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Hydrodynamics: Unsteady Bernoulli's Equation

1) The document derives the unsteady Bernoulli's equation from conservation of momentum and the total derivative of velocity for irrotational flow. 2) It shows that for irrotational flow, the total derivative of velocity can be written in terms of the velocity potential. 3) The momentum equation is then written in terms of pressure, gravity, and the velocity potential, and integrating the spatial derivatives yields the unsteady Bernoulli's equation.

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Ihab Omar
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478 views1 page

Hydrodynamics: Unsteady Bernoulli's Equation

1) The document derives the unsteady Bernoulli's equation from conservation of momentum and the total derivative of velocity for irrotational flow. 2) It shows that for irrotational flow, the total derivative of velocity can be written in terms of the velocity potential. 3) The momentum equation is then written in terms of pressure, gravity, and the velocity potential, and integrating the spatial derivatives yields the unsteady Bernoulli's equation.

Uploaded by

Ihab Omar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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2.

016 Hydrodynamics
Professor A.H. Techet

0.1 Derivation of unsteady Bernoullis Equation


Conservation of Momentum says
ma = F
so
DV F
a = =
Dt V
This is the acceleration and forces acting on Bob the Fluid Blob. The total derivative of the velocity is
expanded like this:
(t, x, y, z)
DV
V x V
V y V z
= + + +
Dt t x
t y
t z
t
u v w
DV
V V V
V
= +u +v +w
Dt t x y z

V




= + u +v +w V
t x y z

V



= + (u, v, w) , ,
V
t x y z
DV
V
= + (V )
V
Dt t
V
For irrotational ow, ( = 0), so (V )
V = (
1V V ) and
2

DV V
1

= + V V
Dt t 2
= ,
Also for irrotational ow, we can use the velocity potential V and we have

DV


1
= +

Dt t 2
The forces acting on Bob are pressure and gravity, so the momentum equation becomes




1

+
= p g k = p (gz)

t 2
d
dz
(gz)

1

+ + p + gz = 0
t 2
And in one last glorious step, we integrate all the spacial derivatives (i.e. knock the nabla out), and we
have the unsteady Bernoullis Equation;
1

+ + p + gz = F (t)
t 2
where F (t) is some function of t (is the constant of integration).

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