Newtonian plane Couette flow with dynamic wall slip

Meccanica - The occurrence of slip complicates the estimation of the viscosity in rheometric flows. Thus, special analyses and experimental protocols are needed in order to obtain reliable...     

Cessation of Newtonian circular and plane Couette flows with wall slip and non-zero slip yield stress

We solve analytically the cessation flows of a Newtonian fluid in circular and plane Couette geometries assuming that wall slip occurs provided that the wall shear stress exceeds a critical threshold, the slip yield stress. In steady-state, slip occurs only beyond a critical value of the angular velocity of the rotating inner cylinder in circular Couette flow or of the speed of the moving upper plate in plane Couette flow. Hence, in cessation, the classical no-slip solution holds if the corresponding wall speed is below the critical value. Otherwise, slip occurs only initially along both walls. Beyond a first critical time, slip along the fixed wall ceases, and beyond a second critical time slip ceases also along the initially moving wall. Beyond this second critical time no slip is observed and the decay of the velocity is faster. The velocity decays exponentially in all regimes and the decay is reduced with slip. The effects of slip and the slip yield stress are discussed.     

Perturbation solutions of weakly compressible Newtonian Poiseuille flows with Navier slip at the wall

We consider both the planar and axisymmetric steady, laminar Poiseuille flows of a weakly compressible Newtonian fluid assuming that slip occurs along the wall following Navier’s slip equation and that the density obeys a linear equation of state. A perturbation analysis is performed in terms of the primary flow variables using the dimensionless isothermal compressibility as the perturbation parameter. Solutions up to the second order are derived and compared with available analytical results. The combined effects of slip, compressibility, and inertia are discussed with emphasis on the required pressure drop and the average Darcy friction factor.     

Incompressible Poiseuille flows of Newtonian liquids with a pressure-dependent viscosity

The pressure-dependence of the viscosity becomes important in flows where high pressures are encountered. Applications include many polymer processing applications, microfluidics, fluid film lubrication, as well as simulations of geophysical flows. Under the assumption of unidirectional flow, we derive analytical solutions for plane, round, and annular Poiseuille flow of a Newtonian liquid, the viscosity of which increases linearly with pressure. These flows may serve as prototypes in applications involving tubes with small radius-to-length ratios. It is demonstrated that, the velocity tends from a parabolic to a triangular profile as the viscosity coefficient is increased. The pressure gradient near the exit is the same as that of the classical fully developed flow. This increases exponentially upstream and thus the pressure required to drive the flow increases dramatically.     

Slip yield stress effects in start-up Newtonian Poiseuille flows

Analytical solutions are derived for various start-up Newtonian Poiseuille flows assuming that slip at the wall occurs when the wall shear stress exceeds a critical value, known as the slip yield stress. Two distinct regimes characterise the steady axisymmetric and planar flows, which are defined by a critical value of the pressure gradient. If the imposed pressure gradient is below this critical value, the classical no-slip, start-up solution holds. Otherwise, no-slip flow occurs only initially, for a finite time interval determined by a critical time, after which slip does occur. For the annular case, there is an additional intermediate (steady) flow regime where slip occurs only at the inner wall, and hence, there exist two critical values of the pressure gradient. If the applied pressure gradient exceeds both critical values, the velocity evolves initially with no-slip at both walls up to the first critical time, then with slip only along the inner wall up to the second critical time and finally with slip at both walls.     

Confined viscoplastic flows with heterogeneous wall slip

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 G ₁, the fluid moves unyielded with a uniform velocity, and thus, the two slip velocities are equal. In an intermediate regime between G ₁ and a second critical pressure gradient G ₂, 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 are 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.     

Cessation of Couette and Poiseuille flows of a Bingham plastic and finite stopping times

We solve the one-dimensional cessation Couette and Poiseuille flows of a Bingham plastic using the regularized constitutive equation proposed by Papanastasiou and employing finite elements in space and a fully implicit scheme in time. The numerical calculations confirm previous theoretical findings that the stopping times are finite when the yield stress is nonzero. The decay of the volumetric flow rate, which is exponential in the Newtonian case, is accelerated and eventually becomes linear as the yield stress is increased. In all flows studied, the calculated stopping times are just below the theoretical upper bounds, which indicates that the latter are tight.     

Newtonian flow in a triangular duct with slip at the wall

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.     

Numerical simulation of the transition from adhesion to slip with friction in generalized Newtonian Poiseuille flow

The axisymmetric Poiseuille flow of a purely viscous generalized Newtonian fluid under rate of flow controlled conditions is studied with a change in the boundary conditions at a transition point from an adhesive to a slip condition with friction at the wall. The friction law used originates from an experimental study by (J.M. Piau and N. El Kissi, J. Non-Newtonian Fluid Mech. 54 (1994) 121–142) using a capillary made of steel and a silicone fluid, and is based also on a molecular dynamics theory by (Yu. B. Chernyak, A.I. Leonov, Wear, 108 (1986) 105–138). It gives a non-linear multivalued dependance of the wall shear stress to the velocity at the wall. Moreover, wall shear stress values may become smaller than values obtained when adhesion prevails in the capillary. The shear stress must over-step some limiting stress level to trigger the wall slip. After checking slip boundary condition implementation for the case of Poiseuille flow with slip along the entire wall, the convergence and the validity of the computation was studied. Important morphologic changes of the flow field and the stress field appear around the transition point from adhesion to slip boundary condition. Slip at the wall allows the principal stress difference to be drastically reduced, except in the vicinity of the transition point where this difference is maximum. A peak in shear stress located upstream of the transition, and a peak in elongational stress located downstream of the transition, are observed at the wall. Fully developed near plug-like flows are obtained within about 1D only downstream of the transition point. It is concluded that the effect of slip on extrudates distorsion should appear clearly even when the exit slippery zone is reduced to 1D.     

The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

On the modeling of fluidity loss phenomena in Couette and Poiseuille flows of elastic liquids

Some effects of the possible relaxation transition from viscoelastic liquid state to highly elastic solid state were theoretically and numerically investigated in the shear situations, within the approach proposed in papers [1, 2, 5, 16]. It was found that for a single Maxwellian model the constitutive equations developed in [1, 2, 5] are not valid at elevated shear stresses. Some new aspects of the possible rheological behavior of elastic liquids in subcritical (before transition) and supercritical (after transition) regimes were demonstrated. The mechanism of fluidity loss studied in this paper could serve as a possible trigger mechanism for the melt flow instabilities.     

Extrusion of a compressible Newtonian fluid with periodic inflow and slip at the wall

We explore a mechanism of extrusion instability, based on the combination of nonlinear slip and compressibility. We consider the time-dependent compressible Newtonian extrudate swell problem with slip at the wall. Steady-state solutions are unstable in regimes where the shear stress is a decreasing function of the velocity at the wall. Compressibility provides the means for the alternate storage and release of elastic energy, and, consequently, gives rise to periodic solutions. The added novelty in the present work is the assumption of periodic volumetric flow rate at the inlet of the die. This leads to more involved periodic responses and to free surface oscillations similar to those observed experimentally with the stick-slip instability. To numerically simulate the flow, we use finite elements in space and a fully-implicit scheme in time.Dedicated to the memory of Prof. Tasos Papanastasiou     

Couette flow with slip and jump boundary conditions

The steady plane Couette flow is analyzed within the framework of the five field equations of mass, momentum and energy for a Newtonian viscous heat conducting ideal gas in which slip and jump boundary conditions are considered. The results obtained are compared with those that follow from the direct simulation Monte Carlo method. Received July 10, 2000     

An incompressible smoothed particle hydrodynamics scheme for Newtonian/non-Newtonian multiphase flows including semi-analytical solutions for two-phase inelastic Poiseuille flows

An incompressible smoothed particle hydrodynamics (ISPH) method is developed for the modeling of multiphase Newtonian and inelastic non-Newtonian flows at low density ratios. This new method is the multiphase extension of Xenakis et al, J. Non-Newtonian Fluid Mech., 218, 1-15, which has been shown to be stable and accurate, with a virtually noise-free pressure field for single-phase non-Newtonian flows. For the validation of the method a semi-analytical solution of a two-phase Newtonian/non-Newtonian (inelastic) Poiseuille flow is derived. The developed method is also compared with the benchmark multiphase case of the Rayleigh Taylor instability and a submarine landslide, thereby demonstrating capability in both Newtonian/Newtonian and Newtonian/non-Newtonian two-phase applications. Comparisons with analytical solutions, experimental and previously published results are conducted and show that the proposed methodology can accurately predict the free-surface and interface profiles of complex incompressible multi-phase flows at low-density ratios relevant, for example, to geophysical environmental applications.     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

Unsteady Couette flow of a micropolar fluid with slip

The unsteady Couette flow of an isothermal incompressible micropolar fluid between two infinite parallel plates is investigated. The motion of the fluid is produced by a time-dependent impulsive motion of the lower plate while the upper plate is set at rest. A linear slip, of Basset type, boundary condition on both plates is used. Two particular cases are discussed; in the first case we have assumed that the plate moves with constant speed and in the second case we have supposed that the plate oscillates tangentially. The solution of the problem is obtained in the Laplace transform domain. The inversion of the Laplace transform is carried out numerically using a numerical method based on Fourier series expansion. Numerical results are represented graphically for the velocity, microrotation, and volume flux for various values of the time, slip and micropolar parameters.     

Performance of iterative methods for Newtonian and generalized Newtonian flows

We consider the use of accelerated gradient-type iterative methods for solution of Newtonian and certain non-Newtonian (power-law and Bingham models) viscous flow problems. The formulations are based on penalty and mixed finite element methods, and such factors as the effect of the penalty parameter, asymmetry, continuation and preconditioning are examined.     

Analysis on creeping channel flows of compressible fluids subject to wall slip

Creeping channel flows of compressible fluids subject to wall slip are widely encountered in industries. This paper analyzes such flows driven by pressure in planar as well as circular channels. The analysis elucidates unsteady flows of Newtonian fluids subject to the Navier slip condition, followed by steady flows of viscoplastic fluids, in particular, Herschel–Bulkley fluids and their simplifications including power law and Newtonian fluids, that slip at wall with a constant coefficient or a coefficient inversely proportional to pressure. Under the lubrication assumption, analytical solutions are derived, validated, and discussed over a wide range of parameters. Analysis based on the derived solutions indicates that unsteadiness alters cross-section velocity profiles. It is demonstrated that compressibility of the fluids gives rise to a concave pressure distribution in the longitudinal direction, whereas wall slip with a slip coefficient that is inversely proportional to pressure leads to a convex pressure distribution. Energy dissipation resulting from slippage can be a significant portion in the total dissipation of such a flow. A distinctive feature of the flow is that, in case of the pressure-dependent slip coefficient, the slip velocity increases rapidly in the flow direction and the flow can evolve into a pure plug flow at the exit.