During 17th
century, Daniel Bernoulli investigated the forces present in a moving fluid,
derived an equation and named it as an Bernoulli’s Equation. Below image shows
one of many forms of Bernoulli’s equation.
The Bernoulli equation gives an approximate equation that is valid only in
inviscid regions of flow where net viscous forces are negligibly small compared
to inertial, gravitational or pressure forces. Such regions occur outside of
boundary layers and waves
The Bernoulli Equation can be considered to be a statement
of the conservation of energy principle appropriate for flowing fluids. The
qualitative behavior that is usually labeled with the term "Bernoulli
effect" is the lowering of fluid pressure in regions where the flow
velocity is increased. This lowering of pressure in a constriction of a flow
path may seem counterintuitive, but seems less so when you consider pressure to
be energy density. In the high velocity flow through the constriction, kinetic
energy must increase at the expense of pressure energy.
Steady-state flow caveat: While the Bernoulli equation is stated in
terms of universally valid ideas like conservation of energy and the ideas of
pressure, kinetic energy and potential energy, its application in the above
form is limited to cases of steady flow. For flow through a tube, such flow can
be visualized as laminar flow, which is still an idealization, but if the flow
is to a good approximation laminar, then the kinetic energy of flow at any
point of the fluid can be modeled and calculated. The kinetic energy per unit
volume term in the equation is the one which requires strict constraints for
the Bernoulli equation to apply - it basically is the assumption that all the
kinetic energy of the fluid is contributing directly to the forward flow
process of the fluid. That should make it evident that the existence of
turbulence or any chaotic fluid motion would involve some kinetic energy which
is not contributing to the advancement of the fluid through the tube.
It should also be said that while conservation of energy always applies,
this form of parsing out that energy certainly does not describe how that
energy is distributed under transient conditions. A good visualization of the
Bernoulli effect is the flow through a constriction, but that neat picture does
not describe the fluid when you first turn on the flow.
Another approximation involved in the statement of the Bernoulli equation
above is the neglect of losses from fluid friction. Idealized laminar flow
through a pipe can be modeled by Poiseuille's law, which does include viscous
losses resulting in a lowering of the pressure as you progress along the pipe.
The statement of the Bernoulli equation above would lead to the expectation
that the pressure would return to the value P
1 past the constriction
since the radius returns to its original value. This is not the case because of
the loss of some energy from the active flow process by friction into
disordered molecular motion (thermal energy). More accurate modeling can be
done by combining the Bernoulli equation with Poiseuille's law. A real example
which might help visualize the process is the pressure monitoring of the flow
through a constricted tube.
Despite
its simplicity, Bernoulli’s Principle has proven to be a very powerful
tool in fluid mechanics.
Care must be taken when applying the Bernoulli equation since it is an
approximation that applies only to inviscid regions of flow. In general,
frictional effects are always important very close to solid walls and directly
downstream of bodies.
The motion of a particle and the path it follows are described by the velocity
vector as a function of time and space coordinates and the initial position of
the particle. When the flow is steady, all particles that pass through the same
point follow the same path and the velocity vectors remain tangent to the path
at every point.
During 17th
century, Daniel Bernoulli investigated the forces present in a moving
fluid, derived an equation and named it as an Bernoulli’s Equation.
Below image shows one of many forms of Bernoulli’s equation.
The
Bernoulli equation gives an approximate equation that is valid only in
inviscid regions of flow where net viscous forces are negligibly small
compared to inertial, gravitational or pressure forces. Such regions
occur outside of boundary layers and waves - See more at:
http://www.me-mechanicalengineering.com/bernoullis-principle-and-equation/#sthash.2lIUkDNX.dpuf