Bernoulli's equation

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See Bernoulli differential equation for an unrelated topic in ordinary differential equations.

In fluid dynamics, Bernoulli's equation describes the behavior of a fluid moving along a streamline. There are typically two different formulations of the equations; one applies to incompressible flow and the other applies to compressible flow.

The original form, for incompressible flow in a uniform gravitational field (such as on Earth), is:

<math> {v^2 \over 2}+gh+{p \over \rho}=\mathrm{constant} </math>

v = fluid velocity along the streamline

g = acceleration due to gravity on Earth

h = height from an arbitrary point in the direction of gravity

p = pressure along the streamline

<math>\rho</math> = fluid density

These assumptions must be met for the equation to apply:

Inviscid flow − viscosity (internal friction) = 0
Steady flow
Incompressible flow − <math>\rho</math> = constant along a streamline. Density may vary from streamline to streamline, however.
Generally, the equation applies along a streamline. For constant-density potential flow, it applies throughout the entire flow field.

The decrease in pressure simultaneous with an increase in velocity, as predicted by the equation, is often called Bernoulli's principle.

The equation is named for Daniel Bernoulli although it was first presented in the above form by Leonhard Euler.

A second, more general form of Bernoulli's equation may be written for compressible fluids, in which case, following a streamline:

<math> {v^2 \over 2}+ \phi + w =\mathrm{constant} </math>

Here, <math>\phi</math> is the gravitational potential energy per unit mass, which is just <math> \phi = gh </math> in the case of a uniform gravitational field, and <math> w </math> is the fluid enthalpy per unit mass, which is also often written as <math> h </math> (which conflicts with the use of <math> h</math> in this article for "height"). Note that <math> w = \epsilon + \frac{p}{\rho} </math> where <math> \epsilon </math> is the fluid thermodynamic energy per unit mass, also known as the specific internal energy or "sie".

The constant on the right hand side is often called the Bernoulli constant and denoted <math> b </math>. For steady inviscid adiabatic flow with no additional sources or sinks of energy, <math> b </math> is constant along any given streamline. More generally, when <math> b </math> may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When shock waves are present, in a reference frame moving with a shock, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter itself, however, remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.

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Derivation

Image:BernoullisLawDerivationDiagram.png

The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects.

The work done by the forces in the fluid + decrease in potential energy = increase in kinetic energy.

The work done by the forces is

<math>F_{1} s_{1}-F_{2} s_{2}=p_{1} A_{1} v_

{1}\Delta t-p_{2} A_{2} v_{2}\Delta t. \;</math>

The decrease of potential energy is

<math>m g h_{1}-m g h_{2}=\rho g A

_{1} v_{1}\Delta t h_{1}-\rho g A_{2} v_{2} \Delta t h_{2} \;</math>

The increase in kinetic energy is

<math>\frac{1}{2} m v_{2}^{2}-\frac{1}{2} m v_{1}^{2}=\frac{1}{2}\rho A_{2} v_{2}\Delta t v_{2}

^{2}-\frac{1}{2}\rho A_{1} v_{1}\Delta t v_{1}^{2}.</math>

Putting these together,

<math>p_{1} A_{1} v_{1}\Delta t-p_{2} A_{2} v_{2}\Delta t+\rho g A_{1} v_{1}\Delta t h_{1}-\rho g A_{2} v_{2}\Delta t h_{2}=\frac{1}{2}\rho A_{2} v_{2}\Delta t v_{2}^{2}-\frac{1}{2}\rho A_{1} v_{1}\Delta t v_{1}^{2}</math>

<math>\frac{\rho A_{1} v_{1}\Delta t v_{1}^{

2}}{2}+\rho g A_{1} v_{1}\Delta t h_{1}+p_{1} A_{1 } v_{1}\Delta t=\frac{\rho A_{2} v_{2}\Delta t v_{ 2}^{2}}{2}+\rho g A_{2} v_{2}\Delta t h_{2}+p_{2} A_{2} v_{2}\Delta t.</math>

After dividing by <math>\Delta t</math>, <math>\rho</math> and <math>A_{1} v_{1}</math> (= rate of fluid flow = <math>A_{2} v_{2}</math> as the fluid is incompressible):

or <math>\frac{v^{2}}{2}+g h+\frac{p}{\rho}=C</math> (as stated in the first paragraph).

Further division by g implies

A free falling mass from a height h (in vacuum), will reach a velocity

<math>v=\sqrt{{2 g}{h}},</math> or <math>h=\frac{v^{2}}{2 g}</math>.

The term <math>\frac{v^2}{2 g}</math> is called the velocity head.

The hydrostatic pressure or static head is defined as

<math>p=\rho g h </math>, or <math>h=\frac{p}{\rho g}</math>.

The term <math>\frac{p}{\rho g}</math> is also called the pressure head.

A way to see how this relates to conservation of energy directly is to multiply by density and by unit volume (which is allowed since both are constant) yielding:

<math>v^2 \rho + P = constant </math> and

<math>mV^2 + P*volume = constant </math>

The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time <math> \Delta t </math>, the amount of mass passing through the boundary defined by the area <math> A_1 </math> is equal to the amount of mass passing outwards through the boundary defined by the area <math> A_2 </math>:

<math> 0 = \Delta M_1 - \Delta M_2 = \rho_1 A_1 v_1 \, \Delta t - \rho_2 A_2 v_2 \, \Delta t </math>.

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 <math> A_1 </math> and <math> A_2 </math> 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,

<math> 0 = \Delta E_1 - \Delta E_2 </math>

where <math> \Delta E_1 </math> and <math> \Delta E_2 </math> are the energy entering through <math> A_1 </math> and leaving through <math> A_2 </math>, respectively.

The energy entering through <math> A_1 </math> is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic energy entering, and the energy entering in the form of mechanical <math> p\,dV </math> work:

<math> \Delta E_1 = \left[ \frac{1}{2} \rho_1 v_1^2 + \phi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t </math>

A similar expression for <math> \Delta E_2 </math> may easily be constructed. So now setting <math> 0 = \Delta E_1 - \Delta E_2 </math>:

<math> 0 = \left[ \frac{1}{2} \rho_1 v_1^2+ \phi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t - \left[ \frac{1}{2} \rho_2 v_2^2 + \phi_2\rho_2 + \epsilon_2 \rho_2 + p_2 \right] A_2 v_2 \, \Delta t </math>

which can be rewritten as:

<math> 0 = \left[ \frac{1}{2} v_1^2 + \phi_1 + \epsilon_1 + \frac{p_1}{\rho_1} \right] \rho_1 A_1 v_1 \, \Delta t - \left[ \frac{1}{2} v_2^2 + \phi_2 + \epsilon_2 + \frac{p_2}{\rho_2} \right] \rho_2 A_2 v_2 \, \Delta t </math>

Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain

<math> \frac{1}{2}v^2 + \phi + \epsilon + \frac{p}{\rho} = {\rm constant} \equiv b </math>

which is the sought solution.

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