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Mathematics of fluid dynamics University of Bath 25-26, MA32051/MA52129

 

Section 4.3 The complex potential

In the previous two sections, we studied the properties and utility of the velocity potential \(\phi\) and streamfunction \(\psi\) in the context of two-dimensional potential flows (inviscid, incompressible, irrotational). We did this with techniques from real-valued Vector Calculus. As it turns out, there is a much more elegant and powerful framework for studying two-dimensional flow which leverages the significant power of complex analysis and complex variables.
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In fact, you may have already noticed this on an intuitive level, given the intimate relationships between \(\phi\) and \(\psi\text{,}\) seeming to exhibit a certain kind of symmetry in formulae and operations. Complex analysis is the language in which we can make this kind of "symmetry" transparent.
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We begin by reviewing (or in some cases, introducing) you to some key theorems about the properties of well-behaved (analytic) complex functions.
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In his section, we will refer to a complex function generically as
\begin{equation*} f(z) = u(x, y) + \im v(x, y) \end{equation*}
where \(u\) and \(v\) and the real-valued decompositions of the complex-valued function \(f\text{.}\) Also, \(z = x + \im y\text{.}\)
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A complex function is differentiated in much the same way as real-valued functions, with the definition that
\begin{equation} f'(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z}.\tag{4.3.1} \end{equation}
The crucial difference with real-valued differentiation is that the above limit is required to hold whilst approaching the point \(z\) in any direction of the complex plane.
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Remark 4.3.1.

As long as you stay away from exceptional points of a function, the "calculus" of complex functions is largely the same as for real-valued functions, e.g.
\begin{gather*} \dd{(z^m)}{z} = mz^{m-1}, \qquad \dd{\e^z}{z} = \e^z,\\ \dd{(\log (z+1))}{z} = \frac{1}{z + 1}, \quad \dd{\sin(z^2)}{z} = 2z\cos(z^2). \end{gather*}
and the usual rules of algebraic manipulations hold. There are some caveats, however, which are dealt with on an individual manner.
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Remark 4.3.2. Examinable content.

The proofs of all theorems from this section are non-examinable, but you are expected to understand the theorems and their relevance to the theory of potential flow.
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Below, we will often use susbcripts for partial differentiation, e.g. \(u_x = \pd{u}{x}\text{.}\)
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Subsection 4.3.1 Cauchy’s theorem and harmonic functions

We will follow the reference text by [2] and [6] and introduce the basic notions of complex functions that we will need.
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Definition 4.3.3. Analyticity.

A function \(f: \mathbb{C} \to \mathbb{C}\) is said to be analytic in a domain if \(f(z)\) is defined and differentiable at all points in the domain. The function is said to be analytic at a point \(z = z_0\) if it is analytic in a neighbourhood of \(z_0\text{.}\)
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When we refer to an analytic function, we mean a function that is analytic in some domain of \(\mathbb{C}\) (often clear by context).
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The function is holomorphic if it is analytic; the terms are synonyms.
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An analytic function is entire if its region of analyticity includes all points in \(\mathbb{C}\text{,}\) including infinity.
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Note that above, we have stated that a function is analytic if it is well-defined and differentiable once(!) As it turns out, the requirement that a complex-valued function is differentiable is a strong condition. An analytic function turns out to be infinitely differentiable by consequence!
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Remark 4.3.4.

In this module, we will not be concerned with formalities when they are not relevant. For example, in our definition of analyticity, we do not specify if the relevant domains are open or closed. The functions we work with are generally non-pathological---they may have isolated singularities or exceptional points, but generally the application and context will make it clear the limits of our results.
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Now we have one of the most important theorems of complex analysis.
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Proof.

(Non-examinable)
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This follows by considering the definition of the derivative (4.3.1) when the point \(z\) is approached from the \(x\) or \(y\) directions.
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With \(\Delta z = \Delta x\) real, we can verify from applying the definition that
\begin{equation*} f'(z) = \pd{u}{x} + \im \pd{v}{x}. \end{equation*}
On the other hand, approaching with \(\Delta z = \im \Delta y\) gives
\begin{equation*} f'(z) = -\im \pd{u}{y} + \pd{v}{y}. \end{equation*}
Equating the two results then yields the Cauchy-Riemann equations.
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It turns out that the Cauchy-Riemann equations are not only necessary to an analytic function, but are actually sufficient as well.
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Proof.

(Non-examinable)
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The proof is not difficult, but we will refer students to [2] for its proof. It relies on constructing the derivative of \(f\) along any direction using the decomposition into the two Cartesian directions.
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Our last step involves relating complex functions to the solution of Laplace’s equation(s), i.e. (4.1.5) or (4.2.5), that govern potential flow.
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Proof.

Since \(f\) is analytic, then it follows from Theorem 4.3.5 that \(u_x = v_y\) and \(u_y = -v_x\text{.}\) It furthermore follows from Theorem 4.3.7 that \(f\) is differentiable to all orders; therefore, we can take derivatives to obtain
\begin{equation*} u_{xx} = v_{yx} \quad \text{and} \quad -u_{yy} = v_{xy}. \end{equation*}
The second derivatives are continuous and therefore \(v_{xy} = v_{yx}\text{.}\) Therefore \(u_{xx} + u_{yy} = 0\text{.}\) Laplace’s equation is analogously proved for \(v\text{.}\)
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The above theorem is truly a remarkable result; it would not be a stretch to state that this single result was at the forefront of why complex analysis played such an important role in the development of applied mathematics and physics in the 18th, 19th, and 20th centuries.
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Since Laplace’s equation, \(\nabla^2 \phi = 0\text{,}\) is such an important equation in physics, occuring in theories of gravitation, electrostatics, fluid mechanics, etc. the theorem establishes that there exists a parallel theory in the language of complex variables for the specific case of two-dimensional applications.
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A potential fluid, for example, can be studied by manipulating complex functions of the form
\begin{equation*} f(z) = \phi(x, y) + \im \psi(x, y). \end{equation*}
One can envision all kinds of different forms of \(f\)---polynomials, cosines and sines, exponentials, logarithms, etc. As long as the function is locally differentiable, it is thus analytic and therefore can be associated with (some kind of) fluid flow.
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Subsection 4.3.2 Examples of elementary flows

Let us return to the examples of flows studied in Section 4.1 and re-interpret them using the theory of analytic functions.
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We will associate the velocity potential, \(\phi\text{,}\) and streamfunction \(\psi\) with an analytic function in the following way.
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Definition 4.3.9.

Let \(\phi(x,y)\) and \(\psi(x, y)\) be the respective velocity potential and streamfunction for some potential flow. We define
\begin{equation} f(z) = \phi(x, y) + \im \psi(x, y),\tag{4.3.3} \end{equation}
and call \(f\) the complex potential.
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Note that by differentiating in the horizontal and vertical directions, we have
\begin{equation*} \dd{f}{z} = \pd{}{x} (\phi + \im \psi) = \pd{}{y} (\phi + \im \psi). \end{equation*}
Therefore it follows that the horizontal and vertical velocity components of the flow, related via \(\bu = [u, v]\text{,}\) are given by
\begin{equation} \dd{f}{z} = u - \im v.\tag{4.3.4} \end{equation}
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Example 4.3.10. Uniform flow.

We can verify that the complex potential for uniform flow is given from
\begin{equation*} f(z) = U \e^{-\im \alpha} z. \end{equation*}
This can be compared to Example 4.1.3.
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Example 4.3.11. Stagnation point flow.

We can verify that the complex potential for a stagnation point flow is given by
\begin{equation*} f(z) = \frac{z^2}{2}. \end{equation*}
This can be compared to Example 4.1.5.
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Example 4.3.12. Line source.

We can verify that the complex potential for a line source flow is given by
\begin{equation} f(z) = \frac{Q}{2\pi} \log z\tag{4.3.5} \end{equation}
This can be compared to Example 4.1.7.
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The complex logarithm is an example of a function that is not analytic at the isolated point \(z = 0\) where it possesess a branch point. However, it still provides a permissible analytic function away from the origin.
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The evaluation of the complex logarithm can be performed via the definition
\begin{equation} \log z = \log r + \im \theta,\tag{4.3.6} \end{equation}
where \(z = r\e^{\im \theta}\) is the polar form representation of \(z\text{.}\) In particular \(log z\) is a multi-function with a branch point at the origin.
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With the above decomposition of the logarithm in mind, notice that we can then conclude that the velocity potential and streamfunction are given by
\begin{equation*} \phi = \Re[f] = \frac{Q}{2\pi} \log r \quad \text{and} \quad \psi = \Im[f] = \frac{Q}{2\pi} \theta. \end{equation*}
Indeed the streamlines are along the rays \(\theta\) constant.
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From (4.3.5), we can also compute the velocities using the relationship (4.3.4). We thus have
\begin{equation*} u - \im v = \frac{Q}{2\pi} \frac{1}{z} = \frac{Q}{2\pi} \frac{x - \im y}{(x^2 + y^2)}, \end{equation*}
once we have multiplied the top and bottom by the conjugate of \(z\text{.}\)
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Example 4.3.13. Line vortex flow.

We can verify that the complex potential for a line vortex is given by
\begin{equation} f(z) = -\frac{\im \Gamma}{2\pi} \log z.\tag{4.3.7} \end{equation}
This can be compared to Example 4.2.10.
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Again, using the definition of the complex logarithm, via (4.3.6), we can write \(f\) in terms of its real and complex components as
\begin{equation*} f(z) = \frac{\Gamma}{2\pi} (\theta - \im \log r), \end{equation*}
where \(z = r\e^{\im\theta}\text{.}\)
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Therefore, the streamfunction is given by \(\psi = -(\Gamma/2\pi) \log r\) and is constant along circular trajectories with constant distance from the origin, \(r\text{.}\)
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