Navier-Stokes Equation for Viscous Flow | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
8. Navier-Stoke's Equation for Viscous 
Flow 
8.1 Find Navier-Stokes equation for a steady laminar flow of a viscous 
incompressible fluid between two infinite parallel plates. 
(2014 : 20 Marks) 
Solution: 
By Laminar flow, we mean that fluid moves in layers parallel to the plates. 
We suppose that an incompressible fluid with constant viscosity is confined between two 
parallel plates, 
?? =
?? 2
 
?? =-
?? 2
 
Let, the fluid be moving with velocity is parallel to ?? -axis with laminar flow. In order to 
maintain such a motion, the difference of pressure in ?? -direction must be balanced by 
shearing stresses. 
Here:                                                                 ?? =?? (?? ,0,0) 
Equation of continuity 
?? =0; so that ?? =?? (?? ,?? ) 
Navier-strokes equation in absence of external force is 
????
????
=
??? ??? +(?? ·?)?? =-
1
?? ??? +?
2
?? or, ??? ??? ??? +????
??? ??? =-
1
?? ??? +???? ?
2
?? or, ??? ??? ??? =-
1
?? ??? +???? ?
2
?? as 
??? ??? =0
 This, 
??? ??? =
1
?? ??? ??? +?? ?
2
?? ?? =·?? (?? ,?? )
?? =?? ?? =?? (?? ,?? ),?? =?? (?? ,?? )
 
Page 2


Edurev123 
8. Navier-Stoke's Equation for Viscous 
Flow 
8.1 Find Navier-Stokes equation for a steady laminar flow of a viscous 
incompressible fluid between two infinite parallel plates. 
(2014 : 20 Marks) 
Solution: 
By Laminar flow, we mean that fluid moves in layers parallel to the plates. 
We suppose that an incompressible fluid with constant viscosity is confined between two 
parallel plates, 
?? =
?? 2
 
?? =-
?? 2
 
Let, the fluid be moving with velocity is parallel to ?? -axis with laminar flow. In order to 
maintain such a motion, the difference of pressure in ?? -direction must be balanced by 
shearing stresses. 
Here:                                                                 ?? =?? (?? ,0,0) 
Equation of continuity 
?? =0; so that ?? =?? (?? ,?? ) 
Navier-strokes equation in absence of external force is 
????
????
=
??? ??? +(?? ·?)?? =-
1
?? ??? +?
2
?? or, ??? ??? ??? +????
??? ??? =-
1
?? ??? +???? ?
2
?? or, ??? ??? ??? =-
1
?? ??? +???? ?
2
?? as 
??? ??? =0
 This, 
??? ??? =
1
?? ??? ??? +?? ?
2
?? ?? =·?? (?? ,?? )
?? =?? ?? =?? (?? ,?? ),?? =?? (?? ,?? )
 
Consequently (iii) declares that either 
??? ??? is constant or function If ' ?? . Now consider the 
case of steady motion so that (iii) becomes. 
?? ?
2
?? ??? 2
=
??? ??? =
????
????
 or 
?
2
?? ??? 2
=
1
?? ????
????
????
????
=
?? ?? ????
????
+?? ?? =
?? 2
2?? ????
????
+???? +?? 
Case i: Plane Couette Flow; 
In this case 
????
????
=0; the lower plate is stationary while the upper is moving with uniform 
velocity ' ?? parallel to ?? -axis. The boundary conditions are 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =?? = constant: ?? =
h
2
 
Substituting (Iv) and (i) and (ii), we get. 
and 
0?=
h
2
8?? ·0+?? (-
h
2
)+?? ?? ?=
h
2
8?? ·0+?? ·
h
2
+?? -?? h+2?? ?=0;??? h+2?? =2?? 2?? ?=?? ;?-?? h+?? =0
 
Now, (iv) becomes, 
?? =
?? h
?? +
?? 2
(?? ) 
Evidently; velocity distribution is linear. 
Case II. Plane Poiseuille flow : 
In this case 
????
????
= constant =?? ?0 and both the walls are at rest. 
The boundary conditions are: 
Page 3


Edurev123 
8. Navier-Stoke's Equation for Viscous 
Flow 
8.1 Find Navier-Stokes equation for a steady laminar flow of a viscous 
incompressible fluid between two infinite parallel plates. 
(2014 : 20 Marks) 
Solution: 
By Laminar flow, we mean that fluid moves in layers parallel to the plates. 
We suppose that an incompressible fluid with constant viscosity is confined between two 
parallel plates, 
?? =
?? 2
 
?? =-
?? 2
 
Let, the fluid be moving with velocity is parallel to ?? -axis with laminar flow. In order to 
maintain such a motion, the difference of pressure in ?? -direction must be balanced by 
shearing stresses. 
Here:                                                                 ?? =?? (?? ,0,0) 
Equation of continuity 
?? =0; so that ?? =?? (?? ,?? ) 
Navier-strokes equation in absence of external force is 
????
????
=
??? ??? +(?? ·?)?? =-
1
?? ??? +?
2
?? or, ??? ??? ??? +????
??? ??? =-
1
?? ??? +???? ?
2
?? or, ??? ??? ??? =-
1
?? ??? +???? ?
2
?? as 
??? ??? =0
 This, 
??? ??? =
1
?? ??? ??? +?? ?
2
?? ?? =·?? (?? ,?? )
?? =?? ?? =?? (?? ,?? ),?? =?? (?? ,?? )
 
Consequently (iii) declares that either 
??? ??? is constant or function If ' ?? . Now consider the 
case of steady motion so that (iii) becomes. 
?? ?
2
?? ??? 2
=
??? ??? =
????
????
 or 
?
2
?? ??? 2
=
1
?? ????
????
????
????
=
?? ?? ????
????
+?? ?? =
?? 2
2?? ????
????
+???? +?? 
Case i: Plane Couette Flow; 
In this case 
????
????
=0; the lower plate is stationary while the upper is moving with uniform 
velocity ' ?? parallel to ?? -axis. The boundary conditions are 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =?? = constant: ?? =
h
2
 
Substituting (Iv) and (i) and (ii), we get. 
and 
0?=
h
2
8?? ·0+?? (-
h
2
)+?? ?? ?=
h
2
8?? ·0+?? ·
h
2
+?? -?? h+2?? ?=0;??? h+2?? =2?? 2?? ?=?? ;?-?? h+?? =0
 
Now, (iv) becomes, 
?? =
?? h
?? +
?? 2
(?? ) 
Evidently; velocity distribution is linear. 
Case II. Plane Poiseuille flow : 
In this case 
????
????
= constant =?? ?0 and both the walls are at rest. 
The boundary conditions are: 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =0;?? =
h
2
 
Subjecting (iv) to condition (i) and (ii), 
and 
?? h
2
8?? +?? (-
h
2
)+?? =0
?? h
2
8?? +?? (
h
2
)+?? =0
 
Subtracting we get, 
?? =0 and ?? =
-?? h
2
8?? . 
Now (iv) becomes, 
?? =
?? ?? 2
2?? -
?? h
2
8?? =-
h
2
8?? (1-
4?? 2
h
2
)
????
????
(???? )
?? =?? ?? (1-
4?? 2
h
2
) (???? )
?? ?? =-
h
2
8?? ????
????
(?????? )
 
Where, 
Is the maximum velocity in the flow accruing at ?? =0; evidently, velocity distribution is 
parabolic. Drag (shear stress) at lower plate. 
=(?? ????
????
)
?? =-
h
2
=?? (-
8?? h
2
?? ?? )
?? =-
h
2
=
4?? ?? ?? h
 
? The average velocity distribution for the present flow is given by 
or 
Page 4


Edurev123 
8. Navier-Stoke's Equation for Viscous 
Flow 
8.1 Find Navier-Stokes equation for a steady laminar flow of a viscous 
incompressible fluid between two infinite parallel plates. 
(2014 : 20 Marks) 
Solution: 
By Laminar flow, we mean that fluid moves in layers parallel to the plates. 
We suppose that an incompressible fluid with constant viscosity is confined between two 
parallel plates, 
?? =
?? 2
 
?? =-
?? 2
 
Let, the fluid be moving with velocity is parallel to ?? -axis with laminar flow. In order to 
maintain such a motion, the difference of pressure in ?? -direction must be balanced by 
shearing stresses. 
Here:                                                                 ?? =?? (?? ,0,0) 
Equation of continuity 
?? =0; so that ?? =?? (?? ,?? ) 
Navier-strokes equation in absence of external force is 
????
????
=
??? ??? +(?? ·?)?? =-
1
?? ??? +?
2
?? or, ??? ??? ??? +????
??? ??? =-
1
?? ??? +???? ?
2
?? or, ??? ??? ??? =-
1
?? ??? +???? ?
2
?? as 
??? ??? =0
 This, 
??? ??? =
1
?? ??? ??? +?? ?
2
?? ?? =·?? (?? ,?? )
?? =?? ?? =?? (?? ,?? ),?? =?? (?? ,?? )
 
Consequently (iii) declares that either 
??? ??? is constant or function If ' ?? . Now consider the 
case of steady motion so that (iii) becomes. 
?? ?
2
?? ??? 2
=
??? ??? =
????
????
 or 
?
2
?? ??? 2
=
1
?? ????
????
????
????
=
?? ?? ????
????
+?? ?? =
?? 2
2?? ????
????
+???? +?? 
Case i: Plane Couette Flow; 
In this case 
????
????
=0; the lower plate is stationary while the upper is moving with uniform 
velocity ' ?? parallel to ?? -axis. The boundary conditions are 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =?? = constant: ?? =
h
2
 
Substituting (Iv) and (i) and (ii), we get. 
and 
0?=
h
2
8?? ·0+?? (-
h
2
)+?? ?? ?=
h
2
8?? ·0+?? ·
h
2
+?? -?? h+2?? ?=0;??? h+2?? =2?? 2?? ?=?? ;?-?? h+?? =0
 
Now, (iv) becomes, 
?? =
?? h
?? +
?? 2
(?? ) 
Evidently; velocity distribution is linear. 
Case II. Plane Poiseuille flow : 
In this case 
????
????
= constant =?? ?0 and both the walls are at rest. 
The boundary conditions are: 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =0;?? =
h
2
 
Subjecting (iv) to condition (i) and (ii), 
and 
?? h
2
8?? +?? (-
h
2
)+?? =0
?? h
2
8?? +?? (
h
2
)+?? =0
 
Subtracting we get, 
?? =0 and ?? =
-?? h
2
8?? . 
Now (iv) becomes, 
?? =
?? ?? 2
2?? -
?? h
2
8?? =-
h
2
8?? (1-
4?? 2
h
2
)
????
????
(???? )
?? =?? ?? (1-
4?? 2
h
2
) (???? )
?? ?? =-
h
2
8?? ????
????
(?????? )
 
Where, 
Is the maximum velocity in the flow accruing at ?? =0; evidently, velocity distribution is 
parabolic. Drag (shear stress) at lower plate. 
=(?? ????
????
)
?? =-
h
2
=?? (-
8?? h
2
?? ?? )
?? =-
h
2
=
4?? ?? ?? h
 
? The average velocity distribution for the present flow is given by 
or 
?? ?? =
1
h
? ?
h/2
-h/2
??? ·???? (?? sing?(6)); we get 
?? ?? =
1
h
?? ?? ? ?
h/2
-h/2
?(1-
4?? 2
h
2
)???? =
2
h
4?? ? ?
h/2
h
?(1-
4?? 2
h
2
)????
?? ?? =
2
h
(-
h
2
8?? ·?? )[
h
2
(-
4
h
2
)·
1
3
(
h
2
)
3
]=(-
h?? 4?? )(
h
3
)
?? ?? =
2
3
(-
h
2
?? 8?? )=
2
3
?? ?? 
where; 
?? ?? ?= average velocity 
?? ?=
????
????
= constant 
?? ?? ?= maximum velocity. 
 
Case III : Generalised Plane Couette flow: 
In this case 
????
????
= constant =?? ?0; the lower plate is at rest while the upper plate is in 
motion with velocity ?? . The boundary condition are 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =?? ;?? =
h
2
 
Substituting in (iv) from (i) and (ii) 
?? h
2
8?? +?? (-
h
2
)+?? ?=0
h
2
8?? +?? (
h
2
)+?? ?=?? 
This, 
?? =
?? 2
-
h
2
8?? ;?? =
?? h
 
Now (iv) becomes, 
?? =
?? ?? 2
2?? +
?? h
?? +
?? 2
-
?? h
2
8?? ?? =
?? 8?? (4?? 2
-h
2
)+
?? 2
(1+
2?? h
)
 
Evidently : velocity distribution is parabolic 
Page 5


Edurev123 
8. Navier-Stoke's Equation for Viscous 
Flow 
8.1 Find Navier-Stokes equation for a steady laminar flow of a viscous 
incompressible fluid between two infinite parallel plates. 
(2014 : 20 Marks) 
Solution: 
By Laminar flow, we mean that fluid moves in layers parallel to the plates. 
We suppose that an incompressible fluid with constant viscosity is confined between two 
parallel plates, 
?? =
?? 2
 
?? =-
?? 2
 
Let, the fluid be moving with velocity is parallel to ?? -axis with laminar flow. In order to 
maintain such a motion, the difference of pressure in ?? -direction must be balanced by 
shearing stresses. 
Here:                                                                 ?? =?? (?? ,0,0) 
Equation of continuity 
?? =0; so that ?? =?? (?? ,?? ) 
Navier-strokes equation in absence of external force is 
????
????
=
??? ??? +(?? ·?)?? =-
1
?? ??? +?
2
?? or, ??? ??? ??? +????
??? ??? =-
1
?? ??? +???? ?
2
?? or, ??? ??? ??? =-
1
?? ??? +???? ?
2
?? as 
??? ??? =0
 This, 
??? ??? =
1
?? ??? ??? +?? ?
2
?? ?? =·?? (?? ,?? )
?? =?? ?? =?? (?? ,?? ),?? =?? (?? ,?? )
 
Consequently (iii) declares that either 
??? ??? is constant or function If ' ?? . Now consider the 
case of steady motion so that (iii) becomes. 
?? ?
2
?? ??? 2
=
??? ??? =
????
????
 or 
?
2
?? ??? 2
=
1
?? ????
????
????
????
=
?? ?? ????
????
+?? ?? =
?? 2
2?? ????
????
+???? +?? 
Case i: Plane Couette Flow; 
In this case 
????
????
=0; the lower plate is stationary while the upper is moving with uniform 
velocity ' ?? parallel to ?? -axis. The boundary conditions are 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =?? = constant: ?? =
h
2
 
Substituting (Iv) and (i) and (ii), we get. 
and 
0?=
h
2
8?? ·0+?? (-
h
2
)+?? ?? ?=
h
2
8?? ·0+?? ·
h
2
+?? -?? h+2?? ?=0;??? h+2?? =2?? 2?? ?=?? ;?-?? h+?? =0
 
Now, (iv) becomes, 
?? =
?? h
?? +
?? 2
(?? ) 
Evidently; velocity distribution is linear. 
Case II. Plane Poiseuille flow : 
In this case 
????
????
= constant =?? ?0 and both the walls are at rest. 
The boundary conditions are: 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =0;?? =
h
2
 
Subjecting (iv) to condition (i) and (ii), 
and 
?? h
2
8?? +?? (-
h
2
)+?? =0
?? h
2
8?? +?? (
h
2
)+?? =0
 
Subtracting we get, 
?? =0 and ?? =
-?? h
2
8?? . 
Now (iv) becomes, 
?? =
?? ?? 2
2?? -
?? h
2
8?? =-
h
2
8?? (1-
4?? 2
h
2
)
????
????
(???? )
?? =?? ?? (1-
4?? 2
h
2
) (???? )
?? ?? =-
h
2
8?? ????
????
(?????? )
 
Where, 
Is the maximum velocity in the flow accruing at ?? =0; evidently, velocity distribution is 
parabolic. Drag (shear stress) at lower plate. 
=(?? ????
????
)
?? =-
h
2
=?? (-
8?? h
2
?? ?? )
?? =-
h
2
=
4?? ?? ?? h
 
? The average velocity distribution for the present flow is given by 
or 
?? ?? =
1
h
? ?
h/2
-h/2
??? ·???? (?? sing?(6)); we get 
?? ?? =
1
h
?? ?? ? ?
h/2
-h/2
?(1-
4?? 2
h
2
)???? =
2
h
4?? ? ?
h/2
h
?(1-
4?? 2
h
2
)????
?? ?? =
2
h
(-
h
2
8?? ·?? )[
h
2
(-
4
h
2
)·
1
3
(
h
2
)
3
]=(-
h?? 4?? )(
h
3
)
?? ?? =
2
3
(-
h
2
?? 8?? )=
2
3
?? ?? 
where; 
?? ?? ?= average velocity 
?? ?=
????
????
= constant 
?? ?? ?= maximum velocity. 
 
Case III : Generalised Plane Couette flow: 
In this case 
????
????
= constant =?? ?0; the lower plate is at rest while the upper plate is in 
motion with velocity ?? . The boundary condition are 
(i) ?? =0;?? =-
h
2
 
(ii) ?? =?? ;?? =
h
2
 
Substituting in (iv) from (i) and (ii) 
?? h
2
8?? +?? (-
h
2
)+?? ?=0
h
2
8?? +?? (
h
2
)+?? ?=?? 
This, 
?? =
?? 2
-
h
2
8?? ;?? =
?? h
 
Now (iv) becomes, 
?? =
?? ?? 2
2?? +
?? h
?? +
?? 2
-
?? h
2
8?? ?? =
?? 8?? (4?? 2
-h
2
)+
?? 2
(1+
2?? h
)
 
Evidently : velocity distribution is parabolic 
?? ????
????
?=
?? 8?? (8?? -0)+?? ·
?? 2
(0+
2
h
) 
=???? +
?? h
·?? 
Drag per unit area on boundaries. 
?=?? ????
????
 at ??? =±
h
2
?=?? ?? h
±
h
2
????
????
 
Total flux (flow) per unit breadth across a plane perpendicular to ?? -axis is ?? . 
or, 
=? ?
h/2
-h/2
?????? =[
?? 8?? (
4
3
?? 3
-h
2
?? )+
?? 2
(?? +
?? 3
h
)]
?? =
-h
2
?? =
h
2
 
Vorticity ?? (?? ,?? ,?? ) at any point is given by 
?? =?? ·
h
2
-
h
3
12
·
?? ?? 
?? =0;?? =0
?? =
1
2
(
??? ??? -
??? ??? )=-
1
2
??? ??? =-
1
2
????
????
 
?? =-
1
2
(
????
?? +
?? h
) by (ix)  
Rate ?? of dissipation of energy per unity area is given by 
?? =4?? ? ?
h/2
-h/2
??? 2
???? =?? ? ?
h/2
-h/2
?[
????
?? +
?? h
]
2
????
?? =?? ? ?
h/2
-h/2
?(
?? 2
?? 2
?? 2
+
?? 2
h
2
+
2?????? ?? h
)????
?? =
?? 2
h
3
12?? +
?? 2
?? h