A free boundary problem: steady axisymmetric potential flow

Appendix

To determine the velocity vector field v, the following Picard iteration [3, 15] can be used. This procedure, similar to Matthew’s method [18], is an alternative to the method proposed by Valiani and Caleffi [23]. Valiani and Caleffi’s procedure is based on a power expansion of the harmonic stream function associated with the velocity vector field v [2, 4, 12, 17, 20].

Setting as a first approximation:

$$ v_{r} = \frac{q}{2\pi rh};\qquad v_{z} = 0 $$

(A.1)

the irrotationality condition is automatically satisfied. The solenoidality condition is verified if and only if the following first-order term is considered nil:

$$ \frac{\mathrm{d}h}{\mathrm{d}r} = 0 $$

(A.2)

Equation (A.2) defines the approximation order associated with Eq. (A.1). The second approximation of the velocity vector field v can be deduced by setting:

$$ v_{r} = \frac{q}{2\pi rh};\qquad v_{z} = f_{2} ( r,z ) $$

(A.3)

where f ₂(r,z) a function unknown a priori. From the solenoidality condition, it follows that:

$$ \frac{\mathrm{d}f_{2}}{\mathrm{d}z} = \frac{q}{2\pi rh^{2}}\frac{\mathrm{d}h}{\mathrm{d}r} $$

(A.4)

from which:

$$ f_{2} = \frac{q}{2\pi rh^{2}}\frac{\mathrm {d}h}{\mathrm{d}r}z + g_{2} ( r ) $$

(A.5)

The kinematic condition at the base (v _z =0 at EA) corresponds to g ₂(r)=0. In conclusion, the second approximation of the velocity vector field v is given as follows:

$$ v_{r} = \frac{q}{2\pi rh}; \qquad v_{z} = \frac{q}{2\pi rh^{2}}\frac{\mathrm{d}h}{\mathrm{d}r}z $$

(A.6)

The velocity vector field expressed by Eq. (A.6) is solenoidal. The velocity vector field itself is irrotational if and only if the following second-order terms are considered nil:

$$ \frac{\mathrm{d}}{\mathrm{d}r} \biggl( \frac {1}{rh^{2}}\frac{\mathrm{d}h}{\mathrm{d}r} \biggr) = 0 $$

(A.7)

Equation (A.7) defines the approximation order associated with Eq. (A.6). The third approximation of the velocity vector field v can be deduced by setting:

$$ v_{r} = \frac{q}{2\pi rh} + f_{3} ( r,z ) ; \qquad v_{z} = \frac{q}{2\pi rh^{2}}\frac{\mathrm{d}h}{\mathrm{d}r}z $$

(A.8)

From the irrotationality condition, it follows that:

$$ f_{3} = \frac{q}{2\pi} \frac{z^{2}}{2} \frac{\mathrm{d}}{\mathrm{d}r} \biggl( \frac{1}{rh^{2}}\frac {\mathrm{d}h}{\mathrm{d}r} \biggr) + g_{3} ( r ) $$

(A.9)

Given that:

$$ q = \int_{0}^{h ( r )} 2\pi rv_{r}dz $$

(A.10)

the function g ₃(r) takes the following form:

$$ g_{3} = - \frac{q}{2\pi} \frac{h^{2}}{6} \frac{\mathrm{d}}{\mathrm{d}r} \biggl( \frac{1}{rh^{2}}\frac {\mathrm{d}h}{\mathrm{d}r} \biggr) $$

(A.11)

and the third approximation of the velocity vector field v is expressed by Eqs. (3) and (4). This procedure can be extended to any required degree of accuracy. For example, it is easy to verify that the fourth approximation of the velocity vector field v is given as:

$$ v_{r} = \frac{q}{2\pi rh} + \frac{q}{2\pi} \frac{\mathrm{d}}{\mathrm{d}r} \biggl( \frac{1}{rh^{2}}\frac {\mathrm{d}h}{\mathrm{d}r} \biggr) \biggl( \frac{z^{2}}{2} - \frac{h^{2}}{6} \biggr) $$

(A.12)

(A.13)

while the closure hypotheses associated with Eqs. (A.12) and (A.13) are given as follows:

$$ \everymath{\displaystyle }\begin{array}{@{}lll} \frac{\mathrm{d}}{\mathrm{d}r} \biggl[ h\frac{\mathrm{d}}{\mathrm {d}r} \biggl( \frac{1}{rh^{2}}\frac{\mathrm{d}h}{\mathrm{d}r} \biggr)\frac {\mathrm{d}h}{\mathrm{d}r} \biggr] = 0 \cr\noalign{\vspace{6pt}} r \frac{\mathrm{d}}{\mathrm{d}r} \biggl[ \frac{1}{r}\frac{\mathrm {d}}{\mathrm{d}r} \biggl( \frac{1}{rh^{2}}\frac{\mathrm{d}h}{\mathrm{d}r} \biggr) \biggr] = 0 \end{array} $$

(A.14)