Chap 10. Stream function and complex potential

10.1 Stream Function

Continuity equation

$\nabla\cdot\vec{u}=0$

by

\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\]

Let

\[u=\frac{\partial\psi}{\partial y},\quad v=-\frac{\partial\psi}{\partial x}\]

to make $\psi$ satisfy continuity equation

10.2 Complex Potential

In a complex domain, define a function such that

\[f(z)=\phi(x,y)+i\psi(x,y)\]

If $f(z)$ is analytic function (differentiable for $\forall z$) the necessary condition is

\[\frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial y},\quad\frac{\partial\psi}{\partial x}=-\frac{\partial\phi}{\partial y}\]

So-called Cauchy-Riemann condition

$z=x+iy$, $\frac{\partial z}{\partial x}=1,\quad\frac{\partial z}{\partial y}=i$

\[\frac{\partial f}{\partial x}=\frac{\partial\phi}{\partial x}+i\frac{\partial\psi}{\partial x}\] \[=\frac{\partial z}{\partial x}\left(\frac{\partial\phi}{\partial z}+i\frac{\partial\psi}{\partial z}\right)\to\frac{\partial f}{\partial z}=\frac{\partial\phi}{\partial x}+i\frac{\partial\psi}{\partial x}\] \[\frac{\partial f}{\partial y}=\frac{\partial\phi}{\partial y}+i\frac{\partial\psi}{\partial y}=\frac{\partial z}{\partial y}\frac{\partial f}{\partial z}\] \[\frac{\partial f}{\partial z}=\frac{\partial\psi}{\partial y}+i\frac{\partial\phi}{\partial x}\] \[\to\frac{\partial\phi}{\partial x}=-\frac{\partial\psi}{\partial y},\quad\frac{\partial\psi}{\partial x}=\frac{\partial\phi}{\partial y}\]

Differentiate with $x$,

\[\frac{\partial^2\phi}{\partial x^2}=\frac{\partial}{\partial x}\left(\frac{\partial\psi}{\partial y}\right)=\frac{\partial}{\partial y}\left(-\frac{\partial\psi}{\partial x}\right)=-\frac{\partial^2\phi}{\partial y^2}\] \[\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}=0\]

$\nabla^2\phi=0$

Similarly, $\nabla^2\psi=0$

\[f(z)=\phi(x,y)+i\psi(x,y)\] \[u=\frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial y}\] \[v=\frac{\partial\phi}{\partial y}=-\frac{\partial\psi}{\partial x}\] \[\Rightarrow \Phi(x,y)=\phi(x,y)+i\psi(x,y)\]

10.3 Streamline

\[\frac{dx}{u}=\frac{dy}{v} \quad \text{ or }\quad v\,dx-u\,dy=0\]

Value of stream function is constant through streamline

\[u=\frac{\partial\psi}{\partial y},\quad v=-\frac{\partial\psi}{\partial x}\] \[\frac{\partial\psi}{\partial x}dx+\frac{\partial\psi}{\partial y}dy=d\psi=0\]

All velocities are tangent to the line

\[\frac{\phi_2-\phi_1}{\delta x}=u\] \[u=\frac{\psi_1-\psi_2}{\delta y}\]

• Problems of Streamfunction : Cannot be defined in 3D

Some representative Stream function

• Uniform Flow

\[u=U, v=V, \psi = -Vx+Uy\]

• 2D Source

\[\psi = {m\over 2 \pi}\theta\]

• 2D Source with uniform flow

\[\psi = Ur\sin\theta + {m \over 2\pi } \theta\]

• 2D doublet

\[\psi = -{\mu y \over x^2+y^2} = -{\mu \sin\theta \over r}\]

• 2D Point vortex

\[\psi = -{\Gamma \over 2 \pi} \ln r\]

• 2D source in Uniform Flow

\[\psi = Ur\sin \theta + {m\over 2\pi} \theta\]

• 3D axi-symmetric problem for source in uniform flow

\[\psi = {1\over 2} Ur^2 \sin^2 \theta - {m\over 4\pi}\cos\theta\]

10.4 Some Representative Complex Potential

• Uniform Flow

\[\Phi = Az\]

• Source

\[\Phi = c\ln z\]

• Vortex

\[\Phi = -ic\ln z\]

• Corner flow

\[\Phi = Uz^n, n={\pi \over \alpha}\]

$\alpha$ is angle of corner (from x+ direction)

• Dipole with angle $\alpha$

\[\Phi = {c\over z}e^{i\alpha}\]

• Circular cylinder + Point vortex

\[\Phi = U(z+{R^2\over z})+{i\Gamma \over 2\pi}\ln {z\over R}\]

where

\[u_r = U(1-{R^2\over r^2}) \cos\theta\] \[u_\theta = -U(1+{R^2\over r^2})\sin\theta - {\Gamma \over 2\pi r}\]

$r=R, u_r=0$

\[u_{\theta} = -U(1+{R^2\over r^2})\sin\theta - {\Gamma \over 2\pi r}\]

To find stagnation pnt, $u_\theta = 0$

\[\theta = \sin^{-1}\big({-\Gamma\over 4U\pi R}\big)\]

If the absolute value inside the parenthesis <1, Two stagnation pnt generates

=1 : 1

1 : No stagnation pnt on body