On the use of the Lambert function in solving non-Newtonian flow problems

We consider unidirectional flows of ideal or regularized Bingham fluids as well as viscoelastic fluids for which analytical solutions can be derived in terms of the Lambert W function. Explicit expressions are derived for the radius of the yielded region in partially yielded circular and axial Couette flows. Analytical solutions are also derived for the velocity and the volumetric flow rate in the plane and axisymmetric Poiseuille flows of a Windhab fluid, which is a combination of the Bingham and Papanastasiou models, and for the shear stress in the plane Couette flow of an exponential Phan–Thien–Tanner fluid. Finally, the Lambert function is used to solve the Poiseuille flow of a power-law fluid and the Newtonian circular Couette flow with wall slip and non-zero slip yield stress by means of a regularized slip equation, which is valid for any value of the wall shear stress.