Two questions occurred to me on
first seeing this diagram which
incidentally is taken from:
Fluid Mechanics Fundamentals and
Applications
Y. A. Cengel and J. M. Cimbala.
McGraw Hill
1.
If the Navier-Stokes equations really are the corner stone of Fluid Mechanics shouldn't
they be inserted as part of the Fluid Mechanics foundation course.
2. Now come I had manages to teach fluid mechanics all these years without needing to make
use of these strange equations.
A talk by T. Swann
Mathematics From Fluids -1
Dr Anthony B Swann
Dept of CEE University of Auckland, Auckland, New Zealand
Mathematics From Fluids
- A review of Gauss' Theorem
The successful use Computational Fluid
Dynamic software such as Flexpde5 to
model complex flow problems requires an
appreciation of three dimensional field
theory and in particular of Gauss' theorem.
Traditionally, Gauss' theorem has been
derived mathematically from a
consideration of a general three
dimensional field. The aim of this seminar
is to explore an alternative "Fluid
Mechanics" approach to teaching Gauss'
Theorem in which a consideration of the
well known principle due to Archimedes
provides the context. Once established,
the approach can be
readily adapted to the application of the
conservation laws of mass and momentum
to fluid flows as well as to a derivation of
the well known Navier–Stokes equations.
Introduction
Archimedes principle deals with the
upthrust or buoyancy force which
acts on a body immersed in a fluid.
To establish a link with Gauss'
theorem we will reinterpret it as the
resultant force acting on a body
immersed in a fluid when linearly
varying pressure field (the
hydrostatic field) exists in the fluid.
•
•
Diagram 1 shows the force of the
water on the upper surface of the
discretised body
Diagram 4 shows the hydrostatic pressure force
acting on a small element of the surface of the
body resolved in the vertical direction.
Integrating this over the whole of the surface
gives the buoyancy force.
Diagram 3 shows how integration over
the region is used to find the volume
of the body.
z2
Also since pg z dz   P and the total
1
pressure change can be found by integrating
the vertical gradient of the pressure from the
bottom to the top of the column it follows that
z1
Taking the water surface as datum
k
F1 = - PδaR = -gz1aR
z1
z2
P
n
γ


z2
z1
P
dz
z
and hence that
  g 
FBuoyancy
Column
area = δaR
Area of surface element δaS

R
FBuoyancy 
and hence
•
Vertical height of the elementary volume = z2 - z1

RU
  gz1 daR
Diagram 2 shows the force of the
water on the lower surface of the body
z2
RL
= (z2 - z1)δaR
The total volume of the body
Hence
 z
R
2
 z1  daR
FBuoyancy   g 

RL
PdaR 

RU
PdaR


Surface
 gz2 daR
Diagram 5 is similar to diagram 3.It shows a small
element of the elementary column. By combining
an inner integration w.r.t z with an outer regional
integration we can integrate over the entire
volume of the body.
FBuoyancy 
volume

P
d
z

Volume
grad  P  d
Conclusions
Though rarely used, it is the region
integration that provides the link between
the surface and volume integrations and
hence provides the theoretical basis for
Gauss' theorem. The situation is
conveniently illustrated by the diagram:

z1
RL
PdaR 

RU
PdaR

z2
z
born 1598, Milan [Italy]
died Nov. 30, 1647, Bologna, Papal States
Resultant buoyancy force
FBuoyancy

which reduces to
 P k  n daS
Bonaventura Cavalieri
Cavalieri, in 1629, developed his method of
indivisibles, a means of determining the size of
   PdaR   PdaR   g   z2  z1  daR geometric figures similar to the methods of integral
RU
RL
R
calculus.

FBuoyancy
Integration over the Volume
So far so good but instead of stopping here we
consider calculation the buoyancy force by
integrating over the surface of the body. To do
this we first introduce two unit vectors k in the
vertical direction and n in the direction normal to
the surface on the fluid side. (called the
outward unit normal).
F2 = P δaR = gz2aR

= δ
Therefore volume of the element
 
FHyd 
Surface
P
dz daR
z
N.B. -ve because a positive buoyancy force requires a negative gradient
 P k  n daS
This establishes an equivalence relation
between the region and the surface integral,
namely:
Region R on a horizontal
plane which is the
shadow of the object
Total Force is obtained by F 
integrating over the region Hyd

z2
z1
or as a volume integral
Archimedes principle states that:
When a body is placed in a fluid it
experiences an upward force which
is equal the weight of the fluid
displaced
Integration over the Surface
Integration over the Region
Abstract
This leads to the double integral expression
FBuoyancy


R

Surface
 P k  n daS

Volume
grad  P  d
This result is at the very heart of our use of
control volumes in the study fluid dynamics of
flows. This is because it shows that, in
analysing the dynamics of a flow, we can, in
many cases, consider only the flow conditions
that exist at the surface of the control volume
and can ignore those that exist within.
z2
 g  dz daR
z1
Further Work
 ABS October 2007
Mathematics From Fluids - 2
Dr Anthony B Swann
Dept of CEE University of Auckland, Auckland, New Zealand
Integration over the Region
Abstract
Application to the mass conservation
law of Fluid Mechanics
The application of conservation laws to
fluid flows requires the use of a control
volume to define the extent of the body
of fluid under consideration. Such a
control volume can be conceptualised
by first considering Archimedes's
submerged body and then removing its
substance. This leaves the essence of
the body in much the same way that the
removal of the "Cheshire cat" leaves
only the "Grin". Being only essence, the
flow is not affected by the presence of
the C.V. We can however use it to
determine whether the C.V. has a net
inflow or a net outflow or whether the
flow is balanced.
•
Diagram 6 is based on diagram 3 except that
body is now an insubstantial shape (the C.V.).
•
The moving fluid, now passes through it
unhindered. The moving fluid constitutes a 3-D
flow field described by lines called streamlines
which are tangential to the velocity vector at
every point along their length.
•
Integration over the Surface
To get the total flow crossing the surface we
must include the flows across the two sets of
risers. Diagram 8 shows all three integration
regions.
Diagram 9 shows a patch of the surface above a
single tread. Here n is the unit normal to the patch
and V is the local velocity vector.
n
V
Surface
Patch
Region Ry
Region Rx
Rate of flow through this
 Q  V  n  as
surface element
Control Surface
Qnet (total ) 
and hence
Control Volume

surface
Rx
V  n  aS
V  i daRx

  V  j daRy
surface
Ry
Introduction
V  n  aS
Integration over the Volume

V  i daRx


V  j daRy


V  k daRz
Qnet (total ) 
V
Vz
Vy

R( far  near )
R( far  near )
R( far  near )
Area = δaR
Vx
 QU  Vz   aR
 V  k  aR

RU
V  k daR 
R
Vx
dx daR 
x
y2
Vy
y1
x

Vol

dy daR 
Vol
Vx
d
x
Vy
y
d
Diagram 10 is similar to diagram 6 except that it
shows a small element of the elementary column.
Now since integrating the gradient of Vz over the
height of the column gives the total change in Vz
across the column we can use it to substitute for
the expression under the regional integral in the
previous result. This leads to a double integral
with the inner integration being performed over
the entire height of the column.

Vol
 Vx Vy Vz 



 d
y
z 
 x
is called the divergence of the vector field div(V)
and hence that
Qnet (total ) 

Volume
div V d 
Conclusions
Again we see that the regional integration
provides the link between the surface and
volume integrations and hence it is another
example of Gauss' theorem at work. The
situation in this case is:
m 3
 ( far  near )m V  m daR
m 1
z
Gauss
z2
born 1777, Braunschweig, [Germany]
died 1855, Göttingen, Hanover, {Germany]
The net outflow flow through all the treads is
obtained by integrating over the entire region for
both upper and lower control surfaces and
subtracting thus:
Qnet ( z ) 
R x1
Vy
 Vx
Vz 




y
z 
 x
Next we calculate the flow by integrating over
the surface of the C.V.
The flow through this element
Qu is given by:
x2
We may note that the expression
Rz
The result for the three regions is:
Diagram 7 shows flow through a single
tread element of area ar on the upper
control surface.
Q y  
  V  k daRz
Region Rz
•

Qnet (total ) 
Region
The continuity law of incompressible fluid
flow effectively says that in the absence of
any sources or sinks, any net inflow to the
C.V. through one part of its surface must be
just balanced by a net outflow through
another part of its surface. So if we first
divide the surface into small elements then
multiply the area of each element by the
normal component of the flow velocity
where the flow crosses the element and
finally sum the results for all the elements
then the result should be zero. This concept
forms the basis of an important theorem
called the "Divergence Theorem". It is in
effect Gauss' Theorem applied to the
continuity problem.
Q x  
Now we can add the three expressions to get
the net outflow from the U.C.V. The result is:
This establishes an equivalence relation
between the region and the surface integrals,
namely:

The expressions for the other directions are.

RL
V  k daR
Carl Friedrich Gauss, in 1813, developed the
divergence theorem (also called Gauss' Theorem)
which states that "The net flux of any vector field
through any closed surface is equal to the divergence
of the field integrated throughout the volume enclosed
by the surface".
z1
The resulting expression for the z-direction velocity
component is:
Q z  

z2
R z1
Vz
dz daR 
z

Vol
Vz
d
z

surface
V  n  aS

Volume
div V d 
The divergence of the velocity field is a
measure of the overall rate at which fluid is
either approaching or retreating from a point in
the flow.
 ABS January 2008
Mathematics From Fluids - 3
Dr Anthony B Swann
Dept of CEE University of Auckland, Auckland, New Zealand
Abstract
Application to the linear momentum
conservation law of Fluid Mechanics
The linear momentum conservation law
is simply an extension of the law of
continuity with the linear momentum
vector replacing the velocity vector.
The U.C.V. is used as before to control
the calculation. In all other respects the
derivation is the same
Introduction
Integration over the
Surface
When integrated over the entire surface, these
expressions give the change of momentum of
the component in each direction. Further, by
the momentum principle this change will be
equal to the sum of all external forces resolved
in that direction. thus:

Fx 

surface
 V n Vx das
Poster 1 introduced Gauss, theorem
Fy 
 V n Vy das
as it applies to forces acting at the
surface
surface of a body (real or imagined)
while poster 2 applied Gauss' theorem
Fz 
 V n Vz das
a quantity (namely volume) which the
surface
field is convecting through an
imaginary body (referred to as the
Integration over the
Control Volume). In this poster we will
Volume
use Gauss' theorem to apply the
momentum principle to the C.V. using
However, our template also tells us that :
the previous applications as templates.
As momentum is convected by the flow Q 
V n das 
divV d 
S
surface
volume
we will use the surface integral from
poster 2 diagram 9 as our template and Which implies that for instance for the xsimply replace volume by momentum.
direction momentum :
Thus:
 Fx  volume div  Vx V  d 
 Q  V  n  as




Becomes:
 M x   V n Vx  as
Where  Mx is the x-direction
component of the momentum
convected through the surface
patch.
Similarly:
 M y   V n Vy  as
 M z   V n Vz  as

Expanding this expression we get:
 Fx
vol Vx div   V  d 
  V  grad  Vx  d 
vol

But conservation of mass requires that the
divergence term be zero leaving only the 2nd
term whence:
 Fx

vol V  grad  Vx  d 
Gravity Force
To handle the forces active on the control
surface we turn to our poster 1 template.
Further, since gravity is a body force it is
logical to consider the volume integral from
our template which is:
Fz 

Volume
grad  P  d
However, for bodies on or near the earths
surface, the gradient of the gravity field will be
sensibly constant, directed in the vertical and
equal to g times the mass of the body . Thus
we can substitute ρg for -grad(P) to get:

Fz 
Volume
 g d
According to Newton, Viscous forces in a fluid
can be characterised as "The resistance which
arises from the lack of slipperiness originating in
a fluid which, other things being equal, is
proportional to the velocity by which the parts of
the fluid are being separated from each other.
"In other words, viscous stress is proportional to
the rate of strain of the fluid.
The rate at which fluid particles are separating
from each other at the tread surface is simply
Vz/z or in other words grad(Vz)k. The viscous
force on the step is obtained by multiplying by
the tread area aR and the coefficient of
proportionality .. Of course in this case other
forces will be acting on the risers.
as the gravity force acting on the fluid inside the
C.V., the other components being zero. A
more general result is obtained by letting X, Y
and Z be the component of the gravity force per
unit mass in the x, y and z directions
respectively. This gives three components of
the gravity force namely:
F x  grav 
vol  X d 
F y  grav 
vol
F z  grav 
vol  Z d 
The pressure force comes directly from the poster 1
template except that we need to consider each
coordinate direction separately. For instance, for
the x-direction we can write:
Fx ( presure ) 


Surface

Volume
  P  i  ndaS
 grad  P   i d 
Newton
Born 1642, at Woolsthorpe in Lincolnshire (UK)
died 1727, Kensington, London
In Book II of his Principia, Newton laid the
foundations of a theory of fluid mechanics which
included the effects of viscosity on a flow.
Similarly for the x and y direction components
Fz ( visc ) x
and
Fz ( visc ) y

•z2
•
z
x
Vol
 grad Vz  y


 grad Vz  x

 

y
Vol
d
d
Since these are all z-direction components of
the viscous force the total component will be
   grad Vz  
x

x


  grad Vz   y

 


Volume
y


  grad Vz   z



z
Fz (Viscosity )

•Grad(Vz2 )z2





 d




Which is simply:
•Grad(Vz1)z1
Fz (Viscosity )

•z1

div  grad Vz   d 

Volume
or:
Fz (Viscosity )
Y d 
Pressure Force
Volume Integration
Viscous Force
 
Volume
2 Vx  d 
• Area element daR Similar results for the other velocity
•Region
Diagram 11 is similar to diagram 10 except that
the velocity vectors must be replaced by the
component gradient vectors since these give the
magnitude and direction of the viscous stress
components. This lets us assess the total for the
C.V. For instance, for the z-direction component,
of the gradient we have :
Fz ( visc )



 grad Vz  z
z2
z
R z1
dz daR
Again the double integration can be replaced by
an integration over the volume of the C.V.
giving:
Fz ( visc )



Vol
 grad Vz  z
z
d
component scalar fields leads to the final
result :
F( visc )   
Volume
2  V  i  V  j  V  k  d 
Conclusions
In this poster we have obtained the ingredients
required to apply the momentum principle to the
C.V. The result for say the z-direction in volume
integral form is:

vol
V  grad  Vz d  

vol

 Zd

vol
grad  P   k d 
   div  grad Vz  d 
vol
 ABS January 2008
Mathematics From Fluids - 4
Dr Anthony B Swann
Dept of CEE University of Auckland, Auckland, New Zealand
RHS Terms
Abstract
In part 1, the formula embodying the
application of the momentum principle
to the Universal Control Volume is
presented in different forms .
Part 2 investigates the implications of
the formula for the gravity, pressure
and viscous forces which must act on a
fluid particle having the same shape as
the Control Volume used previously.
The result is a set of equations known
collectively as The Constitutive
Equations
Introduction
Part 1. Versions of the basic momentum
principle formula:

vol
V  grad  Vu d  

vol

U d 

vol
grad  P   l d 
   div  grad Vu  d 
vol
u  x ; y ; z : l  i ; j; k 
We might for instance write the formula for a
control volume of infinitesimal volume d we
can then omit the volume integrals thus:
V   Vu   U +    P   l       Vu  
u  x ; y ; z : l  i ; j; k 
The above can now be expanded in partial
differential form. In this case the LHS
becomes:
V
V
V  Vx
 Vy x
 Vz x 
 x
x
y
z 


  Vy
 Vy
 Vy 
 Vx
 Vy
 Vz
x
y
z 




V
V
 Vx  Vz
 Vy z
 Vz z 
x
y
z 

The first term on the RHS which is the gravity
force term becomes:
X 
 
 
 Y 
 
 
 
Z 
The Constitutive equations
Part 2. The constitutive equations relate the
forces acting on a fluid particle to the state of
stress generated inside the fluid. To derive
the constitutive equations first consider the
net forces acting on the surface of an oblong
fluid particle at point P:
 z
z
While the third or viscosity term becomes: :
  2Vx

 2Vx
 2Vx


2
2

x
y
z 2 


2
2
  2V


V

V
y
y
 y 2 
2 
2
x
y
z 

 2

2
2
  Vz 2   Vz 2   Vz 2 
x
y
 z 

Back substitution then gives the well known
Navier-Stokes equation (see background):
 y
z
  yz
y
y 
zx  
xz
z x
  yx
y
The second or pressure term becomes:
 P 
 x 
 
 P 
 
 y 
 
 P 
  z 
  zy
Now because, for an incompressible flow,
μ div(V) is zero we may add it to the LHS of the
equation The result is
  2V
 2Vx  2Vx 
P
    2x 


x
y2
z 2 
 x
  2V
 2Vy
 2Vz
   2x 

 x
 y x  z  x

x
 x
x
 Fy
 Fz
 xy
 V

 x  yx  zx 
  X 


 x y z
x
y
z 

 xy  y
 zy 

  Y 


 x y z
x
y
z 


 xz  yz  z 
  Z 


 x y z
x
y
z 




If we now equate the expressions for say Fx we get:
V
 2  2V
  2Vx
 2Vy    2Vx  2Vz  
P
x
 



 2


2
x
 x y    z 2  x z  
  x
 y
Combining terms having a common denominator
we get :
2
 P
 x
V 
 
 2 2x 
x
x 
 x
  yx
  2Vx
 2Vy 
  2 

 y
 x y 

y

  x   yx   zx 


 X 
 x  y  z
  2V
 zx
 2Vz 
x
y
z 

   2x 

z
xz 

  2Vx  2Vx  2Vx  
 z
P
  X   2 

 x  y  z
x
y2
 z 2  
 x

Then integrating both sides of each equation
Navier, Claude-Louis
w.r.t. its denominator gives the constitutive
equations:
Vx 
 x 
 Vz 
 V
  x 
 x 
 z


 x    P  2
•Borne 1785 in Dijon France died 1836 in Paris
•Although he made great contributions relating
to elasticity and stress in solids, His major
contribution however remains the Navier-Stokes
equations(1822), central to Fluid Mechanics.
V 

 z    P  2 z 
z 

V 
 V
 yz    z  y 
z 
 y
  x   yx   zx


x
y
z
 -
 zx
V 
 zy    y  z 
y 
 z
Associating the terms in brackets, the equation
reduces to
•From the diagram
 Vy 

y 

 Vy Vx 
 


y 
 x

 y    P  2
  x   yx   zx


x
y
z
  xy
P
 Fx
Thus:
V
 xz    z  x 
z 
 x
Conclusions
The Navier-Stokes equations are derived from
the volume integral form of the momentum
principle equations applied to an elementary
oblong control volume.
This is in contrast with the conventional method
which is to use the surface integral form of the
equations and integrate over a piecewise
continuous control surface chosen to make the
integration process as straight forward as
possible.
The practical difference is that the Navier-Stokes
solution will give information about velocities,
pressures etc everywhere in the flow field. In
other words to “model” the flow. Conventional
momentum principle solutions yield results only
bulk results which apply to the whole control
volume.
The constitutive equations allow us the use
velocity gradient information from the Navies yx
Stokes solution to find the distribution of normal
and shear forces throughout the field.
the other six constitutive equations can be easily
generated by the cyclic interchange of suffices
technique.
 Vy 
 V
  x 

x 
 y
 ABS January 2008