= Unbuttressed Newtonian grounding-line stress
{title2=$4\mu H u_x=\rho g'H^2/2$}
For a two-dimensional <Newtonian fluid> <ice> sheet attached to an unbuttressed <ice shelf>, <ice-sheet flotation> gives $\rho H=\rho_wb$. The difference between the depth-integrated <ice> and ocean <hydrostatic pressures> is $\rho gH^2/2-\rho_wgb^2/2=\rho g'H^2/2$, where $g'=g(1-\rho/\rho_w)$. Shelf force balance integrates $4\mu Hu_x-\rho g'H^2/2$ to a constant; absence of a buttressing force makes that constant zero. The <shallow-shelf approximation> therefore gives the displayed boundary condition, or $u_x=g'H/(8\nu)$ with <kinematic viscosity> $\nu=\mu/\rho$.
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