# River Crossing

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## River Crossing.

Consider the steamline flow of water, viscosity coefficient n, down a rectangular section of river bed of width W, depth of water Y, Y < W, and length-1, "under a pressure P. Let the velocity of the water at a distance y from the river floor and remote from the river banks he [unclear: u]. Assume pressure is constant over any given cross-section of river and that the water in contact with the river floor is stationary. When steady conditions are reached the velocity gradient is du/dy and the viscous drag per unit area is n.du/dy. This acts over the lower surface (neglecting drag caused by contact with the atmosphere and with the river bonks) of the upper rectangular box of water. The force, tending to accelerate this water box is PW(Y-y) and for steady conditions

Consider a leg of width s in the river, the foot having a [unclear: solid] grin on the river floor. A morent aacts about the foot tending to push its comer over. Assuming that the drag on an element strip of log at distance y fren the river floor is given by

The noment due to this drag is

The total movment is

Thus if a river doubles its depth duo to flooding the forces tending to push a tranper over are increased 26 = 64 times or the force is doubled by a 21/6 = 1.12 increase in depth.

In this treatment we have specified streamline flow. In page break real rivers is always turbulent and consequently trampers are subject to fluctuating forces with a grater danger of losing balance.

An approximate expression for the mean velocity of water in a turbulent river is

Substituting this progression in the above derivation me obtain

This equation predicts a far loss dranatic increase in danger with increase, in depth. However the equation, U=A+By, only holds for flow in which the values of y considered are much less then for flow in which the values y considered are much less then the dimensions erf eddies and disturbences of the water. Thus if boulders, in the river are of size more than a small fraction of the depthh, the equation for break down. Rivers with large bounders require considerations of the dynamics of the tramper rather than just a consideration of static forces. The turbulence. The turbulence would probably require a statistical treatment. The maths would be complex, the number of variables great. It might be possible to derive an expression for the prebedility of the tramper making a safe crossing as a function of Y and other variables.

Kevin Pearce, B.Sc., V.U.M.T.C.

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The mountains send down
Their cold tribute of snow
And the birch makes brown,
The rivulets running down.

from 'Arawata Bill' by Denis Glover.

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