Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

7: Two Dimensional Hydrodynamics and Complex Potentials

  • Page ID
    6514
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Laplace’s equation and harmonic functions show up in many physical models. As we have just seen, harmonic functions in two dimensions are closely linked with complex analytic functions. In this section we will exploit this connection to look at two dimensional hydrodynamics, i.e. fluid flow. Since static electric fields and steady state temperature distributions are also harmonic, the ideas and pictures we use can be repurposed to cover these topics as well.

    • 7.1: Velocity Fields
      A velocity field is a vector function that denotes the velocity as a function of the spatial and time coordinates.
    • 7.2: Stationary Flows
      If the velocity field is unchanging in time we call the flow a stationary flow. In this case, we can drop t as an argument.
    • 7.3: Physical Assumptions and Mathematical Consequences
      We will make some standard physical assumptions. These don’t apply to all flows, but they do apply to a good number of them and they are a good starting point for understanding fluid flow more generally. More importantly, these are the flows that are readily susceptible to complex analysis. Here are the assumptions about the flow, we’ll discuss them further below: (1) The flow is stationary, (2) The flow is incompressible, and (3) The flow is irrotational.
    • 7.4: Complex Potentials
      We’ll start by seeing that every complex analytic function leads to an irrotational, incompressible flow. Then we’ll go backwards and see that all such flows lead to an analytic function. We will learn to call the analytic function the complex potential of the flow.
    • 7.5: Stream Functions
      In everything we did above poor old ψ just tagged along as the harmonic conjugate of the potential function ϕ . Let’s turn our attention to it and see why it’s called the stream function.
    • 7.6: More Examples with Pretty Pictures

    • Was this article helpful?