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Weissenberg number

About: Weissenberg number is a(n) research topic. Over the lifetime, 1334 publication(s) have been published within this topic receiving 29265 citation(s).

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Open accessJournal ArticleDOI: 10.1126/SCIENCE.283.5408.1724
12 Mar 1999-Science
Abstract: The conformational dynamics of individual, flexible polymers in steady shear flow were directly observed by the use of video fluorescence microscopy. The probability distribution for the molecular extension was determined as a function of shear rate, gamma;, for two different polymer relaxation times, tau. In contrast to the behavior in pure elongational flow, the average polymer extension in shear flow does not display a sharp coil-stretch transition. Large, aperiodic temporal fluctuations were observed, consistent with end-over-end tumbling of the molecule. The rate of these fluctuations (relative to the relaxation rate) increased as the Weissenberg number, gamma;tau, was increased.

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  • Fig. 5. Probability distributions for polymer extension in shear flow for several values of Wi. Histograms were calculated with 1-mm bins and normalized by the total number of points (;2000 to 6000 points).
    Fig. 5. Probability distributions for polymer extension in shear flow for several values of Wi. Histograms were calculated with 1-mm bins and normalized by the total number of points (;2000 to 6000 points).
  • Fig. 6. (A) Plots of the power spectral density per unit time (PSD) of the extension fluctuations for various values of Wi. Here we have expressed the PSD in dimensionless units on the basis of known values of RG,0 and t. At equilibrium (Wi 5 0) the shape of the PSD is close to that of a Lorentzian function (dashed line), PSD ; 1/(a 1 f 2), where a is a constant. When the shear flow is applied the PSD increasingly deviates from this form; it rolls off faster at higher frequencies. The other dashed line shows the behavior of a function proportional to 1/(a 1 f 3.7). (B) Dimensionless PSD plot for two data sets with different shear rates and relaxation times (different solvent viscosities) but the same value of Wi. The data sets overlap as expected theoretically. (C) The autocorrelation function ^x(t)x(t 1 T )& plotted versus time delay, T, for various Wi. ^x(t)x(t 1 T )& was calculated from each data set after subtracting the mean extension value. The products of each pair of data points were collected in 1-s time interval bins by rounding the time difference between each pair of points to the nearest second. The results are normalized so that ^x(t)x(t)& 5 1.0.
    Fig. 6. (A) Plots of the power spectral density per unit time (PSD) of the extension fluctuations for various values of Wi. Here we have expressed the PSD in dimensionless units on the basis of known values of RG,0 and t. At equilibrium (Wi 5 0) the shape of the PSD is close to that of a Lorentzian function (dashed line), PSD ; 1/(a 1 f 2), where a is a constant. When the shear flow is applied the PSD increasingly deviates from this form; it rolls off faster at higher frequencies. The other dashed line shows the behavior of a function proportional to 1/(a 1 f 3.7). (B) Dimensionless PSD plot for two data sets with different shear rates and relaxation times (different solvent viscosities) but the same value of Wi. The data sets overlap as expected theoretically. (C) The autocorrelation function ^x(t)x(t 1 T )& plotted versus time delay, T, for various Wi. ^x(t)x(t 1 T )& was calculated from each data set after subtracting the mean extension value. The products of each pair of data points were collected in 1-s time interval bins by rounding the time difference between each pair of points to the nearest second. The results are normalized so that ^x(t)x(t)& 5 1.0.
  • Fig. 4. The mean fractional extension was calculated by time averaging over each data set. The number of separate molecules on which the measurements for each data set were made varied from 6 to 40, depending on Wi. The results for both shear flow (open circles, h 5 60 cP; open squares, h 5 220 cP) and elongational flow (filled circles) are shown. Lines are to guide the eye. Error bars represent the standard error.
    Fig. 4. The mean fractional extension was calculated by time averaging over each data set. The number of separate molecules on which the measurements for each data set were made varied from 6 to 40, depending on Wi. The results for both shear flow (open circles, h 5 60 cP; open squares, h 5 220 cP) and elongational flow (filled circles) are shown. Lines are to guide the eye. Error bars represent the standard error.
  • Fig. 3. Images of individual polymer chains undergoing conformational changes in steady shear with Wi 5 19 and ġ 5 1. Each row of images is a series in time proceeding from left to right. The vertical bar at the left of the top row indicates 5 mm. (A) Example of a molecule that went from being coiled, to stretched, back to coiled, and then stretched again. The time interval between images is 6 s. (B) Example of a lump of mass density that propagated down the contour of a partially stretched chain. The time interval between images is 0.84 s. (C) Example of a molecule that became folded into an upward U shape, then unraveled like a rope sliding over a pulley, then tumbled end-over-end and became folded in an inverted U shape. The time interval between images is 6 s.
    Fig. 3. Images of individual polymer chains undergoing conformational changes in steady shear with Wi 5 19 and ġ 5 1. Each row of images is a series in time proceeding from left to right. The vertical bar at the left of the top row indicates 5 mm. (A) Example of a molecule that went from being coiled, to stretched, back to coiled, and then stretched again. The time interval between images is 6 s. (B) Example of a lump of mass density that propagated down the contour of a partially stretched chain. The time interval between images is 0.84 s. (C) Example of a molecule that became folded into an upward U shape, then unraveled like a rope sliding over a pulley, then tumbled end-over-end and became folded in an inverted U shape. The time interval between images is 6 s.
  • Fig. 2. Examples of typical extension versus time data for different Weissenberg numbers. Many data sets were recorded at each W i with different molecules until a suitable amount of data for statistical analysis was obtained. The finite length of the shear channel limited the length of each data set. The vertical dotted lines in the W i 5 25.2 plot indicate where individual data sets begin and end.
    Fig. 2. Examples of typical extension versus time data for different Weissenberg numbers. Many data sets were recorded at each W i with different molecules until a suitable amount of data for statistical analysis was obtained. The finite length of the shear channel limited the length of each data set. The vertical dotted lines in the W i 5 25.2 plot indicate where individual data sets begin and end.
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Topics: Shear flow (63%), Shear rate (60%), Weissenberg number (59%) ...read more

663 Citations


Journal ArticleDOI: 10.1016/J.JNNFM.2004.08.008
Raanan Fattal1, Raz Kupferman1Institutions (1)
Abstract: We show how to transform a large class of differential constitutive models into an equation for the (matrix) logarithm of the conformation tensor. Under this transformation, the extensional components of the deformation field act additively, rather than multiplicatively. This transformation is motivated by numerical evidence that the high Weissenberg number problem may be caused by the failure of polynomial-based approximations to properly represent exponential profiles developed by the conformation tensor. The potential merits of the new formulation are demonstrated for a finitely-extensible fluid in a two-dimensional lid-driven cavity at Weissenberg number W i = 5 .

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Topics: Cauchy elastic material (62%), Tensor density (60%), Tensor contraction (59%) ...read more

340 Citations


Open accessJournal ArticleDOI: 10.1016/J.JNNFM.2005.01.002
Abstract: The log conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] has been implemented in a FEM context using the DEVSS/DG formulation for viscoelastic fluid flow. We present a stability analysis in 1D and identify the failure of the numerical scheme to balance exponential growth as a possible source for numerical instabilities at high Weissenberg numbers. A different derivation of the log-based evolution equation than in [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] is also presented. We show numerical results for the flow around a cylinder for an Oldroyd-B and a Giesekus model. With the log conformation representation, we are able to obtain solutions beyond the limiting Weissenberg numbers in the standard scheme. In particular, for the Giesekus model the improvement is rather dramatic: there does not seem to be a limit for the chosen model parameter (α = 0.01). However, it turns out that although in large parts of the flow the solution converges, we have not been able to obtain convergence in localized regions of the flow. Possible reasons include artefacts of the model and unresolved small scales. More work is necessary, including the use of more refined meshes and/or higher order schemes, before any conclusion can be made on the local convergence problems. © 2005 Elsevier B.V. All rights reserved.

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  • Fig. 14. The value of cxx for Wi = 5 on the centerline and on the wall of the cylinder for the Giesekus model with α = 0.01 for various meshes. Also shown are the results for meshes M3 and M4 using the 1D procedure as explained in Sec. 6.2.3.
    Fig. 14. The value of cxx for Wi = 5 on the centerline and on the wall of the cylinder for the Giesekus model with α = 0.01 for various meshes. Also shown are the results for meshes M3 and M4 using the 1D procedure as explained in Sec. 6.2.3.
  • Fig. 3. The value of C for various Wi on the centerline in the wake of the cylinder for the Oldroyd-B model. The mesh is M4. The values shown are in the (two) Gauss integration points and connected by a line.
    Fig. 3. The value of C for various Wi on the centerline in the wake of the cylinder for the Oldroyd-B model. The mesh is M4. The values shown are in the (two) Gauss integration points and connected by a line.
  • Fig. 10. The value of log(det c) as a function of the coordinate x on the center line in front of the cylinder, along the cylinder surface, and the center line in the wake of the cylinder.
    Fig. 10. The value of log(det c) as a function of the coordinate x on the center line in front of the cylinder, along the cylinder surface, and the center line in the wake of the cylinder.
  • Fig. 9. The stress component τxx as a function of the curve coordinate s along the cylinder surface and the center line in the wake of the cylinder. In the front stagnation point s = 0 and at the back stagnation point s = π. Left figure: Wi = 1.4, right figure: Wi = 1.6.
    Fig. 9. The stress component τxx as a function of the curve coordinate s along the cylinder surface and the center line in the wake of the cylinder. In the front stagnation point s = 0 and at the back stagnation point s = π. Left figure: Wi = 1.4, right figure: Wi = 1.6.
  • Fig. 5. The stress component τxx as a function of the curve coordinate s along the cylinder surface and the center line in the wake of the cylinder. In the front stagnation point s = 0 and at the back stagnation point s = π. Left figure: Wi = 0.6, right figure: Wi = 0.7.
    Fig. 5. The stress component τxx as a function of the curve coordinate s along the cylinder surface and the center line in the wake of the cylinder. In the front stagnation point s = 0 and at the back stagnation point s = π. Left figure: Wi = 0.6, right figure: Wi = 0.7.
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Topics: Weissenberg number (58%), Flow (mathematics) (51%)

337 Citations


Open accessJournal ArticleDOI: 10.1016/J.JNNFM.2005.04.006
Lucy E. Rodd1, Lucy E. Rodd2, Timothy P. Scott1, David V. Boger2  +3 moreInstitutions (3)
Abstract: The non-Newtonian flow of dilute aqueous polyethylene oxide (PEO) solutions through micro-fabricated planar abrupt contraction-expansions is investigated. The small lengthscales and high deformation rates in the contraction throat lead to significant extensional flow effects even with dilute polymer solutions having time constants on the order of milliseconds. By considering the definition of the elasticity number, El = Wi/Re, we show that the lengthscale of the geometry is key to the generation of strong viscoelastic effects, such that the same flow behaviour cannot be reproduced using the equivalent macro-scale geometry using the same fluid. We observe significant vortex growth upstream of the contraction plane, which is accompanied by an increase of more than 200% in the dimensionless extra pressure drop across the contraction. Streak photography and video-microscopy using epifluorescent particles shows that the flow ultimately becomes unstable and three-dimensional. The moderate Reynolds numbers (0.44 ≤ Re ≤ 64) associated with these high Weissenberg number (0 ≤ Wi ≤ 548) micro-fluidic flows results in the exploration of new regions of the Re-Wi parameter space in which the effects of both elasticity and inertia can be observed. Understanding such interactions will be increasingly important in micro-fluidic applications involving complex fluids and can best be interpreted in terms of the elasticity number, El = Wi/Re, which is independent of the flow kinematics and depends only on the fluid rheology and the characteristic size of the device.

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Topics: Weissenberg number (57%), Reynolds number (56%), Pipe flow (55%) ...read more

300 Citations


Journal ArticleDOI: 10.1016/J.JNNFM.2004.12.003
Raanan Fattal1, Raz Kupferman1Institutions (1)
Abstract: We present a second-order finite-dierence scheme for viscoelastic flows based on a recent reformulation of the constitutive laws as equations for the matrix logarithm of the conformation tensor. We present a simple analysis that clarifies how the passage to logarithmic variables remedies the high-Weissenberg numerical instability. As a stringent test, we simulate an Oldroyd-B fluid in a lid-driven cavity. The scheme is found to be stable at large values of the Weissenberg number. These results support our claim that the high Weissenberg numerical instability may be overcome by the use of logarithmic variables. Remaining issues are rather concerned with accuracy, which degrades with insucient resolution.

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Topics: Weissenberg effect (67%), Weissenberg number (65%), Logarithm (53%) ...read more

255 Citations


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No. of papers in the topic in previous years
YearPapers
20224
202191
2020118
2019105
201889
201789

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Topic's top 5 most impactful authors

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Radhakrishna Sureshkumar

13 papers, 524 citations

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