4.3 Fluid Dynamics
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equations of state for systems in equilibrium. We also need to know how the energy
is transported across its boundaries, and how material is carried across boundaries.
Some marvelous properties of fluids in motion are predicted by the Navier–
Stokes Eq. (4.43). The creation of vortices and turbulence comes about mathematically by the effect of the innocent-looking second term on the left. You will notice
that all the other terms are either first power in the speed v or free of the speed. The
second term, having v·∇v, is second power in the speed. This makes the differential
equation non-linear in the unknown velocity. In such systems, one can interpret the
non-linear term as the system acting back on itself.
Non-linear differential equations are often notoriously difficult to solve. But the
solutions of such systems are also often fascinating. In fact, some people can spend
hours watching a babbling brook. Babbling brooks are one class of solutions of
the Navier–Stokes equation. Hurricanes are another. People watch them also, but
usually with the motive to learn how to avoid them. 9
4.3.6 Chaotic Evolution of Fluids
Non-linear systems can show chaos. The idea behind chaotic systems was first
described by Poincaré in the early 1890s in his work on the celestial three-body
problem. 10 The idea was revived in the early 1960s by Edward Lorenz 11 in his
attempts to predict the weather using none other than the Navier–Stokes equation,
with the atmosphere as the fluid. Lorenz found that his computed solutions could
differ widely when the initial conditions were changed only infinitesimally. He
allowed his observation to be called the “butterfly effect”. The flapping of the wings
of a butterfly in Brazil could make the difference in whether a tornado occurs
in Texas months later. In general, ‘chaotic systems’ have the property that two
solutions which are arbitrarily close to one another at one time may diverge from
each other exponentially at later times. 12
In principle, the solutions of Navier–Stokes equation are completely determined
by the initial conditions. If we had exact knowledge of initial states and exact
solutions, we could make exact predictions. But if we do not know the exact initial
9 I was once on a hiking trip near Boulder, Colorado, with a group of theoretical physicists. One
half-jokingly said, after looking over at a distant landscape, “That is almost as beautiful as a
Wightman function.” A friend in the group responded that if he expressed further audacities in
the face of the gods, his ‘Wightman’ may no longer function.
10 Henri Poincaré, Les mèthods nouvelles de la mècanique cèleste, [Paris: Gauthier-Villars] Vol I
(1892), Vol II (1893), Vol III (1899).
11 Edward N. Lorenz, Deterministic non-periodic flow, J Atmos Sci 20, 130–141 (1963).
12 Mathematically, this is expressed as follows. Suppose a dynamical system depends on a
parameter r. Express the solutions of the system as y(t, r). Let δr be a small change in the
parameter r. If the solutions diverges in time as [y(t, r + δr) − y(t, r)] ∝ exp (λt), the system
is exhibiting chaotic behavior. The constant λ is referred to as the Lyapunov exponent.
97
equations of state for systems in equilibrium. We also need to know how the energy
is transported across its boundaries, and how material is carried across boundaries.
Some marvelous properties of fluids in motion are predicted by the Navier–
Stokes Eq. (4.43). The creation of vortices and turbulence comes about mathematically by the effect of the innocent-looking second term on the left. You will notice
that all the other terms are either first power in the speed v or free of the speed. The
second term, having v·∇v, is second power in the speed. This makes the differential
equation non-linear in the unknown velocity. In such systems, one can interpret the
non-linear term as the system acting back on itself.
Non-linear differential equations are often notoriously difficult to solve. But the
solutions of such systems are also often fascinating. In fact, some people can spend
hours watching a babbling brook. Babbling brooks are one class of solutions of
the Navier–Stokes equation. Hurricanes are another. People watch them also, but
usually with the motive to learn how to avoid them. 9
4.3.6 Chaotic Evolution of Fluids
Non-linear systems can show chaos. The idea behind chaotic systems was first
described by Poincaré in the early 1890s in his work on the celestial three-body
problem. 10 The idea was revived in the early 1960s by Edward Lorenz 11 in his
attempts to predict the weather using none other than the Navier–Stokes equation,
with the atmosphere as the fluid. Lorenz found that his computed solutions could
differ widely when the initial conditions were changed only infinitesimally. He
allowed his observation to be called the “butterfly effect”. The flapping of the wings
of a butterfly in Brazil could make the difference in whether a tornado occurs
in Texas months later. In general, ‘chaotic systems’ have the property that two
solutions which are arbitrarily close to one another at one time may diverge from
each other exponentially at later times. 12
In principle, the solutions of Navier–Stokes equation are completely determined
by the initial conditions. If we had exact knowledge of initial states and exact
solutions, we could make exact predictions. But if we do not know the exact initial
9 I was once on a hiking trip near Boulder, Colorado, with a group of theoretical physicists. One
half-jokingly said, after looking over at a distant landscape, “That is almost as beautiful as a
Wightman function.” A friend in the group responded that if he expressed further audacities in
the face of the gods, his ‘Wightman’ may no longer function.
10 Henri Poincaré, Les mèthods nouvelles de la mècanique cèleste, [Paris: Gauthier-Villars] Vol I
(1892), Vol II (1893), Vol III (1899).
11 Edward N. Lorenz, Deterministic non-periodic flow, J Atmos Sci 20, 130–141 (1963).
12 Mathematically, this is expressed as follows. Suppose a dynamical system depends on a
parameter r. Express the solutions of the system as y(t, r). Let δr be a small change in the
parameter r. If the solutions diverges in time as [y(t, r + δr) − y(t, r)] ∝ exp (λt), the system
is exhibiting chaotic behavior. The constant λ is referred to as the Lyapunov exponent.