6.3: Velocity Potential

We introduce the vector field \(\phi (\vec{x}, \, t)\) to satisfy the following relation:

\[ \vec{V} \, = \begin{Bmatrix} u \\[4pt] v \\[4pt] w \end{Bmatrix} = \left[ \dfrac{\partial \phi}{\partial x} \quad \dfrac{\partial \phi}{\partial y} \quad \dfrac{\partial \phi}{\partial z} \right] ^T = \, \nabla \phi. \]

The conservation of mass is transformed to

\[ \dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial^2 \phi}{\partial y^2} + \dfrac{\partial^2 \phi}{\partial z^2} \, = \, \nabla^2 \cdot \phi \, = \, 0. \]

Considering Newton’s law, the first force balance (\(x\)-direction) equation that we gave above is

\[ \rho \left[ \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + w \dfrac{\partial u}{\partial z} \right] \, = \, -\dfrac{\partial p}{\partial x}; \]

this becomes, substituting the velocity potential \(\phi\),

\[ \rho \left[ \dfrac{\partial^2 \phi}{\partial t \partial x} + \dfrac{\partial \phi}{\partial x} \dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial \phi}{\partial y} \dfrac{\partial ^2 \phi}{\partial y \partial x} + \dfrac{\partial \phi}{\partial z} \dfrac{\partial^2 \phi}{\partial z \partial x} \right] \, = \, -\dfrac{\partial \rho}{\partial x}. \]

Integrating on \(x\) we find

\[ p + \rho \dfrac{\partial \phi}{\partial t} + \dfrac{1}{2} \rho (u^2 + v^2 + w^2) \, = \, C, \]

where \(C\) is a constant. The other two force balance equations are precisely the same but with the addition of gravity effects in the \(z\)-direction. Hence a single equation for the whole field is

\[ p + \rho \dfrac{\partial \phi}{\partial t} + \dfrac{1}{2} \rho (u^2 + v^2 + w^2) + \rho g z \, = \, C. \]

This is the Bernoulli equation.