The Bernoulli equation of fluid dynamics

I has been a while since I did some work in fluid dynamics but I stumbled across this b/c of a job interview so I thought that I maybe share it here.

The Bernoulli equation of fluid-dynamics is basically a law of conservation. In physics there are certain conserved quantities that stay the same during a physical transformation. An example would be energy, or momentum. The Equation derived below is basically a statement of energy conservation, however it requires a so-called ideal fluid.

An ideal fluid has the following properties:

it is stationary and laminar (so there is no turbulence, and velocity at one point does not change) $\frac{\partial v}{\partial t} = 0$
it has no viscosity (so there is no internal friction) $\eta = 0$
it has no friction (with outside walls or anything else)
it is incompressible, so the density is the same everywhere $\rho = \text{const.}$

Imagine a tube that is filled with this ideal fluid and its radius is shrinking in the middle in a conical fashion. Because we already know the equation of state ( $\rho = 0$ ), we can make an assumption about the mass-flow within the tube.

Considering the difference in mass $\Delta m_1$ , that is the mass that enters the pipe in the beginning in a certain amount of time we can easily see that this must be the $\Delta m_2$ , which is the mass that leaves the pipe at the end in the same amount of time. This balance, also called continuity-equation, we can write as

(1) $\begin{equation*} \Delta m \equiv \rho_1 A_1 v_1 \Delta t= \rho_2 A_2 v_2 \Delta t \end{equation*}$

When we consider the total work done to the fluid we have to be careful about the sign and can write it as

(2) $\begin{equation*} \Delta W = p_1 A_1 v_1 \Delta t - p_2 A_2 v_2 \Delta t \end{equation*}$

This total work must be equal to all other energy representations in the fluid, which we write in a way that is normalized by the mass difference

(3) $\begin{equation*} \Delta W &=& \Delta m (E_2-E_1) \ \end{equation*}$

(4) $\begin{equation*} \frac{p_1 A_1 v_1 \Delta t}{\Delta m} - \frac{p_2 A_2 v_2 \Delta t}{\Delta m} &=& \frac{1}{2}v_2^2 + \phi_2 + U_2 - \left( \frac{1}{2}v_1^2 + \phi_1 + U_1 \right), \end{equation*}$

where $\frac{1}{2}v_2^2$ represents the kinetic energy $\phi_2$ some form of potential (in our case $gz$ ) and $U_2$ is a representation of internal energy. We disregard internal energy and fill in equation $ref{eq:massflow}$ for $\Delta m$ we get this nice little expression:

(5) $\begin{equation*} \frac{1}{2}v_1^2 + \phi_1 + \frac{p_1}{\rho_1} + U_1 = \frac{1}{2}v_2^2 + \phi_2 + \frac{p_2}{\rho_2} + U_2 \end{equation*}$

which basically states

(6) $\begin{equation*} \left[ \frac{1}{2}v^2 + \phi + \frac{p}{\rho} \right]_{\text{streamline}} = \text{const.} \end{equation*}$

This is Bernoulli’s equation of fluid dynamics, it states that within a flow of constant energy the velocity of a fluid rises in fields of low pressure and lowers in fields of low pressure.

Even though we simplified the underlying physics massively there’s still a lot of phenomena to be explained using this simple equation. For example aeroplanes flying can be partially explained with this equation.