Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 168

of γ xy = ∂v/∂x + ∂u/∂y and regarded as 2μ∂ u /∂ x . Also, an effect of
distortion (term of volume change) by pressure is added.

u
2
u

u

T N M

-2 p (5.28)

xx

x

u
2
u

v

T N M

-2 p

(5.29)

yy

y

u
2
u

w

T N M

-2 p (5.30)

zz

z

Here, volume change Θ is
2 u

¥
§ ¦

u u u u

µ
· ¶

u
x

v
y

w
z

div v (5.31)

u

Θ

= 0. Although positive direction of
pressure p is set to the direction of the surface, positive direction
of stress is generally coincident with the direction of coordinate
axis and negative sign is added on right hand of Eq. (5.31). We sum
equations above and consider p = - ( σxx + σyy + σzz )/3.

In an incompressible luid

2
- 3

M – N

(5.32)

Here, μ and λ are irst viscosity coeficient and second viscosity
coeficient, respectively.
Now, we could make a correlation between stress deriving
from viscosity and motion of luid (deformation rate), we build this
relationship into motion equation. Here, we consider a force in x
direction. From Fig. (5.4), normal stress

−

σ xx dydz works on x -plane
and also σ xx + (( ∂σ xx /∂x ) ∂x ) dydz on ( x + dx )-plane and the balance of
them comes out (( ∂σ xx /∂x ) dx ) dydz . Next, stress in the direction of x
axis on y -plane is − τ yx dxdz and x component on ( y + dy )-plane is τ yx
+ (( ∂τ yx /∂y ) dy ) dxdz . Their difference is (( ∂τ yx /∂y ) dx ) dydz in the end.
The x component of stress on two faces perpendicular to the z axis
is similarly (( ∂τ zx /∂z ) dx ) dydz . Stress per unit mass in x axis on this
cube will be

¥
§ ¦

u

U µ

· ¶

u T

u
u

U

1
S

yx

xx zx

(5.33)

u u

x y z

A similar relation is obtained in other components. Equation 5.33
is added to Euler's equation, and motion equation for viscosity luid
is obtained. For example, as for the x component of the coordinate
axis,