*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.

*History*

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

∂ρ
div(ρv) + -- = 0
∂t

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*