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In a parallel two dimensional flow in the positive x-direction, the velocity varies linearly from zero at y=0 to 32 m/s at y=1m in perpendicular direction

Determine the expression for stream function $(\psi)$ and plot streamline at interval of $d\psi=3m^2/s.$ Is the flow is irrational. Consider unit width of flow.

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Data:-

Velocity in x-direction at y=0 is zero at y=1 m is 32 m/s.

To find:-

(i) $\psi=?$

(ii) Streamlines at $d\psi =3m^2/s$

(iii) Flow rotational or not.

Solution:-

Velocity component 'u' along x-direction at y distance may be written as

$\frac{32}{u}=\frac{1}{y}$

$\therefore u=32y$

As there is no velocity along y-direction

$\therefore v=0$

Stream function $\frac{\partial \psi}{\partial y}=u$ and $\frac{\partial \psi}{\partial x}=-v$

Let $\frac{\partial \psi}{\partial y}=u$

$\therefore \partial \psi=u.\partial y=32y.\partial y$

Integrating

$\psi=\frac{32y^2}{2}+f(u)$

$\psi=16y^2+f(x)$.................(1)

Differentiating equation (1) w.r.t. 'x'

$\frac{\partial \psi}{\partial x}=0+f'(x)$

But $\frac{\partial \psi}{\partial x}=-v=0$

$\therefore f'(x)=0$

integrating we get

$f(X)=0$

$\therefore$ equation (1) will becomes

$y=16y^2$..........(Ans)

-Considering unit width of flow

$\partial Q=\partial \psi \times 1$

Let $d\psi$ be the difference in $\psi$ at y=y and y=0

then $\partial \psi=\psi _y-\psi _0$

$\psi _0=0$...........at y=0 and

$\psi _y=16y^2$

$\psi _y=3m^2/s$

$\therefore 3=16y^2-0$

$\therefore y=0.433 m$

-The first streamline will at y=0 and parallel to x-axis

-The second streamline at y=0.4333 and parallel to x-axis.

-The third streamline may be obtained as $\partial \psi=\psi _y-\psi_{0.433}$

$\therefore 3=16y^2-16\times (0.433)^2$

$\therefore y=0.6125 m$

-Fourth streamline

$3=16y^2-16\times (0.6125)^2$

$\therefore=y=0.75 m$

-Fifth streamline

$3=16y^2-16\times (0.75)^2$

$y=0.866m$

-Sixth streamline

$3=16y^2-16\times (0.866)^2$

$y=0.968m$

-Seventh streamline

$3=16y^2-16\times (0.968)^2$

$y=1.061m$

and we can calculate $8^{th}$,$9^{th}$ etc.

(iii) To check whether flow is rotational or irrotational

We have,

$W_z=\frac{1}{2}[\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}]$

We know u=32y and v=0

$\frac{\partial u}{\partial y}=32$ and $\frac{\partial v}{\partial x}=0$

$w_t=\frac{1}{2}(0-32) \ne 0$

$\therefore$ Flow is not irrotational.

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