Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 23

dF -
dA -

τ n 5 ð 2 : 10 Þ

To relate these equations to the stress tensor ( Equation 2.6 ), you must define a coordinate
system and define how the areas and forces relate to this system. In the most simplistic
form, when the axes are directly aligned with the fluid element, the normal stress σ xx can

be represented as Equation 2.11 from Equation 2.9 and the shear stress σ xy can be repre-

sented as Equation 2.12 from Equation 2.10 (this is for a simplified two-dimensional case):

dF -
dA -

σ 5 ð 2 : 11 Þ

xx

dF -
dA -

τ 5σ 5 ð 2 : 12 Þ

xy xy

For our analysis, we will normally use an orthogonal coordinate system. In Cartesian
coordinates, this is the standard x, y and z directions, which you should be familiar with.
When we begin to discuss flow through blood vessels, we will begin to adopt the cylindri-
cal coordinate system for ease. Although, if you are asked to solve a problem involving
two-dimensional blood flow through a vessel, you may choose to solve the problem in
rectangular coordinates, if you assume that the flow is symmetrical about the centerline
(although this would mimic a rectangular channel solution instead of a circular cylinder).
If you consider a differential element in three dimensions, we can see how the compo-
nents of the stress tensor fall on this element ( Figure 2.14 ) giving us the standard nomen-
clature used within the stress tensor. The components take the form of normal stresses

FIGURE 2.14

Y The nine components of the

stress tensor on a differential fluid volume.
For momentum balance, only six of these values
are independent.

σ YY

τ YX

τ YZ

σ ZZ

τ XY

τ ZX

τ XZ

σ XX

τ ZY

τ ZY

σ XX

τ XZ

τ ZX

τ XY

σ ZZ

X

τ YZ

τ YX

σ YY

Z