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Page 1
Edurev123
5. Euler's Equation of Motion for Inviscid
Flow
5.1 Show that ?? =???? (?? ) is a possible form for the velocity potential for an
incompressible fluid motion. If the fluid velocity ???? ??? as ?? ?8, find the surfaces
of constant speed.
(2012: 30 Marks)
Solution:
Given, the velocity potential is
?? ?=???? (?? ) (??)
?? ?=-??? =-?[???? (?? )] (???? )
?=-[?? (?? )??? +?? ??? (?? )] (???? )
?????????????????????????????????????????????????????????????? =-??? =-?[???? (?? )]
Now ?? 2
=?? 2
+?? 2
+?? 2
?2?? ????
????
=2??
??? ??? =
?? ??
Similarly,
??? ??? =
?? ?? ,
??? ??? =
?? ?? (?????? )
But
??? =[??
?
??? +??
?
??? +???
?
??? ]?? =??
??? (?? )?=[??
?
??? +??
?
??? +???
?
??? ]?? (?? )
?=?? ?? '
(?? )(
??? ??? )+?? ?? '
(?? )(
??? ??? )+???
?? '
(?? )(
??? ??? )
?=?? ?? '
(?? )(
?? ?? )+?? ?? '
(?? )(
?? ?? )+???
?? '
(?? )(
?? ?? )
?=
1
?? ?? '
(?? )(?? ?? +?? ?? +???
?? )=
1
?? ?? '
(?? )??
???? =-?? (?? )?? -
?? ?? ?? '
(?? )?? (from (ii) (???? )
Page 2
Edurev123
5. Euler's Equation of Motion for Inviscid
Flow
5.1 Show that ?? =???? (?? ) is a possible form for the velocity potential for an
incompressible fluid motion. If the fluid velocity ???? ??? as ?? ?8, find the surfaces
of constant speed.
(2012: 30 Marks)
Solution:
Given, the velocity potential is
?? ?=???? (?? ) (??)
?? ?=-??? =-?[???? (?? )] (???? )
?=-[?? (?? )??? +?? ??? (?? )] (???? )
?????????????????????????????????????????????????????????????? =-??? =-?[???? (?? )]
Now ?? 2
=?? 2
+?? 2
+?? 2
?2?? ????
????
=2??
??? ??? =
?? ??
Similarly,
??? ??? =
?? ?? ,
??? ??? =
?? ?? (?????? )
But
??? =[??
?
??? +??
?
??? +???
?
??? ]?? =??
??? (?? )?=[??
?
??? +??
?
??? +???
?
??? ]?? (?? )
?=?? ?? '
(?? )(
??? ??? )+?? ?? '
(?? )(
??? ??? )+???
?? '
(?? )(
??? ??? )
?=?? ?? '
(?? )(
?? ?? )+?? ?? '
(?? )(
?? ?? )+???
?? '
(?? )(
?? ?? )
?=
1
?? ?? '
(?? )(?? ?? +?? ?? +???
?? )=
1
?? ?? '
(?? )??
???? =-?? (?? )?? -
?? ?? ?? '
(?? )?? (from (ii) (???? )
For a possible motion of an incompressible fluid, we have
?·?? =0??(-??? )=0??
2
?? =0
??(
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
)(???? (?? )) =0 (trom (i)) [ Note :?
2
=
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
]
Now,
?
2
??? 2
[???? (?? )]?=
?
??? [
?
??? (???? (?? )]
?=
?
??? [?? (?? )+?? ??? (?? )
??? ]
?=
??? ??? +
??? ??? +?? ?
2
?? ??? 2
?=2
??? ??? +?? ?
2
?? ??? 2
Similarly,
?
2
??? 2
[???? (?? )]=?? ?
2
?? ??? 2
and
?
2
??? 2
[???? (?? )]=?? ?
2
?? ??? 2
??(
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
)(???? (?? ))=0
??2
??? ??? +?? (
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
)=0 (?? )
Using (iii),
??? ??? ?=
????
????
·
??? ??? =?? '
·
?? ?? ?=
?? '
?? +?? ?
??? (
?? '
?? )
?=
?? '
?? +?? ?? ????
(
?? '
?? )·
??? ??? ?=
?? '
?? +?? ·
?? ''
-?? '
?? 2
·
?? ??
??
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
Similarly,
Page 3
Edurev123
5. Euler's Equation of Motion for Inviscid
Flow
5.1 Show that ?? =???? (?? ) is a possible form for the velocity potential for an
incompressible fluid motion. If the fluid velocity ???? ??? as ?? ?8, find the surfaces
of constant speed.
(2012: 30 Marks)
Solution:
Given, the velocity potential is
?? ?=???? (?? ) (??)
?? ?=-??? =-?[???? (?? )] (???? )
?=-[?? (?? )??? +?? ??? (?? )] (???? )
?????????????????????????????????????????????????????????????? =-??? =-?[???? (?? )]
Now ?? 2
=?? 2
+?? 2
+?? 2
?2?? ????
????
=2??
??? ??? =
?? ??
Similarly,
??? ??? =
?? ?? ,
??? ??? =
?? ?? (?????? )
But
??? =[??
?
??? +??
?
??? +???
?
??? ]?? =??
??? (?? )?=[??
?
??? +??
?
??? +???
?
??? ]?? (?? )
?=?? ?? '
(?? )(
??? ??? )+?? ?? '
(?? )(
??? ??? )+???
?? '
(?? )(
??? ??? )
?=?? ?? '
(?? )(
?? ?? )+?? ?? '
(?? )(
?? ?? )+???
?? '
(?? )(
?? ?? )
?=
1
?? ?? '
(?? )(?? ?? +?? ?? +???
?? )=
1
?? ?? '
(?? )??
???? =-?? (?? )?? -
?? ?? ?? '
(?? )?? (from (ii) (???? )
For a possible motion of an incompressible fluid, we have
?·?? =0??(-??? )=0??
2
?? =0
??(
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
)(???? (?? )) =0 (trom (i)) [ Note :?
2
=
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
]
Now,
?
2
??? 2
[???? (?? )]?=
?
??? [
?
??? (???? (?? )]
?=
?
??? [?? (?? )+?? ??? (?? )
??? ]
?=
??? ??? +
??? ??? +?? ?
2
?? ??? 2
?=2
??? ??? +?? ?
2
?? ??? 2
Similarly,
?
2
??? 2
[???? (?? )]=?? ?
2
?? ??? 2
and
?
2
??? 2
[???? (?? )]=?? ?
2
?? ??? 2
??(
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
)(???? (?? ))=0
??2
??? ??? +?? (
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
)=0 (?? )
Using (iii),
??? ??? ?=
????
????
·
??? ??? =?? '
·
?? ?? ?=
?? '
?? +?? ?
??? (
?? '
?? )
?=
?? '
?? +?? ?? ????
(
?? '
?? )·
??? ??? ?=
?? '
?? +?? ·
?? ''
-?? '
?? 2
·
?? ??
??
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
Similarly,
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
and
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
??=
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=3
?? '
?? +
?? 2
+?? 2
+?? 2
?? 2
·?? ''
-
?? 2
+?? 2
+?? 2
?? 3
?? '
=3
?? '
?? +?? ''
-
?? '
?? =2
?? '
?? +?? ''
? from (v), we have
2?? '
?? ?? +?? (
2?? '
?? +?? ''
)=0
? ?? ''
+4
?? '
?? =0
?
?? ''
?? '
+
4
?? '
=0
? log??? +4log??? =log??? 1
? ?? '
=?? 1
?? 4
? ?? =-
?? 1
3
?? -3
+?? 2
? from (iv),
?? =[
?? 1
3?? 3
-?? 2
]?? -
?? 1
?? ?? 5
??
But, it is given that ?? ?0 as ?? ?8,
Page 4
Edurev123
5. Euler's Equation of Motion for Inviscid
Flow
5.1 Show that ?? =???? (?? ) is a possible form for the velocity potential for an
incompressible fluid motion. If the fluid velocity ???? ??? as ?? ?8, find the surfaces
of constant speed.
(2012: 30 Marks)
Solution:
Given, the velocity potential is
?? ?=???? (?? ) (??)
?? ?=-??? =-?[???? (?? )] (???? )
?=-[?? (?? )??? +?? ??? (?? )] (???? )
?????????????????????????????????????????????????????????????? =-??? =-?[???? (?? )]
Now ?? 2
=?? 2
+?? 2
+?? 2
?2?? ????
????
=2??
??? ??? =
?? ??
Similarly,
??? ??? =
?? ?? ,
??? ??? =
?? ?? (?????? )
But
??? =[??
?
??? +??
?
??? +???
?
??? ]?? =??
??? (?? )?=[??
?
??? +??
?
??? +???
?
??? ]?? (?? )
?=?? ?? '
(?? )(
??? ??? )+?? ?? '
(?? )(
??? ??? )+???
?? '
(?? )(
??? ??? )
?=?? ?? '
(?? )(
?? ?? )+?? ?? '
(?? )(
?? ?? )+???
?? '
(?? )(
?? ?? )
?=
1
?? ?? '
(?? )(?? ?? +?? ?? +???
?? )=
1
?? ?? '
(?? )??
???? =-?? (?? )?? -
?? ?? ?? '
(?? )?? (from (ii) (???? )
For a possible motion of an incompressible fluid, we have
?·?? =0??(-??? )=0??
2
?? =0
??(
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
)(???? (?? )) =0 (trom (i)) [ Note :?
2
=
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
]
Now,
?
2
??? 2
[???? (?? )]?=
?
??? [
?
??? (???? (?? )]
?=
?
??? [?? (?? )+?? ??? (?? )
??? ]
?=
??? ??? +
??? ??? +?? ?
2
?? ??? 2
?=2
??? ??? +?? ?
2
?? ??? 2
Similarly,
?
2
??? 2
[???? (?? )]=?? ?
2
?? ??? 2
and
?
2
??? 2
[???? (?? )]=?? ?
2
?? ??? 2
??(
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
)(???? (?? ))=0
??2
??? ??? +?? (
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
)=0 (?? )
Using (iii),
??? ??? ?=
????
????
·
??? ??? =?? '
·
?? ?? ?=
?? '
?? +?? ?
??? (
?? '
?? )
?=
?? '
?? +?? ?? ????
(
?? '
?? )·
??? ??? ?=
?? '
?? +?? ·
?? ''
-?? '
?? 2
·
?? ??
??
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
Similarly,
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
and
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
??=
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=3
?? '
?? +
?? 2
+?? 2
+?? 2
?? 2
·?? ''
-
?? 2
+?? 2
+?? 2
?? 3
?? '
=3
?? '
?? +?? ''
-
?? '
?? =2
?? '
?? +?? ''
? from (v), we have
2?? '
?? ?? +?? (
2?? '
?? +?? ''
)=0
? ?? ''
+4
?? '
?? =0
?
?? ''
?? '
+
4
?? '
=0
? log??? +4log??? =log??? 1
? ?? '
=?? 1
?? 4
? ?? =-
?? 1
3
?? -3
+?? 2
? from (iv),
?? =[
?? 1
3?? 3
-?? 2
]?? -
?? 1
?? ?? 5
??
But, it is given that ?? ?0 as ?? ?8,
??
?? 2
?=0
?? ?=
?? 1
3?? 3
(?? -
3?? ??
?? 2
)
?? 2
?=?? ·?? =
?? 1
2
9?? 6
(?? -
3?? ??
?? 2
)(?? -
3?? ??
?? 2
)
?=
?? 1
2
9?? 6
(?? ·?? -
6?? ?? 2
?? ·?? +
9?? 2
?? 4
·?? ·?? )?(??? ·?? =?? 2
and ?? ·?? =?? )
?=
?? 1
2
9?? 6
(1-
6?? 2
?? 2
+
9?? 2
?? 2
?? 4
)?(1+
3?? 2
?? 2
)
?=
?? 1
2
9?? 6
(1)?(1+?? 2
)
Hence, the required surfaces of constant speed are:
?? 2
= Constant
or
?? 1
2
9?? 6
(1+
3?? 2
?? 2
)= Constant
or ?
?? 2
+3?? 2
?? 8
= Constant
5.2 Consider a uniform flow ?? ?? in the positive ?? direction. A cylinder of radius ?? is
located at the origin. Find the stream function and the velocity potential. Find also,
the stagnation points.
(2015 : 10 Marks)
Solution:
Figure shows flow past the fixed cylinder. Let the fluid be inviscid and incompressible.
So, it is equivalent to superposition of uniform flow and a doublet.
As
Velocity =?? ?? ??ˆ
? Velocity potential,
Page 5
Edurev123
5. Euler's Equation of Motion for Inviscid
Flow
5.1 Show that ?? =???? (?? ) is a possible form for the velocity potential for an
incompressible fluid motion. If the fluid velocity ???? ??? as ?? ?8, find the surfaces
of constant speed.
(2012: 30 Marks)
Solution:
Given, the velocity potential is
?? ?=???? (?? ) (??)
?? ?=-??? =-?[???? (?? )] (???? )
?=-[?? (?? )??? +?? ??? (?? )] (???? )
?????????????????????????????????????????????????????????????? =-??? =-?[???? (?? )]
Now ?? 2
=?? 2
+?? 2
+?? 2
?2?? ????
????
=2??
??? ??? =
?? ??
Similarly,
??? ??? =
?? ?? ,
??? ??? =
?? ?? (?????? )
But
??? =[??
?
??? +??
?
??? +???
?
??? ]?? =??
??? (?? )?=[??
?
??? +??
?
??? +???
?
??? ]?? (?? )
?=?? ?? '
(?? )(
??? ??? )+?? ?? '
(?? )(
??? ??? )+???
?? '
(?? )(
??? ??? )
?=?? ?? '
(?? )(
?? ?? )+?? ?? '
(?? )(
?? ?? )+???
?? '
(?? )(
?? ?? )
?=
1
?? ?? '
(?? )(?? ?? +?? ?? +???
?? )=
1
?? ?? '
(?? )??
???? =-?? (?? )?? -
?? ?? ?? '
(?? )?? (from (ii) (???? )
For a possible motion of an incompressible fluid, we have
?·?? =0??(-??? )=0??
2
?? =0
??(
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
)(???? (?? )) =0 (trom (i)) [ Note :?
2
=
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
]
Now,
?
2
??? 2
[???? (?? )]?=
?
??? [
?
??? (???? (?? )]
?=
?
??? [?? (?? )+?? ??? (?? )
??? ]
?=
??? ??? +
??? ??? +?? ?
2
?? ??? 2
?=2
??? ??? +?? ?
2
?? ??? 2
Similarly,
?
2
??? 2
[???? (?? )]=?? ?
2
?? ??? 2
and
?
2
??? 2
[???? (?? )]=?? ?
2
?? ??? 2
??(
?
2
??? 2
+
?
2
??? 2
+
?
2
??? 2
)(???? (?? ))=0
??2
??? ??? +?? (
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
)=0 (?? )
Using (iii),
??? ??? ?=
????
????
·
??? ??? =?? '
·
?? ?? ?=
?? '
?? +?? ?
??? (
?? '
?? )
?=
?? '
?? +?? ?? ????
(
?? '
?? )·
??? ??? ?=
?? '
?? +?? ·
?? ''
-?? '
?? 2
·
?? ??
??
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
Similarly,
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
and
?
2
?? ??? 2
=
?? '
?? +
?? 2
?? 2
?? ''
-
?? 2
?? 3
?? '
??=
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=3
?? '
?? +
?? 2
+?? 2
+?? 2
?? 2
·?? ''
-
?? 2
+?? 2
+?? 2
?? 3
?? '
=3
?? '
?? +?? ''
-
?? '
?? =2
?? '
?? +?? ''
? from (v), we have
2?? '
?? ?? +?? (
2?? '
?? +?? ''
)=0
? ?? ''
+4
?? '
?? =0
?
?? ''
?? '
+
4
?? '
=0
? log??? +4log??? =log??? 1
? ?? '
=?? 1
?? 4
? ?? =-
?? 1
3
?? -3
+?? 2
? from (iv),
?? =[
?? 1
3?? 3
-?? 2
]?? -
?? 1
?? ?? 5
??
But, it is given that ?? ?0 as ?? ?8,
??
?? 2
?=0
?? ?=
?? 1
3?? 3
(?? -
3?? ??
?? 2
)
?? 2
?=?? ·?? =
?? 1
2
9?? 6
(?? -
3?? ??
?? 2
)(?? -
3?? ??
?? 2
)
?=
?? 1
2
9?? 6
(?? ·?? -
6?? ?? 2
?? ·?? +
9?? 2
?? 4
·?? ·?? )?(??? ·?? =?? 2
and ?? ·?? =?? )
?=
?? 1
2
9?? 6
(1-
6?? 2
?? 2
+
9?? 2
?? 2
?? 4
)?(1+
3?? 2
?? 2
)
?=
?? 1
2
9?? 6
(1)?(1+?? 2
)
Hence, the required surfaces of constant speed are:
?? 2
= Constant
or
?? 1
2
9?? 6
(1+
3?? 2
?? 2
)= Constant
or ?
?? 2
+3?? 2
?? 8
= Constant
5.2 Consider a uniform flow ?? ?? in the positive ?? direction. A cylinder of radius ?? is
located at the origin. Find the stream function and the velocity potential. Find also,
the stagnation points.
(2015 : 10 Marks)
Solution:
Figure shows flow past the fixed cylinder. Let the fluid be inviscid and incompressible.
So, it is equivalent to superposition of uniform flow and a doublet.
As
Velocity =?? ?? ??ˆ
? Velocity potential,
?? =?? ?? ?? +
?? cos??? ??
As ?? and ?? satisfy Cauchy-Riemann equation
????? ?? =?? ?? ? ( ?? is stream function)
????? =?? 0
?? -
?? sin??? ?? ????? =?? +???? =?? 0
?? -
?? cos??? ?? +?? (?? 0?? -
?? sin??? ?? )
????? =?? 0
(?? +???? )+
?? ?? (cos??? -??sin??? )
????? =?? 0
?? +
?? ?? 2
(???)=?? 0
?? +
(?? +???)
2?? ·???
·???
????? =?? 0
?? +
1
2
(1+
???
?? )
????? =?? 0
?? +
1
2
(1+
1
?? 2
)
?? At stagnation points, ?
????
????
=0
????? 0
+0-
2
2?? 3
=0
???
1
?? 3
=?? 0
????? =(
1
?? 0
)
1/3
5.3 The space between two concentric spherical shells of radii ?? ,?? (?? <?? ) is filled
with a liquid of density p. If the shells are set in motion, the inner one with velocity
?? in the ?? -direction and the outer one with velocity ?? in the ?? -direction, then show
that the initial motion of the liquid is given by velocity potential
?? ={
?? ?? ?? (?? +
?? ?? ?? ?? ?? -?? )?? (-?? ?? ?? )(?? +
?? ?? ?? ?? ?? -?? )?? (?? ?? -?? ?? )
}
where ?? ?? =?? ?? +?? ?? +?? ?? , the coordinates being rectangular. Evaluate the velocity
at any point of the liquid.
(2016 : 20 Marks)
Solution:
As shown in figure, let ?? be the common centre and ?? be velocity potential of initial
motion. ?? and ?? be initial velocities of inner and outer shells in ?? and ?? direction
respectively. The following two conditions should be satisfied.
(i) ?? satisfies Laplace's equation
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Oct 31, 2024 Last updated |
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Information about Euler's Equation of Motion for Inviscid Flow
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Additional Information about Euler's Equation of Motion for Inviscid Flow for UPSC Preparation
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