The lubrication flow of a Herschel-Bulkley fluid in a symmetric long channel of varying width, 2h(x), is modeled extending the approach proposed by 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)] for a Bingham plastic. Moreover, both the consistency index and the yield stress are assumed to be pressure-dependent. Under the lubrication approximation, the pressure at zero order depends only on x and the semi-width of the unyielded core is found to be given by σ(x) = (1 + 1/n)h(x) + C, where n is the power-law exponent and the constant C depends on the Bingham number and the consistency-index and yield-stress growth numbers. Hence, in a channel of constant width, the width of the unyielded core is also constant, despite the pressure dependence of the yield stress, and the pressure distribution is not affected by the yield-stress function. With the present model, the pressure is calculated numerically solving an integro-differential equation and then the position of the yield surface and the two velocity components are computed using analytical expressions. Some analytical solutions are also derived for channels of constant and linearly varying widths. The lubrication solutions for other geometries are calculated numerically. The implications of the pressure-dependence of the material parameters and the limitations of the method are discussed.

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The steady, pressure-driven flow of a Herschel-Bulkley fluid in a microchannel is considered assuming that different power-law slip equations apply at the two walls due to slip heterogeneities, allowing the velocity profile to be asymmetric. Three different flow regimes are observed as the pressure gradient is increased. Below a first critical pressure gradient 1 G , the fluid moves unyielded with a uniform velocity and thus the two slip velocities are equal. In an intermediate regime between 1 G and a second critical pressure gradient 2 G , the fluid yields in a zone near the weak-slip wall and flows with uniform velocity near the stronger-slip wall. Beyond this regime, the fluid yields near both walls and the velocity is uniform only in the central unyielded core. It is demonstrated that the central unyielded region tends towards the midplane only if the power-law exponent is less than unity; otherwise, this region rends towards the weak-slip wall, and asymmetry is enhanced. The extension of the different flow regimes depends on the channel gap; in particular the intermediate asymmetric flow regime dominates when the gap becomes smaller than a characteristic length which incorporates the wall slip coefficients and the fluid properties. The theoretical results compare well with available experimental data on soft glassy suspensions. These results open new routes in manipulating the flow of viscoplastic materials in applications where the flow behavior depends not only on the bulk rheology of the material but also on the wall properties.

The effect of pressure-dependent slip at the wall in steady, isothermal, incompressible Poiseuille flows of a Newtonian liquid is investigated. Exponential dependence of the slip coefficient on the pressure is assumed and the flow problems are solved using a regular perturbation scheme in terms of the exponential decay parameter of the slip coefficient. The sequence of partial differential equations resulting from the perturbation procedure is solved analytically up to second order. The two-dimensional solution reveals the effects of the slip decay coefficient and the other dimensionless numbers and parameters, in the flow. The average pressure drop and the skin friction factor are also derived and discussed. V

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.

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The steady-state Couette flow of a yield-stress material obeying the Bingham-plastic constitutive equation is analyzed assuming that slip occurs when the wall shear stress exceeds a threshold value, the slip (or sliding) yield stress. The case of Navier slip (zero slip yield stress) is studied first in order to facilitate the analysis and the discussion of the results. The different flow regimes that arise depending on the relative values of the yield stress and the slip yield stress are identified and the various critical angular velocities defining those regimes are determined. Analytical solutions for all the regimes are presented and the implications for this important rheometric flow are discussed.

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