Elastic vs . inertial instability in a polymer solution flow (original) (raw)
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Mechanism of elastic instability in Couette flow of polymer solutions: Experiment
Physics of Fluids - PHYS FLUIDS, 1998
Experiments on flow stability and pattern formation in Couette flow between two cylinders with highly elastic polymer solutions are reported. It is found that the flow instabilities are determined by the elastic Deborah number, De, and the polymer concentration only, while the Reynolds number becomes completely irrelevant. A mechanism of such ``purely elastic'' instability was suggested a few years ago by Larson, Shaqfeh, and Muller [J. Fluid Mech. 218, 573 (1990)], referred to as LMS. It is based on the Oldroyd-B rheological model and implies a certain functional relation between De at the instability threshold and the polymer contribution to the solution viscosity, εp/ε, that depends on the polymer concentration. The elastic force driving the instability arises when perturbative elongational flow in radial direction is coupled to the strong primary azimuthal shear. This force is provided by the ``hoop stress'' that develops due to stretching of the polymer mole...
2001
Previous experimental measurements and linear stability analyses of curvilinear shearing flows of viscoelastic fluids have shown that the combination of streamwise curvature and elastic normal stresses can lead to flow destabilization. Torsional shear flows of highly elastic fluids with closed streamlines can also accumulate heat from viscous dissipation resulting in nonuniformity in the temperature profile within the flow and nonlinearity in the viscometric properties of the fluid. Recently, it has been shown by Al-Mubaiyedh et al. ͓Phys. Fluids 11, 3217 ͑1999͔͒ that the inclusion of energetics in the linear stability analysis of viscoelastic Taylor-Couette flow can change the dominant mode of the purely elastic instability from a nonaxisymmetric and time-dependent secondary flow to an axisymmetric stationary Taylor-type toroidal vortex that more closely agrees with the stability characteristics observed experimentally. In this work, we present a detailed experimental study of the effect of viscous heating on the torsional steady shearing of elastic fluids between a rotating cone and plate and between two rotating coaxial parallel plates. Elastic effects in the flow are characterized by the Deborah number, De, while the magnitude of the viscous heating is characterized by the Nahme-Griffith number, Na. We show that the relative importance of these two competing effects can be quantified by a new dimensionless thermoelastic parameter, ⌰ϭNa 1/2 /De, which is a material property of a given viscoelastic fluid independent of the rate of deformation. By utilizing this thermoelastic number, experimental observations of viscoelastic flow stability in three different fluids and two different geometries over a range of temperatures can be rationalized and the critical conditions unified into a single flow stability diagram. The thermoelastic number is a function of the molecular weight of the polymer, the flow geometry, and the temperature of the test fluid. The experiments presented here were performed using test fluids consisting of three different high molecular weight monodisperse polystyrene solutions in various flow geometries and over a large range of temperatures. By systematically varying the temperature of the test fluid or the configuration of the test geometry, the thermoelastic number can be adjusted appreciably. When the characteristic time scale for viscous heating is much longer than the relaxation time of the test fluid ͑⌰Ӷ1͒ the critical conditions for the onset of the elastic instability are in good agreement with the predictions of isothermal linear stability analyses. As the thermoelastic number approaches a critical value, the strong temperature gradients induced by viscous heating reduce the elasticity of the test fluid and delay the onset of the instability. At even larger values of the thermoelastic parameter, viscous heating stabilizes the flow completely.
A purely elastic instability in Taylor–Couette flow
Journal of Fluid Mechanics, 1990
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) €7, 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 B of 0.14, and a Taylor number of 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 coneand-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.
Journal of Non-Newtonian Fluid Mechanics, 2006
Nonlinear dynamics that ensue after the inception of viscoelastic flow instabilities in homogeneous, curvilinear shear flows remain largely unexplored. In this work, we have developed an efficient, operator splitting influence matrix spectral (OSIMS) algorithm for the simulation of three-dimensional and transient viscoelastic flows. The OSIMS algorithm is applied to explore, for the first time, the post-critical dynamics of viscoelastic Taylor-Couette flow of dilute polymeric solutions utilizing the Oldroyd-B constitutive equation. Linear stability theory predicts that the flow is unstable to non-axisymmetric and time-dependent disturbances with critical conditions depending on the flow elasticity, E, defined as the ratio of the characteristic time scales of fluid relaxation to viscous diffusion. Two types of secondary flow patterns emerge near the bifurcation point, namely, ribbons and spirals. We have demonstrated via time-dependent simulations for narrow and moderate gap widths, ribbon-like patterns are generally stable at and above the linear stability threshold for 0.05 ≤ E ≤ 0.15. For an inner to outer cylinder radius ratio of 0.8, the bifurcation to ribbons at E = 0.1 and 0.125 occurs through a subcritical transition while the transition is supercritical at smaller E values.
Couette-Taylor Flow in a Dilute Polymer Solution
Physical Review Letters, 1996
We present experimental evidence of the striking influence of small additions of high molecular weight polymers on stability and pattern selection in Couette-Taylor flow. Two novel oscillatory flow patterns were observed. One of them is essentially due to the fluid elasticity. The other results from inertial instability modified by the elasticity. [S0031-9007(96)00915-5]
Elastic turbulence in curvilinear flows of polymer solutions
2004
Following our first report (A. Groisman and V. Steinberg, slNature\sl NatureslNature bf405\bf 405bf405, 53 (2000)) we present an extended account of experimental observations of elasticity induced turbulence in three different systems: a swirling flow between two plates, a Couette-Taylor (CT) flow between two cylinders, and a flow in a curvilinear channel (Dean flow). All three set-ups had high ratio of width of the region available for flow to radius of curvature of the streamlines. The experiments were carried out with dilute solutions of high molecular weight polyacrylamide in concentrated sugar syrups. High polymer relaxation time and solution viscosity ensured prevalence of non-linear elastic effects over inertial non-linearity, and development of purely elastic instabilities at low Reynolds number (Re) in all three flows. Above the elastic instability threshold, flows in all three systems exhibit features of developed turbulence. Those include: (i)randomly fluctuating fluid motion excited in a broa...
Journal of Physics: Conference Series, 2008
We have characterized the transition to turbulence in a flow with semi-dilute solutions (concentration of 0.07%) of polyethylene oxide (PEO) in the Couette-Taylor system with rotating only the inner cylinder. The first instability mode occurs via a supercritical Hopf bifurcation in form of interacting right and left spirals. For higher values of the rotation velocity, turbulent spots appear in the
Polymer conformations and hysteretic stresses in nonstationary flows of polymer solutions
EPL (Europhysics Letters), 2009
The low Reynolds number flow of a polymer solution around a cylinder engenders a nonlinear drag force vs. the flow velocity. A velocity quench of such a flow gives rise to a long time relaxation and hysteresis of the stress due to history-dependent elastic effects. Our results suggest that such hysteretic behavior has its origin in the long time relaxation dynamics and hysteresis of the polymer conformations.
Onset of transition in the flow of polymer solutions through deformable tubes
Physics of Fluids
Experiments are performed to investigate laminar-turbulent transition in the flow of Newtonian and viscoelastic fluids in soft-walled microtubes of diameter ∼400 μm by using the micro-particle image velocimetry technique. The Newtonian fluids used are water and water-glycerine mixtures, while the polymer solutions used are prepared by dissolving polyacrylamide in water. Using different tube diameters, elastic moduli of the tube wall, and polymer concentrations, we probe a wide range of dimensionless wall elasticity parameter Σ and dimensionless fluid elasticity number E. Here, Σ = (ρGR 2)/η 2 , where ρ is the fluid density, G is the shear modulus of the soft wall, R is the radius of the tube, and η is the solution viscosity. The elasticity of the polymer solution is characterized by E = (λη 0)/R 2 ρ, where λ is the zero-shear relaxation time, η 0 is the zero-shear viscosity, ρ is the solution density, and R is the tube radius. The onset of transition is detected by a shift in the ratio of centerline peak to average velocity. A jump in the normalized centerline velocity fluctuations and the flattening of the velocity profile are also used to corroborate the onset of instability. Transition for the flow of Newtonian fluid through deformable tubes (of shear modulus ∼50 kPa) is observed at a transition Reynolds number of Ret ∼ 700, which is much lower than Ret ∼ 2000 for a rigid tube. For tubes of lowest shear modulus ∼30 kPa, Ret for Newtonian fluid is as low as 250. For the flow of polymer solutions in a deformable tube (of shear modulus ∼50 kPa), Ret ∼ 100, which is much lower than that for Newtonian flow in a deformable tube with the same shear modulus, indicating a destabilizing effect of polymer elasticity on the transition already present for Newtonian fluids. Conversely, we also find instances where flow of a polymer solution in a rigid tube is stable, but wall elasticity destabilizes the flow in a deformable tube. The jump in normalized velocity fluctuations for the flow of both Newtonian and polymer solutions in soft-walled tubes is much gentler compared to that for Newtonian transition in rigid tubes. Hence, the mechanism underlying the soft-wall transition for the flow of both Newtonian fluids and polymer solutions could be very different as compared to the transition of Newtonian flows in rigid pipes. When Ret is plotted with the wall elasticity parameter Σ for different moduli of the tube wall, by taking Newtonian fluids of different viscosities and polymer solutions of different concentrations, we observed a data collapse, with Ret following a scaling relation of Ret ∼ Σ 0.7. Thus, both fluid elasticity and wall elasticity combine to trigger a transition at Re as low as 100 in the flow of polymer solutions through deformable tubes.