# [Physics] Bernoulli equation, energy conservation and unsteady flow

bernoulli-equationfluid dynamics

The Bernoulli equation for unsteady flow is given by:
$$\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\langle #1 \rangle} \p{\phi}{t}+\f{1}{2}\vec u\cdot \vec u+gh+\f{p}{\rho}=0$$
In steady state the first term is zero with the equation been a statement of the conservation of energy:

• 2nd term is kinetic energy
• 3rd term is potential energy
• 4th term is pressure energy (described e.g. here)

Assuming that in the unsteady case the equation still represents a statement of the conservation of energy, what type of energy/phenomenon does:
$$\p{\phi}{t}$$
represent?

There is a hypothesis left over here: the flow is irrotational, i.e. the so-called vorticity

$$\vec{\omega}=\nabla\times\vec{v}$$

is zero. As a result, the velocity field is a gradient,

$$\vec{v}=\nabla\phi,$$

for some scalar field $\phi$: the field appearing in the Bernoulli equation you quoted. Note that the right-hand side is not necessarily zero but it can be taken as a constant, which does not depend either on position or on time.

This is a simple consequence of (i) Euler equation for an incompressible fluid,

$$\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot\nabla) v + \nabla\left(\frac{p}{\rho}+\psi\right)=0,$$

where $\psi=gh$ but it could be the potential for any other conservative force field; and (ii) the identity

$$\vec{v}\times\vec{\omega} = \nabla\left(\frac{v^2}{2}\right) - (\vec{v}\cdot\nabla) v.$$

$$\nabla\left(\frac{\partial\phi}{\partial t}+\frac{v^2}{2}+\frac{p}{\rho}+\psi\right)=0.$$
Therefore the term between parentheses is function of time only but we can absorb any time dependence into $\phi$, and therefore we end up with
$$\frac{\partial\phi}{\partial t}+\frac{v^2}{2}+\frac{p}{\rho}+\psi=C$$
where $C$ is constant over space and time. This is therefore a stronger result than the traditional Bernoulli theorem where the constancy is only along each streamline, with a priori a different constant for each streamline. Here the constancy is over the entire fluid.