Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 21

about, the bulk viscosity can be a good approximation; however, it may be more realistic
to discuss the apparent viscosity. It is also true that viscosity is not only a function of
space, but also of time and temperature. Think about the oil in your car engine. On a cold
winter morning when you start your car the oil will initially be very viscous. However,
after the car has been running for 5 to 10 minutes, the oil heats up and the viscosity
reduces. Over that 5-minute period the viscosity decreased due to an increase in engine
temperature. Most biological fluids in humans see an approximate constant temperature
(approximately 37 C), but individual property values can change in time based on other
physical or mechanical properties. For instance, an increase in shear rate between particu-
lar joints can decrease the viscosity of the fluid within the joint (see Chapter 11).
Sometimes, temperature can be critical to a particular problem, such that if you were to
analyze the blood flow throughout your arm/hand when throwing snowballs barehanded
at your classmates, you might want to include temperature changes. Be cautioned that
temperature fluctuations can arise in biological problems, and these fluctuations can alter
the fluid properties.

2.5 ELEMENTAL STRESS AND PRESSURE

You may recall from engineering mechanics that there are a number of different ways
to define stress, including the Cauchy stress, the first Piola-Kirchhoff stress and the second
Piola-Kirchhoff stress. The most useful definition for stress in fluid mechanics is the
Cauchy stress ( ), which is a measure of all of the forces acting on a volume oriented in

space in the current configuration. This is in contrast to the Piola-Kirchhoff stresses, which
are related to a reference configuration, which may be different from the current configu-
ration. The reference configuration is normally a cube in space oriented along the assumed
Cartesian coordinate axes, whereas the current configuration could be that same cube
rotated to any degree off any of the three Cartesian axes. In general, we can resolve the
forces that act on a cube in three-dimensional space in terms of nine stress components,
otherwise known as the stress tensor ( Equation 2.6 ). The stress tensor is represented as a
3

σ

), with subscript entries representing first the face the force is acting upon
and second the direction that the force acts. As an example, the first stress component

3 3 matrix ( σ

σ xx
is the stress that acts on the x-face of the cube (the x-face is parallel to the yz-plane) and
acts in the x-direction. The stress

σ xz acts on the x-face in the z-direction. Only six of the
nine components in the stress tensor are independent because angular momentum must
be conserved on each differential volume element. The six independent values of stress
are σ xx , σ xy , σ xz , σ yy , σ yz and σ zz . To conserve angular momentum σ yx must be equal to σ xy ;

σ zx must be equal to σ xz ; and σ zy must be equal to σ yz .

2 3

σ σ σ

xx xy xz

4 5

σ5 σ σ σ ð 2 : 6 Þ

yx yy yz

σ σ σ

zx zy zz

σ ð face Þð direction Þ