Unit 14: Fluid Flow, Bernoulli’s Principle

Definition of circulation, curl, vorticity, irrotational flow.

Steady flow. Bernoulli’s equation/principle, some illustrations.

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Recall the discussion on directional derivative

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s 0

ˆ

ˆ lim

, tiny

increament

, differential

increament

d uds

r drus ds

s r

ds dr

δ

ψ ψ

δδ

δ δ

→

= •∇

= =

=

=

r

uur uur

uur

uur

Gradient: direction in which the function varies fastest / most rapidly.

F ψ= −∇ur r

Force: Negative gradient of the potential

‘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.

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F ma=ur r

F ψ= −∇ur r

Is it obvious that the ‘force’ defined by these 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 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.

ψ

ψ−∇r

0

b

a

F dr

F dr is

INDEPENDENTof the patha to b

• =

•

∫

∫

ur uur

ur uur

Consistency in these relations exists only for ‘conservative’forces.

PATH INTEGRAL“CIRCULATION”

0

b

a

F dr

F dr is

INDEPENDENTof the patha to b

• =

•

∫

∫

ur uur

ur uur

It is only when the line integral of the work done is path independent that the force is conservative and accounts for the acceleration it generates when it acts on a particle of mass m through the causality relation of Newtonian mechanics: F ma=

ur r

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 , denoted as .F

urF∇×

ur ur

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F∇×ur ur

is a vector point functionDefinition of Curl of a vector:

0

( )ˆ ( ) ( ) lim ,

where the path integral is taken over a closed path C, taken over a tiny closed loop C which bounds an elemental vector

ˆ ( ), d

i S

i

F r dru r F r

s

surface area S S u r the

Δ →

••∇× =

Δ

Δ = Δ

∫ur r uur

r ur ur r

uuur rirection of the unit vector

being that in which a right-hand screw would propogate forward when turned along the sense in which the path integral is determined.

( )F rur r

{ }ˆ ( ), 1, 2,3 ,iu r i =r

of the vector field such that for an orthonormal 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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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.

0

( )ˆ ( ) ( ) limi S

F r dru r F r

sΔ →

••∇× =

Δ∫ur r uur

r ur ur r

{ }ˆ ( ), 1, 2,3iu r i =r

general, the unit vectors may depend on the particular point under discussion, and hence written as

ˆfunctions ( ) of .i

In

u r rr r

Cartesian unit vectors, of course, do not change from point to point.

They are constant vectors.

Mean (average) ‘circulation’ per unit area taken at the point when the elemental area becomes infinitesimally small.

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( ).C

circulation A r dl= ∫rr r

circulation and curl

0

( ).ˆlim ( ).C

S

A r dlA n

Sδ δ→= ∇ ×

∫rr r

rr

limiting circulation per unit area

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 pointfunction.

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.

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 { }1 2 3ˆ ˆ ˆ, ,n n n

C

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Curl measures how much the vector “curls” around at a point

0

( ).ˆ ˆ( ). lim ( ).C

S

A r dlcurl A n A n

Sδ δ→= = ∇ ×

∫rr r

r rr

limiting circulation per unit area

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If in a region then there would be no curliness/rotation, and the field is called irrotational.

0 F∇× =ur rr

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.

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If in a region, then there would be no curliness (rotation), and the field is called irrotational.

0 F∇× =ur rr

Conservative force fields are IRROTATIONAL

Examples for irrotational fields: electrostatic, gravitational

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0

( ).ˆ ˆ( ). lim ( ).C

S

A r dlcurl A n A n

Sδ δ→= = ∇ ×

∫rr r

r rr

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

equivalent to the condition that 0 F∇× =ur rr

x 0 0 0 x 0 0 0

y 0 0 0 y 0 0 0

Circulation over the peremeter of an elemental surface ˆto which the normal is is

A , , A , ,2 2

A , , A , ,2 2

ze

y yx y z x y z x

x xx y z x y z y

δ δ δ

δ δ δ

⎡ ⎤⎛ ⎞ ⎛ ⎞− − + +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞+ − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Curl in Cartesian Co-ordinate0 0 0Consider a point ( , , ).

Consider a vector field ( ) in some region of space.

P x y z

A rur r

Z

X

Y

xδ

xδ

yδyδ

1st leg

2nd

leg

3rd leg

( ). yx

C

AAA r dl y x x yy xδ δ δ δ

∂∂= − +

∂ ∂∫rr r

ˆ( ). ( )yxz z

AAcurl A e curlAy x

∂∂⇒ = − + =

∂ ∂

r r

4th

legˆze

Closed path C which bounds an elemental surface

1st leg

2nd

leg

3rd leg4th

leg

Make sure that you understand the signs ±

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Similarly if we get circulation per unit area along other two orthogonal closed paths and add up, we get:

ˆ ˆ ˆy yx xz zx y z

A AA AA ACurl A e e ey z z x x y

∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

ur

( ) yx

C

AAA r dl y x x yy xδ δ δ δ

∂∂• = − +

∂ ∂∫rr r

Determining now the net circulation per unit area:

ˆ( ) ( )yxz z

AAcurl A e Ay x

∂∂• = − + = ∇×

∂ ∂

urr r

Color coded arrows are unit vectors orthogonal to the three mutually orthogonal surface elements bounded by their perimeters.

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The Cartesian expression for curl of a vector field can be expressed as a determinant; but it is, of course, not a determinant!

ˆ ˆ ˆy yx xz zx y z

A AA AA ACurl A e e ey z z x x y

∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

ur

ˆ ˆ ˆx y z

x y z

e e e

Ax y z

A A A

∂ ∂ ∂∇× =

∂ ∂ ∂

urrcurl A =

ur

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

yx exeyyxV ˆˆ),( +−=r

zeV ˆ2=×∇rr

xˆ ˆA(x,y)=(x-y)e ( ) yx y e+ +r

zˆA 2e∇× =rr

Curl Curl here is directed along the positive z-axis

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x

y

z

yex ˆv =

zev =×∇r

example of rotational fields, nonzero curl

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Reference: Berkeley Physics Course, Volume I

What is the DIVERGENCE and the CURL of the following vector field?

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Reference: Berkeley Physics Course, Volume I

What is the DIVERGENCE and the CURL of the following vector field?

Reference: Berkeley Physics Course, Volume IPCD-08

What is the DIVERGENCE and the CURL of the following vector field?

Reference: Berkeley Physics Course, Volume IPCD-08

What is the DIVERGENCE and the CURL of the following vector field?

Reference: Berkeley Physics Course, Volume IPCD-08

What is the DIVERGENCE and the CURL of the following vector field?

Reference: Berkeley Physics Course, Volume IPCD-08

What is the DIVERGENCE and the CURL of the following vector field?

ˆ ˆ ˆx y ze e ex y zφ φ φφ ∂ ∂ ∂

∇ = + +∂ ∂ ∂

ur

zyx

zyx

eee zyx

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=∇×∇

φφφ

φ

ˆˆˆ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂−

∂∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂−

∂∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂−

∂∂∂

=∇×∇xyyx

ezxxz

eyzzy

e zyxφφφφφφφ

222222

ˆˆˆ

0r

=

curl of a gradient is zero

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x I R

d ddt dt

ω⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

ur

x I R

d dr r rdt dt

ω⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

r r ur r

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Refer: Page33 of Unit 7

When =0,R

d rdt

⎛ ⎞⎜ ⎟⎝ ⎠

r r

Iv = x I

d r rdt

ω⎛ ⎞ =⎜ ⎟⎝ ⎠

r r ur r

x I

d r rdt

ω⎛ ⎞ =⎜ ⎟⎝ ⎠

r ur r

v rω= ×rr r

v ( )rω∇× = ∇× × =r ur rr r ˆ ˆ ˆx y z

x y z

e e e

x y zω ω ω∇×

r

ˆ ˆ ˆ( ) ( ) ( )y z z x y x y zz y i x z e y x eω ω ω ω ω ω⎡ ⎤= ∇× − + − + −⎣ ⎦r

ˆ ˆ ˆ

( ) ( ) ( )

x y z

y z z x x y

e e e

x y zz y x z y xω ω ω ω ω ω

∂ ∂ ∂=

∂ ∂ ∂− − −

ˆ ˆ ˆ2( ) 2x x y y z ze e eω ω ω ω+ + =r

v∇× =r r

Iv = x I

d r rdt

ω⎛ ⎞ =⎜ ⎟⎝ ⎠

r r ur r

The ‘curl’ of the linear velocity gives you a measure of the angular velocity; thus justifying the term ‘curl’.

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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 aboutthe axis , or the amount to which a particle being carried by the vector field is being rotated about .

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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/

William Thomson, 1st Baron Kelvin

(1824-1907)

George Gabriel Stokes

(1819–1903)

In that 0th lecture, we mentioned that we shall lead you to the STOKES THEOREM:

Note! It is STOKES’THEOREM not STOKE’S THEOREM

0K temperature

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( ) ( )A r dl A dS• = ∇ × •∫ ∫∫uurrr rrr

0

( )ˆ ˆ: ( ) lim ( )C

S

A r dlDefinition curl A n A n

Sδ δ→

•• = = ∇ × •

∫rr r

r rr

For a path δC, which binds an area δS,

ˆ( )C

A dl S curl A nδ

δ• = •∫rr r

1 1( ) ( ) ( )

( ) ( )i

n n

i ic c

c

A r dl A r dl curl A r dS

A r dl curl A r dS

δ= =

• = • = •

• = •

∑ ∑∫ ∫ ∫∫

∫ ∫∫

ur r uur ur r uur ur r uur

ur r uur ur r uur

A finite area S is enclosed by C, we can split up S into infinitesimal bits δSn bounded by tiny curves δCn

Stokes’ theorem

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Proof of Stokes’ theorem follows from the very definition of the curl:

Stokes’ theoremconsider a surface S enclosed by a curve C

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!

( ) ( )c

A r dl curl A r dS• = •∫ ∫∫ur r uur ur r uur

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.

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Stokes’ theoremconsider a surface S enclosed by a curve C

( ) ( )c

A r dl curl A r dS• = •∫ ∫∫ur r uur ur r uur

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The direction of the vector surface element that appears in the right hand side of the above equation must be defined in a manner that is consistentwith the sense in which the closed path integral in the left hand side is evaluated.

The right-hand-screw convention must be followed.

C traversed one way

C traversed the other way

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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?

Are there non-orientable surfaces?

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?

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The surface under consideration, however, better be a ‘well-behaved’ surface!

1ˆ ˆ ˆ ˆ ˆ ˆe e e e ( , , ) e ( , , ) e ( , , )z z z

A

A z A z A zzρ ϕ ρ ρ ϕ ϕρ ϕ ρ ϕ ρ ϕ

ρ ρ ϕ

∇× =

⎡ ⎤∂ ∂ ∂ ⎡ ⎤+ + × + +⎢ ⎥ ⎣ ⎦∂ ∂ ∂⎣ ⎦

rr

ˆ ˆ ˆ ˆe e ( , , ) e ( , , ) e ( , , )

1ˆ ˆ ˆ ˆe e ( , , ) e ( , , ) e ( , , )

ˆ ˆ ˆ ˆe e ( , , ) e ( , , ) e ( , , )

z z

z z

z z z

A z A z A z

A z A z A z

A z A z A zz

ρ ρ ρ ϕ ϕ

ϕ ρ ρ ϕ ϕ

ρ ρ ϕ ϕ

ρ ϕ ρ ϕ ρ ϕρ

ρ ϕ ρ ϕ ρ ϕρ ϕ

ρ ϕ ρ ϕ ρ ϕ

⎡ ⎤∂ ⎡ ⎤× + + +⎢ ⎥ ⎣ ⎦∂⎣ ⎦⎡ ⎤∂ ⎡ ⎤× + + +⎢ ⎥ ⎣ ⎦∂⎣ ⎦

∂⎡ ⎤ ⎡ ⎤× + +⎣ ⎦⎢ ⎥∂⎣ ⎦

A∇× =rr

Expression for ‘curl’ in cylindrical polar coordinate system

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{ }ˆ ˆ ˆ, , ze e eρ ϕ

[ ]

ˆ ˆ ˆ ˆe e ( , , ) e ( , , ) e ( , , )

1ˆ ˆ ˆ ˆ e e ( , , ) e ( , , ) e ( , , )

ˆ ˆ ˆ ˆ e e ( , , ) e ( , , ) e ( , , )

z z

z z

z z z

A A z A z A z

A z A z A z

A z A z A zz

ρ ρ ρ ϕ ϕ

ϕ ρ ρ ϕ ϕ

ρ ρ ϕ ϕ

ρ ϕ ρ ϕ ρ ϕρ

ρ ϕ ρ ϕ ρ ϕρ ϕ

ρ ϕ ρ ϕ ρ ϕ

∂⎡ ⎤ ⎡ ⎤∇× = × + + +⎣ ⎦ ⎣ ⎦∂

⎛ ⎞∂⎡ ⎤ ⎡ ⎤× + + +⎜ ⎟⎣ ⎦ ⎣ ⎦∂⎝ ⎠∂ ⎡ ⎤× + +⎣ ⎦∂

rr

( )1 1ˆ ˆ ˆ = e e ez zz

AA A AA Az z

ϕϕ ρ ρρ ϕ

ρ

ρ ϕ ρ ρ ρ ϕ

⎡ ⎤∂∂ ∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂− + − + ⎢ − ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦

Expression for ‘curl’ in cylindrical polar coordinate system

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{ }ˆ ˆ ˆ, , ze e eρ ϕ

( )

=

1 1ˆ ˆ ˆ ˆ ˆ ˆ= e e e e ( , , ) e ( , , ) e ( , , )sinr r r

A

A r A r A rr r rθ ϕ θ θ ϕ ϕθ ϕ θ ϕ θ ϕ

θ θ ϕ

∇×

⎛ ⎞∂ ∂ ∂+ + × + +⎜ ⎟∂ ∂ ∂⎝ ⎠

rr

Expression for ‘curl’ in spherical polar coordinate system

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{ }ˆ ˆ ˆ, ,re e eθ ϕ

( )

( )

( )

1ˆ=e sinsin

1 1ˆ +esin1ˆ e

r

r

r

AA Ar

A rAr r

ArAr r

θϕ

θ ϕ

ϕ θ

θθ θ ϕ

θ ϕ

θ

⎧ ⎫∂∂∇× −⎨ ⎬∂ ∂⎩ ⎭

⎧ ⎫∂ ∂−⎨ ⎬∂ ∂⎩ ⎭

∂∂⎧ ⎫+ −⎨ ⎬∂ ∂⎩ ⎭

rr

Do you like the earring?

It is not at all obvious….

It is MOBIUS !!!

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The Möbius strip used to be 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 belt was spread out more 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

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http://www.mcescher.com/

“The laws of mathematics are not merely human inventions or creations. They 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. ”

MauritsCornelius

Escher1898 - 1972

By divergence theorem SdAdAV S

rrrrrr⋅×∇=×∇⋅∇∫ ∫ )()( τ

V

S S1

S2

c1 c2

Applying Stoke’s theorem

2211

21

ˆ)(ˆ)()( dSnAdSnASdASSS∫∫∫∫∫∫ ⋅×∇+⋅×∇=⋅×∇

rrrrrrr

01 2

=⋅+⋅= ∫ ∫C C

ldAldArrrr

Another important identity: divergence of a curl is zero

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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.

Bernoulli’s Principle

Posed the brachistrocrone problem

Bernoulli brothers

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.

Reference:http://www.york.ac.uk/depts/maths/histstat/people/bernoulli_tree.htmPCD-08

NicolausBernoulli’s 2 sons

Bernoulli Family Math/Phys Tree

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

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http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Bernoulli_Daniel.html

Daniel Bernoulli1700 - 1782

“...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

φρ

∇−∇−

==⎥⎦⎤

⎢⎣⎡

∂∂

+∇•)(

),(v),(vvrp

dttrdtr

tResult of page 47,Unit 11+12+13:

Use now the following vector identity:

( )( ) ( ) ( ) ( )ABBAABBA

BA

×∇×+×∇×+∇•+∇•

=•∇

( )( ) ( ) ( ) ( )( ) ( ) ( )vvvvvv

21 ..

vvvvvvvv

vv

×∇×+∇•=•∇

×∇×+×∇×+∇•+∇•

=•∇

ei

( ) ( ) φρ

∇−∇−

=∂∂

+×∇×−•∇)(

vvvvv21

rp

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( ) ( )

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∇≈∇

∇−∇−

=∂∂

+×∇×−•∇

ρρ

φρ

)()()( ,

)(vvvvv

21

rprrpNow

rp

t

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( ) ( )

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+∇−=∂∂

+×∇×−•∇

∇−⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∇−=∂∂

+×∇×−•∇

φρ

φρ

)(vvvvv21

)(vvvvv21

rpt

rpt

{ }

RI I

RI

Recall that : x

v v x , where v is just the velocity that is employed in the equation of motion for the fluid.

v v x

R

R

I R

d dr r rdt dt

r

r

ω

ω

ω

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= +

∴ ∇× = ∇× + ∇×

r r ur r

r r ur r r

ur r ur r ur ur r

To determine we now use another vector Identity, for the curl of cross-product of two vectors:

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( ) ( ) ( ) )()( ABBABAABBA •∇−•∇+∇•−∇•=××∇

{ }Rx rω∇×ur ur r

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+∇−=∂∂

+×∇×−•∇ φρ

)(vvvvv21 rp

t

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( ) ( ) ( )( ) ( )

ωχ

ω

ωωωω

ωωωωω

2v VORTICITY, the,

,0v,frame rotating In the

2vv

2)(

)()(

==×∇

=

+×∇=×∇

=•∇+∇•−=××∇

•∇−•∇+∇•−∇•=××∇

I

R

RI

hence

rrr

rrrrr

RRR

RRRRR

( ) ( ) ( ) )()( ABBABAABBA •∇−•∇+∇•−∇•=××∇

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+∇−=∂∂

+×∇×−•∇ φρ

)(vvvvv21 rp

t

PCD-08

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧++∇−=×−

∂∂

∇−⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+∇−=×−∂∂

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+∇−=∂∂

+×−∇

2

v)(vv

v21)(vv

)(vvv21

2

2

2

φρ

χ

φρ

χ

φρ

χ

rpt

rpt

rpt

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧++∇−=×−

2

v)(v

2

φρ

χ rp

0v=

∂∂

t

For ‘STEADY STATE’

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧++∇•=

2

v)(v0

2

φρrp

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+∇−=∂∂

+×∇×−•∇ φρ

)(vvvvv21 rp

t

PCD-08

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧++∇•=

2

v)(v0

2

φρrp

SSTREAMLINErpei

rp

toORTHOGONAL bemust 2

v)(.,.

,v toORTHOGONAL bemust 2

v)(

2

2

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧++∇

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧++∇⇒

φρ

φρ

.2

v)(

, toORTHOGONAL bemust 2

++=Ψ

Ψ∇⇒

φρrpwhere

sstreamline

streamlinegiven afor constant 2

v)(2

=++=Ψ⇒ φρrp

Daniel Bernoulli’s Theorem

2

v)(2

++=Ψ⇒ φρrp

PCD-08

streamlinegiven afor constant 2

v)(2

=++=Ψ⇒ φρrp

is constant for the entire velocity field in the liquid.

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧++∇−=×−

2

v)(v

2

φρ

χ rp

We derived the above result for a ‘STEADY STATE’ and made use of the relation

If the fluid flow is both ‘steady state’ and ‘irrotational’,

0v ==×∇ χ

Daniel Bernoulli’s Theorem

PCD-08

P

12:),(v),(),(−−

=

TMLDimensionstrtrtrJ ρ

r A

B

CD

E

F

G

H

Mass Current Density Vector

{ }volume surfaceregion enclosing

thatregion

( ) ( ) 0, for STEADY STATE as 0t

d J r J r dS ρτ ∂∇• = • = =

∂∫∫∫ ∫∫ur ur ur r uurr

since,area sectional-cross:A

,constant vAFlow, StateSteady For

=ρ

tApsAps

tApsAps

δδδδ

δδδδ

2222222

1111111

vFW

vFW

===

===

Work done by the fluid on Face 2 is:

Work done on the fluid by the pressure that the fluid exerts on Face 1 is:

Net work done on the fluid in the parallelepiped by the pressure that the fluid exerts on Faces 1 & 2 is:

tAptAp δδδδ 22211121 vvWW −=−

PCD-08

Net work done on the fluid in the parallelepiped by the pressure that the fluid exerts at Faces 1 & 2 :

tAptAp δδδδ 22211121 vvWW −=−

[ ]m

tApApδ

δ22211112

vvEE −=−

Energy gained per unit mass by the fluid as it traverses the x-axis of the parallelepiped across the Faces 1 & 2 :

[ ]

[ ]( ) 1

int2

2int

2222111

1int

2

2int

2

22211112

v21v

21vv

,

v21v

21

vvEE

⎥⎦⎤

⎢⎣⎡ ++−⎥⎦

⎤⎢⎣⎡ ++=

−

⎥⎦⎤

⎢⎣⎡ ++−⎥⎦

⎤⎢⎣⎡ ++=

−=−

ernalernal

ernalernal

UUsA

tApApThus

UU

mtApAp

φφδρ

δ

φφ

δδ

PCD-08

[ ]( ) 1

int2

2int

2222111 v21v

21vv

⎥⎦⎤

⎢⎣⎡ ++−⎥⎦

⎤⎢⎣⎡ ++=

−ernalernal UU

sAtApAp φφ

δρδ

( ) ( ) 1int

2

2int

2

22

222

11

111 v21v

21

vv

vv

⎥⎦⎤

⎢⎣⎡ ++−⎥⎦

⎤⎢⎣⎡ ++=⎥

⎦

⎤⎢⎣

⎡− ernalernal UUt

AtAp

AtAp φφδ

δρδρ

constantv21 ..

v21v

210

v21v

21

internal2

1int

2

2int

2

1int

2

2int

221

=+++

⎥⎦

⎤⎢⎣

⎡+++−⎥

⎦

⎤⎢⎣

⎡+++=

⎥⎦⎤

⎢⎣⎡ ++−⎥⎦

⎤⎢⎣⎡ ++=⎥

⎦

⎤⎢⎣

⎡−

ρφ

ρφ

ρφ

φφρρ

pUei

pUpU

UUpp

ernalernal

ernalernal

is constant for the entire velocity field in the liquid.

2

v)(

:47 page From2

++=Ψ φρrpDaniel Bernoulli’s Theorem

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.

Debashsish MohantyInswing / Outswing

bowler

constant2

v)(2

=++φρrp

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.

PCD-08

PCD-08

00

20

0

1

1

0

εμ

μ

ερ

=

∂∂

+∂∂

=×∇

∂∂

−=×∇

=•∇

=•∇

c

tE

ctJB

tBE

B

E

Helmholtz TheoremPH102: PHYSICS II

James Clerk Maxwell1831-1879

Divergence and Curl of

( )BE,