Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
Navier-Stokes Equations { 2d case
SOE3211/2 Fluid Mechanics lecture 3
Navier-Stokes
Equations {
2d case
NSE (A)
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Continuity equation :
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
conservation of mass, momentum.
often written as set of pde's
dierential form { uid ow at a point
2d case, incompressible ow :
@ ux @ uy
+
=0
@x
@y
conservation of mass
seen before { potential ow
Navier-Stokes
Equations {
2d case
Momentum equations :
@ ux
@u
@u
+ ux x + uy x =
@t
@x
@y
NSE (A)
Equation
analysis
+ Equation
analysis
@ uy
@u
@u
+ ux y + uy y
@t
@x
@y
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
(x
1 @p
@x
2
@ ux
@2u
+ 2x + fx
@y
@x 2
1 @p
=
@y
2
@ uy @ 2 uy
+ 2 + fy
+ @x 2
@y
and y cmpts)
3 variables, ux , uy , p
linked equations
need to simplfy by considering details of problem
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
The NSE are
Non-linear { terms involving ux @@uxx
Partial dierential equations { ux , p
functions of x , y , t
2
2nd order { highest order derivatives @@ xu2x
Coupled { momentum equation involves p , ux , uy
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
The NSE are
Non-linear { terms involving ux @@uxx
Partial dierential equations { ux , p
functions of x , y , t
2
2nd order { highest order derivatives @@ xu2x
Coupled { momentum equation involves p , ux , uy
Two ways to solve these equations
1 Apply to simple cases { simple geometry, simple conditions
{ and reduce equations until we can solve them
2 Use computational methods { CFD
(SOE3212/3)
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation analysis (A)
Consider the various terms :
@u
@u
@ ux
+ ux x + uy x =
@t
@x
@y
1 @p
@x
+ Equation
analysis
2
@ ux
@x 2
@2u
+ 2x + fx
@y
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
@ ux
@t
change of ux at a point
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation analysis (A)
Consider the various terms :
@u
@u
@ ux
+ ux x + uy x =
@t
@x
@y
1 @p
@x
+ Equation
analysis
2
@ ux
@x 2
@2u
+ 2x + fx
@y
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
ux
@ ux
@u
+ uy x
@x
@y
term
how does ow (ux ; uy )
move ux ?
non-linear
transport/advection
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation analysis (A)
Consider the various terms :
@u
@u
@ ux
+ ux x + uy x =
@t
@x
@y
1 @p
@x
+ Equation
analysis
2
@ ux
@x 2
@2u
+ 2x + fx
@y
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
1 @p
@x
pressure gradient {
usually drives ow
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation analysis (A)
Consider the various terms :
@u
@u
@ ux
+ ux x + uy x =
@t
@x
@y
Equation
analysis
Flow down
inclined plane
(A)
Tips (A)
@x
+ Equation
analysis
Laminar ow
between plates
(A)
1 @p
2
@ ux
@x 2
@2u
+ 2x
@y
2
@ ux
@x 2
@2u
+ 2x + fx
@y
viscous term { eect of
viscosity on ow
has a diusive eect
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation analysis (A)
Consider the various terms :
@u
@u
@ ux
+ ux x + uy x =
@t
@x
@y
1 @p
@x
+ Equation
analysis
2
@ ux
@x 2
@2u
+ 2x + fx
@y
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
fx
external body forces { eg.
gravity
Navier-Stokes
Equations {
2d case
Laminar ow between plates (A)
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
Fully developed laminar ow
between innite plates at
y = a
What do we expect from the
ow?
= 0 at walls
Flow symmetric around y = 0
Flow parallel to walls
u
y
+a
x
-a
Navier-Stokes
Equations {
2d case
NSE (A)
@u
@ ux
@u
+ ux x + uy x =
@t
@x
@y
+ Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
@u
@u
@ uy
+ ux y + uy y
@t
@x
@y
1 @p
@x
2
@ ux
@2u
+ 2x
@y
@x 2
1 @p
=
@y
2
@ uy @ 2 uy
+ 2
+ @x 2
@y
Navier-Stokes
Equations {
2d case
NSE (A)
@u
@ ux
@u
+ ux x + uy x =
@t
@x
@y
+ Equation
analysis
Equation
analysis
Equation
analysis
@u
@u
@ uy
+ ux y + uy y
@t
@x
@y
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
1 @p
@x
2
@ ux
@2u
+ 2x
@y
@x 2
1 @p
=
@y
2
@ uy @ 2 uy
+ 2
+ @x 2
@y
Flow parallel to walls { we expect
uy = 0;
dp
=0
dy
and
ux = ux (y )
Navier-Stokes
Equations {
2d case
NSE (A)
@u
@ ux
+ ux x
@t
@x
=
+ Equation
analysis
Equation
analysis
1 @p
@x
2 @ ux
@y 2
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
Flow parallel to walls { we expect
uy = 0;
dp
=0
dy
and
ux = ux (y )
Navier-Stokes
Equations {
2d case
NSE (A)
@u
@ ux
+ ux x
@t
@x
1 @p
=
@x
+ Equation
analysis
2 @ ux
Equation
analysis
@y 2
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
Flow fully developed { no change in prole in streamwise direction
@
@
i.e.
= 0;
=0
@t
@x
Navier-Stokes
Equations {
2d case
1 @p
=
NSE (A)
@x
+ Equation
analysis
2 @ ux
Equation
analysis
@y 2
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
Flow fully developed { no change in prole in streamwise direction
@
@
i.e.
= 0;
=0
@t
@x
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
So momentum equation becomes
0=
1 dp
dx
+
d 2 ux
dy 2
Navier-Stokes
Equations {
2d case
So momentum equation becomes
0=
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
Integrate once :
y
1 dp
dx
+
d 2 ux
dy 2
dp
du
= x + C1
dx
dy
Navier-Stokes
Equations {
2d case
So momentum equation becomes
0=
NSE (A)
Equation
analysis
Equation
analysis
Integrate once :
y
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
But at y = 0,
dux
dy
1 dp
dx
+
d 2 ux
dy 2
dp
du
= x + C1
dx
dy
= 0 (symmetry), so C1 = 0.
Navier-Stokes
Equations {
2d case
So momentum equation becomes
0=
NSE (A)
Equation
analysis
Equation
analysis
Integrate once :
y
Equation
analysis
Equation
analysis
But at y = 0,
Equation
analysis
Integrate again
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
dux
dy
1 dp
dx
+
d 2 ux
dy 2
dp
du
= x + C1
dx
dy
= 0 (symmetry), so C1 = 0.
1 2 dp
y
= ux + C2
2 dx
Navier-Stokes
Equations {
2d case
So momentum equation becomes
0=
NSE (A)
Equation
analysis
Equation
analysis
Integrate once :
y
Equation
analysis
Equation
analysis
But at y = 0,
Equation
analysis
Integrate again
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
dux
dy
1 dp
dx
+
d 2 ux
dy 2
dp
du
= x + C1
dx
dy
= 0 (symmetry), so C1 = 0.
1 2 dp
y
= ux + C2
2 dx
But at y = a, ux = 0, so
1 dp
2 dx
C2 = a 2
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
Final solution
ux (y ) =
1
y2
2
{ equation of a parabola
Also, remember that
=
@ ux
@y
So from this we see that in this case
=y
dp
dx
a2
dp
dx
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Flow down inclined plane (A)
{ Flow of liquid down inclined plane
y
Equation
analysis
Equation
analysis
Equation
analysis
ux
h
x
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
α
Take x -component momentum equation
@ ux
@u
@u
+ ux x + uy x =
@t
@x
@y
+ 1 @p
@x
2
@ ux
@x 2
@2u
+ 2x + fx
@y
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Note :
1 Steady ow
2 ux (y ) only
Equation
analysis
3
Equation
analysis
4
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
No pressure gradient
fx = g sin Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Note :
1 Steady ow
2 ux (y ) only
Equation
analysis
3
Equation
analysis
4
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
No pressure gradient
fx = g sin Equation becomes
d 2 ux
=
dy 2
which we can integrate easilly.
g
sin Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Boundary conditions :
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
lower surface { ux (0) = 0
x
upper surface { du
dy = 0
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Boundary conditions :
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
lower surface { ux (0) = 0
x
upper surface { du
dy = 0
Solution
ux =
g
sin hy
y2
2
Navier-Stokes
Equations {
2d case
NSE (A)
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Equation
analysis
Laminar ow
between plates
(A)
Flow down
inclined plane
(A)
Tips (A)
Tips (A)
Most NSE problems will be time-independent. They will
probably only involve one direction of ow, and one coordinate
direction. They will probably be either pressure driven (so no
viscous term) or shear driven (ie. viscous related, so no
pressure term).
Thus, most NSE problems will lead to a 2nd order ODE for a
velocity component (ux or uy ) as a function of one coordinate
(x or y ).
Thus we would expect to integrate twice, and to impose two
boundary conditions.
A wall boundary condition produces a xed value : eg.
ux = 0.
A free surface produces a zero gradient condition, eg.
dux
dy = 0.