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An ideal fluid is rotating under gravity g with a constant angular velocity about the z-axis, which is vertical, so that in the fixed Cartesian axes u(x,y,z,t)=(-wy,wx,0). Find the form of the fluid surface.

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Essentially, the surface of your fluid is orthogonal to the resultant force acting on the fluid. Since this is the sum of gravitation and the centrifugal force (when looked at from the rotating frame) it should not be too hard to compute the shape of the surface.

The centrifugal force is easily computed by taking the derivative of the speed $u$, namely

$$-a=-\frac{du}{dt}=-(-w\frac{dy}{dt},w\frac{dx}{dt},0)=-(-w^2x,-w^2y,0)=(w^2x,w^2y,0)$$

Add to that the effect of gravity $(0,0,-g)$ and you get that the gradient of the implicit equation of the surface ($f(x,y,z)=0$) should be propotional to

$$\nabla f(x,y,z) \sim (w^2x,w^2y,-g)$$

This leads to a set of partial differential equations that can be solved to give

$$f(x,y,z)=k\left(\frac{1}{2}w^2x^2 + \frac{1}{2}w^2y^2 - gz + c\right)$$

in which $k$ is some proportionality factor that does not really matter.

Thus, the equation of the surface is that of a paraboloid:

$$z= \frac{w^2}{2g}x^2 + \frac{w^2}{2g}y^2 + \kappa$$

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