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In an irrotational flow pattern, we can automatically satisfy the constraint
by writing
![$\displaystyle {\bf v} = -\nabla\phi.$](img1584.png) |
(7.11) |
Suppose, however, that, in addition to being irrotational, the flow pattern is also incompressible: that is,
. In
this case, Equation (7.11) yields
![$\displaystyle \nabla^{\,2}\phi = 0.$](img1588.png) |
(7.12) |
In spherical coordinates, assuming that the flow pattern is axisymmetric, so that
,
the previous equation leads to (see Section C.4)
![$\displaystyle \frac{1}{r^{\,2}}\,\frac{\partial}{\partial r}\left(r^{\,2}\,\fra...
...{\partial\theta}\left(\sin\theta\,\frac{\partial\phi}{\partial\theta}\right)=0.$](img2572.png) |
(7.13) |
Let us search for a separable solution of Equation (7.13) of the form
![$\displaystyle \phi(r,\theta)= R(r)\,{\mit\Theta}(\theta).$](img2573.png) |
(7.14) |
It is easily seen that
![$\displaystyle \frac{1}{R}\,\frac{d}{dr}\left(r^{\,2}\,\frac{dR}{dr}\right) = -\...
...heta}\,\frac{d}{d\theta}\left(\sin\theta\,\frac{d{\mit\Theta}}{d\theta}\right),$](img2574.png) |
(7.15) |
which can only be satisfied provided
where
, and
is a constant. The solutions to Equation (7.17) that are well behaved for
in the range
to
are known as the Legendre polynomials, and are denoted the
), where
is a non-negative integer (Jackson 1962). (If
is non-integer then the solutions are singular at
) In fact,
![$\displaystyle P_l(\mu)= \frac{(-1)^{\,l}}{2^{\,l}\,l!}\,\frac{d^{\,l}}{d\mu^{\,l}}\left(1-\mu^{\,2}\right)^l.$](img2583.png) |
(7.18) |
Hence,
![$\displaystyle P_0(\mu)$](img2584.png) |
![$\displaystyle = 1,$](img2585.png) |
(7.19) |
![$\displaystyle P_1(\mu)$](img2586.png) |
![$\displaystyle =\mu,$](img2587.png) |
(7.20) |
![$\displaystyle P_2(\mu)$](img2588.png) |
![$\displaystyle = \frac{1}{2}\left(3\,\mu^{\,2}-1\right),$](img2589.png) |
(7.21) |
![$\displaystyle P_3(\mu)$](img2590.png) |
![$\displaystyle = \frac{1}{2}\left(5\,\mu^{\,3}-3\,\mu\right),$](img2591.png) |
(7.22) |
et cetera. The general solution of Equation (7.16) is a linear combination of
and
factors.
Thus, the general axisymmetric solution of Equation (7.12) is written
![$\displaystyle \phi(r,\theta) = \sum_{l=0,\infty} \left[\alpha_l\,r^{\,l} + \beta_l\,r^{-(l+1)}\right]P_l(\cos\theta),$](img2594.png) |
(7.23) |
where the
and
are arbitrary coefficients. It follows from Equations (7.4) that the
corresponding expression for the Stokes stream function is
![$\displaystyle \psi(r,\mu)= \beta_0\,\mu+\sum_{l=1,\infty}\left(\frac{\alpha_l}{...
...1}-\frac{\beta_l}{l}\,r^{\,-l}\right)\left(1-\mu^{\,2}\right)\frac{dP_l}{d\mu},$](img2597.png) |
(7.24) |
where
.
Next: Uniform Flow
Up: Axisymmetric Incompressible Inviscid Flow
Previous: Axisymmetric Velocity Fields
Richard Fitzpatrick
2016-03-31