Question

A stream function is given by   5x  6y . calculate the velocity components and also magnitude with direction of resultant velocity at any point

Answer 1

A stream function is given by   5x  6y . calculate the velocity components and also magnitude with direction of resultant velocity at any point

Answer 2

Question: Stream function is given by ω = 5x - 6y.

Answer:

Velocity components are $u=-6$ and $v=-5$.

Magnitude of resultant velocity is $\sqrt{61}$ in positive x - axis direction.

Explanation:

The velocity components at any point for stream functions are,

$u=\frac{\partial \omega}{\partial y}$,

$v=-\frac{\partial \omega}{\partial x}$

Consider the stream function as follows:

ω = 5x - 6y ...... (1)

Partial differentiation of function (1) with respect to y as follows:

$\frac{\partial \omega}{\partial y} = -6$

⇒ $u=-6$ (Since $u=\frac{\partial \omega}{\partial y}$)

Partial differentiation of function (1) with respect to x as follows:

$\frac{\partial \omega}{\partial x} = 5$

⇒ $-v=5$ (Since $-v=\frac{\partial \omega}{\partial x}$)

⇒ $v=-5$

Thus, the velocity components are $u=-6$ and $v=-5$.

Magnitude of velocity = $\sqrt{u^2+v^2}$

                                     = $\sqrt{(-6)^2+(-5)^2}$

                                     = $\sqrt{36+25}$

                                     = $\sqrt{61}$

Also,

tan$\theta$ = $\frac{v}{u}$

       = $\frac{-5}{-6}$

       = $\frac{5}{6}$

$\theta = tan^{-1}(\frac{5}{6} )$

  = 39.8°

The magnitude of resultant velocity is $\sqrt{61}$at any point is at the angle 39.8° in the positive direction with the x - axis.

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