98
4 Fluid Mechanics Applied to Biosystems
temperatures, pressures, etc. across the fluid, our solutions grow in uncertainty.
This makes long-time predictions difficult, both for meteorologists in predicting the
weather, and for fluid biomechanics in predicting how a bumble bee flies. 13
These days, we resort to computers to follow the development of the solutions.
When chaos applies, the initial conditions would have to be known and the solutions
would have to be tracked with extreme precision to be certain about how the system
might evolve far into the future. As our computers have a finite memory capacity,
we cannot store all the initial conditions nor follow solutions with infinite precision.
With finite precision, we are forced to truncate answers at each iterative step of a
calculation. So we lose more information about the solutions the more we iterate,
even with perfectly known initial conditions. 14 We will return to the topic of chaos
in Sect. (13.2.2).
Do not despair! Chaos is organized. 15 And, under the right conditions, some
solutions of the Navier–Stokes equation are relatively simple. Also, we know
of general conditions which the Navier–Stokes equations must satisfy, such as
local energy, linear momentum, angular momentum, and mass conservation, which
confines the kinds of solutions and their future behavior, chaotic or not.
4.3.7 Scaling Law for Fluid Flows
Notice that the Navier–Stokes equation (4.43) for an incompressible fluid has one
parameter which depends on the intrinsic fluid properties, η/ρ, the specific viscosity,
and an implicit constitutive equation giving how the pressure varies throughout the
fluid.
Now consider the problem of finding the fluid motion around a stationary object
of given shape in an incompressible fluid with speed v f far from the object. The
scale of size of this object is fixed by a single length, say L. For a given pressure
distribution, the solutions for the fluid motion can only depend on three physical
quantities, η/ρ, v f , and L. These three numbers determine a unitless quantity, the
Reynolds number:
13 In Sect. 13.2, we will see that models of population growth and other physical models also can
show chaotic behavior.
14 Of course, we are assuming that Nature is smooth across space and time when we use a
differential equation to describe her. If she were discrete (!), we might have difference equations
instead of differential equations, and these difference equations might involve only rational
numbers. Rational numbers which fit into available memory can be handled by computers without
truncation loss, and, for such discrete equations, computers might be able to track solutions without
errors, limited only by memory capacity. There is still a fundamental limitation: Tracking the whole
universe would require a computer at least as big as the universe, which is a contradiction, as this
computer must be in our universe!
15 Not to be mixed up with ‘organized confusion’.
4 Fluid Mechanics Applied to Biosystems
temperatures, pressures, etc. across the fluid, our solutions grow in uncertainty.
This makes long-time predictions difficult, both for meteorologists in predicting the
weather, and for fluid biomechanics in predicting how a bumble bee flies. 13
These days, we resort to computers to follow the development of the solutions.
When chaos applies, the initial conditions would have to be known and the solutions
would have to be tracked with extreme precision to be certain about how the system
might evolve far into the future. As our computers have a finite memory capacity,
we cannot store all the initial conditions nor follow solutions with infinite precision.
With finite precision, we are forced to truncate answers at each iterative step of a
calculation. So we lose more information about the solutions the more we iterate,
even with perfectly known initial conditions. 14 We will return to the topic of chaos
in Sect. (13.2.2).
Do not despair! Chaos is organized. 15 And, under the right conditions, some
solutions of the Navier–Stokes equation are relatively simple. Also, we know
of general conditions which the Navier–Stokes equations must satisfy, such as
local energy, linear momentum, angular momentum, and mass conservation, which
confines the kinds of solutions and their future behavior, chaotic or not.
4.3.7 Scaling Law for Fluid Flows
Notice that the Navier–Stokes equation (4.43) for an incompressible fluid has one
parameter which depends on the intrinsic fluid properties, η/ρ, the specific viscosity,
and an implicit constitutive equation giving how the pressure varies throughout the
fluid.
Now consider the problem of finding the fluid motion around a stationary object
of given shape in an incompressible fluid with speed v f far from the object. The
scale of size of this object is fixed by a single length, say L. For a given pressure
distribution, the solutions for the fluid motion can only depend on three physical
quantities, η/ρ, v f , and L. These three numbers determine a unitless quantity, the
Reynolds number:
13 In Sect. 13.2, we will see that models of population growth and other physical models also can
show chaotic behavior.
14 Of course, we are assuming that Nature is smooth across space and time when we use a
differential equation to describe her. If she were discrete (!), we might have difference equations
instead of differential equations, and these difference equations might involve only rational
numbers. Rational numbers which fit into available memory can be handled by computers without
truncation loss, and, for such discrete equations, computers might be able to track solutions without
errors, limited only by memory capacity. There is still a fundamental limitation: Tracking the whole
universe would require a computer at least as big as the universe, which is a contradiction, as this
computer must be in our universe!
15 Not to be mixed up with ‘organized confusion’.