δ xδ y
∂y
∂x
X
3rd leg
Z
4th
leg
2nd
leg
1st leg
Closed path C
which bounds
an elemental
surface
r
r
∂Ax ∂Ay
⇒ (curl A).eˆz = −
+
= (curlA) z Make sure that you
∂y
∂x
understand the signs ±
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r
r r
∂Ay
∂Ax
∫C A(r ) • dl = − ∂y δ yδ x + ∂x δ xδ y
Determining now the net circulation per unit area:
ur r
r
∂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:
ur
⎛ ∂Az ∂Ay ⎞
⎛ ∂Ay ∂Ax ⎞
⎛ ∂Ax ∂Az ⎞
−
−
−
Curl A = eˆx ⎜
⎟ + eˆy ⎜
⎟
⎟ + eˆz ⎜
⎝ ∂z ∂x ⎠
⎝ ∂y ∂z ⎠
⎝ ∂x ∂y ⎠
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ur
⎛ ∂Az ∂Ay ⎞
⎛ ∂Ay ∂Ax ⎞
⎛ ∂Ax ∂Az ⎞
−
−
−
Curl A = eˆx ⎜
⎟ + eˆy ⎜
⎟
⎟ + eˆz ⎜
∂
∂
∂
∂
∂
∂
z ⎠
x ⎠
y ⎠
⎝ z
⎝ y
⎝ x
The Cartesian expression for curl of a vector
field can be expressed as a determinant; but it
is, of course, not a determinant!
curl
eˆx
ur r ur ∂
A = ∇× A =
∂x
Ax
eˆy
∂
∂y
Ay
eˆz
∂
∂z
Az
For example, can you just interchange the 2nd and the 3rd row and
change the sign of this ‘determinant’? Besides, the curl is not a
cross product of two vectors; the gradient is a vector operator!
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examples of rotational fields, nonzero curl
z
r
V ( x, y ) = − yeˆx + xeˆ y
r
A(x,y)=(x-y)eˆ x + ( x + y )eˆy
r r
∇ × V = 2eˆz
r r
∇ × A = 2eˆ z
Curl here is directed along the positive z-axis
PCD-08
example of rotational fields, nonzero curl
z
y
x
PCD-08
v = xeˆ y
r
∇ × v = êz
What is the DIVERGENCE and the CURL of the following
vector field?
Reference: Berkeley Physics Course, Volume I
PCD-08
What is the DIVERGENCE and the CURL of the following
vector field?
Reference: Berkeley Physics Course, Volume I
PCD-08
What is the DIVERGENCE and the CURL of the following
vector field?
Reference: Berkeley Physics Course, Volume I
PCD-08
What is the DIVERGENCE and the CURL of the following
vector field?
Reference: Berkeley Physics Course, Volume I
PCD-08
What is the DIVERGENCE and the CURL of the following
vector field?
Reference: Berkeley Physics Course, Volume I
PCD-08
What is the DIVERGENCE and the CURL of the following
vector field?
Reference: Berkeley Physics Course, Volume I
PCD-08
ur
∂φ
∂φ
∂φ
∇φ = eˆx + eˆy + eˆz
∂x
∂y
∂z
curl of a gradient is zero
eˆ x
∂
∇ × ∇φ =
∂x
∂φ
∂x
eˆ y
∂
∂y
∂φ
∂y
eˆ z
∂
∂z
∂φ
∂z
⎛ ∂2φ ∂2φ ⎞ ⎛ ∂2φ ∂2φ ⎞ ⎛ ∂2φ ∂2φ ⎞
⎟⎟ + eˆy ⎜⎜
⎟⎟ + eˆz ⎜⎜
⎟⎟
∇×∇φ = eˆx ⎜⎜
−
−
−
⎝ ∂y∂z ∂z∂y ⎠ ⎝ ∂z∂x ∂x∂z ⎠ ⎝ ∂x∂y ∂y∂x ⎠
r
=0
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ur
⎛d ⎞ ⎛d ⎞
⎜ ⎟ =⎜ ⎟ + ω x
⎝ dt ⎠ I ⎝ dt ⎠ R
Refer: Page33 of Unit 7
ur r
⎛d ⎞ r ⎛d ⎞ r
⎜ ⎟ r =⎜ ⎟ r + ω x r
⎝ dt ⎠ I
⎝ dt ⎠ R
⎛d ⎞ r r
When ⎜ ⎟ r =0,
⎝ dt ⎠ R
ur r
⎛d ⎞ r r
⎜ ⎟ r = vI = ω x r
⎝ dt ⎠ I
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⎛ d ⎞ r ur r
⎜ ⎟ r =ω x r
⎝ dt ⎠ I
ur r
⎛d ⎞ r r
⎜ ⎟ r = vI = ω x r
⎝ dt ⎠ I
r r r
v =ω×r
eˆx eˆy eˆz
r r r ur r
r
∇ × v = ∇ × (ω × r ) = ∇ × ωx ω y ωz
x
y
z
r
= ∇ × ⎡⎣(ω y z − ωz y )iˆ + (ωz x − ωx z )eˆy + (ωx y − ω y x)eˆz ⎤⎦
eˆx
eˆy
eˆz
∂
∂
∂
=
∂x
∂y
∂z
(ω y z − ωz y ) (ω z x − ω x z ) (ω x y − ω y x)
r r
r
∇ × v = 2(ωx eˆx + ω y eˆy + ωz eˆz ) = 2ω
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The ‘curl’ of
the linear
velocity gives
you a
measure of
the angular
velocity; thus
justifying the
term ‘curl’.
Remind yourself of the preview we had shown in L0,
the 0th lecture:
The component of the curl of a vector field in the direction
is the circulation of the vector field per unit area about
the axis
,
or the amount to which a particle being carried by the vector
field is being rotated about
.
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In that 0th lecture, we mentioned that we shall lead
you to the STOKES THEOREM:
William Thomson,
∫
r
r r
A ( r ) • dl =
r r uur
∫∫ ( ∇ × A ) • dS
1st Baron Kelvin
(1824-1907)
Note!
It is STOKES’
THEOREM
not
STOKE’S THEOREM
0
K temperature
George Gabriel
Stokes
(1819–1903)
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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.
Reference: http://www.123exp-math.com/t/01704066342/
Proof of Stokes’ theorem follows from the very definition of the curl:
r
D efinition : ( curl A ) • nˆ = lim
∫
r
r r
A ( r ) • dl
C
δ S→0
δS
r r
= ( ∇ × A ) • nˆ
For a path δC, which binds an area δS,
∫δ
r r
r
A • dl = δ S (curl A) • nˆ
C
A finite area S is enclosed by C, we can split up S into
infinitesimal bits δSn bounded by tiny curves δCn
ur r uur n
∫ A(r ) • dl = ∑
c
i =1
∫δ
ci
ur r uur n
ur r uur
A(r ) • dl = ∑ ∫∫ curl A(r ) • dS
i =1
ur r uur
ur r uur
∫ A(r ) • dl = ∫∫ curl A(r ) • dS Stokes’ theorem
c
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consider a surface S enclosed by a curve C
Stokes’ theorem
ur r uur
ur r uur
A
(
r
)
•
dl
=
curl
A
(
r
)
•
dS
∫
∫∫
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 bounded by the closed curve will work; you
can pinch the butterfly net and distort the shape of the
net any which way – it won’t matter!
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consider a surface S enclosed by a curve C
Stokes’ theorem
ur r uur
ur r uur
A
(
r
)
•
dl
=
curl
A
(
r
)
•
dS
∫
∫∫
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.
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C traversed
the other way
Are there 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-08
Consider a rectangular
strip of paper, spread flat
at first, and given two
colors on opposite sides.
Now, flip it and paste the
short edges on each other
as shown.
Is the resulting object
three-dimensional?
How many ‘sides’ does it
have?
How many ‘edges’ does
it have?
PCD-08
The surface under
consideration, however, better
be a ‘well-behaved’ surface!
Expression for ‘curl’ in cylindrical polar coordinate system
{eˆ , eˆ , eˆ }
ρ
ϕ
z
r r
∇× A =
⎡ ∂
1 ∂
∂⎤
⎢eˆ ρ ∂ρ + eˆ ϕ ρ ∂ϕ + eˆ z ∂z ⎥ × ⎡⎣ eˆ ρ Aρ ( ρ , ϕ , z ) + eˆ ϕ Aϕ ( ρ , ϕ , z ) + eˆ z Az ( ρ , ϕ , z ) ⎤⎦
⎣
⎦
r r
∇ × A = ⎡⎢eˆ ρ ∂ ⎤⎥ × ⎡eˆ ρ Aρ ( ρ , ϕ , z ) + eˆ ϕ Aϕ ( ρ , ϕ , z ) + eˆ z Az ( ρ , ϕ , z ) ⎤ +
⎣
⎦
ρ
∂
⎣
⎦
⎡ 1 ∂ ⎤
⎢ eˆ ϕ ρ ∂ϕ ⎥ × ⎡⎣ eˆ ρ Aρ ( ρ , ϕ , z ) + eˆ ϕ Aϕ ( ρ , ϕ , z ) + eˆ z Az ( ρ , ϕ , z ) ⎤⎦ +
⎣
⎦
⎡ˆ ∂ ⎤ ˆ
⎢⎣e z ∂z ⎥⎦ × ⎡⎣e ρ Aρ ( ρ , ϕ , z ) + eˆ ϕ Aϕ ( ρ , ϕ , z ) + eˆ z Az ( ρ , ϕ , z ) ⎤⎦
PCD-08
Expression for ‘curl’ in cylindrical polar coordinate system
{eˆ , eˆ , eˆ }
ρ
ϕ
z
r r
∂
ˆ
⎡
⎤
⎡ eˆ ρ Aρ ( ρ , ϕ , z ) + eˆ ϕ Aϕ ( ρ , ϕ , z ) + eˆ z Az ( ρ , ϕ , z ) ⎦⎤ +
∇ × A = ⎣e ρ ⎦ ×
⎣
∂ρ
⎛1 ∂ ⎞
⎣⎡ eˆ ϕ ⎦⎤ × ⎜ ρ ∂ϕ ⎟ ⎣⎡ eˆ ρ Aρ ( ρ , ϕ , z ) + eˆ ϕ Aϕ ( ρ , ϕ , z ) + eˆ z Az ( ρ , ϕ , z ) ⎦⎤ +
⎝
⎠
∂
ˆ
[e z ] × ⎡⎣eˆ ρ Aρ ( ρ , ϕ , z ) + eˆ ϕ Aϕ ( ρ , ϕ , z ) + eˆ z Az ( ρ , ϕ , z ) ⎤⎦
∂z
⎛ 1 ∂Az ∂Aϕ ⎞
⎛ ∂Aρ ∂Az ⎞
1 ⎡ ∂ ( ρ Aϕ ) ∂Aρ ⎤
−
−
−
⎥
= eˆ ρ ⎜
⎟ + eˆ ϕ ⎜
⎟ + eˆ z ⎢
∂z ⎠
∂ρ ⎠
∂ϕ ⎥
ρ ⎢⎣ ∂ρ
⎝ ρ ∂ϕ
⎝ ∂z
⎦
PCD-08
Expression for ‘curl’ in spherical polar coordinate system
{eˆ , eˆ , eˆ }
r
θ
ϕ
r r
∇ × A=
⎛ ∂
1 ∂
1
∂ ⎞
ˆ
ˆ
ˆ
= ⎜ er
+ eθ
+ eϕ
⎟ × ( eˆ r Ar (r , θ , ϕ ) + eˆ θ Aθ (r ,θ , ϕ ) + eˆ ϕ Aϕ (r ,θ , ϕ ) )
r ∂θ
r sin θ ∂ϕ ⎠
⎝ ∂r
r r
∇ × A=eˆ r
1
r sin θ
⎧ ∂
∂Aθ ⎫
⎨ ( sin θ Aϕ ) −
⎬
∂ϕ ⎭
⎩ ∂θ
⎫
1 ⎧ 1 ∂Ar ∂
+eˆ θ ⎨
− ( rAϕ ) ⎬
r ⎩ sin θ ∂ϕ ∂r
⎭
PCD-08
∂Ar ⎫
1⎧ ∂
+ eˆ ϕ ⎨ ( rAθ ) −
⎬
∂θ ⎭
r ⎩ ∂r
Do you like the earring? The Möbius strip used to be
It is
It is MOBIUS !!!
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common in belt drives (like a
car fan belt). With an ordinary
belt only the inside of the belt
was in contact with the
wheels, so it would wear out
before the outside did. Since
a Möbius strip has only one
side, the wear and tear on
the at
beltall
wasobvious….
spread out more
not
evenly and they would last
longer. However, modern
belts are made from several
layers of different materials,
with a definite inside and
outside, and do not have a
twist.
http://mathssquad.questacon.edu.au/mobius_strip.html
M.C. Escher
Maurits
Cornelius
Escher
1898 - 1972
http://www.mcescher.com/
“The laws of mathematics are not merely human inventions or creations. They
PCD-08
simply 'are'; they exist quite independently of the human intellect. The most that
any(one) ... can do is to find that they are there and to take cognizance of them. ”
Another important identity: divergence of a curl is zero
r
r r r
r r
By divergence theorem ∫ ∇ ⋅ (∇ × A) dτ = ∫ (∇ × A) ⋅ dS
V
S
S2
V
S
S1
c1
Applying Stoke’s theorem
r
r r
r r
r r
∫∫ (∇ ×A) ⋅ dS = ∫∫ (∇ × A) ⋅ nˆ1 dS 1 + ∫∫ (∇ × A) ⋅ nˆ 2 dS 2
S
S1
r r
r r
= ∫ A ⋅ dl + ∫ A ⋅ dl = 0
C1
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C2
S2
c2
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 in PH102 to study
Maxwell’s equations which provide the curl and the
divergence of the electromagnetic field.
The ‘curl’ finds direct application also in the derivation
of the Bernoulli’s principle, as shown below.
PCD-08
Nicolaus
Bernoulli’s
2 sons
Bernoulli
brothers
Bernoulli’s
Principle
Bernoulli
Family
Math/Phys Tree
It was Johann’s work that the Marquis de
I;Hospital (1661-1704), under a curious
financial agreement with Johann, assembled in
1696 into the first calculus textbook. In this way,
the familiar method of evaluating the
indeterminate form 0/0 became incorrectly
known in later calculus texts as I'Hospital's rule.
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Posed the
brachistrocron
e problem
Reference:
http://www.york.ac.uk/depts/maths/histstat/people/bernoulli_tree.htm
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-08
∂⎤
d v(r , t ) − ∇ p
⎡
=
− ∇φ
⎢⎣ v • ∇ + ∂t ⎥⎦ v ( r , t ) =
dt
ρ (r )
Result of page
47,
Unit 11+12+13:
(
)
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 +
=
− ∇φ
2
∂t
ρ (r )
PCD-08
( )
(
)
1
∂v − ∇ p
=
− ∇φ
∇ v• v − v× ∇× v +
2
∂t
ρ (r )
⎧⎪ p ( r ) ⎫⎪
Now ,
≈ ∇⎨
⎬
⎪⎩ ρ ⎪⎭
ρ (r )
∇ p (r )
( )
(
)
( )
(
)
⎧⎪ p ( r ) ⎫⎪
∂v
1
∇ v• v − v× ∇× v +
= −∇ ⎨
⎬ − ∇φ
∂t
2
⎪⎩ ρ ⎪⎭
⎧⎪ p ( r )
⎫⎪
∂v
1
= −∇ ⎨
+φ⎬
∇ v• v − v× ∇× v +
∂t
2
⎪⎩ ρ
⎪⎭
PCD-08
( )
(
)
⎧⎪ p ( r )
⎫⎪
∂v
1
∇ v• v − v× ∇× v +
= −∇ ⎨
+φ⎬
∂t
2
⎪⎩ ρ
⎪⎭
ur r
⎛d ⎞ r ⎛d ⎞ r
Recall that : ⎜ ⎟ r = ⎜ ⎟ r + ω x r
⎝ dt ⎠ I
⎝ dt ⎠ R
r
r
ur r
r
v I = v R + ω x r R , where v I is just the
velocity that is employed in the equation
of motion for the fluid.
ur r ur r
ur ur r
∴ ∇ × vI = ∇ × vR + ∇ × ω x r R
ur ur r
To determine ∇ × ω x r R
we now use another vector
{
}
{
}
Identity, for the curl of cross-product of two vectors:
(
) (
) (
)
∇ × A × B = B • ∇ A − A • ∇ B + A (∇ • B ) − B (∇ • A )
PCD-08
(
) ( ) ( )
∇ × (ω × r ) = (r • ∇ )ω − (ω • ∇ )r + ω (∇ • r ) − r (∇ • ω )
∇ × (ω × r ) = − (ω • ∇ )r + ω (∇ • r ) = 2ω
∇ × A × B = B • ∇ A − A • ∇ B + A (∇ • B ) − B (∇ • A )
R
R
R
R
R
R
R
R
∇ × v I = ∇ × v R + 2ω
In the rotating frame , v R = 0,
hence , the VORTICITY, ∇ × v I = χ = 2ω
( )
(
)
⎧⎪ p ( r )
⎫⎪
∂v
1
∇ v• v − v× ∇× v +
= −∇ ⎨
+φ⎬
∂t
2
⎪⎩ ρ
⎪⎭
PCD-08
( )
(
)
⎧⎪ p ( r )
⎫⎪
∂v
1
∇ v• v − v× ∇× v +
= −∇ ⎨
+φ⎬
∂t
2
⎪⎩ ρ
⎪⎭
2
⎧⎪ p ( r )
⎫⎪
1
∂v
∇ v − v× χ +
= −∇ ⎨
+φ⎬
2
∂t
⎪⎩ ρ
⎪⎭
2
⎧⎪ p ( r )
⎫⎪ 1
∂v
− v × χ = −∇ ⎨
+φ⎬− ∇ v
∂t
⎪⎩ ρ
⎪⎭ 2
2
⎧
⎫
v
∂v
⎪ p(r )
⎪
− v × χ = −∇ ⎨
+φ +
⎬
2 ⎪
∂t
⎪ ρ
⎩
⎭
For ‘STEADY STATE’
2
⎧
⎫
v
⎪ p(r )
⎪
− v × χ = −∇ ⎨
+φ +
⎬
2 ⎪
⎪ ρ
⎩
⎭
2
⎧
⎫
v
⎪ p(r )
⎪
0 = v • ∇⎨
+φ +
⎬
2 ⎪
⎪ ρ
⎩
⎭
PCD-08
∂v
=0
∂t
2
⎧
⎫
v
⎪ p(r )
⎪
0 = v • ∇⎨
+φ +
⎬
2 ⎪
⎪ ρ
⎩
⎭
2
⎧
⎫
v
⎪ p (r )
⎪
⇒ ∇⎨
+φ +
⎬ must be ORTHOGONAL to v,
2 ⎪
⎪ ρ
⎩
⎭
2
⎧
⎫
v
⎪ p(r )
⎪
+φ +
i.e., ∇ ⎨
⎬ must be ORTHOGONAL to STREAMLINES
2 ⎪
⎪ ρ
⎩
⎭
⇒ ∇Ψ must be ORTHOGONAL to streamlines,
2
where Ψ =
p(r )
ρ
+φ +
v
2
.
2
⇒Ψ=
PCD-08
p (r )
ρ
+φ +
v
2
= constant for a given streamline
Daniel Bernoulli’s Theorem
2
⇒Ψ=
p (r )
ρ
+φ +
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 × χ = −∇ ⎨
+φ +
⎬
ρ
2
⎪
⎪
⎩
⎭
If the fluid flow is both ‘steady state’ and ‘irrotational’,
2
⇒Ψ=
PCD-08
p(r )
ρ
+φ +
∇× v = χ = 0
v
2
Daniel Bernoulli’s Theorem
is constant for the entire velocity field in the liquid.
Mass Current Density Vector J ( r , t ) =
ρ (r , t )v(r , t )
Dimensions : ML − 2T −1
G
For Steady State Flow,
ρvA = constant ,
A : cross - sectional area
since,
∫∫∫
volume
region
ur ur r
dτ ∇ • J (r ) =
{
} ∫∫
surface
enclosing
that
region
D
H
P
F
E
r
B
A
ur r uur
∂ρ
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:
δ W 1 = F1δ s = p 1 A1δ s = p 1 A1 v 1δ t
Work done by the fluid on Face 2 is:
δ W 2 = F2 δ s = p 2 A 2 δ s = p 2 A 2 v 2 δ t
PCD-08
C
Net work done on the fluid in the
parallelepiped by the pressure
that the fluid exerts on Faces 1
& 2 is:
δW1 − δW2 = p1 A1v1δt − p2 A2 v 2δt
Net work done on the fluid in the
parallelepiped by the pressure
that the fluid exerts at Faces 1 & 2 :
δW1 −δW2 = p1 A1v1δt − p2 A2v2δt
Energy gained per unit mass by the
fluid as it traverses the x-axis of the E 2 − E 1 =
parallelepiped across the Faces 1 & 2 :
[ p1 A1 v 1 −
p 2 A2 v 2 ]δ t
δm
[
p1 A1 v 1 −
=
p 2 A2 v 2 ]δ t
E 2 − E1
δm
⎡1 2
⎤
⎡1 2
⎤
= ⎢ v + φ + U int ernal ⎥ − ⎢ v + φ + U int ernal ⎥
⎣2
⎦2 ⎣2
⎦1
Thus ,
[ p1 A1 v 1 − p 2 A2 v 2 ]δ t
ρ (δ sA )
PCD-08
⎡1 2
⎤
⎡1 2
⎤
= ⎢ v + φ + U int ernal ⎥ − ⎢ v + φ + U int ernal ⎥
⎣2
⎦2 ⎣2
⎦1
[ p1 A1 v 1 − p 2 A2 v 2 ]δ t
ρ (δ sA )
⎤
⎡1 2
⎤
⎡1 2
= ⎢ v + φ + U int ernal ⎥ − ⎢ v + φ + U int ernal ⎥
⎦1
⎦2 ⎣2
⎣2
⎡ p1 A1 v 1
p 2 A2 v 2 ⎤
⎡1 2
⎤
⎡1 2
⎤
δ
t
v
φ
U
v
φ
U
−
=
+
+
−
+
+
int ernal ⎥
int ernal ⎥
⎢
⎥
⎢⎣ 2
⎢
(
)
(
)
ρ
v
δ
t
A
ρ
v
δ
t
A
2
⎦2 ⎣
⎦1
1
1
2
2 ⎦
⎣
⎡ p1 p 2 ⎤ ⎡ 1 2
⎤
⎡1 2
⎤
φ
φ
U
U
v
v
−
+
+
=
+
+
−
int ernal ⎥
int ernal ⎥
⎥ ⎢2
⎢ρ
⎢2
ρ
⎦1
⎣
⎦
⎣
⎦
⎣
2
⎡1
⎡1
p⎤
p⎤
0 = ⎢ v 2 + φ + U int ernal + ⎥ − ⎢ v 2 + φ + U int ernal + ⎥
ρ ⎦2 ⎣2
ρ ⎦1
⎣2
p
1 2
From page 47 :
i .e.
v + φ + U internal +
= constant
ρ
2
Daniel Bernoulli’s Theorem
PCD-08
Ψ=
p(r )
ρ
+φ +
is constant for the entire velocity field in the liquid.
2
v
2
2
p (r )
ρ
+φ +
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.
Debashsish Mohanty
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-08
ρ
∇•E =
ε0
∇•B = 0
∂B
∇× E = −
∂t
∂J 1 ∂E
∇ × B = μ0
+ 2
∂t c ∂t
1
c=
μ 0ε 0
Helmholtz Theorem
PH102: PHYSICS II
PCD-08
James Clerk Maxwell
1831-1879
Divergence
and Curl of
(E, B )