Chapter 5
5.7 Microscopic Momentum Balance
We now consider the general open system or control volume fixed in space and located in a
fluid flow field, as shown in Figure 5.7-1. The streamline of a fluid stream is the curve where
the velocity at any point is tangent to it. For a differential element of area dA on the control
surface, the rate of momentum efflux from this element = (v)(vdAcos), where (dAcos) is
the area dA projected in a direction normal to the velocity vector v,  is the angle between
the velocity vector v and the outward-directed unit normal vector n to dA, and  is the
density.
Control volume
dA

Streamlines of
fluid stream
v
n
Normal to surface dA
Control surface
Figure 5.7-1 Flow through a differential area dA on a control surface.
(v)(vdAcos) is the scalar or dot product of v(vn)dA. Since the normal vector n is
pointing outward, the momentum (efflux) leaving the control volume is positive (  < 90o)
and the momentum (influx) entering the control volume is negative ( > 90o). If we now
integrate this quantity over the entire control surface A, we have the net outflow of mometum
~
across the control surface or the net momentum efflux from the entire control volume V .
 net momentum efflux   rate of

=
control
volume  from
 from
 rate of
 
 from
momentum output 

control
volume 
momentum input 

control
volume
 net momentum efflux 

=
control
volume
 from
 vv cos dA =   v(vn)dA
A
A
Since the rate of momentum accumulation within the control volume is
dM

=
 vdV,
dt
t V
the momentum balance is written as

 vdV = 
t V
  v(vn)dA +  F
A
on C.V.
5-41
(5.7-1)
The above equation can also be written as
lim
xyz 0

vdV
t 
=
xyz
 v( v  n)dA
lim
xyz 0
xyz
+
F
lim
on C.V.
xyz 0
x y z
(5.7-2)
Each of the above terms will be evaluated separately and substituted into equation (5.7-2).
The rate of momentum accumulation within the control volume. The rate of momentum
accumulation within the control volume is given by
lim
xyz 0

vdV


v
( / t ) vxyz
t 
v = 
=
=
+v
t
t
t
xyz
xyz
(5.7-3)
Net rate of momentum transported through the control volume. The net rate of
momentum flux into the control volume illustrated in Figure 5.7-2 is
lim
xyz 0
 v( v  n)dA =
xyz
+
( vv y |y  y  vv y |y ) xz
( vvx |x  x  vvx |x )yz
+
xyz
xyz
( vvz |z  z  vvz |z )xy
xyz
y
vvy|y+y
y
vvx|x
x
vvx|x+x
z
x vv |
y y
z
Figure 5.7-2 Momentum flux through a differential control volume
lim
xyz 0
 v( v  n)dA =
xyz
 ( v v y )
 ( v v x )
 ( v v z )
+
+
x
z
y
(5.7-4)
Applying the product rule to the terms on the right-hand side of the equation yields
5-42
lim
xyz 0
 v( v  n)dA = v  ( v )  ( v )  ( v ) 
xyz
 x

 v
v
v 
+  v x
 vy
 vz 

z 
y
z 
 x
y
x
z
y
The above equation can be simplified by the add of the continuity equation (5.2-2)
 ( v x ) ( v y )  ( v z )

=


t
x
z
y
(5.2-2)
 ( v x ) ( v y ) ( vz ) 

 x  y  z  =  t


Therefore
lim
xyz 0
 v( v  n)dA =  v  +  v
t
xyz


x
v
v
v 
 vy
 vz 
x
y
z 
(5.7-5)
Sum of external forces acting on control volume.
y
yy|y+y
yx|y+y
xy|x+x
xx|x
x
xx|x+x
xy|x
z
yx|y
x
z
y
yy|y
Figure 5.7-3 Forces acting on a differential control volume.
The forces acting on the control volume are those due to surfaces forces such as frictional
forces and pressure force, and body forces such as gravitational force. Summing the forces in
the x direction we obtain
F
x
= (xx|x+x  xx|x)yz + (yx|y+y  yx|y)xz + (zx|z+z  zx|z)xy
on C.V.
+ gxxyz
In this equation gx is the component of the gravitational acceleration in the x direction. In the
limit as xyz 0, we obtain
5-43
F
x
lim
xyz 0
on C.V.
x y z
=
 yx
 xx
 zx
+
+
+ gx
x
z
y
Similarly, we will obtain the following expressions for the summation of the forces in the y
and z directions
F
y
lim
xyz 0
on C.V.
x y z
F
z
lim
xyz 0
=
on C.V.
x y z
=
 xy
x
+
 yy
+
y
 zy
+ gy
z
 yz
 zz
 xz
+
+
+ gz
x
z
y
Therefore
F
lim
on C.V.
xyz 0
x y z
lim
xyz 0
=
F
x
lim
on C.V.
xyz 0
x y z
ex +
lim
xyz 0
F
y
on C.V.
x y z
ey +
F
z
lim
xyz 0
on C.V.
x y z
ez
F
 zy 
 yx
 yy
  xy
 zx 
 
=  xx +
+
e
+
+
ey
x+ 
z 
xyz  x
z 
y
y
 x
on C.V.
 yz
 zz 
 
+  xz +
+
ez + (gxex + gyey + gz ez)
z 
y
 x
F
lim
on C.V.
xyz 0
x y z
=  + g = (P + ) + g
(5.7-6)
The total stress at any point within a fluid is composed of both the isotropic pressure and the
anisotropic stress components. By convention, pressure is considered a negative stress
because it is compressive. The differential momentum equation becomes
lim
xyz 0

 vdV
x  y z
=
lim
xyz 0
 v( v  n)dA
xyz
+
F
lim
on C.V.
xyz 0
x y z
 v


v
v
v 
 vy
 v z  +  + g
+v
=v
  v x
t
t
t
y
z 
 x

v
+ vv = P +  + g
t
5-44
(5.7-7)
The three components of the momentum equation in rectangular coordinates system are
 xx  yx
v
v
v 
 v x
 zx
P
+
+
+
+ gx (5.7-8a)
 v x x  v y x  v z x  = 
x
x
z
x
y
z 
y
 t
 
 xy  yy
 zy
v
v
v 
 v
P
  y  v x y  v y y  v z y  = 
+
+
+
+ gy (5.7-8b)
z
x
y
x
y
z 
y
 t
 xz  yz
 zz
 v z
P
v
v
v 
+
+
+
+ gz (5.7-8c)
 v x z  v y z  v z z  = 
z
x
z
x
y
z 
y
 t
 
In the momentum equation, the terms on the left-hand side represent the time-rate of change
of momentum, and the terms on the right-hand side represent the forces.
v x
represents the time rate of change of vx at a fixed location, and is called the
t
v
v 
 v
local acceleration. The terms  v x x  v y x  v z x  are called the convective
y
z 
 x
acceleration that represents the change in velocity from location to location. The sum of local
and convective acceleration is the total acceleration. The four terms




 vy
 v z  is the derivative following the motion of the fluid. This
  v x
x
y
z 
 t
D
derivative is called the substantial or material derivative and is denoted by
.
Dt
The term
D




=
 vx
 vy
 vz
Dt t
x
y
z
When the substantial derivative notation is used, equations (5.7-8) become

Dv x
 xx  yx
 zx
P
=
+
+
+
+ gx
Dt
x
x
z
y
(5.7-9a)

Dv y
 xy  yy
 zy
P
+
+
+
+ gy
z
y
x
y
(5.7-9b)

 xz  yz
Dv z
 zz
P
=
+
+
+
+ gz
z
x
z
Dt
y
Dt
=
(5.7-9c)
Equations (5.7-9) are valid for any type of fluid since we have not assumed any relationship
between shear stress and the shear rate. For laminar flow and Newtonian fluid we have the
following relations between shear stress components and the strain rates in rectangular
coordinate form
5-45
v 
 v
xy = yx =   x  y 
x 
 y
(5.7-10a)
 v
v 
yz = zy =   y  z 
y 
 z
(5.7-10b)
 v z v x 


z 
 x
(5.7-10c)
zx = xz =  
xx =   2
 v x 2

-   v  P
 x 3

(5.7-10d)
 v 2

yy =   2 y -   v   P
 y 3

(5.7-10e)
 v z 2

-   v  P

z
3


(5.7-10f)
zz =   2
These relations are called the constitutive equations that relate the local stress components to
the flow or deformation of the fluid in laminar flow. If equations (5.7-10) are substituted into
equations (5.7-9) and simplified, we will obtain the following relations for incompressible
flow where v = 0

  2v
Dv x
 2vx  2vx 
P
=
+   2x 
 2  + gx
Dt
x
y 2
z 
 x

Dv y
Dt
=
  2v y  2v y  2v y
P

+   2 
2

x

y
z 2
y


 + gy


  2vz  2vz  2vz 
Dv z
P

=
+   2  2  2  + gz
z
Dt
y
z 
 x
(5.7-11a)
(5.7-11b)
(5.7-11c)
Hence, the momentum equation for incompressible, laminar flow of a Newtonian fluid in
vector notation is given by

Dv
= P + 2v + g
Dt
5-46