Stick-slip flow of high density polyethylene in a transparent slit die investigated by laser Doppler velocimetry

The oscillating flow instability of a molten linear high-density polyethylene is carefully studied using a single screw extruder equipped with a transparent slit die. Experiments are performed using laser Doppler velocimetry in order to obtain the local velocities field across the entire die width. At low flow rate, the extrusion is stable and steady state velocity profiles are obtained. During the instability, the velocity oscillates between two steady state limits, suggesting a periodic stick-slip transition mechanism. At high flow rate, the flow is mainly characterized by a pronounced wall slip. We show that wall slip occurs all along the die land. An investigation of the slip flow conditions shows that wall slip is not homogeneous in a cross section of the slit die, and that pure plug flow occurs only for very high flow rates. A numerical computation of the profile assuming wall slip boundary conditions is done to obtain the true local wall slip velocity. It confirms that slip velocities are of the same order of magnitude as those measured with a capillary rheometer.     

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.     

Slip effects on shearing flows in a porous medium

This paper deals with the magnetohydrodynamic （MHD） flow of an Oldroyd 8-constant fluid in a porous medium when no-slip condition is no longer valid. Modified Darcy＇s law is used in the flow modelling. The non-linear differential equation with non-linear boundary conditions is solved numerically using finite difference scheme in combination with an iterative technique. Numerical results are obtained for the Couette, Poiseuille and generalized Couette flows. The effects of slip parameters on the velocity profile are discussed.     

Empirical slip and viscosity model performance for microscale gas flow

For the simple geometries of Couette and Poiseuille flows, the velocity profile maintains a similar shape from continuum to free molecular flow. Therefore, modifications to the fluid viscosity and slip boundary conditions can improve the continuum based Navier–Stokes solution in the non‐continuum non‐equilibrium regime. In this investigation, the optimal modifications are found by a linear least‐squares fit of the Navier–Stokes solution to the non‐equilibrium solution obtained using the direct simulation Monte Carlo (DSMC) method. Models are then constructed for the Knudsen number dependence of the viscosity correction and the slip model from a database of DSMC solutions for Couette and Poiseuille flows of argon and nitrogen gas, with Knudsen numbers ranging from 0.01 to 10. Finally, the accuracy of the models is measured for non‐equilibrium cases both in and outside the DSMC database. Flows outside the database include: combined Couette and Poiseuille flow, partial wall accommodation, helium gas, and non‐zero convective acceleration. The models reproduce the velocity profiles in the DSMC database within an L₂ error norm of 3% for Couette flows and 7% for Poiseuille flows. However, the errors in the model predictions outside the database are up to five times larger. Copyright © 2005 John Wiley & Sons, Ltd.     

Slow motions of a circular cylinder experiencing slip near a plane wall

A theoretical study is presented for the two-dimensional creeping flow caused by a long circular cylindrical particle translating and rotating in a viscous fluid near a large plane wall parallel to its axis. The fluid is allowed to slip at the surface of the particle. The Stokes equations for the fluid velocity field are solved in the quasi-steady limit using cylindrical bipolar coordinates. Semi-analytical solutions for the drag force and torque acting on the particle by the fluid are obtained for various values of the slip coefficient associated with the particle surface and of the relative separation distance between the particle and the wall. The results indicate that the translation and rotation of the confined cylinder are not coupled with each other. For the motion of a no-slip cylinder near a plane wall, our hydrodynamic drag force and torque results reduce to the closed-form solutions available in the literature. The boundary-corrected drag force and torque acting on the particle decrease with an increase in the slip coefficient for an otherwise specified condition. The plane wall exerts the greatest drag on the particle when its migration occurs normal to it, and the least in the case of motion parallel to it. The enhancement in the hydrodynamic drag force and torque on a translating and rotating particle caused by a nearby plane wall is much more significant for a cylinder than for a sphere.     

Exact solutions in generalized Oldroyd-B fluid

This investigation deals with the influence of slip condition on the magnetohydrodynamic (MHD) and rotating flow of a generalized Oldroyd-B (G.Oldroyd-B)fluid occupying a porous space.Fractional calcul...     

On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

Transient motion of finite amplitude standing waves in acoustic resonators

Extraordinarily high maximum-to-minimum gas pressure ratios appear in an oscillating closed resonator at its resonance frequency for certain resonator shapes. Using a quasi-one-dimensional model based on the compressible Navier–Stokes equations and a finite volume method, we investigate the transient motion of a fluid inside oscillating axisymmetric tubes, from the quiescent condition to the periodic steady motion. We find that the amplitude of the fast oscillations in pressure increases monotonically to the value of its steady state for a cylindrical tube of constant cross-section, while the amplitude undergoes a spiral toward the final steady state value for conical or horn-cone resonators. We discuss the effects of fluid properties on the transient motions. In addition, we compare our numerical results with available experimental results and find good agreement. In particular, for horn-cone resonators driven by large amplitude force, we find a secondary lower peak in pressure waveform within one period of oscillation at the small end of the cavity, matching the findings of the existing experimental result.     

不同润湿性纳米通道内库埃特流动的模拟

利用非平衡分子动力学模拟方法, 模拟了两无限大平行平板组成的纳米通道内的库埃特流动, 并给出了壁面润湿性和速度对流场密度、速度分布及壁面滑移的影响规律.数值模拟中, 统计系综采用微正则系综, 势能函数选用LJ/126模型, 壁面设为刚性原子壁面, 温度校正使用速度定标法, 牛顿运动方程的求解则采用文莱特算法.结果表明, 纳米通道内流体密度呈对称的衰减振荡分布, 且随壁面润湿性的降低, 振荡幅度减小, 振荡周期保持不变;滑移量随壁面润湿性的提高而降低, 甚至在亲水壁面时出现负滑移现象;随壁面速度的增加滑移速度逐渐增大, 且在流体呈现非线性流动阶段其增幅显著加大.另外, 还发现当壁面设置为超疏水性时, 壁面滑移呈现出随润湿性降低而减小的反常现象, 并基于杨氏方程对其进行了解释.     

A timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics

We discuss how matrix-free/timestepper algorithms can efficiently be used with dynamic non-Newtonian fluid mechanics simulators in performing systematic stability/bifurcation analysis. The timestepper approach to bifurcation analysis of large-scale systems is applied to the plane Poiseuille flow of an Oldroyd-B fluid with non-monotonic slip at the wall, in order to further investigate a mechanism of extrusion instability based on the combination of viscoelasticity and non-monotonic slip. Due to the non-monotonicity of the slip equation the resulting steady-state flow curve is non-monotonic and unstable steady states appear in the negative-slope regime. It has been known that self-sustained oscillations of the pressure gradient are obtained when an unstable steady state is perturbed [M.M. Fyrillas, G.C. Georgiou, D. Vlassopoulos, S.G. Hatzikiriakos, A mechanism for extrusion instabilities in polymer melts, Polymer Eng. Sci. 39 (1999) 2498–2504].Treating the simulator of a distributed parameter model describing the dynamics of the above flow as an input–output “black-box” timestepper of the state variables, stable and unstable branches of both equilibrium and periodic oscillating solutions are computed and their stability is examined. It is shown for the first time how equilibrium solutions lose stability to oscillating ones through a subcritical Hopf bifurcation point which generates a branch of unstable limit cycles and how the stable periodic solutions lose their stability through a critical point which marks the onset of the unstable limit cycles. This implicates the coexistence of stable equilibria with stable and unstable periodic solutions in a narrow range of volumetric flow rates.     

Exact solutions of incompressible Couette flow with porous walls for slightly rarefied gases

In this work, the transient incompressible Couette flow and steady-state temperature profiles between two porous parallel plates for slightly rarefied gases are solved exactly. The first-order approximation of slip velocity at the boundaries is used in the formulation. The solution is also applicable for Couette flow in micro-channels under certain circumstances. The influences of mass transfer and a nondimensional slip parameter on slip velocities are discussed. It is also found that the transient slip velocities at the walls are greatly different from the steady-state velocity slips. The influences of velocity slip and temperature slip parameters on the temperature distribution and heat transfer at the walls are analyzed and discussed. It is shown that the slip parameters can greatly change the temperature profiles and heat transfer characteristics at the walls.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

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     

Transient flows of Maxwell fluid with slip conditions

Two fundamental flows, namely, the Stokes and Couette flows in a Maxwell fluid are considered. The exact analytic solutions are derived in the presence of the slip condition. The Laplace transform method is employed for the development of such solutions. Limiting cases of no-slip and viscous fluids can be easily recovered from the present analysis. The behaviors of embedded flow parameters are discussed through graphs.     

The Stokes boundary layer for a power-law fluid

We develop semi-analytical, self-similar solutions for the oscillatory boundary layer (‘Stokes layer’) in a semi-infinite power-law fluid bounded by an oscillating wall (the so-called Stokes problem). These solutions differ significantly from the classical solution for a Newtonian fluid, both in the non-sinusoidal form of the velocity oscillations and in the manner at which their amplitude decays with distance from the wall. In particular, for shear-thickening fluids the velocity reaches zero at a finite distance from the wall, and for shear-thinning fluids it decays algebraically with distance, in contrast to the exponential decay for a Newtonian fluid. We demonstrate numerically that these semi-analytical, self-similar solutions provide a good approximation to the flow driven by a sinusoidally oscillating wall.     

Unsteady hydromagnetic Couette flow through a porous medium in a rotating system

This paper investigates the unsteady hydromagnetic Couette fluid flow through a porous medium between two infinite horizontal plates induced by the non-torsional oscillations of one of the plates in a rotating system using boundary layer approximation. The fluid is assumed to be Newtonian and incompressible. Laplace transform technique is adopted to obtain a unified solution of the velocity fields. Such a flow model is of great interest, not only for its theoretical significance, but also for its wide applications to geophysics and engineering. Analytical expressions for the steady state velocity and shear stress on the plates are obtained, and the case of single oscillating plate is also discussed. The influence of pertinent parameters on the flow is delineated, and appropriate conclusions are drawn.     

Slip at the interface between immiscible polymer melts I: method to measure slip

Slip at the interface between immiscible polymer melts remains poorly understood. A method that relies solely on rheological measurements to obtain the interfacial slip velocity uses the slip-induced deviation in the flow variables. To use the method, accurate estimates of the flow variables under the assumption of no-slip are necessary. Although such estimates can be easily derived under some cases, in general, this is not straightforward. Therefore, methods to determine the interfacial slip velocity without using estimates for the flow variables under no-slip conditions are desirable. In this work, we focus on investigations of slip at the interface between two immiscible polymer melts undergoing two-phase coaxial flow. To enable such investigations, we have adapted the Mooney method, usually used to investigate wall slip, to investigate polymer/polymer interfacial slip. Using this method, we have measured the slip velocity at the interface between polypropylene and polystyrene as a function of the interfacial stress. To determine the validity of the modified Mooney method, we also determine the slip velocity using the slip-induced deviation in the flow variables. To enable this determination, we use polypropylene and polystyrene with almost identical shear rate-dependent viscosities over a range of shear rates. The slip velocity obtained from the modified Mooney method displayed excellent agreement with that determined using the deviation from no-slip. In agreement with prior work, the dependence of the slip velocity on the interfacial stress is a power-law. Our investigation spans a sufficiently wide range of interfacial stress to enable the direct observation of two power-law regimes and also the transition between the two regimes. We also find that the power-law exponent of approximately 3 at low stresses decreases to approximately 2 at high stresses.     

A comparative study of poiseuille flow of a polar fluid under various boundary conditions with applications to blood flow

In this paper, Poiseuille flow of a polar fluid (model of a red blood cell suspension) under various boundary conditions at the wall, viz., slip or no-slip in the axial velocity and couple stresses zero or non-zero at the boundary, is considered from the point of view of its applications to blood flow. Analytic expressions for axial and rotational velocities, flow rate, effective viscosity and stresses are obtained. The magnitudes of the length ratioL and the coupling number N are determined in accordance with concentration and tube radius (in the existing literature, values ofL andN are chosen arbitrarily). Velocity profiles (both axial and rotational) and the variation of the effective viscosity with concentration, tube radius and for various values of the boundary condition parameters are shown graphically. The analytic results obtained are compared with experimental results (for blood flow). It is found that they are in a reasonably good agreement. The effective viscosity exhibits the Inverse Fahraeus-Lindquist Effect in all the cases (including the slip or no-slip in the velocity fields). A method is given for determining the non-zero couple stress boundary condition for a given concentration. Applications of this theory to blood flow are briefly discussed.     

Stokes flow between two confocal rotating spheroids with slip

The steady axisymmetric flow problem of a viscous fluid confined between two confocal spheroids that are rotating about their axis of revolution with different angular velocities is considered. A linear slip, of Basset type, boundary condition on both surfaces of the spheroidal particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in prolate and oblate spheroidal coordinates. The boundary conditions on the particle’s surface and spheroidal container are satisfied by a collocation technique. The torque exerted on the spheroidal particle by the fluid is evaluated with good convergence for various values of the slip parameters, the relative angular velocity and aspect ratios of the spheroids. The limiting case of no-slip is in good agreement with the available values in the literature.