Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 99

As the flow is incompressible, steady and in two dimensions, the density term,
the unsteady term and the z -coordinate term cancel out to give

∂
∂

u
x

+ ∂
∂

v
y

= 0

(5.7)

For Sects. 1 and 3 there is a constant velocity in the x -direction and no variation in
the vertical velocity component in the y -direction ∂∂=

vy / 0 . The constant x -ve-

locity component u means that ∂∂= ux / 0 as well and upon integration the problem

simplifies to = = . The cross-sectional area for Sect. 1 is twice that of

Sect. 3, which means the velocity in Sect. 3 is twice that of Sect. 1.
In Sect. 2 we see that the cross-sectional area reduces in size and the velocity
variation occurs in two dimensions. If we consider the y -velocity component above
the pipe centreline, it is flowing downwards when y > 0, while it points upwards
when y < 0 thus giving a negative velocity gradient in the y -direction,

m ρ Au ρ A u

11 3 3

∂ v

y

< 0

∂

this means that

∂
∂

u
x

> 0

i.e. that the velocity in the x -direction is increasing to satisfy mass conservation in
Eq. (5.7).

5.2.2 Momentum Conservation

The conservation of momentum is derived from Newton's second law of motion ,
Fma

∑ = , where ∑ is the sum of all forces acting on a control volume. Two
types of forces are considered, namely body forces and surface forces . We pres-
ent the momentum equation directly in its differential form. For an incompressible
flow, the force per unit volume is,

D U

ρ = FF + (5.8)

body surface

Dt

where the acceleration is defined as D Dt . This represents the rate of change of
the velocity vector of a moving fluid particle.
Body forces act over the entire volume which includes gravity, centrifugal, Co-
riolis and electromagnetic forces that act at a distance to the control volume. These
effects are usually incorporated by introducing them into the momentum equation
as an additional term to the contribution of the surface forces. In many cases the
only body force that is considered is the gravitational force, and therefore