Fluid Dynamics | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
FLUID DYNAMICS 
1. Introduction 
Fluid dynamics deals with the study of fluids in motion. By the term fluid is meant a 
substance that flows; one which does not is termed as a solid.  Fluids may be divided 
into 1) liquids, which are incompressible, i.e., their volumes do not change and (ii) gases 
which are compressible, i.e., their volumes change whenever pressure changes. 
The term hydrodynamics is used to the science of moving incompressible fluids. Fluids 
are classified as viscous fluids and perfect (inviscid) fluids. Viscous fluids sustain 
tangential stresses, but perfect fluids are incapable of exerting shearing stresses. The 
pressure exerted by a perfect fluid is always normal to the surface of contact. The 
pressure at any point in a perfect fluid is the same in every direction. 
Let ???=?? (???,?? ) be the velocity vector. In component form, we write 
???={?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? )} 
The acceleration vector, ä is defined as 
??? =
????
????
=
????
??? +
????
??? +
??? ??? +
????
??? ??? ???ˆ
+
????
??? ??? ??? 
But ?? =
??? ??? ,?? =
??? ??? ?? =
??? ??? 
Hence ???=
????
???ˆ
+?? ????
??? +?? ????
??? +?? ????
??? 
Noting that the operator 
?? ?
??? +?? ?
??? +?? ?
??? =q.? 
we can express the acceleration as 
??? =
???¨
??? +(???·?)???   … (1) 
 
In component form, (1) reads as  
?? 1
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 2
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
Page 2


Edurev123 
FLUID DYNAMICS 
1. Introduction 
Fluid dynamics deals with the study of fluids in motion. By the term fluid is meant a 
substance that flows; one which does not is termed as a solid.  Fluids may be divided 
into 1) liquids, which are incompressible, i.e., their volumes do not change and (ii) gases 
which are compressible, i.e., their volumes change whenever pressure changes. 
The term hydrodynamics is used to the science of moving incompressible fluids. Fluids 
are classified as viscous fluids and perfect (inviscid) fluids. Viscous fluids sustain 
tangential stresses, but perfect fluids are incapable of exerting shearing stresses. The 
pressure exerted by a perfect fluid is always normal to the surface of contact. The 
pressure at any point in a perfect fluid is the same in every direction. 
Let ???=?? (???,?? ) be the velocity vector. In component form, we write 
???={?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? )} 
The acceleration vector, ä is defined as 
??? =
????
????
=
????
??? +
????
??? +
??? ??? +
????
??? ??? ???ˆ
+
????
??? ??? ??? 
But ?? =
??? ??? ,?? =
??? ??? ?? =
??? ??? 
Hence ???=
????
???ˆ
+?? ????
??? +?? ????
??? +?? ????
??? 
Noting that the operator 
?? ?
??? +?? ?
??? +?? ?
??? =q.? 
we can express the acceleration as 
??? =
???¨
??? +(???·?)???   … (1) 
 
In component form, (1) reads as  
?? 1
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 2
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 3
=
??? ??? +?? ????
??? +?? ??? ??? +?? ??? ??? 
where ???=(?? 1
,?? 2
,?? 3
) 
The first term in (1), namely 
????
??? is called the local acceleration. Physically, it is the 
acceleration observed by a stationery observer at the point (?? ,?? ,?? ) . The second term in 
(1), namely (????
·?)????
 is the convective acceleration. This is due to the variation of ??? with 
the position ???. 
The motion is said to be steady when ??? is not an explicit function of time ?? , that is, ??? is in 
dependent of time for any fixed position. Note that the acceleration is not zero for steady 
motion. Indeed, for steady molion ??? =(???.?)??? The operator ???·? is called the mobile 
operator or convective operator. The operator 
?
??? +???·? is denoted by ?? and stands for 
differentiation following the motion 
Thus ???=
?? ???
????
 For any scalar or vector point function ?? associated with fluid motion, we 
have 
????
????
=
??? ??? +(???·?)?? 
Using the formula 
?(???
·????
)=???
×curl ????
+????
×curl ???
-(???
·?)???
-(???
·?)???
 
We find with ???
=??¨
=??? 
(???·?)???=?? (
?? 2
2
)+(curl ???)×??? 
Hence the acceleration vector can also be written as 
???=
????
?q
+?(
1
2
?? 2
)+(??????? ???
???)×??? (2) 
The vector curl ??? i.e. ?×??? is called the vortically vector. It is a measure of rotation in the 
lluid. If ?×??? =0
??
, we call, the motion irrotational. If ?×????0
¯
, the motion is rotational. 
Since ?×??? =0
¯
 for a scalar ?? , when the motion is irrotational, there exists a scalar 
function ?? , called the velocity potential such that 
???=-??? (3) 
The negative sign in (3) is conventional and it ensures that ??? is in the direction of 
decreasing ?? . 
Page 3


Edurev123 
FLUID DYNAMICS 
1. Introduction 
Fluid dynamics deals with the study of fluids in motion. By the term fluid is meant a 
substance that flows; one which does not is termed as a solid.  Fluids may be divided 
into 1) liquids, which are incompressible, i.e., their volumes do not change and (ii) gases 
which are compressible, i.e., their volumes change whenever pressure changes. 
The term hydrodynamics is used to the science of moving incompressible fluids. Fluids 
are classified as viscous fluids and perfect (inviscid) fluids. Viscous fluids sustain 
tangential stresses, but perfect fluids are incapable of exerting shearing stresses. The 
pressure exerted by a perfect fluid is always normal to the surface of contact. The 
pressure at any point in a perfect fluid is the same in every direction. 
Let ???=?? (???,?? ) be the velocity vector. In component form, we write 
???={?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? )} 
The acceleration vector, ä is defined as 
??? =
????
????
=
????
??? +
????
??? +
??? ??? +
????
??? ??? ???ˆ
+
????
??? ??? ??? 
But ?? =
??? ??? ,?? =
??? ??? ?? =
??? ??? 
Hence ???=
????
???ˆ
+?? ????
??? +?? ????
??? +?? ????
??? 
Noting that the operator 
?? ?
??? +?? ?
??? +?? ?
??? =q.? 
we can express the acceleration as 
??? =
???¨
??? +(???·?)???   … (1) 
 
In component form, (1) reads as  
?? 1
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 2
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 3
=
??? ??? +?? ????
??? +?? ??? ??? +?? ??? ??? 
where ???=(?? 1
,?? 2
,?? 3
) 
The first term in (1), namely 
????
??? is called the local acceleration. Physically, it is the 
acceleration observed by a stationery observer at the point (?? ,?? ,?? ) . The second term in 
(1), namely (????
·?)????
 is the convective acceleration. This is due to the variation of ??? with 
the position ???. 
The motion is said to be steady when ??? is not an explicit function of time ?? , that is, ??? is in 
dependent of time for any fixed position. Note that the acceleration is not zero for steady 
motion. Indeed, for steady molion ??? =(???.?)??? The operator ???·? is called the mobile 
operator or convective operator. The operator 
?
??? +???·? is denoted by ?? and stands for 
differentiation following the motion 
Thus ???=
?? ???
????
 For any scalar or vector point function ?? associated with fluid motion, we 
have 
????
????
=
??? ??? +(???·?)?? 
Using the formula 
?(???
·????
)=???
×curl ????
+????
×curl ???
-(???
·?)???
-(???
·?)???
 
We find with ???
=??¨
=??? 
(???·?)???=?? (
?? 2
2
)+(curl ???)×??? 
Hence the acceleration vector can also be written as 
???=
????
?q
+?(
1
2
?? 2
)+(??????? ???
???)×??? (2) 
The vector curl ??? i.e. ?×??? is called the vortically vector. It is a measure of rotation in the 
lluid. If ?×??? =0
??
, we call, the motion irrotational. If ?×????0
¯
, the motion is rotational. 
Since ?×??? =0
¯
 for a scalar ?? , when the motion is irrotational, there exists a scalar 
function ?? , called the velocity potential such that 
???=-??? (3) 
The negative sign in (3) is conventional and it ensures that ??? is in the direction of 
decreasing ?? . 
Streamlines are curves in the fluid such that the tangents to them are parallel to the 
velocity vector. The streamlines can be obtained by giving the differential equations 
????
?? =
????
?? =
????
?? (4)
 
where (?? ,?? ,?? ) is the velocity 
The path lines are the curves traced by the individual fluid particles and their equations. 
are given by 
????
????
=?? (?? ,?? ,?? ,?? ) (5)
????
????
=?? (?? ,?? ,?? ,?? ) (5)
????
????
=?? (?? ,?? ,?? ,?? ) (5)
 
The path lines traced a particle which was at the point (?? 0
,?? 0
,?? 0
) a time ?? 0
, can be 
obtained by integrating the simultaneous equation (5) with the initial conditions: 
?? =?? 0
,?? =?? 0
,?? =?? 0
,?? =?? 0
 
For steady motion, the streamlines and path lines coincide. For unsteady motion they 
may or may not coincide. 
Vortex lines are curves ?? uch that the tangents to them are parallel to vorticity vector ??? =
?
?
×???. Their equations are given by 
????
?? 1
=
????
?? 2
=
????
?? 3
 
 
2. Equation of Continuity 
Consider a fixed closed surface ?? , bounding a volume ?? . Let ?? be the density of the fluid. 
The fluid entering the surface across a surface element dS with unit outward normal ???, 
per unit time is given by 
-?? ???·???(???? ) (1) 
Hence the total mass entering across the entire surface ?? is the surface integral 
-??
?? ??? ???·??????? =-??
?? ??·(?? ???)·???? (2) 
by the divergence theorem 
Page 4


Edurev123 
FLUID DYNAMICS 
1. Introduction 
Fluid dynamics deals with the study of fluids in motion. By the term fluid is meant a 
substance that flows; one which does not is termed as a solid.  Fluids may be divided 
into 1) liquids, which are incompressible, i.e., their volumes do not change and (ii) gases 
which are compressible, i.e., their volumes change whenever pressure changes. 
The term hydrodynamics is used to the science of moving incompressible fluids. Fluids 
are classified as viscous fluids and perfect (inviscid) fluids. Viscous fluids sustain 
tangential stresses, but perfect fluids are incapable of exerting shearing stresses. The 
pressure exerted by a perfect fluid is always normal to the surface of contact. The 
pressure at any point in a perfect fluid is the same in every direction. 
Let ???=?? (???,?? ) be the velocity vector. In component form, we write 
???={?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? )} 
The acceleration vector, ä is defined as 
??? =
????
????
=
????
??? +
????
??? +
??? ??? +
????
??? ??? ???ˆ
+
????
??? ??? ??? 
But ?? =
??? ??? ,?? =
??? ??? ?? =
??? ??? 
Hence ???=
????
???ˆ
+?? ????
??? +?? ????
??? +?? ????
??? 
Noting that the operator 
?? ?
??? +?? ?
??? +?? ?
??? =q.? 
we can express the acceleration as 
??? =
???¨
??? +(???·?)???   … (1) 
 
In component form, (1) reads as  
?? 1
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 2
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 3
=
??? ??? +?? ????
??? +?? ??? ??? +?? ??? ??? 
where ???=(?? 1
,?? 2
,?? 3
) 
The first term in (1), namely 
????
??? is called the local acceleration. Physically, it is the 
acceleration observed by a stationery observer at the point (?? ,?? ,?? ) . The second term in 
(1), namely (????
·?)????
 is the convective acceleration. This is due to the variation of ??? with 
the position ???. 
The motion is said to be steady when ??? is not an explicit function of time ?? , that is, ??? is in 
dependent of time for any fixed position. Note that the acceleration is not zero for steady 
motion. Indeed, for steady molion ??? =(???.?)??? The operator ???·? is called the mobile 
operator or convective operator. The operator 
?
??? +???·? is denoted by ?? and stands for 
differentiation following the motion 
Thus ???=
?? ???
????
 For any scalar or vector point function ?? associated with fluid motion, we 
have 
????
????
=
??? ??? +(???·?)?? 
Using the formula 
?(???
·????
)=???
×curl ????
+????
×curl ???
-(???
·?)???
-(???
·?)???
 
We find with ???
=??¨
=??? 
(???·?)???=?? (
?? 2
2
)+(curl ???)×??? 
Hence the acceleration vector can also be written as 
???=
????
?q
+?(
1
2
?? 2
)+(??????? ???
???)×??? (2) 
The vector curl ??? i.e. ?×??? is called the vortically vector. It is a measure of rotation in the 
lluid. If ?×??? =0
??
, we call, the motion irrotational. If ?×????0
¯
, the motion is rotational. 
Since ?×??? =0
¯
 for a scalar ?? , when the motion is irrotational, there exists a scalar 
function ?? , called the velocity potential such that 
???=-??? (3) 
The negative sign in (3) is conventional and it ensures that ??? is in the direction of 
decreasing ?? . 
Streamlines are curves in the fluid such that the tangents to them are parallel to the 
velocity vector. The streamlines can be obtained by giving the differential equations 
????
?? =
????
?? =
????
?? (4)
 
where (?? ,?? ,?? ) is the velocity 
The path lines are the curves traced by the individual fluid particles and their equations. 
are given by 
????
????
=?? (?? ,?? ,?? ,?? ) (5)
????
????
=?? (?? ,?? ,?? ,?? ) (5)
????
????
=?? (?? ,?? ,?? ,?? ) (5)
 
The path lines traced a particle which was at the point (?? 0
,?? 0
,?? 0
) a time ?? 0
, can be 
obtained by integrating the simultaneous equation (5) with the initial conditions: 
?? =?? 0
,?? =?? 0
,?? =?? 0
,?? =?? 0
 
For steady motion, the streamlines and path lines coincide. For unsteady motion they 
may or may not coincide. 
Vortex lines are curves ?? uch that the tangents to them are parallel to vorticity vector ??? =
?
?
×???. Their equations are given by 
????
?? 1
=
????
?? 2
=
????
?? 3
 
 
2. Equation of Continuity 
Consider a fixed closed surface ?? , bounding a volume ?? . Let ?? be the density of the fluid. 
The fluid entering the surface across a surface element dS with unit outward normal ???, 
per unit time is given by 
-?? ???·???(???? ) (1) 
Hence the total mass entering across the entire surface ?? is the surface integral 
-??
?? ??? ???·??????? =-??
?? ??·(?? ???)·???? (2) 
by the divergence theorem 
But at any instant of time t, the total mass contained in ?? is ?
?? ??????? and the rate at which 
it changes with time ?? is 
?
??? ?? ??
?? ???? =
?
?? ????      …(3) 
Equating (2) and (3) and noting that the surface is arbitrary, we get 
??? ????
+?·(?? ???)=0   …(4) 
which is called the equation of continuity 
ice ?·(?? ???)=??? ·???+?? ???? we can also express (4) as 
D?? ????
+?? (?·???)=0 (5) 
are ?? =
?? ????
+(???·?) as already introduced. 
Equations (4) and (5) are the general forms of equation of continuity applicable to 
compressible fluid. If the fluids incompressible, does not change during the motion, 
i.e., 
????
????
=0 
Hence ?·??? =0 or 
??? ??? +
??? ??? +
??? ??? =0   … (6) 
The equation of continuity, for, an incompressible fluid (liquid) 
Theorem: If in incompressible the fluid is in irrotational motion, the velocity potential is a 
harmonic function. 
Proof: Since the motion is rotational, there is a velocity potential is such that 
???=-??? 
Hence the fluid is incompressible 
?·???=0 or -?·??? = O or ?
2
.?? =0 
Proving that ?? is a harmonic function. 
Example 1 
?? +???? ,?? =???? +???? ,?? =0 are the velocity components of an incompressible fluid. Find 
streamlines. 
Since the fluid is incompressible,  
Page 5


Edurev123 
FLUID DYNAMICS 
1. Introduction 
Fluid dynamics deals with the study of fluids in motion. By the term fluid is meant a 
substance that flows; one which does not is termed as a solid.  Fluids may be divided 
into 1) liquids, which are incompressible, i.e., their volumes do not change and (ii) gases 
which are compressible, i.e., their volumes change whenever pressure changes. 
The term hydrodynamics is used to the science of moving incompressible fluids. Fluids 
are classified as viscous fluids and perfect (inviscid) fluids. Viscous fluids sustain 
tangential stresses, but perfect fluids are incapable of exerting shearing stresses. The 
pressure exerted by a perfect fluid is always normal to the surface of contact. The 
pressure at any point in a perfect fluid is the same in every direction. 
Let ???=?? (???,?? ) be the velocity vector. In component form, we write 
???={?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? ),?? (?? ,?? ,?? ,?? )} 
The acceleration vector, ä is defined as 
??? =
????
????
=
????
??? +
????
??? +
??? ??? +
????
??? ??? ???ˆ
+
????
??? ??? ??? 
But ?? =
??? ??? ,?? =
??? ??? ?? =
??? ??? 
Hence ???=
????
???ˆ
+?? ????
??? +?? ????
??? +?? ????
??? 
Noting that the operator 
?? ?
??? +?? ?
??? +?? ?
??? =q.? 
we can express the acceleration as 
??? =
???¨
??? +(???·?)???   … (1) 
 
In component form, (1) reads as  
?? 1
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 2
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? 
?? 3
=
??? ??? +?? ????
??? +?? ??? ??? +?? ??? ??? 
where ???=(?? 1
,?? 2
,?? 3
) 
The first term in (1), namely 
????
??? is called the local acceleration. Physically, it is the 
acceleration observed by a stationery observer at the point (?? ,?? ,?? ) . The second term in 
(1), namely (????
·?)????
 is the convective acceleration. This is due to the variation of ??? with 
the position ???. 
The motion is said to be steady when ??? is not an explicit function of time ?? , that is, ??? is in 
dependent of time for any fixed position. Note that the acceleration is not zero for steady 
motion. Indeed, for steady molion ??? =(???.?)??? The operator ???·? is called the mobile 
operator or convective operator. The operator 
?
??? +???·? is denoted by ?? and stands for 
differentiation following the motion 
Thus ???=
?? ???
????
 For any scalar or vector point function ?? associated with fluid motion, we 
have 
????
????
=
??? ??? +(???·?)?? 
Using the formula 
?(???
·????
)=???
×curl ????
+????
×curl ???
-(???
·?)???
-(???
·?)???
 
We find with ???
=??¨
=??? 
(???·?)???=?? (
?? 2
2
)+(curl ???)×??? 
Hence the acceleration vector can also be written as 
???=
????
?q
+?(
1
2
?? 2
)+(??????? ???
???)×??? (2) 
The vector curl ??? i.e. ?×??? is called the vortically vector. It is a measure of rotation in the 
lluid. If ?×??? =0
??
, we call, the motion irrotational. If ?×????0
¯
, the motion is rotational. 
Since ?×??? =0
¯
 for a scalar ?? , when the motion is irrotational, there exists a scalar 
function ?? , called the velocity potential such that 
???=-??? (3) 
The negative sign in (3) is conventional and it ensures that ??? is in the direction of 
decreasing ?? . 
Streamlines are curves in the fluid such that the tangents to them are parallel to the 
velocity vector. The streamlines can be obtained by giving the differential equations 
????
?? =
????
?? =
????
?? (4)
 
where (?? ,?? ,?? ) is the velocity 
The path lines are the curves traced by the individual fluid particles and their equations. 
are given by 
????
????
=?? (?? ,?? ,?? ,?? ) (5)
????
????
=?? (?? ,?? ,?? ,?? ) (5)
????
????
=?? (?? ,?? ,?? ,?? ) (5)
 
The path lines traced a particle which was at the point (?? 0
,?? 0
,?? 0
) a time ?? 0
, can be 
obtained by integrating the simultaneous equation (5) with the initial conditions: 
?? =?? 0
,?? =?? 0
,?? =?? 0
,?? =?? 0
 
For steady motion, the streamlines and path lines coincide. For unsteady motion they 
may or may not coincide. 
Vortex lines are curves ?? uch that the tangents to them are parallel to vorticity vector ??? =
?
?
×???. Their equations are given by 
????
?? 1
=
????
?? 2
=
????
?? 3
 
 
2. Equation of Continuity 
Consider a fixed closed surface ?? , bounding a volume ?? . Let ?? be the density of the fluid. 
The fluid entering the surface across a surface element dS with unit outward normal ???, 
per unit time is given by 
-?? ???·???(???? ) (1) 
Hence the total mass entering across the entire surface ?? is the surface integral 
-??
?? ??? ???·??????? =-??
?? ??·(?? ???)·???? (2) 
by the divergence theorem 
But at any instant of time t, the total mass contained in ?? is ?
?? ??????? and the rate at which 
it changes with time ?? is 
?
??? ?? ??
?? ???? =
?
?? ????      …(3) 
Equating (2) and (3) and noting that the surface is arbitrary, we get 
??? ????
+?·(?? ???)=0   …(4) 
which is called the equation of continuity 
ice ?·(?? ???)=??? ·???+?? ???? we can also express (4) as 
D?? ????
+?? (?·???)=0 (5) 
are ?? =
?? ????
+(???·?) as already introduced. 
Equations (4) and (5) are the general forms of equation of continuity applicable to 
compressible fluid. If the fluids incompressible, does not change during the motion, 
i.e., 
????
????
=0 
Hence ?·??? =0 or 
??? ??? +
??? ??? +
??? ??? =0   … (6) 
The equation of continuity, for, an incompressible fluid (liquid) 
Theorem: If in incompressible the fluid is in irrotational motion, the velocity potential is a 
harmonic function. 
Proof: Since the motion is rotational, there is a velocity potential is such that 
???=-??? 
Hence the fluid is incompressible 
?·???=0 or -?·??? = O or ?
2
.?? =0 
Proving that ?? is a harmonic function. 
Example 1 
?? +???? ,?? =???? +???? ,?? =0 are the velocity components of an incompressible fluid. Find 
streamlines. 
Since the fluid is incompressible,  
??? ??? +
??? ??? +
??? ??? =0 
or ?? +?? =0 i.e., ?? =-?? 
Now the streamlines are given by 
????
???? +????
=
????
???? -????
=
????
0
    …(1) 
From the first equation, we get 
(???? -???? )???? =(???? +???? )???? 
???????? =?? (?????? +?????? )+?????? 
or ?? ?? 2
-2?????? -?? ?? 2
= constant. 
The second equation of (1) implies ?? = constant. Hence the streamlines are conics:  
?? ?? 2
-2?????? -?? ?? 2
= constant in the plane ?? = constant. 
When the motion is irrotational ?×???=0
¯
 or 
??? ??? -
??? ??? =0 
Hence ?? =?? . The streamlines are ?? (?? 2
-?? 2
)-2?????? = constant, which are rectangular 
hyperbolas in planes parallel to ?? =0. 
Example 2 
Find the acceleration when the velocity components are 
?? =?? +?? ,?? =?? -?? 1
 =0 
The acceleration components are given by 
?? 1
 =
??? ??? +?? ?
ˆ
??? +?? ??? ??? +?? ??? ??? =?? +?? +?? -?? =2?? 
?? 2
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? =?? +?? -(?? -?? )=2?? 
?? 3
=
??? ??? +?? ??? ??? +?? ??? ??? +?? ??? ??? =0 
Hence ?? =(2?? ,2?? ,0)=2(?? ,?? ,0)  
 
Vorticity Transport Equation