On the thermal runaway of variable viscosity flows between concentric cylinders

The form of blow-up for nonlinear parabolic equations

“…For this source term F(s) = exp (l/s)-2, which grows exponentially fast as s -»0. The estimate in the third region, F [ A -t + r 1 x-3 exp(-x 2 /4A) + o(x~3exp(-x 2 /4A))], is then exp{l/[A-t + r 1 x" 3 exp(-x 2 /4A)] + o(x 3 exp(x 2 /4A))}-2 for ( = A -O(x" 3 exp (-x 2 /4A)) (A = 1/ln 2); we see that not even the order of magnitude is obtainable from this. The difficulty may be overcome, and an asymptotic expansion for u obtained in this regime, by considering higher order terms in the expansion for u in the previous two regions (t small and x large).…”

“…The Dirichlet boundary condition u = 0 is imposed at x = 0 and zero initial temperature, u = 0 at t = 0, is assumed. A similar problem models the temperature of a liquid flowing around a cylinder when the viscosity of the liquid decreases exponentially with temperature, [1]; here the source term represents the production of heat by viscous dissipation. A simple asymptotic argument implies that as t -* t* = 1, u-*u*, where u* grows quadratically in x for large x, i.e.…”