Vortex Motion | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
7. Vortex Motion 
7.1 In an incompressible fluid, the vorticity at every point is constant in magnitude 
and direction; show that the components of velocity ?? ,?? ,?? are solutions of 
Laplace equation. 
(2010: 12 Marks) 
Solution: 
Let velocity, 
?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
 
? Vorticity, 
???? =?×??  
=|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? ?? | 
=??ˆ(
??? ??? -
??? ??? )-??ˆ(
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )= Constant (given) 
??
??? ??? -
??? ??? =0 or 
??? ??? =
??? ??? 
??
?
2
?? ??? 2
=
?
2
?? ??? ??? 
and 
?
2
?? ??? ??? =
?
2
?? ??? 2
(2) 
Similarly, 
and 
??? ??? ?=
??? ??? ?
?
2
?? ??? 2
=
?
2
!
??? ??? (3)
?
2
?? ??? ??? ?=
?
2
?? ??? 2
(4)
 
Also, 
Page 2


Edurev123 
7. Vortex Motion 
7.1 In an incompressible fluid, the vorticity at every point is constant in magnitude 
and direction; show that the components of velocity ?? ,?? ,?? are solutions of 
Laplace equation. 
(2010: 12 Marks) 
Solution: 
Let velocity, 
?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
 
? Vorticity, 
???? =?×??  
=|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? ?? | 
=??ˆ(
??? ??? -
??? ??? )-??ˆ(
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )= Constant (given) 
??
??? ??? -
??? ??? =0 or 
??? ??? =
??? ??? 
??
?
2
?? ??? 2
=
?
2
?? ??? ??? 
and 
?
2
?? ??? ??? =
?
2
?? ??? 2
(2) 
Similarly, 
and 
??? ??? ?=
??? ??? ?
?
2
?? ??? 2
=
?
2
!
??? ??? (3)
?
2
?? ??? ??? ?=
?
2
?? ??? 2
(4)
 
Also, 
??? ??? -
??? ??? =0?
??? ??? =
??? ??? 
??
?
2
?? ??? ??? =
?
2
?? ??? 2
 
and 
?
2
?? ??? 2
=
??? ??? ??? (6) 
As fluid is incompressible, ? by eqn. of continuity 
??? ??? +
??? ??? +
??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? ??? +
?
2
?? ??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 
So, satisfies Laplace's equation. 
In the same way, 
?
2
?? ??? ??? +
?
2
?? ??? 2
+
?
2
?? ??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 and 
?
2
?? ??? ??? +
?
2
?? ??? ??? +
?
2
?? ??? 2
=0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 
So, ?? and ?? also satisfy Laplace's equation. 
??? ,?? ,?? are solutions of Laplace's equation. 
7.2 When a pair of equal and opposite rectilinear vertices are situated in a long 
circular cylinder at equal distances from its axis, show that the path of each 
vortex is given by the equation 
(?? ?? ?????? ??? ?? -?? ?? )=?? ?? ?? ?? ?? ?? ?? ?????? ?? ??? 
?? being measured from the line through the centre perpendicular to the joint of the 
vertices. 
(2010 : 30 Marks) 
Page 3


Edurev123 
7. Vortex Motion 
7.1 In an incompressible fluid, the vorticity at every point is constant in magnitude 
and direction; show that the components of velocity ?? ,?? ,?? are solutions of 
Laplace equation. 
(2010: 12 Marks) 
Solution: 
Let velocity, 
?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
 
? Vorticity, 
???? =?×??  
=|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? ?? | 
=??ˆ(
??? ??? -
??? ??? )-??ˆ(
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )= Constant (given) 
??
??? ??? -
??? ??? =0 or 
??? ??? =
??? ??? 
??
?
2
?? ??? 2
=
?
2
?? ??? ??? 
and 
?
2
?? ??? ??? =
?
2
?? ??? 2
(2) 
Similarly, 
and 
??? ??? ?=
??? ??? ?
?
2
?? ??? 2
=
?
2
!
??? ??? (3)
?
2
?? ??? ??? ?=
?
2
?? ??? 2
(4)
 
Also, 
??? ??? -
??? ??? =0?
??? ??? =
??? ??? 
??
?
2
?? ??? ??? =
?
2
?? ??? 2
 
and 
?
2
?? ??? 2
=
??? ??? ??? (6) 
As fluid is incompressible, ? by eqn. of continuity 
??? ??? +
??? ??? +
??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? ??? +
?
2
?? ??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 
So, satisfies Laplace's equation. 
In the same way, 
?
2
?? ??? ??? +
?
2
?? ??? 2
+
?
2
?? ??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 and 
?
2
?? ??? ??? +
?
2
?? ??? ??? +
?
2
?? ??? 2
=0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 
So, ?? and ?? also satisfy Laplace's equation. 
??? ,?? ,?? are solutions of Laplace's equation. 
7.2 When a pair of equal and opposite rectilinear vertices are situated in a long 
circular cylinder at equal distances from its axis, show that the path of each 
vortex is given by the equation 
(?? ?? ?????? ??? ?? -?? ?? )=?? ?? ?? ?? ?? ?? ?? ?????? ?? ??? 
?? being measured from the line through the centre perpendicular to the joint of the 
vertices. 
(2010 : 30 Marks) 
Solution: 
Let a pair of equal and opposite vertices of strength +?? and -?? are placed at the point 
?? (?? ,?? ) and ?? (?? ,?? ) respectively. 
The image consists of 
(i) A vertex of strength -?? at an inverse point ?? (
?? 2
?? ,?? ) . 
(ii) A vertex of strength +?? at an inverse point ?? (
?? 2
?? ,-?? ) . 
The complex potential at any point ?? becomes 
 
?? =
????
2?? log?(?? -?? ?? ????
)-
????
2?? log?{?? -(
?? 2
?? )?? ????
}-
????
2?? log?(?? -?? ?? -????
)+
????
2?? log?{?? -(
?? 2
?? )?? -????
} 
The motion of ?? is due to other vertices, thus for the motion ?? , the complex potential at 
any point ?? (?? =?? ?? ????
) becomes 
or 
?? 1
=[-
????
2?? log?{?? -(
?? 2
?? )?? ????
}-
????
2?? log?(?? -?? ?? -????
)-
????
2?? log?{?? -(
?? 2
?? )?? -????
}]
?? -?? ?? ????
 
?? 1
=-
????
2?? [log?{?? ?? ????
-(
?? 2
?? )?? ????
}log?(?? ?? ????
-?? ?? -????
)-log?{?? ?? ????
-(
?? 2
?? )?? -????
}] 
or 
Page 4


Edurev123 
7. Vortex Motion 
7.1 In an incompressible fluid, the vorticity at every point is constant in magnitude 
and direction; show that the components of velocity ?? ,?? ,?? are solutions of 
Laplace equation. 
(2010: 12 Marks) 
Solution: 
Let velocity, 
?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
 
? Vorticity, 
???? =?×??  
=|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? ?? | 
=??ˆ(
??? ??? -
??? ??? )-??ˆ(
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )= Constant (given) 
??
??? ??? -
??? ??? =0 or 
??? ??? =
??? ??? 
??
?
2
?? ??? 2
=
?
2
?? ??? ??? 
and 
?
2
?? ??? ??? =
?
2
?? ??? 2
(2) 
Similarly, 
and 
??? ??? ?=
??? ??? ?
?
2
?? ??? 2
=
?
2
!
??? ??? (3)
?
2
?? ??? ??? ?=
?
2
?? ??? 2
(4)
 
Also, 
??? ??? -
??? ??? =0?
??? ??? =
??? ??? 
??
?
2
?? ??? ??? =
?
2
?? ??? 2
 
and 
?
2
?? ??? 2
=
??? ??? ??? (6) 
As fluid is incompressible, ? by eqn. of continuity 
??? ??? +
??? ??? +
??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? ??? +
?
2
?? ??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 
So, satisfies Laplace's equation. 
In the same way, 
?
2
?? ??? ??? +
?
2
?? ??? 2
+
?
2
?? ??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 and 
?
2
?? ??? ??? +
?
2
?? ??? ??? +
?
2
?? ??? 2
=0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 
So, ?? and ?? also satisfy Laplace's equation. 
??? ,?? ,?? are solutions of Laplace's equation. 
7.2 When a pair of equal and opposite rectilinear vertices are situated in a long 
circular cylinder at equal distances from its axis, show that the path of each 
vortex is given by the equation 
(?? ?? ?????? ??? ?? -?? ?? )=?? ?? ?? ?? ?? ?? ?? ?????? ?? ??? 
?? being measured from the line through the centre perpendicular to the joint of the 
vertices. 
(2010 : 30 Marks) 
Solution: 
Let a pair of equal and opposite vertices of strength +?? and -?? are placed at the point 
?? (?? ,?? ) and ?? (?? ,?? ) respectively. 
The image consists of 
(i) A vertex of strength -?? at an inverse point ?? (
?? 2
?? ,?? ) . 
(ii) A vertex of strength +?? at an inverse point ?? (
?? 2
?? ,-?? ) . 
The complex potential at any point ?? becomes 
 
?? =
????
2?? log?(?? -?? ?? ????
)-
????
2?? log?{?? -(
?? 2
?? )?? ????
}-
????
2?? log?(?? -?? ?? -????
)+
????
2?? log?{?? -(
?? 2
?? )?? -????
} 
The motion of ?? is due to other vertices, thus for the motion ?? , the complex potential at 
any point ?? (?? =?? ?? ????
) becomes 
or 
?? 1
=[-
????
2?? log?{?? -(
?? 2
?? )?? ????
}-
????
2?? log?(?? -?? ?? -????
)-
????
2?? log?{?? -(
?? 2
?? )?? -????
}]
?? -?? ?? ????
 
?? 1
=-
????
2?? [log?{?? ?? ????
-(
?? 2
?? )?? ????
}log?(?? ?? ????
-?? ?? -????
)-log?{?? ?? ????
-(
?? 2
?? )?? -????
}] 
or 
 or ??? 1
=
-????
2?? [log?{(?? cos??? -(
?? 2
?? )cos??? )+?? (?? sin??? -
?? 2
?? sin??? )}+log?(2???? sin??? )
-log?{?? cos??? -(
?? 2
?? )cos??? }+?? (?? sin??? +
?? 2
?? )sin??? ]
 or ??? =
-?? 2?? [log?(?? -
?? 2
?? )+log?(2?? sin??? )-
1
2
log?{?? 2
+
?? 4
?? 2
-2?? 2
cos?2?? }]
 or ??? =
-?? 4?? [log?(?? -
?? 2
?? ·2?? sin??? )
2
-log?{?? 2
+
?? 4
?? 2
-2?? 2
cos?2?? }]
 
The stream lines are given by ?? = constant 
i.e. 
-?? 4?? [log?{4(?? 2
-?? 2
)}sin
2
??? ]-log?{
?? 4
+?? 4
2?? 2
?? 2
cos?2?? ?? 2
}= Constant 
or 
?? 2
(?? 2
-?? 2
)
2
sin
2
??? ?? 4
+?? 4
-2?? 2
?? 2
cos?2?? ?=?? 2
? (let) 
?? 2
{(?? 2
-?? 2
)
2
+2?? 2
?? 2
(1-cos?2?? )}?=?? 2
(?? 2
-?? 2
)sin
2
??? 2?? 2
?? 2
?? 2
(1-cos?2?? )?=(?? 2
-?? 2
)(?? 2
sin
2
??? -?? 2
)
4?? 2
?? 2
?? 2
sin
2
??? ?=(?? 2
-?? 2
)
2
(?? 2
sin?2?? -?? 2
)
 
7.3 An infinite row of equidistant rectilinear vortices are at a distance a apart. The 
vortices are of the same numerical strength ?? but they are alternately of opposite 
signe. Find the complex function that determines the velocity potential and the 
stream function. 
(2011 : 30 Marks) 
Solution: 
Let the row of vertices be taken along the ?? -axis. 
Let the points (0,0),(±2?? ,0),(±4?? ,0),…. have vertices of strength ?? and the points 
(±?? ,0),(±3?? ,0) , (±5?? ,0),….. have vortices of strength -?? . 
The complete potential of the entire system is given by 
?? =
????
2?? [{log??? +log?(?? -2?? )+log?(?? +2?? )+log?(?? -4?? )+log?(?? +4?? )+?.}
?-{log?(?? -?? )+log?(?? +?? )+log?(?? -3?? )+log?(?? +3?? )+?.}]
=
????
2?? log?
?? (?? 2
-2
2
?? 2
)(?? 2
-4
2
?? 2
)…..
(?? 2
-?? 2
)(?? 2
-3
2
?? 2
)…..
=
????
2?? log?
?? 2?? [1-(
?? 2?? )
2
][1-(
?? 4?? )
2
]…..
[1-(
?? ?? )
2
][1-(
?? 3?? )
2
]…..
+?? constant 
 
Page 5


Edurev123 
7. Vortex Motion 
7.1 In an incompressible fluid, the vorticity at every point is constant in magnitude 
and direction; show that the components of velocity ?? ,?? ,?? are solutions of 
Laplace equation. 
(2010: 12 Marks) 
Solution: 
Let velocity, 
?? =?? ??ˆ+?? ??ˆ+?? ??ˆ
 
? Vorticity, 
???? =?×??  
=|
??ˆ ??ˆ ??ˆ
?
??? ?
??? ?
??? ?? ?? ?? | 
=??ˆ(
??? ??? -
??? ??? )-??ˆ(
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )= Constant (given) 
??
??? ??? -
??? ??? =0 or 
??? ??? =
??? ??? 
??
?
2
?? ??? 2
=
?
2
?? ??? ??? 
and 
?
2
?? ??? ??? =
?
2
?? ??? 2
(2) 
Similarly, 
and 
??? ??? ?=
??? ??? ?
?
2
?? ??? 2
=
?
2
!
??? ??? (3)
?
2
?? ??? ??? ?=
?
2
?? ??? 2
(4)
 
Also, 
??? ??? -
??? ??? =0?
??? ??? =
??? ??? 
??
?
2
?? ??? ??? =
?
2
?? ??? 2
 
and 
?
2
?? ??? 2
=
??? ??? ??? (6) 
As fluid is incompressible, ? by eqn. of continuity 
??? ??? +
??? ??? +
??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? ??? +
?
2
?? ??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 
So, satisfies Laplace's equation. 
In the same way, 
?
2
?? ??? ??? +
?
2
?? ??? 2
+
?
2
?? ??? ??? =0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 and 
?
2
?? ??? ??? +
?
2
?? ??? ??? +
?
2
?? ??? 2
=0
???
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
=0
 
So, ?? and ?? also satisfy Laplace's equation. 
??? ,?? ,?? are solutions of Laplace's equation. 
7.2 When a pair of equal and opposite rectilinear vertices are situated in a long 
circular cylinder at equal distances from its axis, show that the path of each 
vortex is given by the equation 
(?? ?? ?????? ??? ?? -?? ?? )=?? ?? ?? ?? ?? ?? ?? ?????? ?? ??? 
?? being measured from the line through the centre perpendicular to the joint of the 
vertices. 
(2010 : 30 Marks) 
Solution: 
Let a pair of equal and opposite vertices of strength +?? and -?? are placed at the point 
?? (?? ,?? ) and ?? (?? ,?? ) respectively. 
The image consists of 
(i) A vertex of strength -?? at an inverse point ?? (
?? 2
?? ,?? ) . 
(ii) A vertex of strength +?? at an inverse point ?? (
?? 2
?? ,-?? ) . 
The complex potential at any point ?? becomes 
 
?? =
????
2?? log?(?? -?? ?? ????
)-
????
2?? log?{?? -(
?? 2
?? )?? ????
}-
????
2?? log?(?? -?? ?? -????
)+
????
2?? log?{?? -(
?? 2
?? )?? -????
} 
The motion of ?? is due to other vertices, thus for the motion ?? , the complex potential at 
any point ?? (?? =?? ?? ????
) becomes 
or 
?? 1
=[-
????
2?? log?{?? -(
?? 2
?? )?? ????
}-
????
2?? log?(?? -?? ?? -????
)-
????
2?? log?{?? -(
?? 2
?? )?? -????
}]
?? -?? ?? ????
 
?? 1
=-
????
2?? [log?{?? ?? ????
-(
?? 2
?? )?? ????
}log?(?? ?? ????
-?? ?? -????
)-log?{?? ?? ????
-(
?? 2
?? )?? -????
}] 
or 
 or ??? 1
=
-????
2?? [log?{(?? cos??? -(
?? 2
?? )cos??? )+?? (?? sin??? -
?? 2
?? sin??? )}+log?(2???? sin??? )
-log?{?? cos??? -(
?? 2
?? )cos??? }+?? (?? sin??? +
?? 2
?? )sin??? ]
 or ??? =
-?? 2?? [log?(?? -
?? 2
?? )+log?(2?? sin??? )-
1
2
log?{?? 2
+
?? 4
?? 2
-2?? 2
cos?2?? }]
 or ??? =
-?? 4?? [log?(?? -
?? 2
?? ·2?? sin??? )
2
-log?{?? 2
+
?? 4
?? 2
-2?? 2
cos?2?? }]
 
The stream lines are given by ?? = constant 
i.e. 
-?? 4?? [log?{4(?? 2
-?? 2
)}sin
2
??? ]-log?{
?? 4
+?? 4
2?? 2
?? 2
cos?2?? ?? 2
}= Constant 
or 
?? 2
(?? 2
-?? 2
)
2
sin
2
??? ?? 4
+?? 4
-2?? 2
?? 2
cos?2?? ?=?? 2
? (let) 
?? 2
{(?? 2
-?? 2
)
2
+2?? 2
?? 2
(1-cos?2?? )}?=?? 2
(?? 2
-?? 2
)sin
2
??? 2?? 2
?? 2
?? 2
(1-cos?2?? )?=(?? 2
-?? 2
)(?? 2
sin
2
??? -?? 2
)
4?? 2
?? 2
?? 2
sin
2
??? ?=(?? 2
-?? 2
)
2
(?? 2
sin?2?? -?? 2
)
 
7.3 An infinite row of equidistant rectilinear vortices are at a distance a apart. The 
vortices are of the same numerical strength ?? but they are alternately of opposite 
signe. Find the complex function that determines the velocity potential and the 
stream function. 
(2011 : 30 Marks) 
Solution: 
Let the row of vertices be taken along the ?? -axis. 
Let the points (0,0),(±2?? ,0),(±4?? ,0),…. have vertices of strength ?? and the points 
(±?? ,0),(±3?? ,0) , (±5?? ,0),….. have vortices of strength -?? . 
The complete potential of the entire system is given by 
?? =
????
2?? [{log??? +log?(?? -2?? )+log?(?? +2?? )+log?(?? -4?? )+log?(?? +4?? )+?.}
?-{log?(?? -?? )+log?(?? +?? )+log?(?? -3?? )+log?(?? +3?? )+?.}]
=
????
2?? log?
?? (?? 2
-2
2
?? 2
)(?? 2
-4
2
?? 2
)…..
(?? 2
-?? 2
)(?? 2
-3
2
?? 2
)…..
=
????
2?? log?
?? 2?? [1-(
?? 2?? )
2
][1-(
?? 4?? )
2
]…..
[1-(
?? ?? )
2
][1-(
?? 3?? )
2
]…..
+?? constant 
 
which is the desired potential function that determines the velocity potential and stream 
function. 
From (i), 
?? +???? =
????
2?? log?tan?
?? 2?? (?? +???? ) (???? )
 
???? -???? =-
????
2?? log?tan?
?? 2?? (?? -???? ) (?????? )
 
Subtracting (iii) from (ii), we have 
2???? ?=
????
2?? [log?tan?
?? 2?? (?? +???? )+log?tan?
?? 2?? (?? -???? )]
?? ?=
?? 4?? log?
sin?
?? 2?? (?? +???? )sin?
?? 2?? (?? -???? )
cos?
?? 2?? (?? +???? )cos?
?? 2?? (?? -???? )
?=
?? 4?? log?
cosh?(
????
?? )-cos?(
????
?? )
cosh?(
????
?? )+cos?(
????
?? )
 
Since the motion of the vortex at the origin is due to other vortices only, the velocity ?? 0
 of 
the vortex at the origin is given by 
?? 0
=-[
?? ????
(
????
2?? log?tan?
????
2?? -
????
2?? log??? )]
?? =0
 
=-
????
2?? [
sec
2
?(
????
2?? )
tan?(
????
2?? )
×
?? 2?? -
1
2
]
?? =0
 
=0 (using L'Hospital's Rule)  
Hence, the vortex at the origin is at rest. Similarly, it can be shown that the remaining 
vortices are also at rest. Thus, we find that the vortex row induces no velocity on itself. 
7.4 Prove that the necessary and sufficient conditions that the vortex lines may be 
at right angles to stream lines are : 
?? ,?? ,?? =?? (
?? ?? ?? ?? ,
?? ?? ?? ?? ,
?? ?? ?? ?? ) 
where ?? and ?? are functions of ?? ,?? ,?? ,?? . 
(2013 : 10 Marks) 
Solution: 
Streamlines are given by