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Equation Of Continuity

PYQs

Equation Of Continuity

1) For an incompressible fluid flow, two components of velocity $(u, v, w)$ are given by $u = x^{2} + 2 y^{2} + 3 z^{2}$ , $v = x^{2} y - y^{2} z + z x$ . Determine the third component $w$ so that they satisfy the equation of continuity. Also, find the $z - c o m p o n e n t$ of acceleration.

[2018, 10M]

Using the continuity equation in cartesian coordinates:

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$
$⟹ 2 x + x^{2} - 2 y z + \frac{\partial w}{\partial z} = 0$
$⟹ \frac{\partial w}{\partial z} = 2 y z - 2 x - x^{2}$
Integrating wrt $z$ , we get:
$w = y z^{2} - 2 x z - x^{2} z + f (x, y)$

$⟹$ The $z$ component of acceleration:
$\begin{aligned} a_{z} & = (q \cdot \nabla) w + \frac{\partial ω}{\partial t} \\ = u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial ω}{\partial z} \end{aligned}$

Therefore,

\begin{aligned} a_{z} & = (x^{2} + 2 y^{2} + 3 z^{2}) (\frac{\partial f}{\partial x} - 2 z - 2 x z) \\ + (x^{2} y - y^{2} z + x z) (\frac{\partial f}{\partial y} + z^{2}) \\ + (y z^{2} - 2 x z - x^{2} z + f (x, y)) (2 y z - 2 z - x^{2}) \end{aligned}

2) Show that $(\frac{x^{2}}{a^{2}}) \cos^{2} t + (\frac{y^{2}}{b^{2}}) \sec^{2} t = 1$ is a possible form for the boundary surface of a liquid.

[2007, 12M]

3) Show that: $u = \frac{- 2 x y z}{{(x^{2} + y^{2})}^{2}}$ , $v = \frac{(x^{2} - y^{2}) z}{{(x^{2} + y^{2})}^{2}}$ , $w = \frac{y}{x^{2} + y^{2}}$ are the velocity components of a possible liquid motion. Is this motion irrotational?

[2002, 15M]

Here, we have:

\begin{aligned} \frac{\partial u}{\partial x} & = - 2 y z \frac{1 \cdot {(x^{2} + y^{2})}^{2} - x \cdot 2 (x^{2} + y^{2}) \cdot 2 x}{{(x^{2} + y^{2})}^{4}} \\ = - 2 y z \frac{x^{2} + y^{2} - 4 x^{2}}{{(x^{2} + y^{2})}^{3}} \\ = - 2 y z \frac{y^{2} - 3 x^{2}}{{(x^{2} + y^{2})}^{3}} \end{aligned}

\begin{aligned} \frac{\partial v}{\partial y} & = z \frac{- 2 y {(x^{2} + y^{2})}^{2} - 2 (x^{2} + y^{2}) \cdot 2 y (x^{2} - y^{2})}{{(x^{2} + y^{2})}^{4}} \\ = - 2 y z \frac{x^{2} + y^{2} + 2 (x^{2} - y^{2})}{{(x^{2} + y^{2})}^{3}} \\ = - 2 y z \frac{3 x^{2} - y^{2}}{{(x^{2} + y^{2})}^{3}} \end{aligned}

and

\partial w / \partial z = 0

Hence, the equation of continuity $\partial u / \partial x + \partial v / \partial y + \partial w / \partial z = 0$ is satisfied and so the liquid motion is possible.

Furthermore, we have

\begin{aligned} Ω_{x} & = \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \\ = \frac{x^{2} - y^{2}}{{(x^{2} + y^{2})}^{2}} - \frac{x^{2} - y^{2}}{{(x^{2} + y^{2})}^{2}} \\ = 0 \\ Ω_{y} & = \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \\ = - \frac{2 x y}{{(x^{2} + y^{2})}^{2}} + \frac{2 x y}{{(x^{2} + y^{2})}^{2}} \\ = 0 \\ Ω_{z} & = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \\ = \frac{2 x z (3 y^{2} - x^{2})}{{(x^{2} + y^{2})}^{3}} - \frac{2 x z (3 y^{2} - x^{2})}{{(x^{2} + y^{2})}^{3}} \\ = 0 \end{aligned}

∴ \partial w / \partial y = \partial v / \partial z, \partial u / \partial z = \partial w / \partial x, \partial v / \partial x = \partial u / \partial y

and hence the motion is irrotational.

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