Equation of Continuity | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
4. Equation of Continuity 
4.1 An infinite mass of fluid is acted on by a force 
?? ?? ?? /?? per unit mass directed to the 
origin. If initially the fluid is at rest and there is a cavity in the form of the sphere 
?? =?? in it, show that the cavity will be filled up after an interval of time (
?? ????
)
?? /?? ?? ?? /?? . 
(2009 : 30 Marks) 
Solution: 
Here the motion of the fluid will take place in such a manner that each element of the 
fluid moves towards the centre. Hence, the free surface will be spherical. Thus, the 
velocity ? will be radial and hence ?? will be a function of ?? (radial distance from the 
centre of the sphere, which is taken as origin) and time ?? . Also, let ?? be the velocity at a 
distance ?? . 
Using equation of continuity 
?? 2
?? =?? 2
?? =?? (?? ) (1)
?? (?? )=
?? 2
??? '
??? (1)
 
Using Euler equation, 
??? '
??? +
?? '
??? '
??? '
?=?? -
1
?? ??? ??? where ?? =
-?? ?? 3/2
 
Integrating with respect to ?? 
-?? '
(?? )
?? '
+
1
2
?? '2
=
-?? ?? -
3
2
+1
-3
2
+1
-
1·?? ?? +?? 
Infinite mass, 
at ?? =8, 
?? ?=0,?? =0
0+0?=0+0+?? ??? =0
 
Now at ?? =?? ,?? =?? ,?? =0 
-?? '
(?? )
?? +
?? 2
2
=
2?? v?? (3) 
Page 2


Edurev123 
4. Equation of Continuity 
4.1 An infinite mass of fluid is acted on by a force 
?? ?? ?? /?? per unit mass directed to the 
origin. If initially the fluid is at rest and there is a cavity in the form of the sphere 
?? =?? in it, show that the cavity will be filled up after an interval of time (
?? ????
)
?? /?? ?? ?? /?? . 
(2009 : 30 Marks) 
Solution: 
Here the motion of the fluid will take place in such a manner that each element of the 
fluid moves towards the centre. Hence, the free surface will be spherical. Thus, the 
velocity ? will be radial and hence ?? will be a function of ?? (radial distance from the 
centre of the sphere, which is taken as origin) and time ?? . Also, let ?? be the velocity at a 
distance ?? . 
Using equation of continuity 
?? 2
?? =?? 2
?? =?? (?? ) (1)
?? (?? )=
?? 2
??? '
??? (1)
 
Using Euler equation, 
??? '
??? +
?? '
??? '
??? '
?=?? -
1
?? ??? ??? where ?? =
-?? ?? 3/2
 
Integrating with respect to ?? 
-?? '
(?? )
?? '
+
1
2
?? '2
=
-?? ?? -
3
2
+1
-3
2
+1
-
1·?? ?? +?? 
Infinite mass, 
at ?? =8, 
?? ?=0,?? =0
0+0?=0+0+?? ??? =0
 
Now at ?? =?? ,?? =?? ,?? =0 
-?? '
(?? )
?? +
?? 2
2
=
2?? v?? (3) 
Now 
?? (?? )?=?? 2
?? =?? 2
????
????
?? '
(?? )?=2?? (
????
????
)
2
+?? 2
?? 2
?? ?? ?? 2
?=2?? ?? 2
+?? 2
?? ????
????
?? '
(?? )
?? ?=2?? 2
+????
????
????
 
(due to cavity) 
(from (1)) 
[????????????????????????????????????????????????????????????
?? 2
?? ?? ?? 2
=
?????? ????
] 
Put this value in (3) 
-(2?? 2
+????
????
????
)+
?? 2
2
?=
2?? v?? -3?? 2
2
-????
????
????
?=
2?? v?? 3?? 2
???? +2???????? ?=-4?? ????
v?? 
Multiply by ?? 2
 
3?? 2
?? 2
???? +2?? 3
?????? ?=
-4?? ?? 2
????
?? 1/2
?? (?? 2
?? 2
)?=-4?? ?? 3/2
????
 
Integrating, we get 
?? 3
?? 2
=
-4?? ?? 3
2
+?? 3
2
+1
+?? '
 
Now, initialiy at ?? =?? ,?? =0 
Put this value in (4) 
So (4) becomes 
?? ?? (?? )
2
=
-8?? ?? 5
2
5
+?? '
??? =
8?? ?? 5
2
5
 
Page 3


Edurev123 
4. Equation of Continuity 
4.1 An infinite mass of fluid is acted on by a force 
?? ?? ?? /?? per unit mass directed to the 
origin. If initially the fluid is at rest and there is a cavity in the form of the sphere 
?? =?? in it, show that the cavity will be filled up after an interval of time (
?? ????
)
?? /?? ?? ?? /?? . 
(2009 : 30 Marks) 
Solution: 
Here the motion of the fluid will take place in such a manner that each element of the 
fluid moves towards the centre. Hence, the free surface will be spherical. Thus, the 
velocity ? will be radial and hence ?? will be a function of ?? (radial distance from the 
centre of the sphere, which is taken as origin) and time ?? . Also, let ?? be the velocity at a 
distance ?? . 
Using equation of continuity 
?? 2
?? =?? 2
?? =?? (?? ) (1)
?? (?? )=
?? 2
??? '
??? (1)
 
Using Euler equation, 
??? '
??? +
?? '
??? '
??? '
?=?? -
1
?? ??? ??? where ?? =
-?? ?? 3/2
 
Integrating with respect to ?? 
-?? '
(?? )
?? '
+
1
2
?? '2
=
-?? ?? -
3
2
+1
-3
2
+1
-
1·?? ?? +?? 
Infinite mass, 
at ?? =8, 
?? ?=0,?? =0
0+0?=0+0+?? ??? =0
 
Now at ?? =?? ,?? =?? ,?? =0 
-?? '
(?? )
?? +
?? 2
2
=
2?? v?? (3) 
Now 
?? (?? )?=?? 2
?? =?? 2
????
????
?? '
(?? )?=2?? (
????
????
)
2
+?? 2
?? 2
?? ?? ?? 2
?=2?? ?? 2
+?? 2
?? ????
????
?? '
(?? )
?? ?=2?? 2
+????
????
????
 
(due to cavity) 
(from (1)) 
[????????????????????????????????????????????????????????????
?? 2
?? ?? ?? 2
=
?????? ????
] 
Put this value in (3) 
-(2?? 2
+????
????
????
)+
?? 2
2
?=
2?? v?? -3?? 2
2
-????
????
????
?=
2?? v?? 3?? 2
???? +2???????? ?=-4?? ????
v?? 
Multiply by ?? 2
 
3?? 2
?? 2
???? +2?? 3
?????? ?=
-4?? ?? 2
????
?? 1/2
?? (?? 2
?? 2
)?=-4?? ?? 3/2
????
 
Integrating, we get 
?? 3
?? 2
=
-4?? ?? 3
2
+?? 3
2
+1
+?? '
 
Now, initialiy at ?? =?? ,?? =0 
Put this value in (4) 
So (4) becomes 
?? ?? (?? )
2
=
-8?? ?? 5
2
5
+?? '
??? =
8?? ?? 5
2
5
 
?? 3
?? 2
?=
8?? 5
(?? 5/2
-?? 5/2
)
?? 2
?=
8?? 5?? 3
(?? 5/2
-?? 5/2
)
?? ?=
v
8?? 5
(
?? 5/2
-?? 5/2
?? 3/2
)
1/2
=
-????
????
 
(Negative sign is taken, because ?? is decreasing with ?? ) 
Put 
? ?
0
?? ?
-?? 3/2
????
(?? 5/2
-?? 1/2
)
1/2
=
v
8?? 5
? ?
?? 0
? (5)
 
So, C
5/2
-R
1/2
=?? 
???? =-
5
2
?? 3/2
???? 
??
2????
5
=-?? 3/2
???? 
So, 
2
5
??
????
?? 1/2
=
4
5
v?? +?? 
So (5) becomes 
|
4
5
(?? 5/2
-?? 5/2
)
1/2
|
?? 0
?=
v8?? v3
?? 4
5v5
?? 5/4
?=
v8?? v5
?? ?? ?=
4?? 5/4
v2
v
?? ×v5
=v
2
5?? ?? 5/4
 
4.2 A rigid sphere of radius ?? is placed in a stream of fluid whose velocity in the 
undisturbed state is ?? . Determine the velocity of the fluid at any point of the 
disturbed stream. 
(2012 : 12 Marks) 
Solution: 
We may take the polar axis O?? to be in the direction of the given velocity. 
Page 4


Edurev123 
4. Equation of Continuity 
4.1 An infinite mass of fluid is acted on by a force 
?? ?? ?? /?? per unit mass directed to the 
origin. If initially the fluid is at rest and there is a cavity in the form of the sphere 
?? =?? in it, show that the cavity will be filled up after an interval of time (
?? ????
)
?? /?? ?? ?? /?? . 
(2009 : 30 Marks) 
Solution: 
Here the motion of the fluid will take place in such a manner that each element of the 
fluid moves towards the centre. Hence, the free surface will be spherical. Thus, the 
velocity ? will be radial and hence ?? will be a function of ?? (radial distance from the 
centre of the sphere, which is taken as origin) and time ?? . Also, let ?? be the velocity at a 
distance ?? . 
Using equation of continuity 
?? 2
?? =?? 2
?? =?? (?? ) (1)
?? (?? )=
?? 2
??? '
??? (1)
 
Using Euler equation, 
??? '
??? +
?? '
??? '
??? '
?=?? -
1
?? ??? ??? where ?? =
-?? ?? 3/2
 
Integrating with respect to ?? 
-?? '
(?? )
?? '
+
1
2
?? '2
=
-?? ?? -
3
2
+1
-3
2
+1
-
1·?? ?? +?? 
Infinite mass, 
at ?? =8, 
?? ?=0,?? =0
0+0?=0+0+?? ??? =0
 
Now at ?? =?? ,?? =?? ,?? =0 
-?? '
(?? )
?? +
?? 2
2
=
2?? v?? (3) 
Now 
?? (?? )?=?? 2
?? =?? 2
????
????
?? '
(?? )?=2?? (
????
????
)
2
+?? 2
?? 2
?? ?? ?? 2
?=2?? ?? 2
+?? 2
?? ????
????
?? '
(?? )
?? ?=2?? 2
+????
????
????
 
(due to cavity) 
(from (1)) 
[????????????????????????????????????????????????????????????
?? 2
?? ?? ?? 2
=
?????? ????
] 
Put this value in (3) 
-(2?? 2
+????
????
????
)+
?? 2
2
?=
2?? v?? -3?? 2
2
-????
????
????
?=
2?? v?? 3?? 2
???? +2???????? ?=-4?? ????
v?? 
Multiply by ?? 2
 
3?? 2
?? 2
???? +2?? 3
?????? ?=
-4?? ?? 2
????
?? 1/2
?? (?? 2
?? 2
)?=-4?? ?? 3/2
????
 
Integrating, we get 
?? 3
?? 2
=
-4?? ?? 3
2
+?? 3
2
+1
+?? '
 
Now, initialiy at ?? =?? ,?? =0 
Put this value in (4) 
So (4) becomes 
?? ?? (?? )
2
=
-8?? ?? 5
2
5
+?? '
??? =
8?? ?? 5
2
5
 
?? 3
?? 2
?=
8?? 5
(?? 5/2
-?? 5/2
)
?? 2
?=
8?? 5?? 3
(?? 5/2
-?? 5/2
)
?? ?=
v
8?? 5
(
?? 5/2
-?? 5/2
?? 3/2
)
1/2
=
-????
????
 
(Negative sign is taken, because ?? is decreasing with ?? ) 
Put 
? ?
0
?? ?
-?? 3/2
????
(?? 5/2
-?? 1/2
)
1/2
=
v
8?? 5
? ?
?? 0
? (5)
 
So, C
5/2
-R
1/2
=?? 
???? =-
5
2
?? 3/2
???? 
??
2????
5
=-?? 3/2
???? 
So, 
2
5
??
????
?? 1/2
=
4
5
v?? +?? 
So (5) becomes 
|
4
5
(?? 5/2
-?? 5/2
)
1/2
|
?? 0
?=
v8?? v3
?? 4
5v5
?? 5/4
?=
v8?? v5
?? ?? ?=
4?? 5/4
v2
v
?? ×v5
=v
2
5?? ?? 5/4
 
4.2 A rigid sphere of radius ?? is placed in a stream of fluid whose velocity in the 
undisturbed state is ?? . Determine the velocity of the fluid at any point of the 
disturbed stream. 
(2012 : 12 Marks) 
Solution: 
We may take the polar axis O?? to be in the direction of the given velocity. 
Let us take the polar co-ordinates (?? ,?? ,?? ) with origin at the center of the fixed sphere. 
The velocity of the fluid is given by 
?? =-grad??? , where                                                         (i) 
?
2
?? ?=0
?? ~-???? cos??? ?=-???? ?? 1
cos??? as ?? ?8
 
The axially symmetrical function 
?? =??
?? =0
(?? ?? ?? ?? +
?? ?? ?? ?? +1
)?? ?? (cos??? ) 
satisfies (i). 
Again, condition (ii) is satisfied if we take 
?? ?? ?? ?? ?? -1
-(?? +1)
?? ?? ?? ?? +2
=0 
i.e., if 
?? ?? =?? ?? 2?? +1
?? ?? ?? +1
 
As ?? ?8, this velocity potential has the asymptotic form 
?~??
8
?? =0
?? ?? ?? ?? ?? ?? (cos??? ) 
So to satisfy (iii), we take ?? 1
=-?? and all the other ?? 's zero. 
Hence, the required velocity potential is 
?? =-?? (?? +
?? 3
2?? 2
)cos??? 
The components of velocity are 
?? ?? =-
????
????
=?? (1-
?? 3
?? 3
)cos??? ?? ?? =-
1??? ?? ??? ??? (1-
?? 3
2?? 3
)sin??? 
4.3 Given the velocity potential ?? =
?? ?? ?????? ?[
(?? +?? )
?? +?? ?? (?? -?? )
?? +?? ?? ], determine the streamlines. 
(2014: 20 marks) 
Page 5


Edurev123 
4. Equation of Continuity 
4.1 An infinite mass of fluid is acted on by a force 
?? ?? ?? /?? per unit mass directed to the 
origin. If initially the fluid is at rest and there is a cavity in the form of the sphere 
?? =?? in it, show that the cavity will be filled up after an interval of time (
?? ????
)
?? /?? ?? ?? /?? . 
(2009 : 30 Marks) 
Solution: 
Here the motion of the fluid will take place in such a manner that each element of the 
fluid moves towards the centre. Hence, the free surface will be spherical. Thus, the 
velocity ? will be radial and hence ?? will be a function of ?? (radial distance from the 
centre of the sphere, which is taken as origin) and time ?? . Also, let ?? be the velocity at a 
distance ?? . 
Using equation of continuity 
?? 2
?? =?? 2
?? =?? (?? ) (1)
?? (?? )=
?? 2
??? '
??? (1)
 
Using Euler equation, 
??? '
??? +
?? '
??? '
??? '
?=?? -
1
?? ??? ??? where ?? =
-?? ?? 3/2
 
Integrating with respect to ?? 
-?? '
(?? )
?? '
+
1
2
?? '2
=
-?? ?? -
3
2
+1
-3
2
+1
-
1·?? ?? +?? 
Infinite mass, 
at ?? =8, 
?? ?=0,?? =0
0+0?=0+0+?? ??? =0
 
Now at ?? =?? ,?? =?? ,?? =0 
-?? '
(?? )
?? +
?? 2
2
=
2?? v?? (3) 
Now 
?? (?? )?=?? 2
?? =?? 2
????
????
?? '
(?? )?=2?? (
????
????
)
2
+?? 2
?? 2
?? ?? ?? 2
?=2?? ?? 2
+?? 2
?? ????
????
?? '
(?? )
?? ?=2?? 2
+????
????
????
 
(due to cavity) 
(from (1)) 
[????????????????????????????????????????????????????????????
?? 2
?? ?? ?? 2
=
?????? ????
] 
Put this value in (3) 
-(2?? 2
+????
????
????
)+
?? 2
2
?=
2?? v?? -3?? 2
2
-????
????
????
?=
2?? v?? 3?? 2
???? +2???????? ?=-4?? ????
v?? 
Multiply by ?? 2
 
3?? 2
?? 2
???? +2?? 3
?????? ?=
-4?? ?? 2
????
?? 1/2
?? (?? 2
?? 2
)?=-4?? ?? 3/2
????
 
Integrating, we get 
?? 3
?? 2
=
-4?? ?? 3
2
+?? 3
2
+1
+?? '
 
Now, initialiy at ?? =?? ,?? =0 
Put this value in (4) 
So (4) becomes 
?? ?? (?? )
2
=
-8?? ?? 5
2
5
+?? '
??? =
8?? ?? 5
2
5
 
?? 3
?? 2
?=
8?? 5
(?? 5/2
-?? 5/2
)
?? 2
?=
8?? 5?? 3
(?? 5/2
-?? 5/2
)
?? ?=
v
8?? 5
(
?? 5/2
-?? 5/2
?? 3/2
)
1/2
=
-????
????
 
(Negative sign is taken, because ?? is decreasing with ?? ) 
Put 
? ?
0
?? ?
-?? 3/2
????
(?? 5/2
-?? 1/2
)
1/2
=
v
8?? 5
? ?
?? 0
? (5)
 
So, C
5/2
-R
1/2
=?? 
???? =-
5
2
?? 3/2
???? 
??
2????
5
=-?? 3/2
???? 
So, 
2
5
??
????
?? 1/2
=
4
5
v?? +?? 
So (5) becomes 
|
4
5
(?? 5/2
-?? 5/2
)
1/2
|
?? 0
?=
v8?? v3
?? 4
5v5
?? 5/4
?=
v8?? v5
?? ?? ?=
4?? 5/4
v2
v
?? ×v5
=v
2
5?? ?? 5/4
 
4.2 A rigid sphere of radius ?? is placed in a stream of fluid whose velocity in the 
undisturbed state is ?? . Determine the velocity of the fluid at any point of the 
disturbed stream. 
(2012 : 12 Marks) 
Solution: 
We may take the polar axis O?? to be in the direction of the given velocity. 
Let us take the polar co-ordinates (?? ,?? ,?? ) with origin at the center of the fixed sphere. 
The velocity of the fluid is given by 
?? =-grad??? , where                                                         (i) 
?
2
?? ?=0
?? ~-???? cos??? ?=-???? ?? 1
cos??? as ?? ?8
 
The axially symmetrical function 
?? =??
?? =0
(?? ?? ?? ?? +
?? ?? ?? ?? +1
)?? ?? (cos??? ) 
satisfies (i). 
Again, condition (ii) is satisfied if we take 
?? ?? ?? ?? ?? -1
-(?? +1)
?? ?? ?? ?? +2
=0 
i.e., if 
?? ?? =?? ?? 2?? +1
?? ?? ?? +1
 
As ?? ?8, this velocity potential has the asymptotic form 
?~??
8
?? =0
?? ?? ?? ?? ?? ?? (cos??? ) 
So to satisfy (iii), we take ?? 1
=-?? and all the other ?? 's zero. 
Hence, the required velocity potential is 
?? =-?? (?? +
?? 3
2?? 2
)cos??? 
The components of velocity are 
?? ?? =-
????
????
=?? (1-
?? 3
?? 3
)cos??? ?? ?? =-
1??? ?? ??? ??? (1-
?? 3
2?? 3
)sin??? 
4.3 Given the velocity potential ?? =
?? ?? ?????? ?[
(?? +?? )
?? +?? ?? (?? -?? )
?? +?? ?? ], determine the streamlines. 
(2014: 20 marks) 
Solution: 
Velocity potential, 
?? =
1
2
log?[
(?? +?? )
2
+?? 2
(?? -?? )
2
+?? 2
] 
To determine stream lines 
Hence, 
-??? ??? =?? =-
??? ??? ;-
??? ??? =?? =
??
??? 
Now, 
??? ??? =
??
??? ,
??? ??? =-
??? ??? ??? ??? =
?? +0
(?? +?? )
2
+?? 2
-
?? -?? (?? -?? )
2
+?? 2
 
Integrating w.r.t. y 
?? =tan
-1
?(
?? ?? +?? )-tan
-1
?(
?? ?? -?? )+?? (?? ) (??) 
where ?? (?? ) is constant of integration. To determine ?? (?? ) 
??? ??? ?=-
??? ??? =
-?? (?? +?? )
2
+?? 2
+
?? (?? -?? )
2
+?? 2
(???? )
? by (i) ?
??? ??? ?=
?? (?? +?? )
2
+?? 2
+
?? (?? +?? )
2
+?? 2
+?? '
(?? ) (???? )
 
Equating (ii) and (iii) 
? ?? (?? )=0
 Integrating this, ?? (?? )= absolute constant hence neglected. 
 
Since, it has no effect on the fluid motion. 
Now (i), becomes, 
Stream lines are given by, 
??=tan
-1
?(
?? ?? +?? )-tan
-1
?(
?? ?? -?? ) 
?? = constant