Hypersonic Boundary Layer theory Notes | EduRev

: Hypersonic Boundary Layer theory Notes | EduRev

 Page 1


NPTEL – Aerospace  
 
Module-5: Hypersonic Boundary Layer theory 
Lecture-20: Hypersonic boundary equation 
20.1 Governing Equations for Viscous Flows 
The Navier-Stokes (NS) equaadtions are the governing equations for the viscous 
compressible flow and hence are the governing equations for hypersonic flows. This 
section deals with the basics of NS equations and its non-dimensionalization.  
Continuity Equation: 
 
Considering steady state conditions we have; 
( ) ()
0
uv
x y
? ? ??
+ =
??
 i.e.        (20.1) 
X Momentum Equation: 
() D u Du uD
Dt Dt Dt
?? ?
= + , now since 
D
Dt
?
= 0; 
() D u Du
Dt Dt
??
= ; therefore the L.H.S. simplifies to give X momentum equation in 
steady state  conditions as; 
u u p v u u u
uv
x y xy xy x x x
? ? µ µ
?? ?? ? ? ? ? ? ? ?? ? ? ? ??
+ =-+ + + +
?? ?? ?? ??
? ? ?? ?? ? ? ?
?? ?? ?? ??
  
(20.2) 
Y Momentum Equation: 
() D v Dv vD
Dt Dt Dt
? ? ?
= + , now since 
D
Dt
?
= 0; 
() D v Dv
Dt Dt
? ?
= ; therefore the L.H.S. simplifies to give Y momentum equation in 
steady State  Conditions as; 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 39 
Page 2


NPTEL – Aerospace  
 
Module-5: Hypersonic Boundary Layer theory 
Lecture-20: Hypersonic boundary equation 
20.1 Governing Equations for Viscous Flows 
The Navier-Stokes (NS) equaadtions are the governing equations for the viscous 
compressible flow and hence are the governing equations for hypersonic flows. This 
section deals with the basics of NS equations and its non-dimensionalization.  
Continuity Equation: 
 
Considering steady state conditions we have; 
( ) ()
0
uv
x y
? ? ??
+ =
??
 i.e.        (20.1) 
X Momentum Equation: 
() D u Du uD
Dt Dt Dt
?? ?
= + , now since 
D
Dt
?
= 0; 
() D u Du
Dt Dt
??
= ; therefore the L.H.S. simplifies to give X momentum equation in 
steady state  conditions as; 
u u p v u u u
uv
x y xy xy x x x
? ? µ µ
?? ?? ? ? ? ? ? ? ?? ? ? ? ??
+ =-+ + + +
?? ?? ?? ??
? ? ?? ?? ? ? ?
?? ?? ?? ??
  
(20.2) 
Y Momentum Equation: 
() D v Dv vD
Dt Dt Dt
? ? ?
= + , now since 
D
Dt
?
= 0; 
() D v Dv
Dt Dt
? ?
= ; therefore the L.H.S. simplifies to give Y momentum equation in 
steady State  Conditions as; 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 39 
NPTEL – Aerospace  
 
v v p v u v v
uv
x y y x x y y yy
?? µ µ
? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ?
+ =-+ + + +
? ? ? ? ? ? ??
? ? ? ? ? ? ? ??
? ? ?? ? ? ? ?
  
(20.3) 
Energy Equation: 
22 22 22
() () ()
2 22
uv uv uv
De De e D
Dt Dt Dt
? ??
?? ?? + ???? ++
+ ++
?? ?? ????
?? ? ? ????
= + ,  
Since 
D
Dt
?
= 0; 
22 22
() ()
2 2
uv uv
De De
Dt Dt
? ?
?? ?? + ?? +
+ +
?? ?? ??
?? ? ? ? ?
= , 
 Therefore the L.H.S. simplifies to give Energy Equation in steady State Conditions 
as; 
2
( ) ()
2
xx xy xy yy
V T T pu pv u u v v
ue q
xx y y x y x y x y
? ? ? t t t t
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??
?+ = + + - + + + + +
? ? ? ? ? ? ? ? ??
?? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ?
?
?
 
(20.4)
 
The above equations are written for steady, compressible, viscous, two dimensional 
flows in Cartesian coordinates. Where u and v are velocities in x and y directions 
respectively; erepresents internal energy per unit mass and q ? represents the 
volumetric heating that might occur. All other notations carry their usual meaning. 
We can simplify the above set of equations via appropriate assumptions, and obtain 
approximate viscous flow results. 
 
 
 
 
 
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 39 
Page 3


NPTEL – Aerospace  
 
Module-5: Hypersonic Boundary Layer theory 
Lecture-20: Hypersonic boundary equation 
20.1 Governing Equations for Viscous Flows 
The Navier-Stokes (NS) equaadtions are the governing equations for the viscous 
compressible flow and hence are the governing equations for hypersonic flows. This 
section deals with the basics of NS equations and its non-dimensionalization.  
Continuity Equation: 
 
Considering steady state conditions we have; 
( ) ()
0
uv
x y
? ? ??
+ =
??
 i.e.        (20.1) 
X Momentum Equation: 
() D u Du uD
Dt Dt Dt
?? ?
= + , now since 
D
Dt
?
= 0; 
() D u Du
Dt Dt
??
= ; therefore the L.H.S. simplifies to give X momentum equation in 
steady state  conditions as; 
u u p v u u u
uv
x y xy xy x x x
? ? µ µ
?? ?? ? ? ? ? ? ? ?? ? ? ? ??
+ =-+ + + +
?? ?? ?? ??
? ? ?? ?? ? ? ?
?? ?? ?? ??
  
(20.2) 
Y Momentum Equation: 
() D v Dv vD
Dt Dt Dt
? ? ?
= + , now since 
D
Dt
?
= 0; 
() D v Dv
Dt Dt
? ?
= ; therefore the L.H.S. simplifies to give Y momentum equation in 
steady State  Conditions as; 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 39 
NPTEL – Aerospace  
 
v v p v u v v
uv
x y y x x y y yy
?? µ µ
? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ?
+ =-+ + + +
? ? ? ? ? ? ??
? ? ? ? ? ? ? ??
? ? ?? ? ? ? ?
  
(20.3) 
Energy Equation: 
22 22 22
() () ()
2 22
uv uv uv
De De e D
Dt Dt Dt
? ??
?? ?? + ???? ++
+ ++
?? ?? ????
?? ? ? ????
= + ,  
Since 
D
Dt
?
= 0; 
22 22
() ()
2 2
uv uv
De De
Dt Dt
? ?
?? ?? + ?? +
+ +
?? ?? ??
?? ? ? ? ?
= , 
 Therefore the L.H.S. simplifies to give Energy Equation in steady State Conditions 
as; 
2
( ) ()
2
xx xy xy yy
V T T pu pv u u v v
ue q
xx y y x y x y x y
? ? ? t t t t
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??
?+ = + + - + + + + +
? ? ? ? ? ? ? ? ??
?? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ?
?
?
 
(20.4)
 
The above equations are written for steady, compressible, viscous, two dimensional 
flows in Cartesian coordinates. Where u and v are velocities in x and y directions 
respectively; erepresents internal energy per unit mass and q ? represents the 
volumetric heating that might occur. All other notations carry their usual meaning. 
We can simplify the above set of equations via appropriate assumptions, and obtain 
approximate viscous flow results. 
 
 
 
 
 
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 39 
NPTEL – Aerospace  
 
20.2 Non Dimensional Form of Governing Equations  
The non dimensional form of Navier-Stokes equations can be obtained as follows. 
Here we have considered a two dimensional steady flow and ignored the normal 
stresses t
xx
 and t
yy
. Reference variables of the flow can be used for non-
dimensionalization.  
Non Dimensional Variables: 
u
u
V 8
= 
v
v
V 8
= 
x
x
c
=  
y
y
c
=  
2
p
p
V ? 8
= 
 
v
e
e
C T 8
= 
µ
µ
µ 8
= 
?
?
? 8
= 
?
?
? 8
= 
Where  V 8 , T 8 , ? 8 , µ 8 , ? 8 are free stream parameters and c reference length. 
Therefore the Non Dimensional Equations are given as: 
Non Dimensional Continuity Equation: 
( ) ()
0
uv
xy
?? ??
+=
??
       
(20.5)
 
Non Dimensional X Momentum Equation: 
2
11
Re
u u p vu
u v
M x y x y xy
?? µ
?88
?? ??
? ? ? ? ??
+ =- + +
?? ??
? ? ? ? ??
??
?? ??
  
(20.6) 
Non Dimensional Y Momentum Equation: 
2
11
Re
v v p vu
u v
M x y y x xy
?? µ
?88
?? ??
? ? ? ? ??
+ =- + +
?? ??
? ? ? ? ??
??
?? ??
  
(20.7) 
 
 
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 39 
Page 4


NPTEL – Aerospace  
 
Module-5: Hypersonic Boundary Layer theory 
Lecture-20: Hypersonic boundary equation 
20.1 Governing Equations for Viscous Flows 
The Navier-Stokes (NS) equaadtions are the governing equations for the viscous 
compressible flow and hence are the governing equations for hypersonic flows. This 
section deals with the basics of NS equations and its non-dimensionalization.  
Continuity Equation: 
 
Considering steady state conditions we have; 
( ) ()
0
uv
x y
? ? ??
+ =
??
 i.e.        (20.1) 
X Momentum Equation: 
() D u Du uD
Dt Dt Dt
?? ?
= + , now since 
D
Dt
?
= 0; 
() D u Du
Dt Dt
??
= ; therefore the L.H.S. simplifies to give X momentum equation in 
steady state  conditions as; 
u u p v u u u
uv
x y xy xy x x x
? ? µ µ
?? ?? ? ? ? ? ? ? ?? ? ? ? ??
+ =-+ + + +
?? ?? ?? ??
? ? ?? ?? ? ? ?
?? ?? ?? ??
  
(20.2) 
Y Momentum Equation: 
() D v Dv vD
Dt Dt Dt
? ? ?
= + , now since 
D
Dt
?
= 0; 
() D v Dv
Dt Dt
? ?
= ; therefore the L.H.S. simplifies to give Y momentum equation in 
steady State  Conditions as; 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 39 
NPTEL – Aerospace  
 
v v p v u v v
uv
x y y x x y y yy
?? µ µ
? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ?
+ =-+ + + +
? ? ? ? ? ? ??
? ? ? ? ? ? ? ??
? ? ?? ? ? ? ?
  
(20.3) 
Energy Equation: 
22 22 22
() () ()
2 22
uv uv uv
De De e D
Dt Dt Dt
? ??
?? ?? + ???? ++
+ ++
?? ?? ????
?? ? ? ????
= + ,  
Since 
D
Dt
?
= 0; 
22 22
() ()
2 2
uv uv
De De
Dt Dt
? ?
?? ?? + ?? +
+ +
?? ?? ??
?? ? ? ? ?
= , 
 Therefore the L.H.S. simplifies to give Energy Equation in steady State Conditions 
as; 
2
( ) ()
2
xx xy xy yy
V T T pu pv u u v v
ue q
xx y y x y x y x y
? ? ? t t t t
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??
?+ = + + - + + + + +
? ? ? ? ? ? ? ? ??
?? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ?
?
?
 
(20.4)
 
The above equations are written for steady, compressible, viscous, two dimensional 
flows in Cartesian coordinates. Where u and v are velocities in x and y directions 
respectively; erepresents internal energy per unit mass and q ? represents the 
volumetric heating that might occur. All other notations carry their usual meaning. 
We can simplify the above set of equations via appropriate assumptions, and obtain 
approximate viscous flow results. 
 
 
 
 
 
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 39 
NPTEL – Aerospace  
 
20.2 Non Dimensional Form of Governing Equations  
The non dimensional form of Navier-Stokes equations can be obtained as follows. 
Here we have considered a two dimensional steady flow and ignored the normal 
stresses t
xx
 and t
yy
. Reference variables of the flow can be used for non-
dimensionalization.  
Non Dimensional Variables: 
u
u
V 8
= 
v
v
V 8
= 
x
x
c
=  
y
y
c
=  
2
p
p
V ? 8
= 
 
v
e
e
C T 8
= 
µ
µ
µ 8
= 
?
?
? 8
= 
?
?
? 8
= 
Where  V 8 , T 8 , ? 8 , µ 8 , ? 8 are free stream parameters and c reference length. 
Therefore the Non Dimensional Equations are given as: 
Non Dimensional Continuity Equation: 
( ) ()
0
uv
xy
?? ??
+=
??
       
(20.5)
 
Non Dimensional X Momentum Equation: 
2
11
Re
u u p vu
u v
M x y x y xy
?? µ
?88
?? ??
? ? ? ? ??
+ =- + +
?? ??
? ? ? ? ??
??
?? ??
  
(20.6) 
Non Dimensional Y Momentum Equation: 
2
11
Re
v v p vu
u v
M x y y x xy
?? µ
?88
?? ??
? ? ? ? ??
+ =- + +
?? ??
? ? ? ? ??
??
?? ??
  
(20.7) 
 
 
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 39 
NPTEL – Aerospace  
 
Non Dimensional Energy Equation: 
( ) ( )
22 22
2
( 1)
Pr Re
ee T T
u v M u uv v uv
x y x y xx y y
?
? ? ?? ? ? ? ? 8
88
?? ?? ?? ?? ? ? ? ? ?? ? ?
+ = - + + + + +
?? ?? ?? ??
? ? ? ? ? ???
?? ?? ?? ?? ??
 
                                 
2
( 1)
Re
M vu vu
u
x xy y xy
? ? µ ? µ
8
8
?? ? ?? ? ?? ??
? ?? ? ?? ??
+- + + +
? ?? ? ?? ?? ??
? ?? ? ??
? ?? ?
?? ?? ?? ? ?? ? ??
(20.8) 
20.3 Process of Non-dimensionlisation of Governing Equations:- 
Continuity Equation:- 
We Have, 
( ) ()
0
uv
x y
? ? ??
+ =
??
,  
With non-dimensional parameters 
( /) ( /)
.0
( / ) ( / )
( ) ()
0
V u V vV
c x c y c
uv
x y
? ? ? ??
? ?
8 8 8 8 8 8 ?? ??
+=
??
??
??
??
+ =
??
 
X & Y momentum Equations:- 
The process of non dimensionlisation is almost the same for both X & Y momentum 
equations, therefore we have for X momentum equation; 
u u p v u u u
uv
x y xy xy x x x
? ? µ µ
?? ?? ? ? ? ? ? ? ?? ? ? ? ??
+ =-+ + + +
?? ?? ?? ??
? ? ?? ?? ? ? ?
?? ?? ?? ??
 
Rewriting the above Equation with Non Dimensional parameters, we get;  
2
( / ) ( / )
( / ) ( / )
V u u V u vV
c V x c V y c
?? ?
??
8 8 8 8
8 8 8 8
?? ??
+=
??
??
??
( / )
( / )
p p p
c x c
88 ?
-
?
 
2
( / ) ( / ) ( / ) ( / )
( / ) ( / ) ( / ) ( / ) ( / ) ( / )
V vV u V vV u V
c y c x c y c x c x c y c
µµ µ
µµ
88 88 88
8 8
? ? ???? ?? ?? ? ?? ? ?? ? ?
+ + + +
? ? ???? ?? ??
? ?? ? ??
?? ?? ? ? ???? ? ?
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 4 of 39 
Page 5


NPTEL – Aerospace  
 
Module-5: Hypersonic Boundary Layer theory 
Lecture-20: Hypersonic boundary equation 
20.1 Governing Equations for Viscous Flows 
The Navier-Stokes (NS) equaadtions are the governing equations for the viscous 
compressible flow and hence are the governing equations for hypersonic flows. This 
section deals with the basics of NS equations and its non-dimensionalization.  
Continuity Equation: 
 
Considering steady state conditions we have; 
( ) ()
0
uv
x y
? ? ??
+ =
??
 i.e.        (20.1) 
X Momentum Equation: 
() D u Du uD
Dt Dt Dt
?? ?
= + , now since 
D
Dt
?
= 0; 
() D u Du
Dt Dt
??
= ; therefore the L.H.S. simplifies to give X momentum equation in 
steady state  conditions as; 
u u p v u u u
uv
x y xy xy x x x
? ? µ µ
?? ?? ? ? ? ? ? ? ?? ? ? ? ??
+ =-+ + + +
?? ?? ?? ??
? ? ?? ?? ? ? ?
?? ?? ?? ??
  
(20.2) 
Y Momentum Equation: 
() D v Dv vD
Dt Dt Dt
? ? ?
= + , now since 
D
Dt
?
= 0; 
() D v Dv
Dt Dt
? ?
= ; therefore the L.H.S. simplifies to give Y momentum equation in 
steady State  Conditions as; 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 39 
NPTEL – Aerospace  
 
v v p v u v v
uv
x y y x x y y yy
?? µ µ
? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ?
+ =-+ + + +
? ? ? ? ? ? ??
? ? ? ? ? ? ? ??
? ? ?? ? ? ? ?
  
(20.3) 
Energy Equation: 
22 22 22
() () ()
2 22
uv uv uv
De De e D
Dt Dt Dt
? ??
?? ?? + ???? ++
+ ++
?? ?? ????
?? ? ? ????
= + ,  
Since 
D
Dt
?
= 0; 
22 22
() ()
2 2
uv uv
De De
Dt Dt
? ?
?? ?? + ?? +
+ +
?? ?? ??
?? ? ? ? ?
= , 
 Therefore the L.H.S. simplifies to give Energy Equation in steady State Conditions 
as; 
2
( ) ()
2
xx xy xy yy
V T T pu pv u u v v
ue q
xx y y x y x y x y
? ? ? t t t t
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??
?+ = + + - + + + + +
? ? ? ? ? ? ? ? ??
?? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ?
?
?
 
(20.4)
 
The above equations are written for steady, compressible, viscous, two dimensional 
flows in Cartesian coordinates. Where u and v are velocities in x and y directions 
respectively; erepresents internal energy per unit mass and q ? represents the 
volumetric heating that might occur. All other notations carry their usual meaning. 
We can simplify the above set of equations via appropriate assumptions, and obtain 
approximate viscous flow results. 
 
 
 
 
 
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 39 
NPTEL – Aerospace  
 
20.2 Non Dimensional Form of Governing Equations  
The non dimensional form of Navier-Stokes equations can be obtained as follows. 
Here we have considered a two dimensional steady flow and ignored the normal 
stresses t
xx
 and t
yy
. Reference variables of the flow can be used for non-
dimensionalization.  
Non Dimensional Variables: 
u
u
V 8
= 
v
v
V 8
= 
x
x
c
=  
y
y
c
=  
2
p
p
V ? 8
= 
 
v
e
e
C T 8
= 
µ
µ
µ 8
= 
?
?
? 8
= 
?
?
? 8
= 
Where  V 8 , T 8 , ? 8 , µ 8 , ? 8 are free stream parameters and c reference length. 
Therefore the Non Dimensional Equations are given as: 
Non Dimensional Continuity Equation: 
( ) ()
0
uv
xy
?? ??
+=
??
       
(20.5)
 
Non Dimensional X Momentum Equation: 
2
11
Re
u u p vu
u v
M x y x y xy
?? µ
?88
?? ??
? ? ? ? ??
+ =- + +
?? ??
? ? ? ? ??
??
?? ??
  
(20.6) 
Non Dimensional Y Momentum Equation: 
2
11
Re
v v p vu
u v
M x y y x xy
?? µ
?88
?? ??
? ? ? ? ??
+ =- + +
?? ??
? ? ? ? ??
??
?? ??
  
(20.7) 
 
 
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 39 
NPTEL – Aerospace  
 
Non Dimensional Energy Equation: 
( ) ( )
22 22
2
( 1)
Pr Re
ee T T
u v M u uv v uv
x y x y xx y y
?
? ? ?? ? ? ? ? 8
88
?? ?? ?? ?? ? ? ? ? ?? ? ?
+ = - + + + + +
?? ?? ?? ??
? ? ? ? ? ???
?? ?? ?? ?? ??
 
                                 
2
( 1)
Re
M vu vu
u
x xy y xy
? ? µ ? µ
8
8
?? ? ?? ? ?? ??
? ?? ? ?? ??
+- + + +
? ?? ? ?? ?? ??
? ?? ? ??
? ?? ?
?? ?? ?? ? ?? ? ??
(20.8) 
20.3 Process of Non-dimensionlisation of Governing Equations:- 
Continuity Equation:- 
We Have, 
( ) ()
0
uv
x y
? ? ??
+ =
??
,  
With non-dimensional parameters 
( /) ( /)
.0
( / ) ( / )
( ) ()
0
V u V vV
c x c y c
uv
x y
? ? ? ??
? ?
8 8 8 8 8 8 ?? ??
+=
??
??
??
??
+ =
??
 
X & Y momentum Equations:- 
The process of non dimensionlisation is almost the same for both X & Y momentum 
equations, therefore we have for X momentum equation; 
u u p v u u u
uv
x y xy xy x x x
? ? µ µ
?? ?? ? ? ? ? ? ? ?? ? ? ? ??
+ =-+ + + +
?? ?? ?? ??
? ? ?? ?? ? ? ?
?? ?? ?? ??
 
Rewriting the above Equation with Non Dimensional parameters, we get;  
2
( / ) ( / )
( / ) ( / )
V u u V u vV
c V x c V y c
?? ?
??
8 8 8 8
8 8 8 8
?? ??
+=
??
??
??
( / )
( / )
p p p
c x c
88 ?
-
?
 
2
( / ) ( / ) ( / ) ( / )
( / ) ( / ) ( / ) ( / ) ( / ) ( / )
V vV u V vV u V
c y c x c y c x c x c y c
µµ µ
µµ
88 88 88
8 8
? ? ???? ?? ?? ? ?? ? ?? ? ?
+ + + +
? ? ???? ?? ??
? ?? ? ??
?? ?? ? ? ???? ? ?
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 4 of 39 
NPTEL – Aerospace  
 
Now dividing the above equation on both sides by the factor 
2
V
c
?8 8
; we get term 
wise 
First, the pressure term on Right Hand side (R.H.S); 
2 2
pc p
cV V ??
88
8 8 8 8
= 
Since Mach Number can be written as 
2
2
2
V
M
a
8
=  therefore; 
2 22
pp
V Ma ??
88
8 8 8
= and furthermore, p RT ? 8= 8 8 and 
2
a RT ? 8 = ; the term reduces to 
22 2 2
1 p RT
M a M RT M ? ??
88
88
= = 
Now the viscous term, 
22
1
Re
V c
c V Vc
µµ
??
8 8 8
8 8 8 8 8
= = 
Thus the X momentum Equation reduces to, 
2
11
Re
u u p vu
u v
M x y x y xy
?? µ
?88
?? ??
? ? ? ? ??
+ =- + +
?? ??
? ? ? ? ??
??
?? ??
 
& similarly Y momentum Equation reduces to, 
2
11
Re
v v p vu
u v
M x y y x xy
?? µ
?88
?? ??
? ? ? ? ??
+ =- + +
?? ??
? ? ? ? ??
??
?? ??
 
Normal stresses t
xx
 and t
yy 
are ignored for the sake of simplicity. 
 
 
 
 
 
 
Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 5 of 39 
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