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Fall Apart

For the Navier-Stokes theory in the case of the cochlear fluid dynamics I can point to the following elucidating introductory explanation placed on-line on the Internet by the Academic Medical Center of Amsterdam that I cite here:

-- The Navier-Stokes equations are a set of equations that describe the motion of fluids (liquids and gases, and even solids of geological sizes and time-scales). These equations establish that changes in momentum (acceleration) of the particles of a fluid are simply the product of changes in pressure and dissipative viscous forces (friction) acting inside the fluid. These viscous forces originate in molecular interactions and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of the fluid.

They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They are useful to model weather, ocean currents (climate), water flow in a pipe, motion of stars inside a galaxy, flow around a wing of an aircraft. They are also used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc.

The Navier-Stokes equations are partial differential equations which describe the motion of a fluid, so they focus on the rates of change or fluxes of these quantities. In mathematical terms these rates correspond to their derivatives. Thus, the Navier-Stokes for the simplest case of an ideal fluid (i.e. incompressible) with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure. Poiseuille’s Law and Bernoulli’s equation are special cases of 1D Navier-Stokes.

The fluid motion is described in 3-D space, and densities and viscosities may be different for the 3 dimensions, may vary in space and time. Since the flow can be laminar as well as turbulent, the mathematics to describe the system is highly complex.

In practice only the simplest cases can be solved and their exact solution is known. These cases often involve non turbulent flow in steady state (flow does not change with time) in which the viscosity of the fluid is large or its velocity is small (small Reynolds number).

For more complex situations, solution of the Navier-Stokes equations must be found with the help of numeric computer models, here called computational fluid dynamics.

Even though turbulence is an everyday experience it is extremely hard to find solutions for this class of problems. Often analytic solutions cannot be found.

Reducing the models to 1D, as is often done in fluid dynamics of blood vessels, makes the problem handsome. --

[See also: edited by N. A. M. Schellart 2005]

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