Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 24

and shear stress ( Equation 2.13 ), which is an extension of Equation 2.6 . Using this differen-
tial element and particular coordinate system, the stress tensor can be written as

2
4

3
5

σ xx τ xy τ xz

σ 5 τ xy σ yy τ yz ð : Þ

2 13

τ xz τ yz σ zz

which has been reduced to its six independent values. As discussed previously, the shear
stress values on different adjoining faces on a differential element must be equal so that
momentum is conserved across the element. There is a sign convention for stress values. If
the force acts in the positive direction on a positive face or a negative direction on a nega-
tive face, then the stress value is positive. If the force acts in a negative direction on a posi-
tive face or a positive direction on a negative face, then the stress is negative. All stresses
drawn on Figure 2.14 are positive stresses. The stresses represented as dashed lines, are
negative stresses acting on the negative faces. The stresses represented as solid lines are
positive stresses acting on the positive faces.
Two different types of forces can account for stresses within the stress tensor: surface
forces and body forces. Surface forces act directly on the boundaries of our volume of
interest, not internally within the volume. These forces also act by direct contact of the
force with our fluid element. These forces are normally constant (they can vary with time)
at the point of contact. Contrary to this, body forces are developed without direct contact
with the fluid element. These forces are distributed throughout the entire fluid and are
normally variable depending on where you look within the volume of interest or even
within the same fluid element. Examples of surface forces are the shear force induced by
the fluid flow of a neighboring volume (i.e., neighboring fluid lamina) or the mechanical
force that the elastic blood vessel applies onto the fluid. An example of a body force is the
force due to gravity, which is dependent on the spatial location within the fluid.
Pressure has many functions in fluid mechanics and is a major constituent of forces that
arises in the stress tensor. First, it can be the driving force for fluid motion. This normally
is discussed as the inflow pressure head or the pressure gradient (change in pressure)
across the volume of interest. However, at rest, fluids still exert an internal pressure. This
is the hydrostatic pressure, which is defined partially by the weight of the fluid above a par-
ticular plane within the fluid. This is an example of a body force because it varies depend-
ing on the location of the plane within the fluid. For instance, the pressure on a free-fluid
surface (open to any atmosphere) must be equal to the atmospheric pressure at the free
surface. However, a differential element of fluid below a free surface must be able to sup-
port the weight of the fluid above it as well as the atmospheric pressure. To think of this,
consider walking around your campus grounds with 1 atmosphere of pressure on your
shoulders. If, however, you are carrying a heavy topic bag while walking to your biofluid
mechanics final examination, it might feel like the weight of the world is on your
shoulders. However, in fact, the only added weight is that of the topic bag; you still are
supporting the weight of the 1 atmosphere of pressure. Also, the pressure from the topic
bag is not added to your head (although again the final examination might seem weighty),
but it is added to everything from your shoulders down (even the floor you are standing
on). One useful relationship between stress and pressure is that, the hydrostatic pressure