If you would like feedback on your work, the tutor will by default concentrate on the exercises highlighted with a yellow background in the online notes (the homework exercises), unless you specifically ask for feedback on a different exercise instead.
There is an excellent website at www.potentialflows.com that allows you to plug in different flow elements (sources/sinks, vortices, etc.) into a potential flow and observe the streamlines and potential-flow lines. In your exercises, use this to help you visualise the flow.
The following question relates to two-dimensional potential flow. Remember that the fluid flux through a surface given by contour \(C\) is given by (2.2.2), or
Hint: the unit normal for the circle is \(\bn = [\cos\theta, \sin\theta]\text{;}\) when converting to polar coordinates remember that \(\de{s} = r\de{\theta}\text{.}\)
Circle. Parametrise the unit circle by \([x, y] = [\cos\theta, \sin\theta]\) for \(0 \leq \theta \lt 2\pi\text{.}\) Explain why the outward unit normal is \(\bn = [\cos\theta, \sin\theta]\) and why \(\de{s} = \de\theta\text{,}\) so that the flux becomes \(\int_0^{2\pi} \bu \cdot \bn \, \de\theta\text{.}\) Before integrating, can you predict the answer?
Plate. Parametrise the plate by \(\br(t) = [t, 1 - t]\) for \(0 \leq t \leq 1\text{.}\) Show that the tangent is \(\br'(t) = [1, -1]\text{,}\) and hence that a unit normal, pointing away from the origin, is \(\bn = [1, 1]/\sqrt{2}\text{.}\) Show also that \(\de{s} = |\br'(t)| \, \de{t} = \sqrt{2} \, \de{t}\text{,}\) so that the flux becomes \(\int_0^1 \bu \cdot \bn \, \sqrt{2} \, \de{t}\text{.}\)
For the plate, use the same parametrisation and normal as in part (a), and write the velocity in Cartesian form, \(\bu = \frac{Q}{2\pi}\frac{[x, y]}{x^2 + y^2}\text{.}\) You may use, without proof, that
For the plate, you should find an integral of the form \(\int_0^1 \frac{2t - 1}{t^2 + (1 - t)^2}\,\de{t}\text{.}\) Compare the numerator with the derivative of the denominator, or consider what happens under the substitution \(t \mapsto 1 - t\text{.}\)
Return to the previous question and, instead of directly calculating the flux via a line integral, use Theorem 4.2.2 to calculate the flux for the two geometries (unit circle and straight plate) for each of the flows given.
Write down the complex potential, \(f(z)\text{.}\) Find \(f'(z)\text{.}\) Locate any stagnation points and derive an equation for the streamlines of the flow. Finally, use a plotter, such as the one at the potential flow simulator to sketch the streamlines.
Let \(a \to 0\) and \(Q \to \infty\) while keeping the product \(aQ\) fixed. This gives the flow due to a doublet. Show that its complex potential is \(\mu/z\) where \(\mu\) is expressed in terms of \(a\) and \(Q\text{.}\) It will be useful for you to use the fact that
considered as an appropriate approximation when \(\alpha\) is small. Show that the streamlines are circles through the origin with centres on the \(y\)-axis.
Verify that a single source of strength \(Q\) placed at the point \(z = d > 0\) is insufficient to describe flow bounded in the semi-infinite region, \(x > 0\text{,}\) with a planar boundary at \(x = 0\text{.}\) What is the horizontal and vertical velocities on the boundary? Find an equation for the streamlines and sketch the flow.
Note: the algebra quickly becomes awkward, whatever form you use. In this part, you are not expected to simplify beyond writing the streamfunction in a form such as \(\psi = \frac{Q}{2\pi}\operatorname{Arg}\big[(x + \im y)^2 - d^2\big]\text{.}\) You may then confirm the shape of the streamlines, \(\psi = \textrm{constant}\text{,}\) with a plot.
Finally, recall that any complex number can be written as \(w = a + \im b = |w|\e^{\im\beta}\text{,}\) where \(\beta = \operatorname{Arg} w\text{.}\) Comparing real and imaginary parts gives \(a = |w|\cos\beta\) and \(b = |w|\sin\beta\text{,}\) so that \(b/a = \tan\beta\text{.}\)
Multiply out the result of part (b), writing \(k = \cot\beta\text{,}\) and rearrange it into the form \(Ax^2 + Bxy + Cy^2 = D\text{.}\) Recall that such a conic is a hyperbola when \(B^2 - 4AC \gt 0\text{.}\)
Consider the situation of two point vorticies of identical strength, but opposite direction, placed at \(z = \pm d\text{,}\) with \(d \gt 0\text{.}\) Develop equations for the complex potential, \(f(z) = \phi + \im \psi\text{,}\) and complex velocity, \(u - \im v\text{.}\)
Incompressible inviscid fluid occupies the region \(y > 0\text{,}\) and there is a rigid plane wall at \(y = 0\text{.}\) There is a uniform flow, speed \(U\text{,}\) in the positive \(x\)-direction, and a line source of strength \(Q\) at \((0, a)\text{,}\) where \(a > 0\text{.}\) Find the complex potential \(f(z)\) and calculate \(f'(z)\text{.}\) Let \(\beta = Q/(2\pi aU)\text{.}\) Show that if \(\beta > 1\) there are two stagnation points, both on the wall, while if \(\beta < 1\) there is only one, in the fluid, a distance \(a\) from the origin. Try to sketch the streamlines in either case
Check where each of the labeled points in the associated diagrams for the exponential map (4.5.4) and (4.5.5) are sent, from the \(z\)-plane to the \(\zeta\)-plane. In each of these cases, verify that an interior (within the fluid domain) is sent to the upper half-plane.
Construct the complex potential that corresponds to a single line source placed within the fluid in each of these cases (an infinite strip of height \(h\) and a semi-infinite strip).
Write down the conformal maps from the wedge \(0 < \textrm{arg}(z) < \alpha\) into the upper half-plane. Find all the points at which the map is not conformal.
Construct a flow, by providing the complex potential \(\phi + \im \psi\text{,}\) that consists of: uniform flow oriented at an angle \(\alpha\) to the horizontal at infinity; a line source of strength \(Q\) placed in the flow; and a horizontal flat plate of length \(4a\) on the axis, \(-2a < z < 2a\text{.}\) Sketch what you believe will be the streamlines of such a flow.
