Transcript F z

Integration over the Volume
Diagram 11 is similar to diagram 7
only now we are concerned with the
change in the gradient of each scalar
velocity component between the top
and bottom surfaces of the column.
Considering the component of the
gradient which is aligned with the long
axis of the column we get for the zdirection gradient:
 grad Vz  z 
 grad Vz  z
z
Grad(Vz2 )z2
z
z2
Grad(Vz1)z1
z
z1
And hence the total change in zdirection gradient of the z-direction
velocity components is:
grad Vz 2  z 2  grad Vz1  z1

z2
 grad Vz  z
z1
z

Region
dz
Area element daR
By combining this inner integration w.r.t z with an outer integration over the region we
can integrate over the entire volume of the body. This leads to the result
Fz ( visc ) z



z2
 grad Vz  z
z
R z1
dz daR 

 grad Vz  z
z
Vol
d
Where Fz(Visc) is the net Force in the z-direction due to the z-direction components of
the z-direction velocity gradients..
Similarly for the components in the x and y directions
Fz ( visc ) x

 

Vol
 grad Vz  x
x
d  and
Fz ( visc ) y



 grad Vz  y
Vol
y
d
Now since the total net outflow Fz(visc)(total) will be given by Fz(visc)x + Fz(visc)y + Fz(visc)z
it follows that:
Fz ( visc )(total )


  grad Vz  x  grad Vz  y  grad Vz  z 

  


 d
Vol

x

y

z




Viscous Force by Volume Integration
 grad Vz  x
Recognising the expression:
x

 grad Vz  y
y

 grad Vz  z
z
as the definition of the divergence of a field we have
Fz ( visc )(total )



Volume
div  grad Vz   d 
The div(grad) operation acting on a scalar field is called the Laplacian of the
field and its symbol is 2 (pronounced Del squared) hence:
Fx ( visc )(total )



Volume
2 Vx  d 
Fy ( visc )(total )



Volume
2 Vy  d 
Fz ( visc )(total )



Volume
2 Vz  d 
This can also be written as a vector in the form:
F( visc )



Volume
2  V  i  V  j  V  k  d 
This additional Diagram is of the
top of the z-direction column and
shows the three gradients associated
with each of the three sets of scalar
velocity components. Similar sets
exist for each of the two horizontal
columns (i.e. Those orientated in
the x and y directions.
Grad(Vz )z
Grad(Vy )z
Area = δaR
Grad(Vx )z