Theorem 4.2.1. The streamfunction is constant along streamlines.
The streamfunction, \(\psi(x, y, t)\text{,}\) is constant along streamlines of the flow (i.e. the trajectory formed by a particle in the flow).
Section 4.2 The streamfunction
Our next task is to introduce the concept of the streamfunction. Remember that in 2D, the irrotational flow led to the equation (4.1.2) and this led to the existence of the potential function. If we begin with incompressibility, however, we have (4.1.4), which can be written as
\begin{equation}
\nabla \times [-v, u] = 0.\tag{4.2.1}
\end{equation}
And we can deduce the existence of an analogous function, the streamfunction, \(\psi(x, y, t)\) satisfying,
\begin{equation}
u = \pd{\psi}{y} \quad \textrm{and} \quad v = -\pd{\psi}{x}.\tag{4.2.2}
\end{equation}
Alternatively and more conveniently, we can write
\begin{equation}
\bu = \nabla \times (\psi \bk).\tag{4.2.3}
\end{equation}
To establish the existence of the streamfunction, we follow a similar proof as in Theorem 4.1.1 but now with the definition that
\begin{equation}
\psi(x, y, t) = \psi_0(t) + \int_0^{\bx} (u \de{y} - v \de{x}),\tag{4.2.4}
\end{equation}
where again \(\psi_0(t)\) is an arbitrary function of \(t\text{.}\) The proof is otherwise identical, relying on establishing the independence of path of the integral using Stokes’ theorem.
Why all this work? The streamfunction has an intuitive intepretation via the following result.
Proof.
The proof follows simply by the fact that
\begin{equation*}
\bu \cdot \nabla \psi = \left(\pd{\psi}{y}, -\pd{\psi}{x}\right) \cdot \left(\pd{\psi}{x}, \pd{\psi}{y}\right) = 0,
\end{equation*}
and for which the substitution in the first equality follows from (4.2.2).
The above equality indicates that the velocity vector, \(\bu\text{,}\) is orthogonal to the vector pointing along \(\nabla \psi\text{.}\) However, it is known from Vector Calculus that \(\nabla \psi\) runs along curves of steepest descent/ascent of \(\psi\)---these must hence be orthogonal to the level sets of \(\psi\text{.}\) Therefore, the level sets of \(\psi\) are tangential to \(\bu\) (the definition of a streamline).
A graphical depiction of the property in Theorem 4.2.1 is shown below.
The streamfunction is thus constant on streamlines. Consider two streamlines. The following theorem relates the flux between the streamlines to the streamline values.
Theorem 4.2.3. Flux between streamlines.
Consider two streamlines \(\psi = \psi_1\) and \(\psi = \psi_2\text{.}\) We assume that the streamlines pass through the points A and B respectively. Consider a contour \(C\) connecting A and B. The flux (net flow of fluid) through \(C\) and hence between the streamlines is
\begin{equation*}
|\textrm{flux}| = |\psi_2 - \psi_1|
\end{equation*}
We have not specified the sign of the flux as it is subject to the considered direction through \(C\text{.}\)
Proof.
By definition, the flux is given by the integral
\begin{equation*}
\int_C \bu \cdot \bn \, \de{s},
\end{equation*}
where \(C\) is any smooth path joining the two streamlines with unit normal \(\bn\text{,}\) as shown in Figure 4.2.2.
Note that given the curve, and in consideration of a small arclength element \(\de{s}\text{,}\) the tangent and normal are given by
\begin{equation*}
\bt = \frac{[\de{x}, \de{y}]}{\sqrt{\de{x}^2 + \de{y}^2}} = \frac{[\de{x}, \de{y}]}{\de{s}}, \qquad
\bn = \frac{[-\de{y}, \de{x}]}{\de{s}}.
\end{equation*}
Therefore we can write \(\bn \de{s} = [\de{y}, -\de{x}]\text{.}\) Then the flux is re-written as the following:
\begin{align*}
\textrm{flux} \amp= \int_C \left[\pd{\psi}{y}, -\pd{\psi}{x}\right] \cdot [\de{y}, \de{x}], \\
\amp= \int_C \left(\pd{\psi}{x} \, \de{x} + \pd{\psi}{y} \right), \\
\amp= \bigl[ \psi \bigr]_C, \\
\amp= \psi(B) - \psi(A), \\
\amp= \psi_2 - \psi_1,
\end{align*}
In the third line above, \([\psi]_C\) is the change of \(\psi\) across the contour. Note that there is somewhat an arbitrary choice of direction for the contour \(C\text{,}\) as related to the selection of the normal direction \(\bn\text{,}\) and the positivity or negativity of the flux. To be safe, we have taken the absolute value in the problem.
An example of the use of the above theorem is given in Exercise 4.7.2.
Example 4.2.4.
Let the velocity of a two-dimensional flow be given by the formula
\begin{equation*}
\bu = [3ax^2 - 3ay^2, \, -6axy]
\end{equation*}
where \(a\) is a positive constant. By finding the streamfunction, \(\psi\text{,}\) calculate the volume flux across a curve connecting points \(A = (0, 0)\) and \(B = (1, 1)\) using Theorem 4.2.3.
Solution.
(Done in problem class) We can check, using the integration of velocity components that
\begin{equation*}
\psi = 3ax^2 y - a y^3 + C.
\end{equation*}
Hence the flux is
\begin{equation*}
Q = \psi(B) - \psi(A) = 2a.
\end{equation*}
We can verify this flux should be positive if the normal is oriented to the right of the curve from A to B.
Finally, note that the velocity potential was governed by Laplace’s equation, \(\nabla^2 \phi = 0\text{.}\) The streamfunction is also governed by the same Laplace’s equation.
Theorem 4.2.5. Streamfunction satisfies Laplace’s equation.
Like the velocity potential, the streamfunction satisfies Laplace’s equation:
\begin{equation}
\pd{^2 \psi}{x^2} + \pd{^2 \psi}{y^2} = \nabla^2 \psi = 0.\tag{4.2.5}
\end{equation}
Like in the situation of certain flows, e.g. the line source in Example 4.1.7, it is easier to work in alternative coordinate systems to study the streamfunction. Since we know that \(\bu = \nabla \times (\psi \bk)\) by (4.2.3), we can use the conversion of the curl to polar coordinates in (A.1.31) to give
\begin{equation}
u_r = \frac{1}{r} \pd{\psi}{\theta} \quad \textrm{and} \quad u_{\theta} = -\pd{\psi}{r}\,\tag{4.2.6}
\end{equation}
which allows us to relate the radial and angular velocities to the streamfunction.
Subsection 4.2.1 Elementary flows
Let us return to each of the examples in Section 4.1 and reconsider their corresponding streamfunctions.
Example 4.2.6. Uniform flow.
\begin{equation*}
\psi = Uy\cos\alpha - Ux\sin\alpha,
\end{equation*}
up to an arbitrary constant. Therefore, lines of constant \(\psi\) correspond to
\begin{equation*}
y = x\tan\alpha + \textrm{constant},
\end{equation*}
which indeed yields the image seen in Figure 4.1.4.
Example 4.2.7. Stagnation point flow.
Now turning to Example 4.1.5, we integrate \(u = y = \psi_x\) and \(v = x = \psi_y\text{.}\) This gives
\begin{equation*}
\psi = xy,
\end{equation*}
up to an arbitrary constant. Indeed curves of constant \(\psi\) match the hyperbole shown in Figure 4.1.6.
Example 4.2.8. Line source.
For the situation of the line source in Example 4.1.7, we want to work with polar coordinates. From the previous work, we have for this situation the potential \(\phi = Q/(2\pi log r\text{.}\) It then follows from the polar version of the gradient in (A.1.30), that the velocity is entirely radial and
\begin{equation}
\bu = \pd{\phi}{r}\be_r + 0 = \frac{Q}{2\pi r} \be_r.\tag{4.2.7}
\end{equation}
We use the formulae in (4.2.6) and integrate \(\frac{1}{r} \pd{\psi}{\theta} = \frac{Q}{2\pi r}\) yielding
\begin{equation*}
\psi = \frac{Q}{2\pi} \theta.
\end{equation*}
Then indeed note that the lines of constant \(\psi\) are given by the radial lines of constant \(\theta\text{,}\) matching the illustration in Figure 4.1.8.
Another remark concerns the fact that \(\psi\) is a multi-valued function in the example of the line source Example 4.2.8, gaining a jump of \(Q\) every time the origin is encircled. This is indeed a warning that the standard proof, analogous to Theorem 4.1.1, leading to the existence of a unique streamfunction, via (4.2.4) would not apply since the velocity field Example 4.2.8 is not defined at the origin. However, despite this, we see that the streamfunction provides well-defined predictions of streamlines on the cut plane with e.g. \(\theta \in [0, 2\pi)\text{.}\)
Example 4.2.9. Line source in a uniform flow.
Like the case of the potential function in Example 4.1.9, the linearity of the equation governing the streamfunction implies that we can consider the combination of those streamfunctions for a line source with a uniform flow. This yields
\begin{equation*}
\psi = Uy + \frac{Q}{2\pi}\theta,
\end{equation*}
for the case of uniform flow of speed \(U\) in the positive \(x\)-direction. The streamlines are then given by
\begin{equation*}
Uy + \frac{Q}{2\pi} \theta = \frac{Q}{2\pi}C,
\end{equation*}
having designed the constant combination on the right hand-side for convenience. Then using \(y = r\sin\theta\text{,}\) we have
\begin{equation*}
r = \left(\frac{Q}{2\pi U}\right) \frac{C - \theta}{\sin\theta}.
\end{equation*}
Our previous examples were reliant on considering the flows generated by the velocity potentials studied previously. However, we can also consider "fundamental solutions" of the equation \(\nabla^2 \psi = 0\) in their own right. Recall that in deriving the velocity potential of the line source in Example 4.1.7, we considered the solution of an axi-symmetric problem, where \(\phi = \phi(r)\) is only dependent on the distance from the origin. A similar argument must imply that the analogous axi-symmetric streamfunction is a permissible solution. And this leads us to the following example.
Example 4.2.10. Line vortex.
The fundamental solution for the streamfunction, in plane polar coordinates, is the axisymmetric solution,
\begin{equation}
\psi = -\frac{\Gamma}{2\pi}\log r,\tag{4.2.8}
\end{equation}
defined up to a constant, and corresponds to a line vortex.
The streamlines of such a flow correspond to circular trajectories with \(r\) constant, and are visualised in
Recall that the radial and angular velocities are given by (4.2.6). Therefore, we see that the velocity vector is given by
\begin{equation}
\bu = -\pd{\psi}{r} \be_{\theta} = \frac{\Gamma}{2\pi r} \be_{\theta},\tag{4.2.9}
\end{equation}
and thus this flow corresponds to entirely circular trajectories orbiting the origin, and with angular velocity increasing as \(r \to 0\text{.}\)
The quantity \(\Gamma\) is called the vortex strength, analogous to the source strength \(Q\) in Example 4.1.7. Let us consider the amount of circulation around a contour \(C\) that contains the origin:
\begin{equation*}
\oint_C \bu \cdot \de{\bx} = \oint_C (u \, \de{x} + v \, \de{y}),
\end{equation*}
i.e. one envisions encircling the origin along \(C\text{,}\) adding up each of the velocity components tangential to the path. This is the circulation. By Stokes’ theorem it is equal to the flux of vorticity of the corresponding bounding surface.
Choosing \(C\) to be the circle of radius \(r = a\text{,}\) we have \(\de{\bx} = a \be_{\theta} \de{\theta}\text{.}\) Then the circulation is given by
\begin{equation}
\oint_C \bu \cdot \de{\bx} = \int_0^{2\pi} \frac{\Gamma}{2\pi a} \be_{\theta} \cdot \be_{\theta} a \de\theta = \Gamma.\tag{4.2.10}
\end{equation}
So indeed this gives us an intuitive understanding of \(\Gamma\text{.}\) Notice that \(\Gamma \gt 0\) corresponds to flow in the anticlockwise sense, and \(\Gamma \lt 0\) corresponds to flow in the clockwise sense.
In Exercise 4.7.1, you will practice the computation of fluxes using line integrals.
