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Variable viscosity and frictionally heated channel and pipe flows are investigated. The solutions are bounded and improved estimates of the critical stress (beyond which there are no steady solutions) developed. The stress first increases, then decreases, with increasing maximum temperature. At this stress maximum there is a neutral solution and neighboring unstable solutions to an associated stability problem. Points of inflection in the velocity profile can develop in Poiseuille flows and must develop in Couette flows. The Poiseuille profiles which develop are inviscidly unstable in channels but stable in pipes.
Daniel D. Joseph (Sun,) studied this question.