Fluid Dynamics | Part IB, 2003

Inviscid fluid issues vertically downwards at speed u0u_{0} from a circular tube of radius a. The fluid falls onto a horizontal plate a distance HH below the end of the tube, where it spreads out axisymmetrically.

Show that while the fluid is falling freely it has speed

u=u0[1+2gu02(Hz)]1/2u=u_{0}\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{1 / 2}

and occupies a circular jet of radius

R=a[1+2gu02(Hz)]1/4,R=a\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{-1 / 4},

where zz is the height above the plate and gg is the acceleration due to gravity.

Show further that along the plate, at radial distances rar \gg a (i.e. far from the falling jet), where the fluid is flowing almost horizontally, it does so as a film of height h(r)h(r), where

a44r2h2=1+2gu02(Hh)\frac{a^{4}}{4 r^{2} h^{2}}=1+\frac{2 g}{u_{0}^{2}}(H-h)

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