Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 166

These are Euler's equations of motion.

u

Du 1 p

F -

u

(5.19)

x

S

Dt x

u

Dv 1 p

F -

u

(5.20)

y

S

Dt y

u

Dw 1 p

F -

u

(5.21)

z

S

Dt z

We deine the symbol and the direction of stress. First, consider
unit surface perpendicular to the x axis. We set the normal line on
the face where force is applied and consider its direction of a face.
This stress applied to unit surface is divided into two forces. One is
normal stress at right angle to the face and another is shear stress
parallel to the face. Moreover, the latter is divided into the component
toward the y axis and the z axis and consists of three components
in the end. These three components are expressed as τ αβ . The irst
additional character α indicates the face what we are focusing on
and the second additional character indicates the force direction of
component. For example, three components of forces on the face
perpendicular to the x axis are τ xx τ xy τ xz ; τ xx is usually written as σ xx .
Force direction at a center point of faces must have three directions
of axis, such as x , y , z . Therefore, stress at the points in luid has nine
components:

¨ ¸

TU U
U TU
U U T

xx yx zx

©
©
©
©

¹
¹
¹
¹

P= (5.22)

xy yy zy

ª º

xz yz zz

We call this a stress tensor. The coordinate axis whose off-
diagonal component of stress tensor has zero is called principal axis.
Temperature and density at a point in luid are stated as quantities
without directional property. Velocity at a point that has amplitude
and direction is called vector. Stress tensor is diagonally tensor
whose diagonal component is identical. This means,
U U U U U U

, , (5.23)
Stress in luid comes from the result of relative movement of
luid. Stress tensor deined by Eq. (5.22) must be associated with
luid motion. Although the motion of the minimal element of the
luid can be divided into displacement (translation ⋅ rotation) and

xy yx yz zy zx xz