How Does Bernoulli's Equation Explain Fluid Dynamics?

[Greg Bernhardt]

Definition/Summary

Bernoulli's principle is a conservation principle along any streamline of an inviscous flow. It can be expressed as conservation of different types of pressure (force per area) or as conservation of different types of energy per mass.

Bernoulli's equation for steady incompressible inviscous flow where gravitational potential energy can be ignored:
total pressure (dynamic pressure plus ordinary pressure) is the same (is conserved) along any streamline of a flow.

Bernoulli's equation for steady incompressible inviscous flow:
total pressure plus potential energy density is the same (is conserved) along any streamline of a flow.

Bernoulli's equation for steady compressible inviscous flow:
kinetic energy per mass plus potential energy per mass plus enthalpy per mass is the same (is conserved) along any streamline of a flow.

Equations

Bernoulli's equation along any streamline of a steady incompressible non-viscous flow:

$$P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ =\ constant$$

Bernoulli's equation along any streamline of a steady non-viscous flow with variable internal energy (and therefore compressible):

$$P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ +\ \rho\,\epsilon\ =\ constant$$

or:

$$\frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ +\ \text{enthalpy per unit mass}\ =\ constant$$

Bernoulli's equation in the whole of an unsteady incompressible irrotational non-viscous flow:

$$\rho\frac{\partial\phi}{\partial t}\ +\ P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ =\ f(t)$$

where $\mathbf{v}\ =\ \mathbf{\nabla}\phi$, and $f(t)$ is a "spatial constant of integration".

Extended explanation

Pressure = work done per displaced volume:

Imagine a particular mass of fluid, occupying a region R, whose surface is S. After a short time, it will occupy a slightly different region R'.

R and R' will mostly overlap, but there will be a region R- at the back of R which has been vacated by the mass, and a new region R+ at the front, into which the mass has moved and displaced other fluid.

The work done by the pressure $P$ on the whole mass is the surface-integral, over every part of the surface, of pressure times area "dot" the distance through which that part has moved:

$$\int_S\,P\,\mathbf{x}\cdot\hat{\mathbf{n}}\,dA$$

which, since the pressure always acts inward, and therefore acts against the displacement on R+, is simply the volume-integral of the pressure over R- minus its volume-integral over R+:

$$\int_{R_-}\,P\,dV\ -\ \int_{R_+}\,P\,dV$$

Pressure difference = (minus) energy difference:

If the flow is steady, using $\eta$ for energy per mass (so $\rho\,\eta$ is energy density), and assuming that there is no viscosity (friction), so that no energy is lost from the fluid, we have:

work done = change in total energy = front energy minus back energy:​

$$\int_{R_-}\,\left(\frac{P}{\rho}\right)\,\rho\,dV\ -\ \int_{R_+}\,\left(\frac{P}{\rho}\right)\,\rho\,dV\ =\ \Delta\,E\ =\ E_+\ -\ E_-\ =\ \int_{R_+}\,\eta\,\rho\,dV\ -\ \int_{R_-}\,\eta\,\rho\,dV$$

We may choose R and R' to be "stalks" in a stationary tube made up of streamlines, as narrow as we like, and the displacement volumes R- and R+ to be as small as we like, so that we can regard $P$ and $\rho$ as constants, $P_-$ $P_+$ $\rho_-$ and $\rho_+$:

$$\left(\frac{P_-}{\rho_-}\ +\ \eta_-\right)\int_{R_-}\,\rho\,dV\ =\ \left(\frac{P_+}{\rho_+}\ +\ \eta_+\right)\int_{R_+}\,\rho\,dV$$

and, since, by conservation of mass, the mass in R- and R+ must be the same ($\int_{R_-}\,\rho\,dV\ =\ \int_{R_+}\,\rho\,dV$), the pressure plus the energy density must be constant along any streamline:

$$P/\rho\ +\ \eta\ =\ constant\text{ , or }P\ +\ \rho\,\eta\ =\ constant$$

Energy density:

The energy density, $\rho\,\eta$, includes:

the macroscopic kinetic energy density (or dynamic pressure), $\frac{1}{2}\,\rho\,v^2$

the gravitational potential energy density, $\rho\,g\,h$, and any other potential energy density due to an external field, $\rho\,\Phi$

and the internal energy density, which is all the other energy density, $\rho\,\epsilon\ =\ \rho\,\eta\ -\ \frac{1}{2}\,\rho\,v^2\ -\ \rho\,g\,h\ -\ \rho\,\Phi$

Energy can be external energy (due to macroscopic motion and external fields) or internal energy, $U$ (including relative motion of molecules and dipole moments and stress)

Internal energy plus pressure times volume equals enthalpy: $H\ =\ U\ +\ P\,V\text{ , or }H/V\ =\ \rho\,\epsilon\ +\ P$

$\epsilon$ is the internal energy per unit mass, or specific internal energy (s.i.e)

Incompressible flow:

Incompressible flow is flow whose density is constant along any streamline. In such flow, internal energy may be omitted from Bernoulli's equation (in other words, enthalpy per unit mass may be omitted, and replaced by pressure).

A real flow which may be treated as incompressible along most of its length may nevertheless lose energy at certain points (for example, the extra pressure drop across a valve, due to turbulence noise heat and cavitation), and at such points it will have to be treated as compressible. Alternatively, the whole flow may be treated as incompressible, but with an extra pressure drop "added in by hand" from a manufacturer's table.

For incompressible flow, internal energy per mass is constant, and so for steady inviscous flow, pressure plus the external energy density must be constant along any streamline:

$$P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ =\ constant$$

Euler's equations for fluid dynamics:

Bernoulli's equation is the integral along a streamline of the momentum components of Euler's equations for inviscous flow: $\mathbf{\nabla}P\ +\ \rho\,(\mathbf{v}\cdot\mathbf{\nabla})\mathbf{v}\ =\ 0$

Unsteady irrotational flow:

If the velocity at any point changes with time, two extra terms must be added to Bernoulli's equation for incompressible non-viscous flow: $\rho\,\partial\phi/\partial t\text{ and }f(t)$.

$\phi$ is a potential of the velocity (in other words, the velocity is the gradient of $\phi$): $\mathbf{v}\ =\ \mathbf{\nabla}\phi$. Of course, this only exists if the flow is irrotational.

$f(t)$, obviously, is the same throughout the whole fluid, and may be thought of as a "spatial constant of integration" of Euler's equations.

The potential $\phi$ can be chosen so that $f(t)\ =\ 0$.

Viscous flow: Navier–Stokes equations:

Bernoulli's equations are not applicable where there are viscous forces (the fluid equivalent of friction forces), for example in a boundary layer.

The Navier–Stokes equations must be used instead. Essentially, these are the Euler equations with an extra term for the divergence of the stress tensor.

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[Aaron Paulo]

Thank you for the detailed explanation of Bernoulli's Principle. It is very helpful in understanding how it works.