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- The Bernoulli equation for incompressible fluids can be derived by either integrating Newton's second law of motion or by applying the law of conservation of energy, ignoring viscosity, compressibility, and thermal effects.
; Derivation by integrating Newton's second law of motion
The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the axis be directed down the axis of the pipe.
Define a parcel of fluid moving through a pipe with cross-sectional area , the length of the parcel is , and the volume of the parcel . If mass density is , the mass of the parcel is density multiplied by its volume . The change in pressure over distance is and flow velocity .
Apply Newton's second law of motion and recognizing that the effective force on the parcel of fluid is . If the pressure decreases along the length of the pipe, is negative but the force resulting in flow is positive along the axis.
In steady flow the velocity field is constant with respect to time, , so itself is not directly a function of time . It is only when the parcel moves through that the cross sectional area changes: depends on only through the cross-sectional position .
With density constant, the equation of motion can be written as
by integrating with respect to
where is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa.
In the above derivation, no external work–energy principle is invoked. Rather, Bernoulli's principle was derived by a simple manipulation of Newton's second law.
thumb|center|600px|A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed, distance (), and cross-sectional area. Note that in this figure elevation is denoted as , contrary to the text where it is given by .
; Derivation by using conservation of energy
Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy. In the form of the work-energy theorem, stating that
Therefore,
The system consists of the volume of fluid, initially between the cross-sections and . In the time interval fluid elements initially at the inflow cross-section move over a distance , while at the outflow cross-section the fluid moves away from cross-section over a distance . The displaced fluid volumes at the inflow and outflow are respectively and . The associated displaced fluid masses are – when is the fluid's mass density – equal to density times volume, so and . By mass conservation, these two masses displaced in the time interval have to be equal, and this displaced mass is denoted by :
The work done by the forces consists of two parts:
* The work done by the pressure acting on the areas and
* The work done by gravity: the gravitational potential energy in the volume is lost, and at the outflow in the volume is gained. So, the change in gravitational potential energy in the time interval is
Now, the work by the force of gravity is opposite to the change in potential energy, : while the force of gravity is in the negative -direction, the work—gravity force times change in elevation—will be negative for a positive elevation change , while the corresponding potential energy change is positive. So:
And therefore the total work done in this time interval is
The increase in kinetic energy is
Putting these together, the work-kinetic energy theorem gives:
or
After dividing by the mass the result is:
or, as stated in the first paragraph:
Further division by produces the following equation. Note that each term can be described in the length dimension . This is the head equation derived from Bernoulli's principle:
The middle term, , represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, is called the elevation head and given the designation .
A free falling mass from an elevation will reach a speed
when arriving at elevation . Or when rearranged as head:
The term is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion.
The hydrostatic pressure p is defined as
with some reference pressure, or when rearranged as head:
The term is also called the pressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. The head due to the flow speed and the head due to static pressure combined with the elevation above a reference plane, a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head is obtained.
If Eqn. 1 is multiplied by the density of the fluid, an equation with three pressure terms is obtained:
Note that the pressure of the system is constant in this form of the Bernoulli equation. If the static pressure of the system increases, and if the pressure due to elevation is constant, then the dynamic pressure must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, it must be due to an increase in the static pressure that is resisting the flow.
All three equations are merely simplified versions of an energy balance on a system. (en)
- The derivation for compressible fluids is similar. Again, the derivation depends upon conservation of mass, and conservation of energy. Conservation of mass implies that in the above figure, in the interval of time , the amount of mass passing through the boundary defined by the area is equal to the amount of mass passing outwards through the boundary defined by the area :
Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume
of the streamtube bounded by and is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero,
where and are the energy entering through and leaving through , respectively. The energy entering through is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic internal energy per unit of mass entering, and the energy entering in the form of mechanical work:
where is a force potential due to the Earth's gravity, is acceleration due to gravity, and is elevation above a reference plane. A similar expression for may easily be constructed.
So now setting :
which can be rewritten as:
Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain
which is the Bernoulli equation for compressible flow.
An equivalent expression can be written in terms of fluid enthalpy : (en)
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