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Chapter 6: Problem 15

For each of the following stream functions, with units of \(\mathrm{m}^{2} / \mathrm{s},\) determine the magnitude and the angle the velocity vector makes with the \(x\) axis at \(x=1 \mathrm{m}, y=2 \mathrm{m} .\) Locate any stagnation points in the flow field. (a) \(\psi=x y\) (b) \(\psi=-2 x^{2}+y\)

Short Answer

Expert verified
The velocity vectors for the stream functions are: for \(\psi = xy\), \(V = \sqrt{5} ms^{-1}\), \(\theta = 180 - 63.4 = 116.6^\circ\), and stagnation at (0,0); for \(\psi = -2x^2 + y\), \(V = 4 ms^{-1}\), \(\theta = -90^\circ\), and stagnation at x = 0 for all y.

Step by step solution

01

Part (a) - Find the velocity components

Given the stream function for part (a) as \(\psi = xy\). Now, calculate the velocity components.\nFor u: \(u = -\partial\psi/\partial y = -x\)\nFor v: \(v = \partial\psi/\partial x = y\)
02

Part (a) - Calculate the magnitude and angle of the velocity vector

The velocity at (1,2) is given by substituting x = 1 m and y = 2 m into the equations for u and v. Hence, \(u = -1 ms^{-1}\) and \(v = 2 ms^{-1}\). The magnitude of the velocity, \(V\), is obtained using \(V = \sqrt{(-1)^2 + 2^2} = \sqrt{5} ms^{-1}\). The direction, \(\theta\), is given by \(\theta = \arctan(2/-1)\) which lies in the second quadrant. Add 90 degrees to the obtained angle to correct the quadrant.
03

Part (a) - Find the stagnation points

Stagnation points are at (x,y) where u = v = 0. For the stream function \(\psi = xy\), we get the stagnation point at (0,0).
04

Part (b) - Find the velocity components

Given the stream function for part (b) as \(\psi = -2x^2 + y\). Now, calculate the velocity components.\nFor u: \(u = -\partial\psi/\partial y = 0\)\nFor v: \(v = \partial\psi/\partial x = -4x\)
05

Part (b) - Calculate the magnitude and angle of the velocity vector

The velocity at (1,2) is given by substituting x = 1 m into the equation for v. Hence, \(v = -4 ms^{-1}\). The magnitude of the velocity, \(V\), is \(\sqrt{0^2 + (-4)^2} = 4 ms^{-1}\). The direction, \(\theta\), is given by \(\theta = \arctan(-4/0)\). As this is undefined, it means the angle is 90 degrees in the negative direction or -90 degrees.
06

Part (b) - Find the stagnation points

Stagnation points are at x where v = 0. For the stream function \(\psi = -2x^2 + y\), we get the stagnation point at x = 0, independent of y.

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