Efficient computation of the spectrum of viscoelastic flows

The understanding of viscoelastic flows in many situations requires not only the steady state solution of the governing equations, but also its sensitivity to small perturbations. Linear stability analysis leads to a generalized eigenvalue problem (GEVP), whose numerical analysis may be challenging, even for Newtonian liquids, because the incompressibility constraint creates singularities that lead to non-physical eigenvalues at infinity. For viscoelastic flows, the difficulties increase due to the presence of continuous spectrum, related to the constitutive equations.The Couette flow of upper convected Maxwell (UCM) liquids has been used as a case study of the stability of viscoelastic flows. The spectrum consists of two discrete eigenvalues and a continuous segment with real part equal to ?1/We (We is the Weissenberg number). Most of the approximations in the literature were obtained using spectral expansions. The eigenvalues close to the continuous part of the spectrum show very slow convergence.In this work, the linear stability of Couette flow of a UCM liquid is studied using a finite element method. A new procedure to eliminate the eigenvalues at infinity from the GEVP is proposed. The procedure takes advantage of the structure of the matrices involved and avoids the computational overhead of the usual mapping techniques. The GEVP is transformed into a non-degenerate GEVP of dimension five times smaller. The computed eigenfunctions related to the continuous spectrum are in good agreement with the analytic solutions obtained by Graham [M.D. Graham, Effect of axial flow on viscoelastic Taylor–Couette instability, J. Fluid Mech. 360 (1998) 341].     

Solitary coherent structures in viscoelastic shear flow: computation and mechanism

Starting from stationary bifurcations in Couette-Dean flow, we compute nontrivial stationary solutions in inertialess viscoelastic circular Couette flow. These solutions are strongly localized vortex pairs, exist at arbitrarily large wavelengths, and show hysteresis in the Weissenberg number, similar to experimentally observed "diwhirl" patterns. Based on the computed velocity and stress fields, we elucidate a heuristic, fully nonlinear mechanism for these flows. We propose that these localized, fully nonlinear structures comprise fundamental building blocks for complex spatiotemporal dynamics in the flow of elastic liquids.     

A Stokesian viscoelastic flow: Transition to oscillations and mixing

To understand observations of low Reynolds number mixing and flow transitions in viscoelastic fluids, we study numerically the dynamics of the Oldroyd-B viscoelastic fluid model. The fluid is driven by a simple time-independent forcing that, in the absence of viscoelastic stresses, creates a cellular flow with extensional stagnation points. We find that at O(1) Weissenberg number, these flows lose their slaving to the forcing geometry of the background force, become oscillatory with multiple frequencies, and show continual formation and destruction of small-scale vortices. This drives flow mixing, the details of which we closely examine. These new flow states are dominated by a single-quadrant vortex, which may be stationary or cycle persistently from cell to cell.     

Viscous heating and the stability of newtonian and viscoelastic taylor-couette flows

The effects of viscous heating on the stability of Taylor-Couette flow were investigated through flow visualization experiments for Newtonian and viscoelastic fluids. For highly viscous Newtonian fluids, viscous heating drives a transition to a new, oscillatory mode of instability at a critical Reynolds number significantly below that at which the inertial transition is observed in isothermal flows. The effects of viscous heating may explain the discrepancies between the observed and predicted critical conditions and the symmetry of the disturbance flow for viscoelastic instabilities.     

Shear instability in fluids with a density-dependent viscosity

We present a shear instability, which can be triggered in compressible fluids with density-dependent viscosity at shear rates above critical. The instability mechanism is generic: It is based on density-dependent viscosity, compressibility, as well as flow two-(three-)dimensionality that provides coupling between streamwise and transversal velocity components and density variations. The only factor stabilizing the instability is fluid elasticity. The corresponding eigenvalue problem for a plane Couette flow is solved analytically in the limiting cases of large and small wave numbers.     

Numerical solution of an extended White–Metzner model for eccentric Taylor–Couette flow

In this study, we have developed a new numerical approach to solve differential-type viscoelastic fluid models for a commonly used benchmark problem, namely, the steady Taylor—Couette flow between eccentric cylinders. The proposed numerical approach is special in that the nonlinear system of discretized algebraic flow equations is solved iteratively using a Newton–Krylov method along with an inverse-based incomplete lower-upper preconditioner. The numerical approach has been validated by solving the benchmark problem for the upper-convected Maxwell model at a large Deborah number. Excellent agreement with the numerical data reported in the literature has been found. In addition, a parameter study was performed for an extended White–Metzner model. A large eccentricity ratio was chosen for the cylinder system in order to allow flow recirculation to occur. We detected several interesting phenomena caused by the large eccentricity ratio of the cylinder system and by the viscoelastic nature of the fluid. Encouraged by the results of this study, we intend to investigate other polymeric fluids having a more complex microstructure in an eccentric annular flow field.     

Hydrodynamic stability and stellar oscillations

Chandrasekhar’s monograph on Hydrodynamic and hydromagnetic stability, published in 1961, is a standard reference on linear stability theory. It gives a detailed account of stability of fluid flow in a variety of circumstances, including convection, stability of Couette flow, Rayleigh–Taylor instability, Kelvin–Helmholtz instability as well as the Jean’s instability for star formation. In most cases he has extended these studies to include effects of rotation and magnetic field. In a later paper he has given a variational formulation for equations of non-radial stellar oscillations. This forms the basis for helioseismic inversion techniques as well as extension to include the effect of rotation, magnetic field and other large-scale flows using a perturbation treatment.     

The structure of a planar nematic liquid crystal in an oscillating Couette flow beyond the stability threshold

The deformation of a nematic liquid crystal with a planar molecule alignment under the effect of an oscillating Couette flow is theoretically described. The case of external action with amplitudes exceeding the instability threshold of the initial crystal structure is considered. The effect is analyzed by the perturbation method on the basis of the nonlinear equations of nematodynamics. The type and magnitude of the NLC structure distortions are determined as functions of the frequency and amplitude of shear.     

Shear-thinning-induced chaos in taylor-couette flow

The effect of weak shear thinning on the stability of the Taylor-Couette flow is explored for a Carreau-Bird fluid in the narrow-gap limit. The Galerkin projection method is used to derive a low-order dynamical system from the conservation of mass and momentum equations. In comparison with the Newtonian system, the present equations include additional nonlinear coupling in the velocity components through the viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of the base (Couette) flow, becomes lower as the shear-thinning effect increases. That is, shear thinning tends to precipitate the onset of Taylor vortex flow. Similar to Newtonian fluids, there is an exchange of stability between the Couette and Taylor vortex flows, which coincides with the onset of a supercritical bifurcation. However, unlike the Newtonian model, the Taylor vortex cellular structure loses its stability in turn as the Taylor number reaches a critical value. At this point, a Hopf bifurcation emerges, which exists only for shear-thinning fluids.     

中等半径比反向旋转圆Couette系统流动稳定性的实验和数值研究

圆Couette系统已成为研究从层流转捩为湍流以及有限几何尺寸对图案选择影响的范例.本文以实验和计算机模拟方法研究中等半径比圆Couette系统的稳定性.考察同轴独立旋转圆筒之间的粘性不可压缩流体运动，推广了经典的Rayleigh离心不稳定性理论，导出稳定性判据，用来定量地确定稳定界限.实验采用了流动显示和激光散射技术.仪器有半径比η=0.699，形状比Γ=18.流动状态相图中的显著特征是新的首次失稳态：当外筒静止或反向旋转时，首次失稳出现具有非零方位角波数的螺旋涡流，在轴向和方位角方向为行进波，而并非与时间无关的Taylor涡.初步实验所得的转捩Reynolds数与数值计算结果一致.实验室和数值实验显示出半径比对图案形成和转捩序列的影响. 关键词：     

Stability of flow between rotating cylinders with axial buoyancy effect

The linear stability of a fluid confined between two coaxial cylinders rotating independently with axial buoyancy induced flow is examined. Buoyancy is included through the Boussinesq approximation. The numerical investigation is restricted to radius ratio 0.5 at Prandtl number 0.709 with co-rotation situation. The outer rotating cylinder’s Couette flow Reynolds number is restricted to 200. Zeroth-order discontinuities are found in the critical surface, which are explained as the result of the competition between the centrifugal and axial buoyancy induced shear instability mechanisms. Due to the competition, the neutral stability curves develop islands of instability, which considerably lower the instability threshold. Specific and robust numerical methods to handle these geometrical complexities are developed.     

Pulsed Gradient Spin Echo Nuclear Magnetic Resonance Measurement and Simulation of Two-Fluid Taylor Vortex Flow in a Vertically Oriented Taylor–Couette Device

Magnetic resonance microscopy and Ansys Fluent^? computational fluid dynamics simulation have been used to classify Taylor vortex flows (TVF) for several single fluid and axially stratified two-fluid systems in a vertically oriented Taylor–Couette device. A Rheo-NMR (nuclear magnetic resonance) Couette system (Magritek Ltd, New Zealand) with a 1.05-mm gap was used to evaluate the transition from Couette flow to TVF in 1.65 cSt silicone oil, 1 cSt deionized water, and 0.65 cSt silicone oil. The rotation rate at which instability onset occurred agreed between experiment and simulation, as did the critical wavelength. Velocities were mapped for axially stratified two-fluid systems. The vortex containing the two-fluid interface was found to form with a significantly longer wavelength than that observed in the pure fluids. For experiments and simulations in the TVF regime, a region with no secondary flows was found at the interface, indicating interface stabilization by surface tension forces.     

Thermal instability in Maxwellian and Oldroydian viscoelastic fluids with suspended particles in porous medium

The effect of suspended particles on thermal instability is considered separately in Maxwellian and Oldroydian viscoelastic fluids in a porous medium. The principle of exchange of stabilities is found to hold well under a condition which is the same for Maxwellian as well as Oldroydian fluid. For stationary convection, both the Maxwellian and Oldroydian fluids behave like Newtonian fluid and the medium permeability and the suspended particles have destabilizing effects on the system. The sufficient conditions for the non-existence of overstability for both Maxwellian and Oldroydian viscoelastic fluids are also obtained.     

Degeneration of flow pattern in acousto-elastic flow through sharp-edge microchannels

Acoustic streaming (AS) is the steady time-averaged flow generated by acoustic field, which has been widely used in enhancing mixing and particle manipulation. Current researches on acoustic streaming mainly focus on Newtonian fluids, while many biological and chemical solutions exhibit non-Newtonian properties. The acoustic streaming in viscoelastic fluids has been studied experimentally for the first time in this paper. We found that the addition of polyethylene oxide (PEO) polymer to the Newtonian fluid significantly altered the flow characteristics in the microchannel. The resulting acousto-elastic flow showed two modes: positive mode and negative mode. Specifically, the viscoelastic fluids under acousto-elastic flow exhibit mixing hysteresis features at low flow rates, and degeneration of flow pattern at high flow rates. Through quantitative analysis, the degeneration of flow pattern is further summarized as time fluctuation and spatial disturbance range reduction. The positive mode in acousto-elastic flow can be used for the mixing enhancement of viscoelastic fluids in the micromixer, while the negative mode provides a potential method for particle/cell manipulation in viscoelastic body fluids such as saliva by suppressing unstable flow.     

基于SPH方法的瞬态粘弹性流体的数值模拟

运用SPH(Smoothed Particle Hydrodynamics)方法模拟基于Oldroyd-B模型的平面突然起动Couette流,通过数值解与解析解的比较,验证SPH方法模拟瞬态粘弹性流动的准确性;且对基于Oldroyd-B模型的方腔驱动流进行SPH模拟.采用一种新的固壁边界处理方法,有效地防止了粒子穿透,提高数值计算的准确性.用数值算例验证SPH方法对粘弹性流体模拟的有效性和稳定性.     

Long-wavelength modulation of turbulent shear flows

The reverse transition from turbulent to laminar flow is studied in very large aspect ratio plane Couette and Taylor–Couette experiments. We show that laminar-turbulence coexistence dynamics (turbulent spots, spiral turbulence, etc.) can be seen as the ultimate stage of a modulation of the turbulent flows present at higher Reynolds number leading to regular, long-wavelength, inclined stripes. This new type of instability, whose originality is to arise within a macroscopically fluctuating state, can be described in the framework of Ginzburg–Landau equations to which noise is heuristically added to take into account the intrinsic fluctuations of the basic state.     

Magnetic fluid behavior in uniform DC,AC, and rotating magnetic fields

New flows and instabilities are demonstrated for magnetic fluids and by dual analogy to dielectric fluids. If a fluid drop is contained in a thin gap between two glass plates (Hele–Shaw cell) with a simultaneously applied in-plane rotating field and a DC axial field, smooth spirals or an abrupt transformation to many small droplets can occur. A preliminary minimum magnetization and surface energy analysis is presented to model the abrupt transformation in ferrofluids. An analysis of effective DC magnetoviscosity is also presented for planar Couette flow with an applied uniform DC field transverse to a duct axis with the effective magnetoviscosity and flow spin velocity calculated as a function of field strength. Related Couette viscometer measurements of ferrofluid viscosity show zero and negative magnetoviscosity values for rotating magnetic fields.     

Parallel flow in hele-shaw cells with ferrofluids

Parallel flow in a Hele-Shaw cell occurs when two immiscible liquids flow with relative velocity parallel to the interface between them. The interface is unstable due to a Kelvin-Helmholtz type of instability in which fluid flow couples with inertial effects to cause an initial small perturbation to grow. Large amplitude disturbances form stable solitons. We consider the effects of applied magnetic fields when one of the two fluids is a ferrofluid. The dispersion relation governing mode growth is modified so that the magnetic field can destabilize the interface even in the absence of inertial effects. However, the magnetic field does not affect the speed of wave propogation for a given wave number. We note that the magnetic field creates an effective interaction between the solitons.     

Rayleigh-taylor instability of viscoelastic fluids with suspended particles in porous medium in hydromagnetics

The instability of the plane interface between two viscoelastic (Oldroydian) superposed conducting fluids permeated with suspended particles in porous medium is studied when the whole system is immersed in a uniform magnetic field. The dispersion relation for the Oldroydian viscoelastic fluid is obtained which also yields dispersion relations for Maxwellian and Newtonian fluids in special cases, in the presence of suspended particles in porous medium in hydromagnelics. The system is found to be stable for potentially stable case. The presence of magnetic field stabilizes certain wave number band whereas the system was unstable for all wave numbers in the absence of magnetic field, for the potentially unstable configuration. The growth rates increase (for certain wave numbers) and decrease (for other wave numbers) with the increase in stress relaxation time, strain retardation time, suspended particles number density and medium permeability.     

Effect of the direction of the electric field on the interfacial instability between a passive fluid and a viscoelastic polymer

In this paper, we study the linear stability of the interface between an Upper Convective Maxwell fluid and a hydrodynamically passive fluid subject to an electric field applied either parallel or normal to the flat interface between the two fluids. The fluids are leaky-dielectric and we apply surface-coupled model. We solve the model equations analytically and study the dispersion and neutral curves for various parameters representing the applied potential, the fluid’s elasticity, the physical and the electrical properties of the fluids, and the heights of the fluids in the presence of both normal and parallel electric fields. It is found that the critical wavenumber is independent of the Weissenberg number. However, increasing the Weissenberg number increases the maximum growth rate for both the normal and the parallel fields. The critical wavenumber increases with the dimensionless applied voltage for the normal field. Lastly for the normal field, for some values of the dimensionless parameters, the growth rate reached very large values representing some type of singularity as has been observed in the literature. However, for the same values of the parameters no singularity is observed for the parallel field.