Past exam of the physics B course of the University of Cambridge / 2010 / Paper 2A / Section C / C10

For steady inviscid incompressible flow subject to a conservative body force, Bernoulli's theorem states that $p/\rho +u^2/2+gz$ is constant along a streamline. The same constant applies across streamlines only with an additional condition, such as an irrotational connected flow with the same upstream head.

Neglect gravity and let the film's free-surface pressure be atmospheric. Its radial speed is then $v$ in the ideal model. Conservation of mass equates the incoming flux $\pi r_1^{2}v$ to $2\pi rt(r)v$, soThe latter value already warns that the thin-film approximation is poor near impact; the radial expression is most useful beyond that region. The incoming vertical momentum flux gives downward force $F_m=\rho \pi r_1^2{v}^2$ when the outgoing flow is horizontal.

In the stipulated stagnant-rear-layer idealization, the excess rear pressure is the stagnation value $\rho v^2/2$. Its upward force is $\rho v^2\pi r_2^2/2$. Thus, with downward positive and ignoring the water's weight,For a large plate, the radial-film volume is approximatelyIts weight therefore changes the model toCancellation occurs nearThe printed claim of negligible force for every radius above this scale does not follow from this model. Near the cancellation radius the force is small; for substantially larger radii the quadratic upward-pressure term dominates. The original PDF, not just its transcription, uses that inequality and wording. Moreover, the rear layer's mass and detailed gravity-dependent pressure field are unspecified. The calculation establishes a cancellation scale under the elementary stated pressure approximation, rather than a general large-radius zero-force theorem. A full free-surface flow model would be needed to extend it consistently.