Streamlines from stream function

Given: $\psi = -6Ax - 8Ay$ with $A = 1$ m/s, so $\psi = -6x - 8y$. Using $u = \partial\psi/\partial y$ and $v = -\partial\psi/\partial x$.

(i) Streamlines

• $\psi = 0$: $-6x - 8y = 0\Rightarrow y = -0.75x$
• $\psi = 8$: $-6x - 8y = 8\Rightarrow y = -0.75x - 1$

Both are parallel straight lines of slope $-0.75$ ($-36.9^\circ$ to the x-axis). The second line cuts the y-axis at $y = -1$ m and the x-axis at $x = -1.333$ m.

 y
   |   \  
   |    \ psi=0  (y = -0.75x)
   |     \
---+------\------------ x
   |\      \
   | \ psi=8  (y = -0.75x - 1)

(ii) Velocity at (0, 0)

$$u = \frac{\partial\psi}{\partial y} = -8A = -8\ \text{m/s},\qquad v = -\frac{\partial\psi}{\partial x} = 6A = 6\ \text{m/s}$$ $$|V| = \sqrt{8^2 + 6^2} = 10\ \text{m/s},\qquad \theta = \tan^{-1}\frac{6}{-8} = 143.1^\circ\ \text{from +x}$$

The velocity is uniform everywhere (the same at every point) and parallel to the streamlines.

(iii) Flow rate between (1, 1) and (4, 1)

$$\psi_{(1,1)} = -6 - 8 = -14\ \text{m}^2/\text{s},\qquad \psi_{(4,1)} = -24 - 8 = -32\ \text{m}^2/\text{s}$$ $$q = |\psi_2 - \psi_1| = |-32 - (-14)| = 18\ \text{m}^2/\text{s per metre depth}$$

Answer: (i) $y = -0.75x$ and $y = -0.75x - 1$; (ii) $\vec V = (-8\hat i + 6\hat j)$ m/s, $10$ m/s at $143.1^\circ$ to +x; (iii) $q = 18$ m²/s per unit width.