Nonlinear laminar pipe flow of fluids with strongly temperature-dependent material properties

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Abstract
We study a one-dimensional model describing the laminar flow of a highly viscous fluid representing glass melt in a pipe. The flow is influenced by Lorentz force and gravity as well as temperature variation due to wall heat loss, electrical heating, advection, and heat diffusion. We take into account the full nonlinear temperature dependence of the viscosity and the electrical conductivity. For high and very low driving forces the mean velocity is found to be proportional to the forces as known from laminar pipe flow with constant material parameters. In between these two regimes, however, we identify a new flow regime. If there are no heat losses through the wall the mean velocity is proportional to the square root of the driving force. In the presence of wall heat loss the solution for the steady flow is even found to be nonunique, and to involve bifurcations for a wide range of parameters. This nonlinear behavior is shown to be a result of the closed-loop interaction between the velocity, temperature, and temperature-dependent material properties.

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