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Suppose you are shooting a water jet in a vacuum vertically onto a flat plate in a gravity-less room. The jet makes a right angle to the plate and water is assumed to be incompressible. The mass density of water is $\rho$, the initial speed of the water jet is $u$, the viscosity of water is $\eta$, the distance between the opening of the water shooter to the plate is $L$, and the opening of the water shooter is a circle of radius $R$. Assume that each water molecule bounces elastically on the surface of the plate in accordance with the law of reflection. Hence, the jet cannot remain in a cylindrical shape. The bounced water is bound to cause disruption within the stream of the jet (due to the viscosity of water).

My question is: at the steady state (where the shape of the jet is time-independent), is it possible to describe the shape of the jet? By symmetry, the cross section at any distance $x$ from the plate is a circle of radius $r(x)$. Clearly, $r(L)=R$. Could you please give an equation, such as a differential equation, that relates $r(x)$ to the parameters $\rho,u,L,\eta,R$? Do we need any more information to make some kind of approximation to this problem?

In addition, is there a threshold value for $u$ such that the flow will be smooth? In real life, you can see that a high-speed water jet will splash on the surface, while a low-speed jet does not splash. I suspect that we might need the surface tension $\gamma$ of water for our calculation as well.

Finally, I could be wrong about the existence of the steady state. The resulting jet could have a wave behavior (i.e., changing periodically with time). If that is the case, then clearly, we need an equation which is time-dependent. All references are very welcome.

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The conditions you describe are the idealized ones when studying hydraulic jumps. What you will have is a radial flow along the plate: pressure forces will convert the momentum in the $z$ direction into momentum in the $r$ direction. This may of course be unstable. See e.g. http://web.mit.edu/lienhard/www/hydraulic_jump.pdf

There are some inconsistencies in your question, e.g. ' Assume that each water molecule bounces elastically on the surface of the plate in accordance with the law of reflection' cannot be put directly in terms of the continuum description that you assume in the rest of your text. At the continuum level, you need surface tension and to describe the interaction with the plate.

Depending on the velocity and wetting properties, the air flow will also be of importance. It is actually very important in splashing processes.

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  • $\begingroup$ I was worried about the interaction with the plate, so I thought the most ideal situation is with the assumption on the elastic bouncing I put into my question. Then, I forgot about the lateral movement of the water along the plate. You could be right that this can cause inconsistencies. About the air flow, although I didn't say it at first (added that to my original question now), I meant that the water jet is shot out in a vacuum. Thanks for your reply. $\endgroup$ – Batominovski Aug 10 '15 at 18:02

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