Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 163

-∂ ( ρv ) /∂yδxδyδzδt and -∂ ( ρw ) /∂zδxδyδzδt , respectively. The sum of
these three terms must be equal to the increase of mass (increase of
density × volume) within rectangle, ( ∂ρ/∂t ) × δxδyδzδt . Therefore,
we obtain an equation of continuity as a law of conservation of
mass.

u

u u

S S S S
t

() () ( ) 0 u

x

u

v
y

u

w
z

(5.3)

u u u

The symbol “ ∇ ” means ∇ = (∂/ ∂x , ∂/ ∂y , ∂/∂ z ) and ∇⋅ v is inner
product of vector.
– u

¥
§ ¦ u u

u
x

u u

v
y

w
z

µ
· ¶

v (5.4)

u

We apply substantial derivative symbol of operation to above
expression and we have

D
Dt

S S –

v 0 (5.5)

When the irst term on the left-hand of Eq. (5.4) is equal to zero,

D
Dt

S S S

y u

u u u u

S

u 0

u v w (5.6)

x y z

The working luid is considered an incompressible luid. If the
luid is not compressible, substantial derivative of ρ is zero.

u

u u

SS S
xyz 0

u u

u

(5.7)

Therefore, we obtain
– v 0 ( or div v 0 ) (5.8)

5.1.3 Law of Conservation of Momentum

Dynamics of mass point and rigid body is based on Newton's second
law.
F = Mα (5.9)
where F is force, M mass and α acceleration. Although luid dynamics
is always discussed on the basis of this law, unlike mass point and
rigid body, luid is sort of ininite form. Therefore, we need to change
this law in the way of luid dynamics. Let us suppose a rectangular
area with plane perpendicular to every coordinate axis (center point:
x , y , z , length of side: δx , δy , δz ). We apply Eq. (5.9) to this area.