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The streamlines are represented by $\psi = x^2 - y^2 = C$ where $C$ is a constant.We need to calculate the velocity and its direction at $(2,2)$, we need to sketch the streamlines and show the direction of flow.We found $q = u\hat{i} + v\hat{j}$ where $u = 4$ and $v = -4$ at point $(2,2)$, the slope is 1 so the velocity vector is oriented $45$ degree angle to the $x -$ axis.

We plot the streamlines that is the family of hyperbolas as below.But the main query is how can we decide the direction of the arrows?, as shown in the diagram one is moving upwards and the other downwards.

enter image description here

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  • $\begingroup$ For a 2D flow, $u = \partial_y\psi$ and $v = -\partial_x\psi$. $\endgroup$ – Chee Han Mar 25 '18 at 20:55
  • $\begingroup$ I'm not entirely sure, but I think you are on the right track here. By definition, streamlines are tangential to the flow field at a fixed time $t$. The flow field in this case is given by $(u,v) = -2(y,x)$ and I believe this yields the direction of the tangent vector for the level sets of streamlines. One last thing: you might also want to consider the case where $\psi = \textrm{negative constant}$. $\endgroup$ – Chee Han Mar 25 '18 at 21:19
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As mentioned by Chee Han, $(u, v) = -2(y, x)$. To get a feeling for the plot of the streamlines it is often easiest to consider $\psi = 0$, which corresponds to $\pm y = \pm x$, so the two lines $y=x$ and $y=-x$.

This hints at the fact that you are missing two families of lines, corresponding to negative $\psi$-values. In fact, every value of $\psi \neq 0$ corresponds to a hyperbola.

A useful fact to remember is that, roughly speaking, streamlines close to each other point in similar directions. Abrupt changes of direction are impossible (due to the solution function depending continuously on the initial conditions).

If you probe the axes now to find the direction of the velocity there you find that it points up on the positive $x$-axis, left on the positive $y$-axis, down on the negative $x$-axis and right on the negative $y$-axis.

This information should be everything you need to plot all the streamlines.

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In continuation to the above comments; in the positive quadrant for instance, you may determine the direction of flux here by considering the velocities at these points.

This might be helpful. https://books.google.co.in/books?id=_Jo-T8nBghYC&pg=PA285&dq=hyperbola+streamline&hl=en&sa=X&ved=0ahUKEwj2speb2fbfAhVJ2LwKHVXtAAYQ6AEIDzAB#v=onepage&q=hyperbola%20streamline&f=false

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