Equation of Continuity
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PYQs
Equation Of Continuity
1) For an incompressible fluid flow, two components of velocity \((u,v,w)\) are given by \(u=x^2+2y^2+3z^2\), \(v=x^2y-y^2z+zx\). Determine the third component \(w\) so that they satisfy the equation of continuity. Also, find the \(z-component\) of acceleration.
[2018, 10M]
Using the continuity equation in cartesian coordinates:
\(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0\)
\(\implies 2 x+x^{2}-2 y z+\frac{\partial w}{\partial z}=0\)
\(\implies \quad \frac{\partial w}{\partial z}=2 y z-2 x-x^{2}\)
Integrating wrt \(z\), we get:
\(w=y z^{2}-2 x z-x^{2} z+f(x, y)\)
\(\implies\) The \(z\) component of acceleration:
\(\begin{aligned} a_{z} &=(q \cdot \nabla) w+\frac{\partial \omega}{\partial t} \\ &=u \frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w \frac{\partial \omega}{\partial z} \end{aligned}\)
Therefore,
\[\begin{aligned} a_{z} &=\left(x^{2}+2 y^{2}+3 z^{2}\right)\left(\frac{\partial f}{\partial x}-2 z-2 x z\right) \\ &+\left(x^{2} y-y^{2} z+x z\right)\left(\frac{\partial f}{\partial y}+z^{2}\right) \\ &+\left(y z^{2}-2 x z-x^{2} z+f(x, y)\right)\left(2 y z-2 z-x^{2}\right) \end{aligned}\]2) Show that \(\left(\dfrac{x^{2}}{a^{2}}\right) \cos ^{2} t+\left(\dfrac{y^{2}}{b^{2}}\right) \sec ^{2} t=1\) is a possible form for the boundary surface of a liquid.
[2007, 12M]
3) Show that: \(u=\dfrac{-2 x y z}{\left(x^{2}+y^{2}\right)^{2}}\), \(v=\dfrac{\left(x^{2}-y^{2}\right) z}{\left(x^{2}+y^{2}\right)^{2}}\), \(w=\dfrac{y}{x^{2}+y^{2}}\) are the velocity components of a possible liquid motion. Is this motion irrotational?
[2002, 15M]
Here, we have:
\[\begin{aligned}\frac{\partial u}{\partial x}&=-2 y z \frac{1 \cdot\left(x^{2}+y^{2}\right)^{2}-x \cdot 2\left(x^{2}+y^{2}\right) \cdot 2 x}{\left(x^{2}+y^{2}\right)^{4}}\\ &=-2 y z \frac{x^{2}+y^{2}-4 x^{2}}{\left(x^{2}+y^{2}\right)^{3}}\\ &=-2 y z \frac{y^{2}-3 x^{2}}{\left(x^{2}+y^{2}\right)^{3}}\end{aligned}\] \[\begin{aligned} \frac{\partial v}{\partial y}&=z \frac{-2 y\left(x^{2}+y^{2}\right)^{2}-2\left(x^{2}+y^{2}\right) \cdot 2 y\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)^{4}} \\&=-2 y z \frac{x^{2}+y^{2}+2\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}}\\ &=-2 y z \frac{3 x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{3}} \end{aligned}\]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
\[\begin{aligned} \Omega_{x}&=\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\\&=\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}}-\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}}\\&=0 \\ \Omega_{y}&=\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\\&=-\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}+\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}\\&=0 \\ \Omega_{z}&=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\\&=\frac{2 x z\left(3 y^{2}-x^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}}-\frac{2 x z\left(3 y^{2}-x^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}}\\&=0 \end{aligned}\] \[\therefore \quad \partial w / \partial y=\partial v / \partial z, \quad \partial u / \partial z=\partial w / \partial x, \quad \partial v / \partial x=\partial u / \partial y\]and hence the motion is irrotational.