Complex Velocity

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

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

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

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

$$\frac{dF}{dz} = 0$$ (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

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

The solutions are

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

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

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