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Complex Velocity

Equations (5.17), (5.20), and (6.15) imply that

$\displaystyle \frac{dF}{dz} = \frac{\partial\phi}{\partial x} + {\rm i}\,\frac{\partial\psi}{\partial x} = -v_x+{\rm i}\,v_y.$ (6.35)

Consequently, $ dF/dz$ is termed the complex velocity. It follows that

$\displaystyle \left\vert\frac{dF}{dz}\right\vert^{\,2} = v_x^{\,2} + v_y^{\,2} =v^{\,2},$ (6.36)

where $ v$ is the flow speed.

A stagnation point is defined as a point in a flow pattern at which the flow speed, $ v$ , falls to zero. (See Section 5.8.) According to the previous expression,

$\displaystyle \frac{dF}{dz} = 0$ (6.37)

at a stagnation point. For instance, the stagnation points of the flow pattern produced when a cylindrical obstacle of radius $ a$ , centered on the origin, is placed in a uniform flow of speed $ V_0$ , directed parallel to the $ x$ -axis, and the circulation of the flow around is cylinder is $ {\mit\Gamma}$ , are found by setting the derivative of the complex potential (6.32) to zero. It follows that the stagnation points satisfy the quadratic equation

$\displaystyle \frac{dF}{dz}= -V_0\left(1-\frac{a^{\,2}}{z^{\,2}}\right) + {\rm i}\,\frac{\mit\Gamma}{2\pi\,z} = 0.$ (6.38)

The solutions are

$\displaystyle \frac{z}{a} = -{\rm i}\,\zeta \pm \sqrt{1-\zeta^{\,2}},$ (6.39)

where $ \zeta=-{\mit\Gamma}/(4\pi\,V_0\,a)$ , with the proviso that $ \vert z\vert/a>1$ , because the region $ \vert z\vert/a<1$ is occupied by the cylinder. Thus, if $ \zeta \leq 1$ then there are two stagnation points on the surface of the cylinder at $ x/a=\pm\sqrt{1-\zeta^{\,2}}$ and $ y/a=-\zeta$ . On the other hand, if $ \zeta>1$ then there is a single stagnation point below the cylinder at $ x/a=0$ and $ y/a=-\zeta-\sqrt{\zeta^{\,2}-1}$ .

According to Section 4.15, Bernoulli's theorem in an steady, irrotational, incompressible fluid takes the form

$\displaystyle p +\frac{1}{2}\,\rho\,v^{\,2} = p_0,$ (6.40)

where $ p_0$ is a uniform constant. Here, gravity (and any other body force) has been neglected. Thus, the pressure distribution in such a fluid can be written

$\displaystyle p = p_0 -\frac{1}{2}\,\rho\left\vert\frac{dF}{dz}\right\vert^{\,2}.$ (6.41)


next up previous
Next: Method of Images Up: Two-Dimensional Potential Flow Previous: Complex Velocity Potential
Richard Fitzpatrick 2016-01-22