We consider the flow of a De-Kèe-Turcotte viscoplastic fluid in two configurations: planar Poiseuille flow in a channel driven by an applied pressure drop and Taylor-Couette flow between two rotating cylinders. After a suitable scaling of the governing equations, we explicitly determine the base 1D flow exploiting the Lambert function. We show that the problem admits two steady-state solutions, one for each real branch of the Lambert function. For the plane Poiseuille flow we also perform a linear stability analysis of the basic velocity profiles. We find that the profile relative to the secondary branch of the Lambert function is unconditionally unstable. For the profile relative to the principal branch we obtain closed loops for the marginal stability curves.
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