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Visiting Bernoulli
Bernoulli's equation
is one of the most famous equations of fluid mechanics and is
applied to a lot of different situations. Unfortunately, it is
also applied in situations where it does not belong. Give some
people a nice and clean equation and they want to stretch it beyond
its limits. Bernoulli's equation is basically defining a relationship
between the pressure and the fluid velocity. The velocity is an
unambiguious concept but pressure is a bit confusing, specially
when people talk about static pressure and stagnation pressure,
etc. So, why not start with some clarification on the meaning
of pressure before diving into explaining the Bernoulli's equation.
Static Pressure in
a Fluid
Let's start with a
simple case of pressure in a static fluid. According to most textbooks,
pressure, P, at a point on a plane surface (inside the fluid or
on the boundaries of its container), is defined as the limiting
value of the ratio of normal force to surface area as the area
approaches zero size. This pressure (or force per unit area) always
acts normal to the surface.
Pascal's Law
Pascal's Law states
that the pressure at a point in a fluid at rest is the same in
all directions.
Variation of static
pressure with depth
For an incompressible
fluid pressure at a point inside a column of fluid may be expressed
as:
It is convenient and
customary to use the gauge pressure instead of absolute pressure.
Gauge pressure is the pressure measured above or below the atmospheric
pressure. In terms of gauge pressure the pressure at depth h is
obtained from
Measuring static pressure
The measurement of
static pressure may be done with a simple device called a manometer.
In its simplest configuration, a manometer is a U-shaped tube.
One side of the tube is connected to the atmosphere and the other
side is places inside the fluid at rest for which a measurement
is sought.
Dynamic Pressure
Judging from its name
this pressure must have something to do with motion and it does.
It is defined as:
There are two reasons
we call this pressure. The first one is the fact that it has the
units of pressure but that is not a very strong case. As you will
see later, when we actually get into the Bernoulli equation, this
pressure plus the static pressure is a constant quantity in a
moving fluid (neglecting the gravitational body force). More on
this later.
Now, let's derive the
Bernoulli's equation.
Bernoulli's Equation
Consider a control
volume in a flow with curved streamlines:
Using Newton's second
law of motion we can study the acceleration of this volume:
The streaming fluid
accelerates as a result of decreasing pressure (i.e. or a negative
pressure gradient). This derivation clearly shows that an acceleration
can never be the cause of decreasing pressure. It is the other
way around.
Rewriting this equation
gives us a very popular form of the Bernoulli's equation:
Consider an inviscid
(non-viscous) flow around a blunt body:
The streamline going
through point 1 is brought to rest at point 0; hence V0
=0. This point of zero velocity is called the Stagnation
Point and the pressure at point zero is called the Stagnation
Pressure or Total Pressure:
Therefore:
For an inviscid
incompressible fluid with negligible body force the stagnation
pressure is equivalent to the sum of the dynamic pressure and
the static pressure, and is constant throughout the fluid.
Please note that we
have put 3 important conditions on this statement:
- The flow must be
inviscid
- The Flow must be
incompressible
- Negligible body
force
Bernoulli equation
can be used to study flow from a tank through a small orifice,
trajectory of a free jet, flow distribution in a closed channel,
etc. People have used it to explain other physical situations.
Read the following articles for two interesting discussions:
http://www.informatik.uni-frankfurt.de/~plass/MIS/mis6.html
http://www.cfcl.com/jef/coanda_effect.html
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