We report a stereolithographic three-dimensional printing approach for polymeric components that uses a mobile liquid interface (a fluorinated oil) to reduce the adhesive forces between the interface and the printed object, thereby allowing for a continuous and rapid print process, regardless of polymeric precursor. The bed area is not size-restricted by thermal limitations because the flowing oil enables direct cooling across the entire print area. Continuous vertical print rates exceeding 430 millimeters per hour with a volumetric throughput of 100 liters per hour have been demonstrated, and proof-of-concept structures made from hard plastics, ceramic precursors, and elastomers have been printed.

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Rapid, large-volume, thermally controlled 3D printing using a mobile liquid interface.

High solid dispersions are soft materials made of colloidal or non-colloidal particles dispersed at high volume fractions in a liquid matrix. They include hard sphere glasses, colloidal pastes, concentrated emulsions, foams, and vesicles. These materials are prone to exhibit different kinds of flow heterogeneities: shear banding, wall slip, and fracture. While wall slip is often considered as a nuisance by experimentalists, it appears to be a fundamental component to the way that high solid dispersions respond to mechanical deformation. Moreover, the ability of soft materials to slip onto surfaces allows them to move readily and efficiently in many natural phenomena and industrial processes. This review surveys recent developments and current research in the field. Topics like wall slip detection and control, microscopic modeling for rigid and soft particles materials, and the relation between wall slip and other flow heterogeneities are discussed. We also identify important open issues for future research.

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A review on wall slip in high solid dispersions

Numerical simulations of viscoplastic fluid flows have provided a better understanding of fundamental properties of yield stress fluids in many applications relevant to natural and engineering sciences. In the first part of this paper, we review the classical numerical methods for the solution of the non-smooth viscoplastic mathematical models, highlight their advantages and drawbacks, and discuss more recent numerical methods that show promises for fast algorithms and accurate solutions. In the second part, we present and analyze a variety of applications and extensions involving viscoplastic flow simulations: yield slip at the wall, heat transfer, thixotropy, granular materials, and combining elasticity, with multiple phases and shallow flow approximations. We illustrate from a physical viewpoint how fascinating the corresponding rich phenomena pointed out by these simulations are.

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Progress in numerical simulation of yield stress fluid flows

Viscoplasticity is characterized by a yield stress, below which the materials will not deform and above which they will deform and flow according to different constitutive relations. Viscoplastic models include the Bingham plastic, the Herschel-Bulkley model and the Casson model. All of these ideal models are discontinuous. Analytical solutions exist for such models in simple flows. For general flow fields, it is necessary to develop numerical techniques to track down yielded/unyielded regions. This can be avoided by introducing into the models a regularization parameter, which facilitates the solution process and produces virtually the same results as the ideal models by the right choice of its value. This work reviews several benchmark problems of viscoplastic flows, such as entry and exit flows from dies, flows around a sphere and a bubble and squeeze flows. Examples are also given for typical processing flows of viscoplastic materials, where the extent and shape of the yielded/unyielded regions are clearly shown. The above-mentioned viscoplastic models leave undetermined the stress and elastic deformation in the solid region. Moreover, deviations have been reported between predictions with these models and experiments for flows around particles using Carbopol, one of the very often used and heretofore widely accepted as a simple “viscoplastic” fluid. These have been partially remedied in very recent studies using the elastoviscoplastic models proposed by Saramito.

Numerical simulations of complex yield-stress fluid flows

We present a comparative experimental study of unsteady laminar flows of a yield stress shear thinning fluid (Carbopol ® 980) in two distinct configurations: a parallel plate rheometric flow and a pressure driven pipe flow. Consistently with the observations in the case of the rheometric flow, the in situ characterisation of the unsteady pipe flow reveals three distinct flow regimes: solid (plug-like) , solid–fluid and fluid . In both configurations and as the flow forcing is gradually increased, the yielding emerges via an irreversible transition. The irreversibility of the deformation states is coupled to the wall slip phenomenon. Particularly, the presence of wall slip nearly suppresses the scaling of the deformation power deficit associated to the rheological hysteresis with the rate at which the material is forced. An universal scaling of the slip velocity with the wall velocity gradients and a slip length which is independent on the degree of the flow steadiness is observed in the pipe flow.

Unsteady laminar flows of a Carbopol® gel in the presence of wall slip

We solve numerically the cessation of axisymmetric Poiseuille flow of a Herschel–Bulkley fluid under the assumption that slip occurs along the wall. The Papanastasiou regularization of the constitutive equation is employed. As for the slip equation, a power-law expression is used to relate the wall shear stress to the slip velocity, assuming that slip occurs only above a critical wall shear stress, known as the slip yield stress. It is shown that, when the latter is zero, the fluid slips at all times, the velocity becomes and remains uniform before complete cessation, and the stopping time is finite only when the slip exponent s 1, the decay is much slower. Analytical expressions of the decay of the flat velocity for any value of s and of the stopping time for s < 1 are also derived. Using a discontinuous slip equation with slip yield stress poses numerical difficulties even in one dimensional time-dependent flows, since the transition times from slip to no-slip and vice versa are not known a priori. This difficulty is overcome by regularizing the slip equation. The numerical results showed that when the slip yield stress is non-zero, slip ceases at a finite critical time, the velocity becomes flat only in complete cessation, and the stopping times are finite, in agreement with theoretical estimates.

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Cessation of viscoplastic Poiseuille flow with wall slip

The Newtonian Poiseuille flow is considered for various geometries, under the assumption that wall slip occurs above a critical value of the wall shear stress known as, the slip yield stress. In the axisymmetric and planar cases, there are two flow regimes defined by a critical value of the pressure gradient above which slip occurs. Two critical pressure gradients characterise the annular and rectangular Poiseuille flows. Below the first critical value no-slip occurs while above the second-one, slip occurs at all walls. In the intermediate regime for the annular problem, slip occurs only at the inner-wall, while for the rectangular problem, there are two intermediate regimes for which there are no analytical solutions. In the first regime slip occurs only in the middle sections of the wider walls and in the second-one partial slip also occurs along the narrower walls. Analytical solutions of all flow problems (with the exception of the two intermediate regimes of the rectangular Poiseuille case) are derived and discussed.

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Newtonian Poiseuille flows with slip and non-zero slip yield stress

We solve numerically the cessation of the pressure-driven Poiseuille flow of a Bingham plastic under the assumption that slip occurs along the wall following a generalized Navier-slip law involving a non-zero slip yield stress. In order to avoid the numerical difficulties caused by their inherent discontinuities, both the constitutive and the slip equations are regularized by means of exponential (Papanastasiou-type) regularizations. As with one-dimensional Poiseuille flows, in the case of Navier slip (zero slip yield stress), the fluid slips at all times, the velocity becomes and remains plug before complete cessation, and the theoretical stopping time is infinite. The cessation of the plug flow is calculated analytically. No stagnant regions appear at the corners when Navier slip is applied. In the case of slip with non-zero slip yield stress, the fluid may slip everywhere or partially at the wall only in the initial stages of cessation depending on the initial condition. Slip ceases at a critical time after which the flow decays exponentially and the stopping times are finite in agreement with theory. The combined effects of viscoplasticity and slip are investigated for wide ranges of the Bingham and slip numbers and results showing the evolution of the yielded and unyielded regions are presented. The numerical results also showed that the use of regularized equations may become problematic near complete cessation or when the velocity profile becomes almost plug.

Cessation of viscoplastic Poiseuille flow in a square duct with wall slip

We consider the Newtonian Poiseuille flow in a tube whose cross-section is an equilateral triangle. It is assumed that boundary slip occurs only above a critical value of the wall shear stress, namely the slip yield stress. It turns out that there are three flow regimes defined by two critical values of the pressure gradient. Below the first critical value, the fluid sticks everywhere and the classical no-slip solution is recovered. In an intermediate regime the fluid slips only around the middle of each boundary side and the flow problem is not amenable to analytical solution. Above the second critical pressure gradient non-uniform slip occurs everywhere at the wall. An analytical solution is derived for this case and the results are discussed.

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Newtonian flow in a triangular duct with slip at the wall

Analytical solutions are derived for the start-up and cessation Newtonian Poiseuille and Couette flows with wall slip obeying a dynamic slip model. This slip equation allows for a relaxation time in the development of wall slip by means of a time-dependent term which forces the eigenvalue parameter to appear in the boundary conditions. The resulting spatial problem corresponds to a Sturm–Liouville problem different from that obtained using the static Navier slip condition. The orthogonality condition of the associated eigenfunctions is derived and the solutions are provided for the axisymmetric and planar Poiseuille flows and for the circular Couette flow. The effect of dynamic slip on the flow development is then discussed.

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George Kaoullas

Papers

Cessation of viscoplastic Poiseuille flow with wall slip

Newtonian Poiseuille flows with slip and non-zero slip yield stress

Cessation of viscoplastic Poiseuille flow in a square duct with wall slip

Newtonian flow in a triangular duct with slip at the wall

Start-up and cessation Newtonian Poiseuille and Couette flows with dynamic wall slip