6.1: Applications of Conformal Mappings

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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 \(ψ\) of the harmonic function \(ϕ\) (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 \(ψ\) have \(V\) as their tangents, so lines of constant \(ψ\) may be interpreted as the lines along which particles of fluid move; hence the name stream function.