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Axisymmetric Irrotational Flow in Spherical Coordinates

In an irrotational flow pattern, we can automatically satisfy the constraint $ \nabla\times {\bf v}={\bf0}$ by writing

$\displaystyle {\bf v} = -\nabla\phi.$ (7.11)

Suppose, however, that, in addition to being irrotational, the flow pattern is also incompressible: that is, $ \nabla\cdot{\bf v}=0$ . In this case, Equation (7.11) yields

$\displaystyle \nabla^{\,2}\phi = 0.$ (7.12)

In spherical coordinates, assuming that the flow pattern is axisymmetric, so that $ \phi=\phi(r,\theta)$ , the previous equation leads to (see Section C.4)

$\displaystyle \frac{1}{r^{\,2}}\,\frac{\partial}{\partial r}\left(r^{\,2}\,\fra...
...{\partial\theta}\left(\sin\theta\,\frac{\partial\phi}{\partial\theta}\right)=0.$ (7.13)

Let us search for a separable solution of Equation (7.13) of the form

$\displaystyle \phi(r,\theta)= R(r)\,{\mit\Theta}(\theta).$ (7.14)

It is easily seen that

$\displaystyle \frac{1}{R}\,\frac{d}{dr}\left(r^{\,2}\,\frac{dR}{dr}\right) = -\...
...heta}\,\frac{d}{d\theta}\left(\sin\theta\,\frac{d{\mit\Theta}}{d\theta}\right),$ (7.15)

which can only be satisfied provided

$\displaystyle \frac{d}{dr}\left(r^{\,2}\,\frac{dR}{dr}\right) -l\,(l+1)\,R$ $\displaystyle = 0,$ (7.16)
$\displaystyle \frac{d}{d\mu}\left[(1-\mu^{\,2})\,\frac{d{\mit\Theta}}{d\mu}\right]+l\,(l+1)\,{\mit\Theta}$ $\displaystyle = 0,$ (7.17)

where $ \mu=\cos\theta$ , and $ l\,(l+1)$ is a constant. The solutions to Equation (7.17) that are well behaved for $ \mu$ in the range $ -1$ to $ +1$ are known as the Legendre polynomials, and are denoted the $ P_l(\mu$ ), where $ l$ is a non-negative integer (Jackson 1962). (If $ l$ is non-integer then the solutions are singular at $ \mu=\pm 1.$ ) In fact,

$\displaystyle P_l(\mu)= \frac{(-1)^{\,l}}{2^{\,l}\,l!}\,\frac{d^{\,l}}{d\mu^{\,l}}\left(1-\mu^{\,2}\right)^l.$ (7.18)

Hence,

$\displaystyle P_0(\mu)$ $\displaystyle = 1,$ (7.19)
$\displaystyle P_1(\mu)$ $\displaystyle =\mu,$ (7.20)
$\displaystyle P_2(\mu)$ $\displaystyle = \frac{1}{2}\left(3\,\mu^{\,2}-1\right),$ (7.21)
$\displaystyle P_3(\mu)$ $\displaystyle = \frac{1}{2}\left(5\,\mu^{\,3}-3\,\mu\right),$ (7.22)

et cetera. The general solution of Equation (7.16) is a linear combination of $ r^{\,l}$ and $ r^{-(l+1)}$ factors. Thus, the general axisymmetric solution of Equation (7.12) is written

$\displaystyle \phi(r,\theta) = \sum_{l=0,\infty} \left[\alpha_l\,r^{\,l} + \beta_l\,r^{-(l+1)}\right]P_l(\cos\theta),$ (7.23)

where the $ \alpha_l$ and $ \beta_l$ are arbitrary coefficients. It follows from Equations (7.4) that the corresponding expression for the Stokes stream function is

$\displaystyle \psi(r,\mu)= \beta_0\,\mu+\sum_{l=1,\infty}\left(\frac{\alpha_l}{...
...1}-\frac{\beta_l}{l}\,r^{\,-l}\right)\left(1-\mu^{\,2}\right)\frac{dP_l}{d\mu},$ (7.24)

where $ \mu=\cos\theta$ .


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Next: Uniform Flow Up: Axisymmetric Incompressible Inviscid Flow Previous: Axisymmetric Velocity Fields
Richard Fitzpatrick 2016-03-31