# Applications of

Conformal Mappings

## Hydrodynamics

If we have a (steady-state) incompressible, nonviscous fluid, we are interested
in finding its velocity field $$\mathbf V (x,y)= \left(u(x,y), v(x,y)\right).$$
From vector analysis we know that 'incompressible' means that the divergence
$\text{div}\,\mathbf V =0.$ (We say $\mathbf V$ is *divergence free.*)
We assume that $\mathbf V$ is also a *potential flow* and hence is circulation
free; that is $\mathbf V = \text{grad } \phi $ for some $\phi$ called the *velocity
potential.* Thus $\phi$ is harmonic because $$\nabla^2\phi = \text{div } \text{grad }\phi =
\text{div } \mathbf V=0.$$
Thus when we solve for $\phi$ we can obtain $\mathbf V$ by taking $\mathbf V = \text{grad } \phi$. That
is
\begin{eqnarray*}
u=\frac{\partial \phi }{\partial x},\quad v=\frac{\partial \phi }{\partial y}.
\end{eqnarray*}

The conjugate $\psi$ of the harmonic function $\phi$ (which will exist on any simple
connected region) is called the *stream function*, and the analytic function
$$F=\phi +i\psi$$ is called the *complex potential*.

The stream function must satisfy
\begin{eqnarray*}
u=\frac{\partial \psi }{\partial y},\quad v=-\frac{\partial \psi }{\partial x}.
\end{eqnarray*}
Finally, lines of constant $\psi$
have $\mathbf V$ as their tangents, so lines of constant $\psi$ may be interpreted as
the *lines along which particles of fluid move*; hence the name stream function.