Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 162

5.1.2 Law of Conservation of Mass

We suppose an imaginary minimal rectangular area centered around
the point ( x , y , z ). We consider ρu as a parameter and Taylor series
of above equation around the point ( x , y , z ) is obtained. Mass of luid
going through two faces perpendicular to x axis and entering and
leaving inspection area between the time t and t + δt is

E E

¥
§ ¦

x

µ
· ¶ –

¥
§ ¦

x

µ
· ¶

S

x , yzt u x , , , yzt , ,

EEE

y zt

2 2

(5.1)

u

S EE

() ()

u
x

¥
§ ¦

x

µ
· ¶ ,,,

S

u

– EEE

yzt

u

2

xyzt

and then, as the luid low out from the face on the right-hand
perpendicular to x axis, we reverse the direction of the Eq. (5.1) as
follows:

¥
§ ¦

E

x

µ
· ¶ –

¥
§ ¦

E

x

µ
· ¶

S EEE

x , yzt u x , , , yzt , , y zt

2 2

(5.2)

u

S

–– ¥

E µ

· ¶

() u

x

x

()

S

u

EEE

yzt

§ ¦

u

2

xyzt ,,,

y

1
2

( v )
y

( v + y ) x t

1
2

( u )
x

( u x ) y t

y

1
2

( u )
x

( u + x ) y t

x

1
2

( u )
y

( u y ) x t

x

Figure 5.2 Fluid low of rectangular area.

From the sum of Eqs. (5.1) and (5.2), the increase of mass from
the face perpendicular to x axis is -∂ ( ρu ) /∂xδxδyδzδt . The increase
of mass through the face of cuboid perpendicular y axis and z axis is