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4 Fluid Mechanics Applied to Biosystems
a shearing force internally on a bounding surface. When you are at rest under
still water, the water can exert pressure on your skin, but will not push your skin
sideways.
As we have described, a colloid, i.e. a mixture containing nanoscale structures
having a significant surface area, can sometimes act as a solid, called a sol, and
other times as a liquid, called a gel. The transition between a sol and a gel depends
on temperature, pressure, and the rate at which a stress is imposed.
4.2 Laws for Fluids at Rest
A fluid may be sometimes modeled as a smoothed distribution of individual masses,
each one able to move relative to others. It should be no surprise to find that the
motion of fluids can be described by the same principles used to follow the behavior
of interacting masses. To do so successfully, however, the segments of the fluid
must be taken small enough that the velocities of components within these segments
do not deviate much from the average over all the components, and large enough
so that averages are meaningful. To use Newton’s laws, the segments must not
move relativistically, and their subparticle quantum waves must not significantly
extend beyond the molecular level, so that quantum dynamics is well approximated
by Newtonian dynamics for the fluid segments. For values of macroscopically
measurable quantities such as pressure within a given segment not to deviate much
from their average over the components of that segment, the number of molecules in
the components must be large. Moreover, these molecules must interact sufficiently
that the spatial averages within a segment approximate averages over short time
spans. 1
4.2.1 Pascal’s Law for Pressure in a Fluid
Those of us who have dived into deep water likely know that water pressure
increases with the depth of the water, based on how our ears react. As we go deeper,
we experience pain in our middle ear from the pressure on our eardrum. We also
may have observed that the extra pressure is felt irrespective of the angle of our
head. A pressure gauge rotated under water reads the same value.
In 1646, Blaise Pascal investigated how pressure acts in fluids, and found that
pressure at any one location in a fluid was independent of the orientation of the
surface used to measure that pressure, and that if the pressure in a fluid is changed,
that change is felt throughout the fluid by the time the fluid returns to rest. This
discovery, known as Pascal’s principle, is the result of the physical properties of a
1 These ‘short’ time spans are the ones used to define time derivatives of the measurable quantities.
4 Fluid Mechanics Applied to Biosystems
a shearing force internally on a bounding surface. When you are at rest under
still water, the water can exert pressure on your skin, but will not push your skin
sideways.
As we have described, a colloid, i.e. a mixture containing nanoscale structures
having a significant surface area, can sometimes act as a solid, called a sol, and
other times as a liquid, called a gel. The transition between a sol and a gel depends
on temperature, pressure, and the rate at which a stress is imposed.
4.2 Laws for Fluids at Rest
A fluid may be sometimes modeled as a smoothed distribution of individual masses,
each one able to move relative to others. It should be no surprise to find that the
motion of fluids can be described by the same principles used to follow the behavior
of interacting masses. To do so successfully, however, the segments of the fluid
must be taken small enough that the velocities of components within these segments
do not deviate much from the average over all the components, and large enough
so that averages are meaningful. To use Newton’s laws, the segments must not
move relativistically, and their subparticle quantum waves must not significantly
extend beyond the molecular level, so that quantum dynamics is well approximated
by Newtonian dynamics for the fluid segments. For values of macroscopically
measurable quantities such as pressure within a given segment not to deviate much
from their average over the components of that segment, the number of molecules in
the components must be large. Moreover, these molecules must interact sufficiently
that the spatial averages within a segment approximate averages over short time
spans. 1
4.2.1 Pascal’s Law for Pressure in a Fluid
Those of us who have dived into deep water likely know that water pressure
increases with the depth of the water, based on how our ears react. As we go deeper,
we experience pain in our middle ear from the pressure on our eardrum. We also
may have observed that the extra pressure is felt irrespective of the angle of our
head. A pressure gauge rotated under water reads the same value.
In 1646, Blaise Pascal investigated how pressure acts in fluids, and found that
pressure at any one location in a fluid was independent of the orientation of the
surface used to measure that pressure, and that if the pressure in a fluid is changed,
that change is felt throughout the fluid by the time the fluid returns to rest. This
discovery, known as Pascal’s principle, is the result of the physical properties of a
1 These ‘short’ time spans are the ones used to define time derivatives of the measurable quantities.