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Willson February 18, 2015 1

Euler’s Equation

( )

zp

dl

d

al

γρ

+−=

assumes only gravity and pressure forces

Gives the pressure variation due to weight and

acceleration

Valid for

inviscid

steady flow

along a streamline

C

r

zp =−+

2

22

ρω

γ

For rotational flows

Bernoulli Equation

0=++ udugdh

dp

ρ

Euler’s equation

0

1=++ ∫∫∫ udugdhdp

ρ

constant

2

2

=++ u

gh

p

ρ

constant

2

2

=++

g

u

h

p

γ

The Bernoulli Equation is a

statement of the conservation

of ____________________

constant

2

2

=++ v

gh

p

ρ

p.e. k.e.

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CE 2200 Fluid Mechanics

Willson February 18, 2015 2

Bernoulli Equation

Assumptions needed for Bernoulli Equation

Eliminate the constant in the Bernoulli equation?

_______________________________________

Bernoulli equation does not include

___________________________

___________________________

Bernoulli’s Equation

(for irrotational flow in two dimensions)

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛+

∂

∂

−=z

p

xg

ax

γ

y

u

v

x

u

uax∂

∂

+

∂

∂

=

By definition h = z + p/ϒ

Euler’s equation for the x-

coordinate system and

incompressible fluid

x

h

g

ax

∂

∂

−=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

=

∂

∂

−y

u

v

x

u

u

gx

h1

y-direction

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

=

∂

∂

−y

v

v

x

v

u

gy

h1

x-direction

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CE 2200 Fluid Mechanics

Willson February 18, 2015 3

Bernoulli’s Equation

(What is irrotational flow?)

u, v, w - velocities in the x, y, and z directions

x

v

y

u

∂

∂

=

∂

∂

x

w

z

u

∂

∂

=

∂

∂

x

v

y

w

∂

∂

=

∂

∂

u

v

x

y

y

y

u

uΔ

∂

∂

+

x

x

v

vΔ

∂

∂

+

Irrotational flow exists only when the average

rates of rotation are zero (Section 4.6)

Bernoulli’s Equation

(for irrotational flow in two dimensions)

and

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

=

∂

∂

−y

u

v

x

u

u

gx

h1

x

v

y

u

∂

∂

=

∂

∂

For irrotational flow

⎟

⎠

⎞

⎜

⎝

⎛

∂

∂

+

∂

∂

=

∂

∂

−x

v

v

x

u

u

gx

h1

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

=

∂

∂

2

2

u

xx

u

u

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

=

∂

∂

2

2

v

xx

v

v

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

=

∂

∂

−22

122 v

x

u

xgx

h

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