SQUEEZE FILM FLOW WITH NONLINEAR BOUNDARY SLIP

A nonlinear boundary slip model consisting of an initial slip length and a critical shear rate was used to study the nonlinear boundary slip of squeeze fluid film confined between two approaching spheres. It is found that the initial slip length controls the slip behavior at small shear rate, but the critical shear rate controls the boundary slip at high shear rate. The boundary slip at the squeeze fluid film of spherical surfaces is a strongly nonlinear function of the radius coordinate. At the center or far from the center of the squeeze film, the slip length equals the initial slip length due to the small shear rate. However, in the high shear rate regime the slip length increases very much. The hydrodynamic force of the spherical squeeze film decreases with increasing the initial slip length and decreasing the critical shear rate. The effect of initial slip length on the hydrodynamic force seems less than that of the critical shear rate. When the critical shear rate is very small the hydrodynamic force increases very slowly with a decrease in minimum film thickness. The theoretical predictions agree well with the experiment measurements.     

Squeeze flow of a Bingham-type fluid with elastic core

The squeeze flow of a Bingham-type material between finite circular disks is considered. The material is modelled assuming that the unyielded region behaves like a linear elastic core. A lubrication approximation is considered. It is shown that no paradox can arise, such as that has been pointed out for many years by various authors when the unyielded region in the fluid is supposed to be perfectly rigid. The unyielded region is shown to be always detached from the axis of symmetry. Some numerical simulations are worked out for different squeezing rates.     

Stokes flow of micro-polar fluids by peristaltic pumping through tube with slip boundary condition

This paper studies the Stokes flow of micro-polar fluids by peristaltic pumping through the cylindrical tube under the effect of the slip boundary condition. The motion of the wall is governed by the sinusoidal wave equation. The analytical and numerical solutions for the axial velocity, the micro-polar vector, the stream function, the pressure gradient, the friction force, and the mechanical efficiency are obtained by using the lubrication theory under the low Reynolds number and long wavelength approximations. The impacts of the emerging parameters, such as the coupling number, the micro-polar parameter, the slip parameter on pumping characteristics, the friction force, the velocity profile, the mechanical efficiency, and the trapping phenomenon are depicted graphically. The numerical results infer that large pressure is required for peristaltic pumping when the coupling number is large, while opposite behaviors are found for the micro-polar parameter and the slip parameter. The size of the trapped bolus reduces with the increase in the coupling number and the micro-polar parameter, whereas it blows up with the increase in the slip parameter.     

Exact solution for compressive flow of viscoplastic fluids under perfect slip wall boundary conditions

Based on the perfect slip condition between rigid walls and fluids, the compressive flow of Herschel-Bulkley fluids and biviscous fluids was studied. The explicit expressions of stresses and fluid velocity were given. To move the rigid walls for a Herschel-Bulkley fluid with the yield stress (τ₀), the mean pressure applied onto the rigid wall should be larger than 2τ₀/. No yield surface exists in the interior of the fluids when flow occurs. For a biviscous fluid, a critical load was given. The fluid behaves like the Bingham fluid when the external applied load onto the wall is larger than the critical load, otherwise the fluid is Newtonian. Received: 10 June 1997 Accepted: 22 September 1997     

Squeeze flow of a power-law fluid between two rigid spheres with wall slip

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Squeeze flow of soft solids between rough surfaces

Newtonian liquids and non-Newtonian soft solids were squeezed between parallel glass plates by a constant force F applied at time t=0. The plate separation h(t) and the squeeze-rate were measured for different amplitudes of plate roughness in the range 0.3–31 m. Newtonian liquids obeyed the relation Vh ³ of Stephan (1874) for large plate separations. Departures from this relation that occurred when h approached the roughness amplitude were attributed to radial liquid permeation through the rough region. Most non-Newtonian materials showed boundary-slip that varied with roughness amplitude. Some showed slip that varied strongly during the squeezing process. Perfect slip (zero boundary shear stress) was not approached by any material, even when squeezed by optically-polished plates. If the plates had sufficient roughness amplitude (e.g. about 30 m), boundary slip was practically absent, and the dependence of V on h was close to that predicted by no-slip theory of a Herschel-Bulkley fluid in squeeze flow (Covey and Stanmore 1981, Adams et al. 1994).     

Stagnation-point flow of a second-grade fluid with slip

The steady two-dimensional stagnation-point flow of a second-grade fluid with slip is examined. The fluid impinges on the wall either orthogonally or obliquely. Numerical solutions are obtained using a quasi-linearization technique.     

Electroosmotic flow of Eyring fluid in slit microchannel with slip boundary condition

In consideration of the electroosmotic flow in a slit microchannel, the con-stitutive relationship of the Eyring fluid model is utilized. Navier＇s slip condition is used as the boundary condition. The governing equations are solved analytically, yielding the velocity distribution. The approximate expressions of the velocity distribution are also given and discussed. Furthermore, the effects of the dimensionless parameters, the electrokinetic parameter, and the slip length on the flow are studied numerically, and appropriate conclusions are drawn.     

Effect of variable slip boundary conditions on flows of pressure driven non-Newtonian fluids

In microfluidic devices it has been suggested a scheme for enhancing the mixing of two fluids is to use patterned, slip boundary conditions. This has been shown to induce significant transverse flow for Newtonian fluids [S.C. Hendy, M. Jasperse, J. Burnell, Effect of patterned slip on micro- and nanofluidic flows, Phys. Rev. E 72 (2005) 016303]. Here we study the effect of patterned slip on non-Newtonian fluids. Using a power-law model it is shown for shear-thickening fluids patterned slip can induce significant transverse flows comparable in size to those produced for Newtonian fluids. However, for shear-thinning fluids this transverse flow is suppressed. We predict a convenient way to increase the transverse flow for shear-thinning fluids is to use a patterned slip boundary condition coupled to a sinusoidally time-varying pressure gradient. This system is studied using a simple linearized White–Metzner model which has a power-law viscosity function [R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, John Wiley & Sons, New York, 1987]. In this case it is shown the two variations combine to produce transverse flow, which can be increased by increasing the frequency of the sinusoidal time-dependent fluctuation.     

SECOND-ORDER MOMENT MODEL FOR DENSE TWO-PHASE TURBULENT FLOW OF BINGHAM FLUID WITH PARTICLES

The USM-θmodel of Bingham fluid for dense two-phase turbulent flow was developed, which combines the second-order moment model for two-phase turbulence with the particle kinetic theory for the inter-particle collision. In this model, phases interaction and the extra term of Bingham fluid yield stress are taken into account. An algorithm for USM-θmodel in dense two-phase flow was proposed, in which the influence of particle volume fraction is accounted for. This model was used to simulate turbulent flow of Bingham fluid single-phase and dense liquid-particle two-phase in pipe. It is shown USM-θmodel has better prediction result than the five-equation model, in which the particle-particle collision is modeled by the particle kinetic theory, while the turbulence of both phase is simulated by the two-equation turbulence model. The USM-θmodel was then used to simulate the dense two-phase turbulent up flow of Bingham fluid with particles. With the increasing of the yield stress, the velocities of Bingham and particle decrease near the pipe centre. Comparing the two-phase flow of Bingham-particle with that of liquid-particle, it is found the source term of yield stress has significant effect on flow.     

宾汉流体阻力近似解公式精度的优化提高

针对前苏联学求解宾汉流体布金汉方程的阻力近似解公式，其与精确解最大偏差为6．7％，首次通过数学分析和三维优化计算，改变公式中的参数，使偏差大幅度降低．偏差是参数和核心流相对半径r^-O的函数，用极限判定了在r^-O闭区间内的连续性和间断点，为降低偏差提供了依据．绘制了偏差三维变化图，应用切片平面解决了多峰曲面的极值问题．最终优化出的参数使公式的最大偏差为2．6％，比6．7％降低了4．1％，优化后的公式，在管道输送阻力计算中更有实用价值．     

Instability of Bingham fluids in Taylor-Dean flow between two concentric cylinders at arbitrary gap spacings

Stability of Bingham fluids is investigated numerically in azimuthal pressure-driven flow between two infinitely long concentric cylinders. An infinitesimal perturbation is introduced to the basic flow and its time evolution is monitored using normal mode linear stability analysis. An eigenvalue problem is obtained which is solved numerically using pseudo-spectral collocation method. Numerical results are obtained for two different cases: (i) the inner cylinder is rotating at constant velocity while the outer cylinder is fixed (i.e., the Taylor-Dean flow) and (ii) both cylinders are fixed (i.e., the Dean flow). The results show that the yield stress always has a stabilizing effect on the Taylor-Dean flow. But, for the Dean flow the effect of the yield stress is predicted to be stabilizing or destabilizing depending on the magnitude of the Bingham number and also the gap size.     

On the flow of a Bingham fluid passing through an electric field

We study the action of an electric field on a Bingham fluid from the point of view of existence and uniqueness of solutions. We also give an upper bound for the stopping time.     

On the no-slip boundary conditions for squeeze flow of viscous fluids

A typical class of boundary conditions for squeeze flow problems in lubrication approximation is the one in which the squeezing rate and the width between the squeezing plates are constant. This hypothesis is justified by claiming that the plates moves so slowly that they can be considered static. In this short note we prove that this assumption leads to a contradiction and hence cannot be used.     

An implicit solution of bi-penalty approximation with orthogonality projection for the numerical simulation of Bingham fluid flow

1.IntroductionItisamajordiffct.encefi-omtheNewtonnuidflowthattheBinghammodelofNonNewtonfluidflowischaracterizedbytwoparameters:ayieldstressandaviscosity.WhenthestressoftheBinghalnmaterialbelowtheyieldstress,materialisrigidotherwisethequasiNewtolliannowresultstll:'71.Hence,therearesomeofthefloating"rigidcores"involvedintheBinghamfluidfloworsomeofthe'rigidcores"attachedtotheboundaries,inwhichthelocationsalldshapesofthese"rigidcores"maychangeforthetransientBinghamfluid,flow.ThisBingllammodelh…     

Time‐dependent flow across a step: the slip with friction boundary condition

The paper studies numerically the slip with friction boundary condition in the time‐dependent incompressible Navier–Stokes equations. Numerical tests on two‐ and three‐dimensional channel flows across a step using this boundary condition on the bottom wall are performed. The influence of the friction parameter on the flow field is studied and the results are explained according to the physics of the flow. Due to the stretching and tilting of vortices, the three‐dimensional results differ in many respects from the two‐dimensional ones. Copyright © 2005 John Wiley & Sons, Ltd.     

The squeeze flow of a bi-viscosity fluid between two rigid spheres with wall slip

In this paper, the squeeze flow between two rigid spheres with a bi-viscosity fluid is examined. Based on lubrication theory, the squeeze force is calculated by deriving the pressure and velocity expressions. The results of the normal squeeze force are discussed, and fitting functions of the squeeze and correction coefficients are given. The squeeze force between the rigid spheres increases linearly or logarithmically with the velocity when most or part of the boundary fluid reaches the yield state, respectively. Furthermore, the slip correction coefficient decreases with the increase in the velocity. The investigation may contribute to the further study of bi-viscosity fluids between rigid spheres with wall slip.     

Finite element analysis of the duct flow of Bingham plastic fluids: an application of the variational inequality

The duct flow of Bingham plastic fluids is analysed with the variational inequality-based finite element method. The problem of tracking the yield surface is solvable through the regularization technique which can be easily incorporated into the existing finite element code. The existence theorem of this method was established through the theory of variational inequalities. A small positive constant is added to the second shear rate invariant, resulting in an apparent viscosity of finite magnitude in the unyielding plug zone. This makes the minimization of the non-differential variational integral possible. In order to achieve convergence at small regularization parameter, a zero-order continuation is employed. It is also shown that a fine tessellation of the flow domain is necessary for tracking the yield surfaces unambiguously. Two classes of duct flow, namely axial flows in eccentric annuli and in an L-shaped duct, were investigated. In both cases it was easy to show the presence of the mobile plugs around the duct centres from the axial velocity profiles; however, the stagnant plugs at the narrow side in eccentric annuli with large eccentricity and near the apex of right-angled corners in an L-shaped duct could only be identified from the calculation of the distributions of the second shear rate or shear stress invariant. © 1997 John Wiley & Sons, Ltd.     

Start-up Flow of a Bingham Fluid in a Pipe

We present an analytical solution of axisymmetric motion for a Bingham fluid initially at rest subjected to a constant pressure gradient applied suddenly. Using the Laplace transform, we obtain expressions which allow the calculation of the instantaneous velocity, plug radius and rate of flow as a function of time. We also give a relation for the shear stress in the plug and in the region where the behaviour of the fluid is Newtonian.     

On slip effect in free coating of non-Newtonian fluids

The failure of the current theories to predict the coating thickness of non-Newtonian fluids in free coating operations is shown to be a result of the effective slip at the moving rigid surface being coated. This slip phenomenon is a consequence of stress induced diffusion occurring in flow of structured liquids in non-homogeneous flow fields. Literature data have been analysed to substantiate the slip hypothesis proposed in this work. The experimentally observed coating thickness is shown to lie between an upper bound, which is estimated by a no-slip condition for homogeneous solution and a lower bound, which is estimated by using solvent properties. Some design considerations have been provided, which will serve as useful guidelines for estimating coating thickness in industrial practice.fa exponent in eq. (15) - b n/(4 –n)(n + 1) - Ca Capillary number - D diffusivity - De Deborah number - g acceleration due to gravity - G Goucher number - h thickness profile - h ₀ final coating thickness - K consistency index - L length available for diffusion - L _t tube length - n power-law index - P pressure drop - Q flow rate - R cylinder radius - R _t tube radius - t time available for diffusion - T ₀ dimensionless thickness without slip - T _s dimensionless thickness with slip - U _c theoretically calculated withdrawal velocity to match the film thickness - u _s slip velocity - U withdrawal velocity - U _w theoretically calculated withdrawal velocity based on solvent properties - U ^* effective withdrawal velocity - x distance in the direction of flow - y distance transverse to the flow direction - curvature coefficient - slip coefficient - curvature coefficient - rate of deformation tensor - u _s /U - relaxation time - density - surface tension - shear stress in tube flow - _w wall shear stress in tube flow - stress tensor - _w wall shear stress - T _s /T ₀ NCL-Communication No. 2818