# 0.6 Solution of the partial differential equations  (Page 9/13)

 Page 9 / 13

The circle theorem. (Batchelor, 1967) The following result, known as the circle theorem (Milne-Thompson, 1940) concerns the complex potential representing the motion of an inviscid fluid of infinite extent in the presence of a single internal boundary of circular form. Suppose first that in the absence of the circular cylinder the complex potential is

${w}^{o}=f\left(z\right)$

and that $f\left(z\right)$ is free from singularities in the region $|z|\le a$ , where $a$ is a real length. If now a stationary circular cylinder of radius $a$ and center at the origin bounds the fluid internally, the flow is modified to the following complex potential:

${w}^{1}=f\left(z\right)+\overline{f}\left({a}^{2}/z\right)$

We show that the surface of the cylinder, $|z|=a$ , is a streamline.

${a}^{2}=z\phantom{\rule{0.166667em}{0ex}}\overline{z}$
$\begin{array}{ccc}\hfill {\left({w}^{1}\left(z\right)|}_{\left|z\right|=a}& =& {\left(f\left(z\right)|}_{\left|z\right|=a}+{\left(\overline{f}\left({a}^{2}/z\right)|}_{\left|z\right|=a}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =& {\left(f\left(z\right)|}_{\left|z\right|=a}+{\left(\overline{f}\left(z\overline{z}/z\right)|}_{\left|z\right|=a}\hfill \\ & =& {\left(f\left(z\right)|}_{\left|z\right|=a}+{\left(\overline{f}\left(\overline{z}\right)|}_{\left|z\right|=a}\hfill \\ & =& 2\phantom{\rule{0.166667em}{0ex}}\mathrm{Real}\left[{\left(f\left(z\right)|}_{\left|z\right|=a}\right]+i\phantom{\rule{0.166667em}{0ex}}0\hfill \\ & =& {\varphi }_{\left|z\right|=a}+i\phantom{\rule{0.277778em}{0ex}}{\psi }_{\left|z\right|=a}\hfill \end{array}$

A complex potential of this form thus has $|z|=a$ as a streamline, $\psi =0$ ; and it has the same singularities outside $|z|=a$ as $f\left(z\right)$ , since if $z$ lies outside $|z|=a$ , ${a}^{2}/z$ lies in the region inside this circle where $f\left(z\right)$ is known to be free from singularities. Consequently the additional term $\overline{f}\left({a}^{2}/z\right)$ in the equation represents fully the modification to the complex potential due to the presence of the circular cylinder. It should be noted that the complex potentials considered, both in the absence and in the presence of the circular cylinder, refer to the flow relative to axis such that the cylinder is stationary.

The simplest possible application of the circle theorem is to the case of a circular cylinder held fixed in a stream whose velocity at infinity is uniform with components $\left(-U,-V\right)$ . In the absence of the cylinder the complex potential is $-\left(U-i\phantom{\rule{0.166667em}{0ex}}V\right)z$ , it is singular at infinity and the circle theorem shows that, with the cylinder present,

$w\left(z\right)=-\left(U-i\phantom{\rule{0.166667em}{0ex}}V\right)z-\left(U+i\phantom{\rule{0.166667em}{0ex}}V\right)\phantom{\rule{0.166667em}{0ex}}{a}^{2}/z$

The potentials and streamlines for the steady translation of an inviscid fluid past a circular cylinder can be viewed with the MATLAB code circle.m .

Conformal Transformation (Batchelor, 1967). We now have the complex potential flow solutions of several problems with fairly simple boundary conditions. These solutions are analytical functions whose real and imaginary parts satisfy the Laplace equation. They also have a streamline that coincides with the boundary to satisfy the condition of zero flux across the boundary. Conformal transformations can be used to obtain solutions for boundaries that are transformed to different shapes. Suppose we have an analytical function $w\left(z\right)$ in the $z=x+iy$ plane. This solution can be transformed to the $\zeta =\xi +i\eta$ plane as another analytical function provided that the relation between these two planes, $\zeta =F\left(z\right)$ is an analytical function. This mapping is a connection between the shape of a curve in the $z$ - plane and the shape of the curve traced out by the corresponding set of points in the $\zeta -plane$ . The solution in the $\zeta -plane$ is analytical, i.e., its derivative defined, because the mapping, $\zeta =F\left(z\right)$ is an analytical function. The inverse transformation is also analytical.

$\begin{array}{c}\frac{dw\left(\varsigma \right)}{d\varsigma }=\frac{d\phantom{\rule{0.166667em}{0ex}}w\left(z\right)}{dz}\phantom{\rule{0.166667em}{0ex}}\frac{dz}{d\varsigma }=\frac{\frac{d\phantom{\rule{0.166667em}{0ex}}w\left(z\right)}{dz}}{\frac{dF\left(z\right)}{dz}}\hfill \\ \frac{d\phantom{\rule{0.166667em}{0ex}}w\left(z\right)}{dz}=\frac{dw\left(\varsigma \right)}{d\varsigma }\phantom{\rule{0.166667em}{0ex}}\frac{dF\left(z\right)}{dz}\hfill \end{array}$

$w\left(\zeta \right)$ is thus the complex potential of an irrotational flow in a certain region of the $\zeta$ -plane, and the flow in the $z$ -plane is said to been 'transformed' into flow in the $\zeta$ -plane. The family of equipotential lines and streamlines in the $z$ -plane given by $\varphi \left(x,y\right)=\text{const.}$ and $\psi \left(x,y\right)=\text{const.}$ transform into families of curves in the $\zeta$ -plane on which $\varphi$ and $\psi$ are constant and which are equipotential lines and streamlines in the $\zeta$ -plane. The two families are orthogonal in their respective plane, except at singular points of the transformation. The velocity components at a point of the flow in the $\zeta$ -plane are given by

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