Eric S. G. Shaqfeh

A Purely Elastic Instability in Taylor-Couette Flow

A non-inertial (zero Taylor number) viscoelastic instability is discovered for Taylor–Couette flow of dilute polymer solutions. A linear stability analysis of the inertialess flow of an Oldroyd-B fluid (using both approximate Galerkin analysis and numerical solution of the relevant small-gap eigenvalue problem) show the growth of an overstable (oscillating) mode when the Deborah number exceeds f ( S ) e −½ , where e is the ratio of the gap to the inner cylinder radius, and f ( S ) is a function of the ratio of solvent to polymer contributions to the solution viscosity. Experiments with a solution of 1000 p.p.m. high-molecular-weight polyisobutylene in a viscous solvent show an onset of secondary toroidal cells when the Deborah number De reaches 20, for e of 0.14, and a Taylor number of 10 −6 , in excellent agreement with the theoretical value of 21. The critical De was observed to increase as e decreases, in agreement with the theory. At long times after onset of the instability, the cells become small in wavelength compared to those that occur in the inertial instability, again in agreement with our linear analysis. For this fluid, a similar instability occurs in cone-and-plate flow, as reported earlier. The driving force for these instabilities is the interaction between a velocity fluctuation and the first normal stress difference in the base state. Instabilities of the kind that we report here are likely to occur in many rotational shearing flows of viscoelastic fluids.

The hydrodynamic stress in a suspension of rods

A theory is presented to describe the momentum transport properties of suspensions containing randomly placed, slender fibers. The theory is based on a diagrammatic representation of the multiple scattering expansion for the averaged Green’s function as developed in the authors’ previous work on the heat and mass transfer properties of fiber dispersions [Phys. Fluids A 1, 3 (1989)]. The ‘‘best one‐body approximation’’ is used to calculate the wavenumber‐dependent, ensemble‐averaged stress for both aligned and isotropically oriented fiber dispersions. Both the dilute and semidilute concentration regimes are considered. The effective viscosity is calculated as a limit unit of the previously obtained wavenumber‐dependent properties. In the semidilute concentration regime the scaling form originally suggested by Batchelor [J. Fluid Mech. 46, 813 (1971)] is recovered for both orientation distributions and its relation to short range ‘‘screening’’ is discussed. Corrections to this result in a ‘‘semidilute expan...

Brownian dynamics simulations of single DNA molecules in shear flow

We present the results of Brownian dynamics simulations of a series of different polymer models which have been used to examine the recent experimental findings of Smith et al. (1999) who studied the dynamics of a single DNA molecule in steady shear flow. The steady average extension at various Weissenberg numbers (Wi) is shown to be well predicted by multimode nonlinear models. Quite surprisingly, the normalized average extension x/L asymptotes to less than 1/2 even for extremely large Wi and we discuss this result on a physical basis. The probability density function of molecular extension at various values of Wi using the Kramer’s chain and the finitely extensible nonlinear elastic dumbbell suggests that the number of internal modes is important in a model designed to capture the dynamics of a real DNA molecule. Three different frequency regimes in the power spectral density observed at finite Wi in the experiments are shown to arise from the coupling of the Brownian fluctuations in the gradient direct...

On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows

Numerical simulations of turbulent polymer solutions using the FENE-P model are used to characterize the action of polymers on turbulence in drag-reduced flows. The energetics of turbulence is investigated by correlating the work done by polymers on the flow with turbulent structures. Polymers are found to store and to release energy to the flow in a well-organized manner. The storage of energy occurs around near-wall vortices as has been anticipated for a long time. Quite unexpectedly, coherent release of energy is observed in the very near-wall region. Large fluctuations of polymer work are shown to re-energize decaying streamwise velocity fluctuations in high-speed streaks just above the viscous sublayer. These distinct behaviours are used to propose an autonomous regeneration cycle of polymer wall turbulence, in the spirit of Jimenez & Pinelli (1999).

Brownian dynamics simulations of bead-rod and bead-spring chains: numerical algorithms and coarse-graining issues

Abstract The efficiency and robustness of various numerical schemes have been evaluated by performing Brownian dynamics simulations of bead-rod and three popular nonlinear bead-spring chain models in uniaxial extension and simple shear flow. The bead-spring models include finitely extensible nonlinear elastic (FENE) springs, worm-like chain (WLC) springs, and Pade approximation to the inverse Langevin function (ILC) springs. For the bead-spring chains two new predictor–corrector algorithms are proposed, which are much superior to commonly used explicit and other fully implicit schemes. In the case of bead-rod chain models, the mid-point algorithm of Liu [J. Chem. Phys. 90 (1989) 5826] is found to be computationally more efficient than a fully implicit Newton’s method. Furthermore, the accuracy and computational efficiency of two different stress expressions for the bead-rod chains, namely the Kramers–Kirkwood and the modified Giesekus have been evaluated under both transient and steady conditions. It is demonstrated that the Kramers–Kirkwood with stochastic filtering is the preferred choice for transient flow while the Giesekus expression is better suited for steady state calculations. The issue of coarse graining from a bead-rod chain to a bead-spring chain has also been investigated. Though bead-spring chains are shown to capture only semi-quantitatively the response of the bead-rod chains in transient flows, a systematic coarse-graining procedure that provides the best description of bead-rod chains via bead-spring chains is presented.

Dynamic simulation of freely draining flexible polymers in steady linear flows

We present a study of the rheological and optical behaviour of Kramers bead{rod chains in dilute solution using stochastic computer simulations. We consider two model linear flows, steady shear and uniaxial extensional flow, in which we calculate the non-Newtonian Brownian and viscous stress contribution of the polymers, their birefringence and a stress-optic coecient. We have developed a computer algorithm to dierentiate the Brownian from the viscous stress contributions which also avoids the order (t) 1=2 noise associated with the Brownian forces. The strain or shear rate is made dimensionless with a molecular relaxation time determined by simulated relaxation of the birefringence and stress after a strong flow is applied. The characteristic long relaxation time obtained from the birefringence and stress are equivalent and shown to scale with N 2 where N is the number of beads in the chain. For small shear or extension rates the viscous contribution to the eective viscosity is constant and scales as N. We obtain an analytic expression which explains the scaling and magnitude of this viscous contribution. In uniaxial extensional flow we nd an increase in the extensional viscosity with the dimensionless flow strength up to a plateau value. Moreover, the Brownian stress also reaches a plateau and we develop an analytic expression which shows that the Brownian stress in an aligned bead{rod chain scales as N 3 . Using scaling arguments we show that the Brownian stress dominates in steady uniaxial extensional flow until large Wi ,Wi 0:06N 2 , where Wi is the chain Weissenberg number. In shear flow the viscosity decays as Wi 1= 2 and the rst normal stress as Wi 4= 3 at moderate Wi. We demonstrate that these scalings can be understood through a quasi-steady balance of shear forces with Brownian forces. For small Wi (in shear and uniaxial extensional flow) and for long times (in stress relaxation) the stress-optic law is found to be valid. We show that the rheology of the bead{rod chain can be qualitatively described by an equivalent FENE dumbbell for small Wi .

A purely elastic transition in Taylor-Couette flow

Experimental evidence of a non-inertial, cellular instability in the Taylor-Couette flow of a viscoelastic fluid is presented. A linear stability analysis for an Oldroyd-B fluid, which is successful in describing many features of the experimental fluid, predicts the critical Deborah number,Dec, at which the instability is observed. The dependence ofDec on the value of the dimensionless gap between the cylinders is also determined.

Relaxation of dilute polymer solutions following extensional flow

Abstract The relaxation of dilute polymer solutions following stretch in uniaxial extensional flow is investigated via Brownian dynamic simulations of a flexible freely-draining bead-rod chain. The bead-rod chain simulations are compared to Brownian dynamic simulations of a FENE dumbbell and numerical calculations of a FENE-PM chain. A universal relaxation curve for the stress decay from steady-state is found by shifting the results to lie on the curve described by the relaxation of an initially straight chain. For all the models investigated, the initial rapid decay of the polymer stress decreases at a rate which scales for large Weissenberg number, Wi as Wi 2 . Our universal curve is in good qualitative and in some cases quantitative agreement with the available experimental data: it is particularly good in predicting decay after stretch at the largest strains. We find hysteresis in comparing the stress versus birefringence during the startup of flow and subsequent relaxation for the bead-rod chain and FENE dumbbell, but not for the FENE-PM chain. The hysteresis in the latter model is lost in the preaveraging of the nonlinear terms. The bead-rod model also displays a configuration hysteresis. The hysteresis observed in these models is in qualitative agreement with recent experiments involving polystyrene-based Boger fluids.

Screening in sedimenting suspensions

A Debye-like screening of a particles velocity disturbance, leading to a finite variance, will occur if the pair probability reflects a net deficit of one particle in the vicinity of each particle. The three-particle ineractions, which determine the structure of a dilute, monodisperse suspension of spheres, lead to a deficit of neighbouring particles

Simulation of profile evolution in silicon reactive ion etching with re-emission and surface diffusion

This article describes a model that simulates etching profiles in reactive ion etching. In particular, models are developed to explain the significant lateral etch rate that is observed in many etch profiles. The total etch rate is considered to consist of two superimposed components: an ion‐assisted rate and a purely ‘‘chemical’’ etch rate, the latter rate being due to etching by radicals in the absence of ion bombardment. The transport of radicals to the evolving interface is studied for two different transport mechanisms: re‐emission from the surface and diffusion along the surface. For the case of transport by surface re‐emission, a reactive sticking coefficient is defined for the radicals, and a formulation is developed to simulate etching for any value (between zero and unity) that this sticking coefficient may assume. When the sticking coefficient approaches either zero or unity, the method of characteristics is shown to be useful for profile simulation. Transport of radicals by surface diffusion i...