Stability of the interface between two dynamic phases in capillary flow of linear polymer melts

McLeish, Tom

doi:10.1002/polb.1987.090251103

Cited by 63 publications

(28 citation statements)

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“…Seldom in bulk systems is it possible to achieve shear rates sufficiently large that the shear stress fails to continue to increase with further increase of the shear rate (a few examples [59][60][61][62][63] are known). A plateau of shear stress, a so-called limiting shear stress, is common to observe in lubricated systems, however.…”

Section: Methodsmentioning

confidence: 99%

Tribology of Confined Fomblin-Z Perfluoropolyalkyl Ethers: Role of Chain-End Chemical Functionality

Ruths

Granick

1999

J. Phys. Chem. B

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Strong dependence of the tribological response upon the nature of the polar end groups of perfluorinated linear chains was observed for Fomblin-Z perfluoropolyalkyl ethers of similar length and composition but terminated by a different polar group at both chain ends. The number-average molecular weight of the polymers was M n ≈ 2000 g mol -1 and the chain-end functionality was either carboxylic acid, hydroxyl, piperonyl, or the p-phenoxyanilinium salt of a carboxylic acid. The method of investigation was a surface forces apparatus modified for dynamic oscillatory shear at variable frequency and effective shear rate. Differences were observed as concerns not only the shear forces but also the minimum thickness to which the films could be compressed under a given normal load and the adhesion measured on separation of the surfaces after prior compression. The shear forces were studied at normal pressures of 1 and 3 MPa, both in the linear viscoelastic regime and at high shear amplitudes corresponding to shear rates of 10 -2 -10 5 s -1 . The carboxylic acid terminated polymer displayed solidlike responses to shear, possibly reflecting dimerization owing to hydrogen bonding. This contrasted with the more fluidlike shear rheology of the hydroxyl-and piperonyl-terminated polymers, in which the association from hydrogen bonding and polar interactions is believed to be weaker and result in a different structure. The sample comprised of the p-phenoxyanilinium salt of a carboxylic acid could not be compressed to less than an exceptionally large film thickness, around 100 Å, and did not appear to solidify at the pressures studied. This study suggests that not only the affinity of the functionalized chain ends to a solid surface but also the self-association of polar end groups in the nonpolar environment of fluorinated chains influences the lubricating performance of these films.

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Section: Methodsmentioning

confidence: 99%

Tribology of Confined Fomblin-Z Perfluoropolyalkyl Ethers: Role of Chain-End Chemical Functionality

Ruths

Granick

1999

J. Phys. Chem. B

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“…As q x increases from zero, this displacement is modulated by a wave of wavevector q xx with an eigenvalue ω max (q x ) = ω 0 + iq x ω 1 + q 2 x ω 2 with ω 2 > 0, signifying instability. A natural question is whether this instability has the same origin as that described by McLeish for the local model [16]. It is not obvious, a priori, that this should be true because, for the base state at least, the limit l → 0 is singular [7].…”

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confidence: 99%

“…Renardy [15] found instability with respect to interfacial fluctuations of high wavevector, q x → ∞, in the local JS model restricted to the case of a thin high shear band. McLeish [16] studied capillary flow, for general band thickness. He demonstrated a long wavelength (q x → 0) instability due to the jump in normal stresses across the interface.…”

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confidence: 99%

Linear Instability of Planar Shear Banded Flow

Fielding

2005

Phys. Rev. Lett.

119

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We study the linear stability of planar shear banded flow with respect to perturbations with wavevector in the plane of the banding interface, within the non local Johnson Segalman model. We find that perturbations grow in time, over a range of wavevectors, rendering the interface linearly unstable. Results for the unstable eigenfunction are used to discuss the nature of the instability. We also comment on the stability of phase separated domains to shear flow in model H. [5] in which the underlying constitutive curve of shear stress vs. shear rate, T xy (γ), is non-monotonic ( Fig. 1), allowing the coexistence of bands of differing shear rate at common shear stress, Fig. 2. However, most theoretical studies have considered only one spatial dimension (1D) [6,7], normal to the interface between the bands (the flow gradient direction, y). The stability of 1D banded profiles in higher dimensions has been implicitly assumed, but is in fact an open question. In this Letter, therefore, we study numerically the linear stability of 1D planar shear banded profiles with respect to perturbations with wavevectors in the interfacial plane (x, z) = (flow, vorticity).We work within the Johnson Segalman (JS) model [8], modified to include non local diffusive terms [9]. These account for gradients in the order parameters across the banding interface, conferring a surface tension. This "dJS" model is often taken as a paradigm of shear banding systems. Our main result will be that interfacial fluctuations typically grow in time, rendering the 1D banded profile linearly unstable. This potentially opens the way to non trivial interfacial dynamics and could form a starting point for understanding an emerging body of data revealing erratic fluctuations of shear banded flows [10]. This work is a timely counterpart to new techniques for measuring interfacial dynamics [11]. It is also relevant industrially, to processing instability and oil extraction.The model is defined as follows. The generalised Navier Stokes equation for a viscoelastic material in a Newtonian solvent of viscosity η and density ρ is:where V(R) is the velocity field and Σ(R) the viscoelastic part of the stress. For homogeneous planar shear, V = yγx, the total shear stress T xy = Σ xy (γ) + ηγ. The pressure P is determined by incompressibility,The viscoelastic stress evolves with dJS dynamics [8, 9]with plateau modulus G and relaxation time τ . The non local diffusive term accounts for spatial gradients across the interface between the bands. It arises naturally in models of liquid crystals, and diffusion of strained polymer molecules [12]. The time derivativein which D and Ω are the symmetric and antisymmetric parts of the velocity gradient tensor, (∇V) αβ ≡ ∂ α v β . The "slip parameter" a measures the non-affinity of deformation of the viscoelastic component [8]. Slip occurs for |a| < 1. The underlying constitutive curve T xy (γ) is then capable of the non-monotonic behaviour of Fig. 1. Within this model we consider planar shear between infinite, flat parallel plat...

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“…McLeish [36] has argued that if the functional relating current stress to strain history is well-behaved, a small perturbation to the flow can only cause a small perturbation to the stress in any fluid element. Thus material can never be transported across a sharp interface since this would cause an order 1 change in the stress for that material.…”

Section: Introductionmentioning

confidence: 99%

“…McLeish [36] considered a DoiEdwards type fluid in capillary flow. He found instability to long waves, provided the high shear rate band is very narrow.…”

Section: Introductionmentioning

confidence: 99%

Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson–Segalman fluids

Wilson

Fielding

2006

Journal of Non-Newtonian Fluid Mechanics

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We consider the linear stability of shear banded planar Couette flow of the JohnsonSegalman fluid, with and without the addition of stress diffusion to regularise the equations. In particular, we investigate the linear stability of an initially onedimensional "base" flow, with a flat interface between the bands, to two-dimensional perturbations representing undulations along the interface. We demonstrate analytically that, for the linear stability problem, the limit in which diffusion tends to zero is mathematically equivalent to a pure (non-diffusive) Johnson-Segalman model with a material interface between the shear bands, provided the wavelength of perturbations being considered is long relative to the (short) diffusion lengthscale.For no diffusion, we find that the flow is unstable to long waves for almost all arrangements of the two shear bands. In particular, for any set of fluid parameters and shear stress there is some arrangement of shear bands that shows this instability. Typically the stable arrangements of bands are those in which one of the two bands is very thin. Weak diffusion provides a small stabilising effect, rendering extremely long waves marginally stable. However, the basic long-wave instability mechanism is not affected by this, and where there would be instability as wavenumber k → 0 in the absence of diffusion, we observe instability for moderate to long waves even with diffusion. This paper is the first full analytical investigation into an instability first documented in the numerical study of [1]. Authors prior to that work have either happened to choose parameters where long waves are stable or used slightly different constitutive equations and Poiseuille flow, for which the parameters for instability appear to be much more restricted.We identify two driving terms that can cause instability: one, a jump in N 1 , as reported previously by Hinch et al. [2]; and the second, a discontinuity in shear rate. The mechanism for instability from the second of these is not thoroughly understood. Preprint submitted to Elsevier Science 3 May 2006We discuss the relevance of this work to recent experimental observations of complex dynamics seen in shear-banded flows.

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Stability of the interface between two dynamic phases in capillary flow of linear polymer melts

Cited by 63 publications

References 9 publications

Tribology of Confined Fomblin-Z Perfluoropolyalkyl Ethers: Role of Chain-End Chemical Functionality

Tribology of Confined Fomblin-Z Perfluoropolyalkyl Ethers: Role of Chain-End Chemical Functionality

Linear Instability of Planar Shear Banded Flow

Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson–Segalman fluids

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