Poiseuille flow in cylindrical and planar geometries with a simplified, pseudoplastic (shear thinning) rheology characterized by constant viscosity plateaus above and below a transition strain rate is considered. Analytical, steady state solutions for velocity profile and volume flux are formulated. Transient flow development is addressed numerically and compared to the theory in the steady state limit. Stationary flow is approached after the momentum diffusion timescale based on the spatially dominant kinematic viscosity. For large viscosity ratio and shear thinning region confined near the domain boundary, velocity distributions are quasi-plug like with large boundary to interior strain rate ratio.
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C. C. Reese (2024) studied this question.
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