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.