Lubrication solution of the flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters in an asymmetric channel

Abstract

A new model for the bouncing regime boundary in binary droplet collisions Physics of Fluids 31, 027105 (2019); https://doi.org/10.1063/1.5085762Post-collision hydrodynamics of droplets on cylindrical bodies of variant convexity and wettability Physics of Fluids 31, 022008 (2019); https://doi. ABSTRACTThe lubrication flow of a Herschel-Bulkley fluid in a long asymmetric channel, the walls of which are described by two arbitrary functions h 1 (x) and h 2 (x) such that h 1 (x) < h 2 (x) and h 1 (x) + h 2 (x) are linear, is solved extending a recently proposed method, which avoids the lubrication paradox approximating satisfactorily the correct shape of the yield surface at zero order [P. Panaseti et al., "Pressure-driven flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters," Phys. Fluids 30, 030701 (2018)]. Both the consistency index and the yield stress are assumed to be pressure-dependent. Under the lubrication approximation, the pressure at zero order is a function of x only, is decoupled from the velocity components, and obeys a first-order integro-differential equation. An interesting feature of the asymmetric flow is that the unyielded core moves not only in the main flow direction but also in the transverse direction. Explicit expressions for the two yield surfaces defining the asymmetric unyielded core are obtained, and the two velocity components in both the yielded and unyielded regions are calculated by means of closed-form expressions in terms of the calculated pressure and the two yield surfaces. The method is applicable in a range of Bingham numbers where the unyielded core extends from the inlet to the outlet plane of the channel. Semi-analytical solutions are derived in the case of an asymmetric channel with h 1 = 0 and linearly varying h 2 . Representative results demonstrating the effects of the Bingham number and the consistency-index and yield-stress growth numbers are discussed.Published under license by AIP Publishing. https://doi.

Lubrication flow of a generalized Casson fluid with pressure-dependent rheological parameters

“…We find that the behavior of the yield surface is opposite to that of the wall function, meaning that the unyielded core expands when the channel width is reduced and vice versa. This is consistent with the results obtained in [7] and [16] for the Bingham and Herschel-Bulkley model respectively.…”

“…Subsequently we consider the case in which the rheological parameters depend on the pressure and solve the problem to investigate the effects that this dependence has on the flow. Besides the increasing complexity of the mathematical problem due to the non constancy of the viscosity and yield stress we find that, differently from the constant case and from the cases studied in [7] and [16], the monotonicity of the yield surface and of the wall function are not necessarily opposite. This means that we can have a reduction of the channel width and of the inner plug at the same time, depending on the particular form of the viscosity and of the yield stress.…”

“…Following [16], without loss of generality we assume that p * o = p * out , where p * out is the constant pressure at the outlet of the channel. Hence the non dimensional inlet pressure becomes…”

“…When the constitutive response exhibits a yield stress the fluid is called viscoplastic, meaning that a critical value of the stress must be exceeded to start the flow. Among visco-plastic models we mention the Bingham model [2], the Herschel-Bulkley model [12], the Casson model [3] and the Heinz-Casson model [10] and [16] we assume that the latter may depend on the pressure.…”