MIT Department of Mechanical Engineering
2.25 Advanced Fluid Mechanics
Problem 4.11
This problem is from “Advanced Fluid Mechanics Problems” by A.H. Shapiro and A.A. Sonin
An incompressible, inviscid liquid flows with speed V vertically downward from the nozzle of the radius R.
The liquid density ρ is high compared with that of the ambient air. The surface tension between the liquid
and the air is σ.
(a) Obtain an expression which relates the local radius r of the liquid stream to the distance x from the
nozzle.
(b) Show that for sufficiently large x,
(
V2
r≈R
2gx
) 1
4
,
(4.11a)
(c) Write down all the criteria which must be satisfied for this expression to be a good approximation.
State each criterion as ‘x must be very large compared with y’, where y is some combination of the
given quantities V , R, g, and σ.
2.25 Advanced Fluid Mechanics
1
c 2008, MIT
Copyright ©
Inviscid Flows
A.H. Shapiro and A.A. Sonin 4.11
Solution:
• (a) Using Bernoulli between 1 and 2,
1 2
1
ρV + Pa = ρV12 − ρgx + Pa ,
2
2
(4.11b)
2 1 2
1
2
( ρV + ρgx) = ρV22 ,
ρ 2
2
ρ
(4.11c)
simplifying,
then,
Also, from mass conservation
V 2 + 2gx = V22 , ⇒ V2 =
πR2 V = πr2 V2 , ⇒
Finally adding the information from Bernoulli,
(
• (b) For
2gx
V2
�1⇒
( )2
r
R
≈
(
V2
2gx
) 21
r
R
)2
r
R
)2
=
V
=
V + 2gx
2
V
.
V2
.
(4.11d)
(4.11e)
(4.11f)
,⇒
r
R
• (c) For the solution to apply,
(
V 2 + 2gx.
=
V2
« 1,
2gx
(
V2
2gx
) 41
OR
.
(4.11g)
ρV 2
« 1,
2ρgx
(4.11h)
then,
x�
2.25 Advanced Fluid Mechanics
2
V2
2g .
(4.11i)
c 2008, MIT
Copyright ©
Inviscid Flows
A.H. Shapiro and A.A. Sonin 4.11
Also, since we neglected surface tension,
σ
ΔPσ
σ
= r « 1, ⇒
« 1.
ρgx
rρgx
ΔPx
(4.11j)
Now, let’s get an estimate of the order of magnitude of r(x) from (b),
r≈R
(
)
1
1
RV 2
V 2 4
,⇒ r ∼
1 ,
2gx
(gx) 4
(4.11k)
now, substituting into the x requirement,
1
σ(gx) 4
1
2
RV ρgx
σ
3
« 1, ⇒ x
4
�
1
3
RV 2
ρg 4
,
(4.11l)
finally,
x%
�
4
σ 3
4
4
2
R 3
gρ 3
V
3
.
(4.11m)
D
Problem Solution by MC, Fall 2008
2.25 Advanced Fluid Mechanics
3
c 2008, MIT
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2.25 Advanced Fluid Mechanics
Fall 2013
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