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 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.
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.
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:
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's 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:
misinterpretations of bernoulli's equation
the coanda effect