Velocity Potentials and Stream Functions

As we have seen, a two-dimensional velocity field in which the flow is everywhere parallel to the $x$ -$y$ plane, and there is no variation along the $z$ -direction, takes the form

$${\bf v} = v_x(x,y,t)\,{\bf e}_x + v_y(x,y,t)\,{\bf e}_y.$$ (5.16)

Moreover, if the flow is irrotational then $\nabla\times {\bf v}={\bf0}$ is automatically satisfied by writing ${\bf v}=-\nabla\phi$ , where $\phi(x,y,t)$ is termed the velocity potential. (See Section 4.15.) Hence,

$$v_x = - \frac{\partial\phi}{\partial x},$$ (5.17)

$$v_y =-\frac{\partial\phi}{\partial y}.$$ (5.18)

On the other hand, if the flow is incompressible then $\nabla\cdot{\bf v}=0$ is automatically satisfied by writing ${\bf v} = \nabla z\times \nabla\psi$ , where $\psi(x,y,t)$ is termed the stream function. (See Section 5.2.) Hence,

$$v_x =-\frac{\partial \psi}{\partial y},$$ (5.19)

$$v_y =\frac{\partial\psi}{\partial x}.$$ (5.20)

Finally, if the flow is both irrotational and incompressible then Equations (5.17)-(5.18) and (5.19)-(5.20) hold simultaneously, which implies that

$$\frac{\partial\phi}{\partial x} =\frac{\partial\psi}{\partial y},$$ (5.21)

$$\frac{\partial\psi}{\partial x} =-\frac{\partial\phi}{\partial y}.$$ (5.22)

It immediately follows, from the previous two expressions, that

$$\frac{\partial^{\,2}\phi}{\partial x^{\,2}} = \frac{\partial^{\,2... ...2}\psi}{\partial y\,\partial x} = -\frac{\partial^{\,2}\phi}{\partial y^{\,2}},$$ (5.23)

or

$$\frac{\partial^{\,2}\phi}{\partial x^{\,2}}+\frac{\partial^{\,2}\phi}{\partial y^{\,2}} = 0.$$ (5.24)

Likewise, it can also be shown that

$$\frac{\partial^{\,2}\psi}{\partial x^{\,2}}+\frac{\partial^{\,2}\psi}{\partial y^{\,2}} = 0.$$ (5.25)

We conclude that, for two-dimensional, irrotational, incompressible flow, the velocity potential and the stream function both satisfy Laplace's equation. Equations (5.21) and (5.22) also imply that

$$\nabla\phi\cdot\nabla\psi = 0.$$ (5.26)

In other words, the contours of the velocity potential and the stream function cross at right-angles.