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1. Dr Patrick Geoghegan
Book: H. Versteeg and W. Malalasekera An Introduction to
Computational Fluid Dynamics: The Finite Volume Method Chapter 2
FEA/CFD for
Biomedical
Engineering
Week 9: CFD –
Momentum

2. Momentum Part 2

3. Consider a moving fluid element
x
δ
y
δ
z
δ
x
z
y
P (point)
u
S
T
E
s
N
E
W
T
B
Note: the element moves with the flow
(it is not stationary at point P)

4. • Newton’s 2nd Law: F = ma
• If we consider the forces acting on the element in the x
direction – there are 2 types:
– Body Forces (per unit mass) - act directly on the entire mass of the fluid element
• gravitational
• coriolis
– Surface Forces - act directly on the surfaces of the fluid
element
• pressure
• normal & shear (friction)
Momentum Equations
z
y
x
f
f
f +
+
=
F x
a
m
fx
= t
u
∂
∂
=
x
a

5.  Body forces act on the entire element
volume
 Surface forces act over an area (i.e. )
Body force in the x direction
z
y
x
f
x
x δ
δ
ρδ
=










direction
in the
element
fluid
on the
acting
force
Body
mass
volume
density =
×
y
xδ
δ

6. When a force acts over an area it is termed a stress. Of
which there are 2 fundamental types – as applied to fluids.
• direct stress
– compress or extend
– Perpendicular/normal
– Pressure
• Shear stress
– shear (like a deck of cards)
– along a surface
– Viscous forces
• Fluid friction
Stresses
zz
yy
xx
σ
σ
σ ,
,
yz
xz
yx
xy σ
σ
σ
σ ,
,
,
( )
yz
xz
yx
xy τ
τ
τ
τ ,
,
,

7. Stresses
Time rate of change of deformation: both are
dependent upon velocity gradients within the flow
y
x
yx
τ
shear Direct (normal)
On plane normal to y
In x direction
y
x
xx
σ
On plane normal to x
In x direction

8. Stress convention
x
δ
y
δ
z
δ
x
z y
T
E
, ij
τ
face direction
+ve stresses act in
+ve direction
ii
σ
δx
x
x
x
xx
xx δ
σ
σ
∂
∂
+
x
xx
∂
∂σ
xx
σ
σ
W
E
W

9. zz
τ
Stress & Force on Surfaces
• stresses have units of pressure N/m2
• so in order to find the force we need to multiply
the stress by the area over which it acts
x
z
y
yx
τ








∂
∂
+ y
y
yx
yx δ
τ
τ
z
xδ
δ
y
xδ
δ
z
xδ
δ

10. Surface Forces (x direction)
S
T
E
z
y
p δ
δ
y
x
z
z
zx
zx δ
δ
δ
τ
τ 





∂
∂
+
y
z
x
z
y
x
x
p
p δ
δ
δ 





∂
∂
+
y
x
zx δ
δ
τ
z
x
yx δ
δ
τ
z
x
y
y
yx
yx δ
δ
δ
τ
τ 







∂
∂
+
z
y
xx δ
δ
σ
z
y
x
x
xx
xx δ
δ
δ
σ
σ 





∂
∂
+

11. Net surface forces (x direction)
.
y
x
z
z
z
x
y
y
z
y
x
x
z
y
x
x
p
p
p
x
zx
zx
zx
yx
yx
yx
xx
xx
xx
δ
δ
τ
δ
τ
τ
δ
δ
τ
δ
τ
τ
δ
δ
σ
δ
σ
σ
δ
δ
δ






−






∂
∂
+
+






−






∂
∂
+
+






−






∂
∂
+
+












∂
∂
+
−
=










direction
the
in
element
fluid
the
on
acting
force
surface
nett
Summing the surface forces and collecting terms allows us to
develop a general expression for the surface forces (pressure
and viscous forces) in the x direction

12. • add body forces and surface forces
Sum of Forces in x
z
y
x
f
z
y
x
z
y
x
x
p
F x
zx
yx
xx
x
δ
δ
ρδ
δ
δ
δ
τ
τ
σ
+






∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
=
total
force
body
force
surface forces

13. Remember Newton’s 2nd!?
total
force
z
y
x
m δ
δ
δ
ρ
=
Dt
Du
ax = (substantial
derivative)
𝐹𝐹𝑥𝑥 = 𝜌𝜌𝜌𝜌𝜌𝜌𝛿𝛿y𝛿𝛿𝑧𝑧
𝐷𝐷𝐷𝐷
𝐷𝐷𝐷𝐷
𝐹𝐹𝑥𝑥 = 𝑚𝑚𝑚𝑚𝑥𝑥

14. • use Fx=max and ÷ through by volume
• The fluid element was moving with the flow: this is the non-
conservation form of the Cauchy equation
• similar equations can be derived in y and z directions
Cauchy Equation in x
x
zx
yx
xx
f
z
y
x
x
p
Dt
Du
ρ
τ
τ
σ
ρ +
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
=
total
force
body
force
surface forces
𝐹𝐹𝑥𝑥 = 𝜌𝜌𝜌𝜌𝜌𝜌𝛿𝛿y𝛿𝛿𝑧𝑧
𝐷𝐷𝐷𝐷
𝐷𝐷𝐷𝐷
z
y
x
f
z
y
x
z
y
x
x
p
F x
zx
yx
xx
x
δ
δ
ρδ
δ
δ
δ
τ
τ
σ
+






∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
=