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The effect of wall slip on the apparent flow curves of viscoplastic materials obtained using torsional parallel plate rheometers is analyzed by considering Herschel–Bulkley fluids and assuming that slip occurs above the slip yield stress τ c , taken to be lower than the yield stress, τ 0 . When the rim shear stress τ R is below τ c , the exerted torque is not sufficient to rotate the disk. When τ c < τ R ≤ τ 0 the material is still unyielded but exhibits wall slip and rotates as a solid at half the angular velocity of the rotating disk. Finally, when τ R > τ 0 , the material exhibits slip everywhere and yields only in the annulus r 0 ≤ r ≤ R , where r 0 is the critical radius at which the shear stress is equal to the yield stress and R is the radius of the disks. In the general case, the slip velocity, which varies with the radial distance, can be calculated numerically and then all quantities of interest, such as the true shear rate, and the two branches of the apparent flow curve can be computed by means of closed form expressions. Analytical solutions have also been obtained for certain values of the power-law exponent. In order to illustrate the effect of wall slip on the apparent flow curve and on the torque, results have been obtained for different gap sizes between the disks choosing the values of the rheological and slip parameters to be similar to reported values for certain colloidal suspensions. The computed apparent flow curves reproduce the patterns observed in the experiments. [ABSTRACT FROM AUTHOR]

We consider the annular Poiseuille flow of a Bingham fluid with wall slip. First, the analytical solution is derived for the case in which Navier-slip conditions are applied at the two cylinders. A sliding (pure plug) regime is observed below a critical pressure gradient, and a yielding regime is eventually encountered above another critical pressure gradient in which the material yields near the two walls and moves as a plug in a core region. An intermediate semi-sliding regime is observed when different slip laws apply at the two walls in which the material yields only near the wall corresponding to weaker slip and the unyielded plug slides along the other. Next, we consider the case where wall slip occurs above a critical wall shear stress, the slip yield stress, which is taken to be less than the yield stress, in agreement with experimental observations. In this case, a no-flow regime is observed below a critical pressure gradient, followed by the sliding and yielding regimes. The critical values of the pressure gradient defining the various flow regimes are determined, and the closed-form solutions are provided for all cases. These are compared with available theoretical and experimental results in the literature. [ABSTRACT FROM AUTHOR]

In this study, we have carried out experiments to characterize the wall slip of colloidal suspensions of kaolinites. To demonstrate slip, the rheological measurements were carried out with parallel-plate geometry with smooth and rough plates. The asperities of the rough surface penetrated the slip layer and created a nearly no-slip region, whereas the smooth plate showed significantly higher slip, a conclusion drawn by comparing the macroscopic flow curves in both cases. Two slip regimes were identified, namely, (i) an elastic slip regime below the yield stress of the suspension where the material slips like a solid and (ii) a slip regime above the yield stress where the material yields and flows. The slip velocity was quantified using a simple phenomenological slip model that seems to capture slip in both flow regimes. The transition from the first slip regime to the other has been resolved numerically as the material starts yielding first at the edge of the parallel-plate geometry with the yield point propagating inwards as the rotational speed is increased. The numerical method also establishes uniquely the yield stress value, which was found to agree with data obtained from parallel-plate, cone-and-plate, and concentric cylinder geometries. [ABSTRACT FROM AUTHOR]

We consider unidirectional flows of ideal or regularized Bingham fluids as well as viscoelastic fluids for which analytical solutions can be derived in terms of the Lambert W function. Explicit expressions are derived for the radius of the yielded region in partially yielded circular and axial Couette flows. Analytical solutions are also derived for the velocity and the volumetric flow rate in the plane and axisymmetric Poiseuille flows of a Windhab fluid, which is a combination of the Bingham and Papanastasiou models, and for the shear stress in the plane Couette flow of an exponential Phan–Thien–Tanner fluid. Finally, the Lambert function is used to solve the Poiseuille flow of a power-law fluid and the Newtonian circular Couette flow with wall slip and non-zero slip yield stress by means of a regularized slip equation, which is valid for any value of the wall shear stress. [ABSTRACT FROM AUTHOR]

We consider various viscometric flows of a Newtonian fluid, i.e., plane, annular, and circular Couette flows and planar and axisymmetric Poiseuille flows, in the presence of wall slip that follows a logarithmic slip law. We derive analytical solutions in terms of the Lambert W function. The effects of logarithmic slip on these flows are discussed, and comparisons of the results with their Navier-slip counterparts are made. [ABSTRACT FROM AUTHOR]

The lubrication flow of a Herschel-Bulkley fluid in a long asymmetric channel, the walls of which are described by two arbitrary functions h1(x) and h2(x) such that h1(x) < h2(x) and h1(x) + h2(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 h1 = 0 and linearly varying h2. Representative results demonstrating the effects of the Bingham number and the consistency-index and yield-stress growth numbers are discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]