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Let
,
,
be standard spherical coordinates. Consider axisymmetric Stokes flow such that
![$\displaystyle {\bf v}({\bf r}) = v_r(r,\theta)\,{\bf e}_r + v_\theta(r,\theta)\,{\bf e}_\theta.$](img3730.png) |
(10.84) |
According to Equations (A.175) and (A.176), we can automatically satisfy the incompressibility constraint (10.80) by writing (see Section 7.4)
![$\displaystyle {\bf v} =\nabla\varphi\times\nabla\psi,$](img3731.png) |
(10.85) |
where
is the Stokes stream function (i.e.,
). It follows that
Moreover, according to Section C.4,
, and
![$\displaystyle \omega_\varphi(r,\theta) = \frac{1}{r}\,\frac{\partial\,(r\,v_\th...
...r}\,\frac{\partial v_r}{\partial\theta} = \frac{{\cal L}(\psi)}{r\,\sin\theta},$](img3735.png) |
(10.88) |
where (see Section 7.4)
![$\displaystyle {\cal L} = \frac{\partial^{\,2}}{\partial r^{\,2}} + \frac{\sin\t...
...artial}{\partial\theta}\,\frac{1}{\sin\theta}\,\frac{\partial}{\partial\theta}.$](img3736.png) |
(10.89) |
Hence, given that
, we can write
![$\displaystyle \mbox{\boldmath$\omega$}$](img808.png) ![$\displaystyle =\nabla\times{\bf v} = {\cal L}(\psi)\,\nabla\varphi.$](img3738.png) |
(10.90) |
It follows from Equations (A.176) and (A.178) that
Thus, by analogy with Equations (10.85) and (10.90), and making use of Equations (A.173) and (A.177),
we obtain
Equation (10.83) implies that
![$\displaystyle {\cal L}^{\,2}(\psi) = 0,$](img3744.png) |
(10.93) |
which is the governing equation for axisymmetric Stokes flow.
In addition, Equations (10.82) and (10.91) yield
![$\displaystyle \nabla P = \mu\,\nabla\varphi\times \nabla[{\cal L}(\psi)].$](img3745.png) |
(10.94) |
Next: Axisymmetric Stokes Flow Around
Up: Incompressible Viscous Flow
Previous: Stokes Flow
Richard Fitzpatrick
2016-03-31