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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

$\displaystyle {\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,

$\displaystyle v_x$ $\displaystyle = - \frac{\partial\phi}{\partial x},$ (5.17)
$\displaystyle v_y$ $\displaystyle =-\frac{\partial\phi}{\partial y}.$ (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,

$\displaystyle v_x$ $\displaystyle =-\frac{\partial \psi}{\partial y},$ (5.19)
$\displaystyle v_y$ $\displaystyle =\frac{\partial\psi}{\partial x}.$ (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

$\displaystyle \frac{\partial\phi}{\partial x}$ $\displaystyle =\frac{\partial\psi}{\partial y},$ (5.21)
$\displaystyle \frac{\partial\psi}{\partial x}$ $\displaystyle =-\frac{\partial\phi}{\partial y}.$ (5.22)

It immediately follows, from the previous two expressions, that

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

or

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

Likewise, it can also be shown that

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

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

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


next up previous
Next: Two-Dimensional Uniform Flow Up: Two-Dimensional Incompressible Inviscid Flow Previous: Two-Dimensional Flow
Richard Fitzpatrick 2016-01-22