Channel flow of viscoplastic fluids with pressure-dependent rheological parameters

We consider the 2D incompressible flow of a Herschel-Bulkley fluid with pressure-dependent rheological moduli in a symmetric channel of non-uniform width and small aspect ratio. Following Fusi et al. [“Pressure-driven lubrication flow of a Bingham fluid in a channel: A novel approach” J. Non-Newtonian Fluid Mech. 221, 66–75 (2015)] and Panaseti et al. [“Pressure-driven flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters,” Phys. Fluids 30, 030701 (2018)], we write the momentum equation of the unyielded domain in an integral form and we focus on the leading order approximation of the lubrication expansion. The use of the integral formulation allows one to overcome the “lubrication paradox,” a well-known inconsistency consisting in a plug speed that is not uniform. Different from the aforementioned papers, here we assume that the flow is driven by a prescribed inlet flux and not by a given pressure drop. As we shall see, this difference is of crucial importance when solving the problem at the leading order of the lubrication expansion. Indeed, when the pressure drop is given, the mathematical problem reduces to a nonlinear integro-differential equation for the pressure, while in the case where the inlet flux is prescribed the mathematical problem reduces to a full nonlinear algebraic equation for the plug speed, a numerical problem which can be far simpler than the determination of the solution of the integro-differential equation. The approach, based on the knowledge of the inlet discharge, can be used even when the pressure gradient is prescribed. Indeed, we determine a relation that links the pressure drop to the flow discharge so that when the pressure drop is assigned we can find the flowdischarge and apply the method presented here. To prove the validity of our approach, we finally show that the solutions obtained with the method proposed here are equal to the ones obtained in Panaseti et al. [“Pressure-driven flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters,” Phys. Fluids 30, 030701 (2018)]. This proves that one can determine the same solutions of Fusi et al. [“Pressure-driven lubrication flow of a Bingham fluid in a channel: A novel approach,” J. Non-Newtonian Fluid Mech. 221, 66 (2015)] and Panaseti et al. [“Pressure-driven flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters,” Phys. Fluids 30, 030701 (2018)] without solving any integro-differential equation.