Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet.
Mathematicians and Physicists believe that an explanation for and the prediction of both: the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the19th Century, our understanding of them remains minimal.
The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier–Stokes Equations.
The knowledge of fluid flow cannot be called a discipline of science until Isaac Newton published his famous work Philosophiae Naturalis Principia Mathematica. In 1687, Newton stated in hisprincipia that, for straight, parallel and uniform flow, the shear stress , between layers is proportional to the velocity gradient in the direction perpendicular to the layers.
With the development of calculus, many problems were solved in the frame of ideal fluid or inviscid fluid. In 1738, Bernoulli proved that, the gradient of pressure is proportional to the acceleration.
Later, the famousdifferential equations, known as Euler’s equations, were derived by Euler.
In 1752, d’Alembert used inviscid theory in the form of potential solution of the incompressible Euler equations, to prove that the drag of a body of any shape moving through an inviscid fluid is zero, which is known as d’Alembert’s paradox. The result was obviously in contradiction to an abundance of evidence of real world;hence mathematical fluid mechanics and engineering hydrodynamics were developed into separated branches.
Many researchers tried to add a friction term into Euler’s differential equations. Navier (1822), Cauchy (1828), Poisson (1829) and Saint-Venant (1843) suggested the function concerning the molecular mechanism. Stokes (1845) first used the absolute viscosity and assumed that,
1. The fluid iscontinuous and the stress tensor has a linear relation with strain tensor .
2. The fluid is isotropic. (the properties are independent of the direction and the frames concerned.
3. The form of the static-fluid pressure can be obtained as the strain rate is zero.
The total stress becomes
is called volume viscosity or dilatational viscosity.
Substitution in the equation of motion and Euler’sequation then gives
The Navier-Stokes equation of motion is finally obtained.
The mathematical history of fluid mechanics begins with Leonhard Euler who was invited by Frederick the Great to Potsdam in 1741. According to a popular story (which we have not been able to corroborate) one of his tasks was to engineer a water fountain. As a true theorist, he began bytrying to understand the laws of motion of fluids. In 1755 he wrote Newton's laws for a fluid which in modern notation reads (for the case of constant density).
Here and are the fluid velocity and pressure at the spatial point at time t. The LHS of this ``Euler equation" for is just the material time derivative of the momentum, and the RHS is the force, which is represented as the gradient of thepressure imposed on the fluid. In fact, trying to build a fountain on the basis of this equation was bound to fail. This equation predicts, for a given gradient of pressure, velocities that are much higher than anything observed. One missing idea was that of the viscous dissipation that is due to the friction of one parcel of fluid against neighboring ones. The appropriate term was added to (1) byNavier in 1827 and by Stokes in 1845. The result is known as the “Navier-Stokes equations”:
Here is the kinematic viscosity, which is about and for water and air at room temperature respectively. Without the term the kinetic energy is conserved; with this term kinetic energy is dissipated and turned into heat. The effect of this term is to stabilize and control the nonlinear energy...