The Flow of a Glacier in a Channel of Rectangular, Elliptic or Parabolic Cross-Section

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Abstract
Numerical solutions are found for the steady rectilinear flow of ice, obeying Glen’s non-linear flow law, down uniform cylindrical channels of rectangular, semi-elliptic and parabolic cross-section. The results are also directly applicable to the pumping of a non-Newtonian fluid down a pipe. There is assumed to be no slip of the ice on the channel surface. Certain results on the centre-line velocity in symmetrical channels may be derived purely from dimensional and symmetry principles. An analytical solution due to Dr. W. Chester is given for a semi-elliptic channel section which departs only slightly from a semi-circle. Contrary to a view sometimes held, the maximum shear stress at the ice surface in a parabolic channel and in some elliptical channels does not always occur at the edge. With the flow law, strain-rate proportional to (stress)3, the velocity averaged across the ice surface, which is easily measured with a line of stakes, is close to the average velocity over the whole section for a wide range of parabolic sections; the hydrological importance of this result is that the discharge may be inferred without the need to measure the velocity at depth. Arguments are given to show that the result still holds when there is slipping on the bed and when the power in the flow law differs somewhat from 3, Depending on the amount of bed slip and the shape of the channel section, the kinematic wave velocity for a range of parabolic channels is between 2.0 and 2.3 times the centre-line velocity of the ice, and between 2.0 and 3.5 times the mean surface velocity of the ice.

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