Navier-Stokes Equations
Now that we have determine the momentum equations we proceed to study two particular cases:
1. Inviscid flow
2. Newtonian fluids
First, considering the flow if inviscid, that is, viscous e↵ect are equal to zero the terms corresponding the
normal and shear stresses are:
xx
=
yy
=
zz
=
p
⌧xy = ⌧xz = ⌧yz = 0
Therefore the momentum equations become:
⇢gx
⇢gy
⇢gz
@p
=⇢
@x
✓
@u
@u
@u
@u
+u
+v
+w
@t
@x
@y
@z
◆
✓
◆
@p
@v
@v
@v
@v
=⇢
+u
+v
+w
@y
@t
@x
@y
@z
✓
◆
@p
@w
@w
@w
@w
=⇢
+u
+v
+w
@z
@t
@x
@y
@z
Or:
⇢~g
rp = ⇢
~
@V
~ · r)V
~
+ (V
@t
!
These set of equations is known as the Euler Equations of Motion
Now, If the fluid under consideration is an incompressible newtonian fluid, the relation between velocity and
normal and shear stresses is given by:
xx
=
p + 2µ
@u
;
@x
yy
=
p + 2µ
⌧xy = ⌧ yx = µ
⌧yz = ⌧ zy = µ
⌧xz = ⌧ zx = µ
1
✓
✓
✓
@v
;
@y
xx
@u @v
+
@y
@x
@v @w
+
@z
@y
@u @w
+
@z
@x
◆
◆
◆
=
p + 2µ
@w
@z
In order to obtain a version of the momentum equations for newtonian fluids, it is necessary substitute to the
previous expressions into the momentum equations. We do it for the x component and extend the results
to the other two directions.
x momentum:
⇢
✓
@u
@u
@u
@u
+u
+v
+w
@t
@x
@y
@z
Thus, by substituting the expressions for
xx ,
◆
= ⇢gx +
✓
@ xx
@⌧yx
@⌧zx
+
+
@x
@y
@z
◆
⌧yx and ⌧zy into the last three terms of the previous equation:
 ✓
◆
 ✓
◆
@u
@
@u @v
@
@u @w
p + 2µ
+
µ
+
+
µ
+
@x
@y
@y
@x
@z
@z
@x
✓
◆
✓
◆
2
2
2
@p
@ u
@ u
@ @v
@ u
@ @w
=
+ 2µ 2 + µ 2 + µ
+µ 2 +µ
@x
@x
@y
@z @x
@z
@z @x
✓ 2
◆

2
2
@p
@ u @ u @ u
@ @u @v
@w
=
+µ
+ 2 + 2 +µ
+
+
@x
@x2
@y
@z
@x @x @y
@z
@ xx
@⌧yx
@⌧zx
@
+
+
=
@x
@y
@z
@x

Since we are taking into consideration an incompressible fluid ⇢ = const, therefore the continuity equation
in general for any fluid is:
@p @(⇢u) @(⇢v) @(⇢w)
+
+
+
=0
@t
@x
@y
@z
Becomes:
@u @v
@w
+
+
=0
@x @y
@z
And therefore:
@ xx
@⌧yx
@⌧zx
+
+
=
@x
@y
@z
@p
+µ
@x
✓
@2u @2u @2u
+ 2 + 2
@x2
@y
@z
◆
@⌧xy
@ yy
@⌧zy
+
+
=
@x
@y
@z
@p
+µ
@y
✓
@2v
@2v
@2v
+
+
@x2
@y 2
@z 2
◆
@⌧xz
@⌧yz
@ zz
+
+
=
@x
@y
@z
@p
+µ
@z
✓
@2v
@2v
@2v
+ 2+ 2
2
@x
@y
@z
◆
In a similar way:
And:
2
Thus, the moment equations are:
◆
@2u @2u @2u
+ 2 + 2
@x2
@y
@z
✓ 2
◆
@v
@v
1 @p
@ v
@2v
@2v
+v
+w
=
+ gy + ⌫
+
+
@y
@z
⇢ @y
@x2
@y 2
@z 2
✓ 2
◆
@w
@w
1 @p
@ v
@2v
@2v
+v
+w
=
+ gz + ⌫
+ 2+ 2
@y
@z
⇢ @z
@x2
@y
@z
@u
@u
@u
@u
+u
+v
+w
=
@t
@x
@y
@z
@v
@v
+u
@t
@x
@w
@w
+u
@t
@x
1 @p
+ gx + ⌫
⇢ @x
✓
Or in vector notation:
~
@V
~ · r)V
~ =
+ (V
@t
1
~
rp + ~g + ⌫r2 V
⇢
3