next up previous
Next: Blasius Theorem Up: Two-Dimensional Potential Flow Previous: Free Streamline Theory


Complex Line Integrals

Consider the line integral of some function $ F(z)$ of the complex variable taken (counter-clockwise) around a closed curve $ C$ in the complex plane:

$\displaystyle J = \oint_C F(z)\,dz.$ (6.158)

Because $ dz=dx+{\rm i}\,dy$ , and writing $ F(z)=\phi(x,y)+{\rm i}\,\psi(x,y)$ , where $ \phi(x,y)$ and $ \psi(x,y)$ are real functions, it follows that $ J=J_r+{\rm i}\,J_i$ , where

$\displaystyle J_r$ $\displaystyle =\oint_C (\phi\,dx-\psi\,dy),$ (6.159)
$\displaystyle J_i$ $\displaystyle =\oint_C (\psi\,dx+\phi\,dy).$ (6.160)

However, we can also write the previous expressions in the two-dimensional vector form

$\displaystyle J_r$ $\displaystyle =\oint_C {\bf A}\cdot d{\bf r},$ (6.161)
$\displaystyle J_i$ $\displaystyle =\oint_C {\bf B}\cdot d{\bf r},$ (6.162)

where $ d{\bf r}=(dx$ , $ dy)$ , $ {\bf A}=(\phi$ , $ -\psi)$ , and $ {\bf B} = (\psi$ , $ \phi)$ . According to the curl theorem (see Section A.22),

$\displaystyle \oint_C {\bf A}\cdot d{\bf r}$ $\displaystyle =\int_S (\nabla\times {\bf A})_z\,dS,$ (6.163)
$\displaystyle \oint_C {\bf B}\cdot d{\bf r}$ $\displaystyle =\int_S (\nabla\times {\bf B})_z\,dS,$ (6.164)

where $ S$ is the region of the $ x$ -$ y$ plane enclosed by $ C$ . Hence, we obtain

$\displaystyle J_r$ $\displaystyle =-\int_S\left(\frac{\partial\psi}{\partial x} + \frac{\partial\phi}{\partial y}\right)dS,$ (6.165)
$\displaystyle J_i$ $\displaystyle =\int_S\left(\frac{\partial\phi}{\partial x} - \frac{\partial\psi}{\partial y}\right)dS.$ (6.166)

Let

$\displaystyle J$ $\displaystyle =\oint_{C}F(z)\,dz,$ (6.167)
$\displaystyle J'$ $\displaystyle =\oint_{C'} F(z)\,dz,$ (6.168)

where $ C'$ is a closed curve in the complex plane that completely surrounds the smaller curve $ C$ . Consider

$\displaystyle {\mit\Delta} J = J-J'.$ (6.169)

Writing $ {\mit\Delta} J={\mit\Delta} J_r+{\rm i}\,{\mit\Delta} J_i$ , a direct generalization of the previous analysis reveals that

$\displaystyle {\mit\Delta} J_r$ $\displaystyle =-\int_{S}\left(\frac{\partial\psi}{\partial x} + \frac{\partial\phi}{\partial y}\right)dS,$ (6.170)
$\displaystyle {\mit\Delta} J_i$ $\displaystyle =\int_{S}\left(\frac{\partial\phi}{\partial x} - \frac{\partial\psi}{\partial y}\right)dS,$ (6.171)

where $ S$ is now the region of the $ x$ -$ y$ plane lying between the curves $ C$ and $ C'$ . Suppose that $ F(z)$ is well-behaved (i.e., finite, single-valued, and differentiable) throughout $ S$ . It immediately follows that its real and imaginary components, $ \phi$ and $ \psi $ , respectively, satisfy the Cauchy-Riemann relations, (6.17)-(6.18), throughout $ S$ . However, if this is the case then it is apparent, from the previous two expressions, that $ {\mit\Delta} J_r={\mit\Delta} J_i=0$ . In other words, if $ F(z)$ is well-behaved throughout $ S$ then $ J=J'$ .

The circulation of the flow about some closed curve $ C$ in the $ x$ -$ y$ plane is defined

$\displaystyle {\mit\Gamma} = \oint_C (v_x\,dx+v_y\,dy) = -{\rm Re}\oint_C \frac{dF}{dz}\,dz,$ (6.172)

where $ F(z)$ is the complex velocity potential of the flow, and use has been made of Equation (6.35). Thus, the circulation can be evaluated by performing a line integral in the complex $ z$ -plane. Moreover, as is clear from the previous discussion, this integral can be performed around any loop that can be continuously deformed into the loop $ C$ while still remaining in the fluid, and not passing over a singularity of the complex velocity, $ dF/dz$ .


next up previous
Next: Blasius Theorem Up: Two-Dimensional Potential Flow Previous: Free Streamline Theory
Richard Fitzpatrick 2016-01-22