# 0.6 Solution of the partial differential equations  (Page 10/13)

 Page 10 / 13
${v}_{\xi }-i{v}_{\eta }=\frac{dw}{d\varsigma }=\frac{dw}{dz}\frac{dz}{d\varsigma }=\left({v}_{x}-i{v}_{y}\right)\frac{dz}{d\varsigma }$

This shows that the magnitude of the velocity is changed, in the transformation from the $z$ -plane to the $\zeta$ -plane, by the reciprocal of the factor by which linear dimensions of small figures are changed. Thus the kinetic energy of the fluid contained within a closed curve in the $z$ -plane is equal to the kinetic energy of the corresponding flow in the region enclosed by the corresponding in the $\zeta$ -plane.

Flow around elliptic cylinder (Batchelor, 1967). The transformation of the region outside of an ellipse in the $z$ -plane into the region outside a circle in the $\zeta$ -plane is given by

$\begin{array}{c}z=\varsigma +\frac{{\lambda }^{2}}{\varsigma }\hfill \\ \varsigma =\frac{1}{2}z+\frac{1}{2}{\left({z}^{2}-4{\lambda }^{2}\right)}^{1/2}\hfill \end{array}$

where $\lambda$ is a real constant so that

$x=\xi \left(1+\frac{{\lambda }^{2}}{{\left|\varsigma \right|}^{2}}\right),\phantom{\rule{1.em}{0ex}}y=\eta \left(1-\frac{{\lambda }^{2}}{{\left|\varsigma \right|}^{2}}\right)$

This converts a circle of radius c with center at the origin in the - plane into the ellipse

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

in the $z$ -plane, where

$\lambda =\frac{1}{2}{\left({a}^{2}-{b}^{2}\right)}^{1/2}$

If the ellipse is mapped into a circle in the $\zeta$ -plane, it is convenient to use polar coordinates $\left(r,\theta \right)$ , especially since the boundary corresponds to a constant radius. The radius that maps to the elliptical boundary is ( ellipse.m in the complex directory)

${r}_{o}=\frac{1}{2}log\left(\frac{a+b}{a-b}\right)$

The transformation from the polar coordinates to the z - plane is defined by

$\begin{array}{c}z=2\lambda \phantom{\rule{0.166667em}{0ex}}cosh\phantom{\rule{0.166667em}{0ex}}\omega \hfill \\ \mathrm{where}\hfill \\ \omega =r+i\phantom{\rule{0.166667em}{0ex}}\theta \hfill \end{array}$ Transformation of cylindrical polar coordinates into orthogonal, elliptical coordinates

The polar coordinates $\left(r,\theta \right)$ , transform to an orthogonal set of curves which are confocal ellipses and conjugate hyperbolae.

This transformation can be substituted into the complex potential expression for the flow of a fluid past a circular cylinder.

$w=-\frac{1}{2}\left(a+b\right)\left[\left(U-iV\right){e}^{\omega -{r}_{o}}+\left(U+iV\right){e}^{{r}_{o}-\omega }\right]$

It is convenient to write - for the angle which the direction of motion of the flow at infinity makes with the x - axis so that

$U+iV={\left({U}^{2}+{V}^{2}\right)}^{1/2}{e}^{-i\alpha }$

The complex potential now becomes

$w=-{\left({U}^{2}+{V}^{2}\right)}^{1/2}\left(a+b\right)cosh\left(\omega -{r}_{o}+i\alpha \right)$

The corresponding velocity potential and stream function are

$\begin{array}{c}\varphi =-{\left({U}^{2}+{V}^{2}\right)}^{1/2}\left(a+b\right)\phantom{\rule{0.166667em}{0ex}}cosh\left(r-{r}_{o}\right)cos\left(\theta +\alpha \right)\hfill \\ \psi =-{\left({U}^{2}+{V}^{2}\right)}^{1/2}\left(a+b\right)sinh\left(r-{r}_{o}\right)sin\left(\theta +\alpha \right)\hfill \end{array}$

The velocity potentials and streamlines are illustrated below for flow past an elliptical cylinder ( fellipse.m in the complex directory). Note the stagnation streamlines on either side of the body. These two stagnation points are regions of maximum pressure and result in a torque on the body. Which way will it rotate? Flow past an ellipse of an inviscid fluid that is in steady translation at infinity.

Pressure distribution. When an object is in a flow field, one may wish to determine the force exerted by the fluid on the object, or the 'drag' on the object. Since the flow field discussed here has assumed an inviscid fluid, it is not possible to determine the viscous drag or skin friction directly from the flow field. It is possible to determine the 'form drag' from the normal stress or pressure distribution around the object. However, one must be critical to determine if the calculated flow field is physically realistic or if some important phenomena such as boundary layer separation may occur but is not allowed in the complex potential solution.

The Bernoulli theorems give the relation between the magnitude of velocity and pressure. We have assumed irrotational, incompressible flow. If in addition we assume the body force can be neglected, then the quantity, $H$ , must be constant along a streamline.

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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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