Basics of Fluid Mechanics, 2014a

10.1. INTRODUCTION 327
where in this case i is x, y, and z. The convective term (not time derivatives) in x
direction of equation (10.3) can be manipulated as
∂U x
U x
∂x + U ∂U x
y
∂y + U ∂U x
z
∂z = 1 ∂ (U x ) 2
2 ∂x +
∂U y
U y
{ }}
∂x
{
1 ∂ (U y ) 2 ∂U y
− U y
}
2 ∂x
{{
∂x
}
=0
0
∂U 1
x
U y @
∂y −∂U y
A
∂x
{ }} {
∂U z
U z
{ }}
∂x
{
∂U x
+U y
∂y + 1 ∂ (U z ) 2
2 ∂x
0
∂U 1
x
U z @
∂z −∂U z
A
∂x
{ }} {
− U z
∂U z
∂x
} {{ }
=0
+ U z
∂U x
∂z
(10.5)
It can be noticed that equation (10.5) several terms were added and subtracted
according to equation (10.4). These two groups are marked with the underbrace and
equal to zero. The two terms in blue of equation (10.5) can be combined (see for the
overbrace). The same can be done for the two terms in the red–violet color. Hence,
equation (10.5) by combining all the “green” terms can be transformed into
∂U x
U x
∂x + U ∂U x
y
∂y + U ∂U x
z
∂z = 1 ∂ (U x ) 2
+ 1 ∂ (U y ) 2
+ 1 ∂ (U z ) 2
2 ∂x 2 ∂x 2 ∂x +
( ∂Ux
U y
∂y − ∂U ) (
y ∂Ux
+ U z
∂x ∂z − ∂U )
z
∂x
(10.6)
The, “green” terms, all the velocity components can be combined because of the
Pythagorean theorem to form
1 ∂ (U x ) 2
+ 1 ∂ (U y ) 2
+ 1 ∂ (U z ) 2
= ∂ (U)2
2 ∂x 2 ∂x 2 ∂x ∂x
Hence, equation (10.6) can be written as
∂U x
U x
∂x + U ∂U x
y
∂y + U ∂U x
z
∂z
= ∂ (U)2
∂x
( ∂Ux
+ U y
∂y − ∂U y
∂x
) ( ∂Ux
+ U z
∂z − ∂U )
z
∂x
(10.7)
(10.8)
In the same fashion equation for y direction can be written as
∂U y
U x
∂x + U ∂U y
y
∂y + U ∂U y
z
∂z
= ∂ (U)2
∂y
( ∂Uy
+ U x
∂x − ∂U x
∂y
) ( ∂Uy
+ U z
∂z − ∂U )
z
∂y
(10.9)