The analytical solution for the flow of a Papanastasiou fluid in ducts with variable geometry

Abstract

We solve the isothermal, steady and creeping flow of the regularized Papanastasiou model in ducts, such as symmetric channels and axisymmetric pipes, with varying rigid walls. The classic lubrication approximation is invoked to simplify the governing equations and to produce an exact analytical solution for the flow field with the aid of the Lambert W function. For specific suitable values of the parameters of the Papanastasiou model, the solution reduces to the corresponding exact analytical solutions for the ideal yield-stress Bingham and Newtonian fluids. The analytical solutions derived here are used to calculate the average pressure drop required to maintain the constant volumetric flowrate through the duct. Results are provided for symmetric undulating or linearly varying channels with the emphasis put on the effects of the Bingham and regularization numbers, and the amplitude of the undulation or the wall variation, respectively. For Bn→0 the yield surface approaches the axis of symmetry, σ(z)→0, i.e., no unyielded region appears in the flow. In this case, the last term in the formula for v is well-defined at y=0 provided that the numerator is zero. This implies that p′=c/h4 where c is constant. With the aid of the total mass balance, we find that c=−16, and thus p′→−16/h4 as Bn→0. Thus, Eqs. (34) and (37) reduce to the corresponding formulas for a Newtonian fluid given in Eq. (25). Noting that f(y,z)>0, and substituting the solution for γ˙ into inequality (52) gives the trivial W(f(y,z))≥0 which holds for any value of y and z because of the properties of the Lambert function. Eq. (53) is used in conjunction with Eq. (8) to find the unknown pressure gradient, p′(z). First, we integrate by parts Eq. (8) and apply the no-slip condition at the upper wall to find ∫0h(z)y2γ˙dy=2. Then, by substituting Eq. (53) into the latter, and performing the integration with respect to y from the axis of symmetry (y=0) to the wall (y=h(z)), gives: Using properties of the Lambert function and denoting with w=w(z) the value of W function at f(y=h(z),z), i.e.: we find the integral I(z) in Eq. (54) in terms of W and w. Then, we set Φ(z)≡−p′(z)h(z)/2>0, rearrange the resulting equation and upon simplification, we derive the strongly non-linear algebraic equation for Φ: The axial velocity component, u=u(y,z), can be found by integrating Eq. (53) with respect to y between y and h(z) and applying the no-slip condition at the wall: