Fluid dynamics - The theory of fluids and how they respond to forces upon them

Applications are astrophysics, geology, medicine, meteorology, and hydraulic engineering. Fluid dynamics is a field of classical physics and a subset of it is hydromechanics, dealing with the fluid water. Another sub-field is aerodynamics deals with gases, most commonly air. One of the remaining unsolved problems of physics, is within the fluid dynamics field; turbulence. For more on the definition of fluids, read Professor Pi's excellent fluid

The first fluid to be scientifically examined was of course water. Most of this concerned water in rest, see Archimedes. Problems that were solved had to do with everyday life, such as irrigation, aqueducts and canals. The purely theoretical studies of fluids were for a long time based upon the four elements introduced by Aristotle, and needless to say, incorrect.  Leonardo da Vinci was one of the first to study water with fresh eyes, and he was among the first to note effects of what is now called continuity. He discovered that the velocity of a flow is inversely proportional to the cross section of the pipe it runs in. After da Vinci's death, Dutch engineer Simon Stevin made important contributions to the study of water, as did Galileo and his students. Soon followed a stream of scientists working with hydraulics; Blaise Pascal, Renée Descartes and finally also Sir Isaac Newton. He looked on water as particles and studied their interactions. 

The first real breakthrough however came when Swiss Johann Bernoulli, his son Daniel Bernoulli and Daniel's friend Leonhard Euler came around, in the 18th century. They published different works that they had come up with together, and they weren't always willing to share the glory with each other. They founded their works on von Leibnitz work on energy conservation, and Euler came up with what today ironically is called Bernoulli's equation or Bernoulli's theorem. Euler's work was further developed by Louis Navier and George Stokes in the 19th century, who included inner friction and effects of viscosity in the equations. 

Still, there were glitches in the theories and one example of this was d'Alembert's paradox. This applied the current theories on a ball in a fluid, and indicated that the ball would move without friction through the fluid, which totally contradicts common knowledge. This was resolved by Ludwig Prandtl who early in the 20th century introduced the concepts of separation and boundary layer, where turbulence occurs. Today, fluid mechanics is almost exclusively an computational science and various numerical methods are used to find approximate solutions to the fundamental equations. 

Important concepts
Essential for technical applications of fluid dynamics is the understanding of the non-linear partial differential equations that are central in this science. Those are 

div(ρv) +  --  =  0
    p         η                    Dv    ∂v               
- ∇(- + gz) -(-) · ∇ X (∇ X v)  =  --  = -- + (v · ∇)v	
    ρ         ρ                    Dt    ∂t                    


Here, ρ is the density and v is the velocity field, p is the pressure and the viscosity, g is the gravitational constant and z is distance.  The term (∇ X v) is also known as vorticity.  

Another important concept is the dimensionless number Reynolds Number, Re. This is important because when the fluid can be approximated as incompressible, the motion of the fluid is dependent on the Reynolds number. If the Re >> 1, then the viscous effects are much smaller than the effects of inertia that affects the fluid. This is usually the case for water in the sink or for ships and airplanes. If Re << 1, then the viscous forces are dominant, as is the case for lubricant oils for instance. 

In the case of large Re, the velocity fields of the fluid in most real cases have a velocity potential, which makes calculations of the pressure and velocity much easier. This is referred to as potential flow, which is an area with well-developed numerical tools. The potential flow theory is very sensitive to turbulence, and therefore the above mentioned boundary layer is important. All bodies in fluids are surrounded by boundary layers. Outside of this, we have potential flow with no turbulence or compression. Inside the boundary layer, we have large effects from viscosity due to adhesion of the fluid to the body. In the case of potential flow, a good way to visualize the flow pattern are streamlines. These are lines in the flow with equal velocity, and these are waht you usually see in schematical drawings of the flow around aircraft wings, for instance. Streamlines are always parallel to the boundary of an object in the steam.

Depending on the Re number, a body in a fluid may invoke turbulence. This is because of separation of the boundary layer, which basically means that the boundary layer is released from the body, and expands into an eddy behind the body. This is fundamental in aerodynamics in the design of wings, for instance. If the boundary layer is still attached closely to the body and no turbulence occurs, the fluid is said to be laminar. The boundary layer and the Reynolds number is therefore important when designing golf balls, see Why are Golf Balls Dimpled?.


Reference: ne.se Britannica

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