Download momentum conservation: cauchy equation

MOMENTUM CONSERVATION: CAUCHY EQUATION
Consider the illustrated control volume, which is fixed in space and
through which momentum can freely flow in and out.
The convective flux in the xj direction of momentum in the i direction is
given as uiuj. The control volume has outward normal nj. The net
inflow velocity into the control volume is thus = - ujnj, and the net
discharge of momentum dQmom,inflow,i into the volume across elemental
surface area dA is
dQmom,inf low ,i  uiu jn jdA
The total inflow rate of momentum into the
control volume is thus
Qmom,inf low,i    uiu jn jdA
ni
dA
dV
S
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MOMENTUM CONSERVATION: CAUCHY EQUATION
There are two types of forces that can operate on a continuous mass:
body forces, which act throughout the body of a control volume
surface forces which act on the surface of any control volume
A body force corresponds to a net source of momentum within a
control volume. As will be shown later, a surface force corresponds to
a net outflow of momentum across the surface of a control volume.
The body force considered in this course is gravity. Where gi denotes
the vector of gravitational acceleration, the gravitational force Fgi acting
on the mass in a control volume fixed in space is given as
Fgi   gidV
V
If x3 is upward vertical than gi = - gi3.
Where m is the mass in the control volume,
then,
Fgi  gi3  dV  mgi3
V
The surface forces act through the stress tensor.
ni
dA
dV
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MOMENTUM CONSERVATION: CAUCHY EQUATION
Again, the control volume is fixed in space, and the fluid is allowed to
freely flow in and out. The surface force dFSi acting on an elemental
surface area dA the control volume is given as
dFSi   jinjdA
Remember the convention First Face Second Stress in the stress
tensor ji. Thus the index j refers to the face and the index i refers to
the stress (surface force per unit area). The outward normal vector nj
makes sure that the sign is correctly accounted for.
The total surface force is thus given as
FSi    jin jdA
S
ni
dA
dV
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MOMENTUM CONSERVATION: CAUCHY EQUATION
To see how the sign convention using nj works, we consider the
elemental cubical control volume illustrated below.
Consider only the component FS1.
FS1    j1n jdA 
S

n dA 
11 1
S

n dA 
x3
21 2
S

x3
n dA
31 3
x1
x1 x2
S
x2
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MOMENTUM CONSERVATION: CAUCHY EQUATION
On the indicated face at x1, nj = (-1, 0, 0).
Thus the contribution from this face is:

n dA   21n2dA   31n3dA 
11 1
S
S
S
 11 x x 2x 3
1
nj
x3
x3
x1
x1 x2
x2
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MOMENTUM CONSERVATION: CAUCHY EQUATION
On the indicated face at x1 + x1, nj = (1, 0, 0).
Thus the contribution from this face is:

n dA   21n2dA   31n3dA 
11 1
S
S
S
 11 x  x x 2x 3
1
1
x3
x3
x1
nj
x1 x2
x2
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MOMENTUM CONSERVATION: CAUCHY EQUATION
On the indicated face at x2 nj = (0, -1, 0).
Thus the contribution from this face is:

n dA   21n2dA   31n3dA 
11 1
S
S
S
 21 x x1x 3
2
x3
nj
x3
x1
x1 x2
x2
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MOMENTUM CONSERVATION: CAUCHY EQUATION
On the indicated face at x2 + x2 nj = (0, 1, 0).
Thus the contribution from this face is:

n dA   21n2dA   31n3dA 
11 1
S
 21 x
S
2  x 2
S
x1x 3
nj
x3
x3
x1
x1 x2
x2
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MOMENTUM CONSERVATION: CAUCHY EQUATION
On the indicated face at x3 nj = (0, 0, -1).
Thus the contribution from this face is:

n dA   21n2dA   31n3dA 
11 1
S
S
S
 31 x x1x 2
3
x3
x3
x1
x1 x2
nj
x2
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MOMENTUM CONSERVATION: CAUCHY EQUATION
On the indicated face at x3 + x3 nj = (0, 0, 1).
Thus the contribution from this face is:

n dA   21n2dA   31n3dA 
11 1
S
 31 x
S
3  x 3
S
nj
x1x 2
x3
x3
x1
x1 x2
x2
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MOMENTUM CONSERVATION: CAUCHY EQUATION
Thus over the entire surface of the control volume,
FS1    j1n jdA 



S
11 x  x
1
1

 11 x x 2x 3 
1
21 x  x
2
2
 21 x
31 x  x
3
3
 31 x
2
3
x x
x x
1
3
1
2
Note how this accounts
correctly for the signs on
every face.
31 x
  21 x
 11 x
1
2
x3
x3
x1
3  x 3
11 x  x
1
x1 x2
1
 31 x
x2
21 x
2  x 2
3
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MOMENTUM CONSERVATION: CAUCHY EQUATION
Again consider the control volume, which is fixed in space and through
which fluid can freely flow in and out.
In words, conservation of momentum can be stated as:
/t(momentum in control volume) = net inflow rate of
momentum + surface force + body (gravitational) force
or

uidV  Qmom,inf low,i  FSi  Fgi

t V
or
ni

uidV    uiujnjdA    jinjdA   gidV

t V
S
S
V
dA
dV
Using the divergence theorem,



u
u
n
dA

(uiu j )dV
i j j


t S
x j
V
,
  jinjdA  
S
V
 ji
x j
dV
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MOMENTUM CONSERVATION: CAUCHY EQUATION
Reducing

uidV    uiujnjdA    jinjdA   gidV

t V
S
S
V
with



u
u
n
dA

i j j
V x j (uiuj )dV
t S
,
 ji
  n dA   x
ji j
S
V
dV
j
results in


 ji

V  t (ui )  x j (uiuj )  x j  gi dV  0


ni
or thus
 ji


(ui ) 
(uiu j ) 
 gi
t
x j
x j
dA
dV
This relation is known as the Cauchy equation. It is generally valid for
any fluid. Further progress is predicated on a specification of the stress
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tensor ij.