A model for viscoelastic fluid behavior which allows non-affine deformation

Johnson, Michelle; Segalman, Daniel J.

doi:10.1016/0377-0257(77)80003-7

Cited by 442 publications

(237 citation statements)

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“…The pressure P is determined by incompressibility, ∇ · v = 0. The viscoelastic stress Σ(r) evolves according to the non-local ("diffusive") Johnson Segalman (DJS) model [12,13] (with plateau modulus G and relaxation time τ . D and Ω are the symmetric and antisymmetric parts of the velocity gradient tensor, (∇v) αβ ≡ ∂ α v β .…”

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confidence: 99%

Nonlinear Dynamics of an Interface between Shear Bands

Fielding

Olmsted

2006

Phys. Rev. Lett.

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We study numerically the nonlinear dynamics of a shear banding interface in two dimensional planar shear flow, within the non-local Johnson Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for sufficiently small ratio of the interfacial width ℓ to cell length Lx. The instability saturates in finite amplitude interfacial fluctuations. For decreasing ℓ/Lx these undergo a non equilibrium transition from simple travelling interfacial waves with constant average wall stress, to periodically rippling waves with a periodic stress response. When multiple shear bands are present we find erratic interfacial dynamics and a stress response suggesting low dimensional chaos.PACS numbers: 47.50.+d, 47.20.-k, 36.20.-r. Complex fluids such as polymers, liquid crystals and surfactant solutions have mesoscopic structure that is readily perturbed by flow [1]. For example, wormlike surfactant micelles with lengths of the order of microns can be induced to stretch, disentangle and entangle, and break or increase in length. Their mechanical response is therefore highly non-Newtonian, with shear flows inducing normal stresses (e.g. σ xx − σ yy ), and with the shear stress σ xy being a nonlinear function of applied shear ratė γ. Recent work on such fluids has led to a fairly consistent picture of "shear banding": coexistence in shear flow of viscously thicker (nascent) and thinner (flow-induced) bands of material flowing at different local shear rates, for an overall average imposed shear rate. This phenomenon can be described by constitutive models for which the shear stress is a non-monotonic function of shear rate for homogeneous flow. This leads to a separation into bands of differing shear rate that coexist at a common shear stress [2,3]. Although most studies have assumed a flat interface between the bands and (with a few exceptions [4]) predicted a timeindependent banded state, an accumulating body of data has demonstrated that the average shear stress, and the banding interface, can fluctuate [5,6,7,8,9,10,11,19]. An important question is then whether these fluctuations resemble small amplitude capillary waves stabilised by surface tension, or whether they arise from an underlying instability, stabilised at large amplitude by nonlinearities. In this Letter we give strong evidence supporting the latter scenario, via the first theoretical study of the nonlinear dynamics of a shear banding interface.The model -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. The pressure P is determined by incompressibility, ∇ · v = 0. The viscoelastic stress Σ(r) evolves according to the non-local ("diffusive") Johnson Segalman (DJS) model [12,13] (with plateau modulus G and relaxation time τ . D and Ω are the symmetric and antisymmetric parts of the velocity gradient tensor, (∇v) αβ ≡ ∂ α v β . For a = 1 and ℓ = 0 this model reduces t...

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confidence: 99%

Nonlinear Dynamics of an Interface between Shear Bands

Fielding

Olmsted

2006

Phys. Rev. Lett.

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“…The fact that the average shear rate at onset of oscillations sits in the flat region of the shear stress curve supports this conjecture. To test this further one should choose a constitutive equation which displays a non-monotonic flow curve (for instance the Johnson-Segalman equation [34,38]) and simulate the falling sphere problem. Such a simulation is currently in progress.…”

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Oscillations of a solid sphere falling through a wormlike micellar fluid

Jayaraman

Belmonte

2003

Phys. Rev. E

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We present an experimental study of the motion of a solid sphere falling through a wormlike micellar fluid. While smaller or lighter spheres quickly reach a terminal velocity, larger or heavier spheres are found to oscillate in the direction of their falling motion. The onset of this instability correlates with a critical value of the velocity gradient scale Γc ∼ 1 s −1 . We relate this condition to the known complex rheology of wormlike micellar fluids, and suggest that the unsteady motion of the sphere is caused by the formation and breaking of flow-induced structures.PACS numbers: 47.50.+d, 83.50.Jf, 83.60.Wc A sphere falling through a viscous Newtonian fluid is a classic problem in fluid dynamics, first solved mathematically by Stokes in 1851 [1]. Stokes provided a formula for the drag force F experienced by a sphere of radius R when moving at constant speed V 0 though a fluid with viscosity µ: F = 6πµRV 0 . The simplicity of the falling sphere experiment has meant that the viscosity can be measured directly from the terminal velocity V 0 , using a modified Stokes drag which takes into account wall effects [2]. The falling sphere experiment has also been used to study the viscoelastic properties of many polymeric (nonNewtonian) fluids [3,4,5]. In general, a falling sphere in a non-Newtonian fluid always approaches a terminal velocity, though sometimes with an oscillating transient [6,7,8]. In this paper we present evidence that a sphere falling in a wormlike micellar solution does not seem to approach a steady terminal velocity; instead it undergoes continual oscillations as it falls, as shown in Fig. 1.A wormlike micellar fluid is an aqueous solution in which amphiphilic (surfactant) molecules self-assemble in the presence of NaSal into long tubelike structures, or worms [9]; these micelles can sometimes be as long as 1µm [10]. Most wormlike micellar solutions are viscoelastic, and at low shear rates their rheological behavior is very similar to that of polymer solutions. However, unlike polymers, which are held together by strong covalent bonds, the micelles are held together by relatively weak entropic and screened electrostatic forces, and hence can break under shear. In fact, under equilibrium conditions these micelles are constantly breaking and reforming, providing a new mechanism for stress relaxation [11].The nonlinear rheology of these micellar fluids can be very different from standard polymer solutions [11,12,13]. Several observations of new phenomena have been reported, including shear thickening [14,15], a stress plateau in steady shear rheology [16,17], and flow instabilities such as shear-banding [18]. Bandyopadhyay et al. have observed chaotic fluctuations in the stress when a wormlike micellar solution is subjected to a step shear rate above a certain critical value (in the plateau region of stress-shear rate curve) [19]. A shear-induced transition from an isotropic to a nematic micellar ordering has also been observed [20]. There is increasing experimental evidence relating the onset of...

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“…(11) and (12) is expressed by means of an upper-convected derivative, which carries the implicit assumption that the polymer molecules are deforming affinely [5], i.e., the molecular strain equals the local macroscopic strain; therefore, Eqs. (11) and (12) do not include constitutive equations that allow slip between the polymer molecules and the surrounding liquid, such as the Johnson-Segalman equation [82], the Phan-Thien and Tanner equation [83,84], and the Larson equation [85].…”

Section: The Approach Of Leonovmentioning

confidence: 99%

“…Jongschaap et al [3] use isotropic representation theorems to write general formulae for the tensors Λ 1 , η, and β, and show that several models of viscoelastic behavior are included in their Matrix Model and can be recovered by specifying particular forms of the tensors Λ 1 and β (e.g, upper-convected Maxwell; Leonov [37]; Johnson and Segalman [82]; Doi [96]; Giesekus [97]; Larson [85]; and finitely extensible nonlinearly elastic dumbbell [12]). Finally, Jongschaap et al remark that because these two tensors are independent of each other, it is legitimate to choose the expression of Λ 1 suggested by one molecular model, and the expression of β suggested by a different model, forming thus new hybrid models.…”

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confidence: 99%

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

Pasquali

Scriven

2004

Journal of Non-Newtonian Fluid Mechanics

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A new method is presented for accounting for microstructural features of flowing complex fluids at the level of mesoscopic, or coarse-grained, models by ensuring compatibility with macroscopic and continuum thermodynamics and classical transport phenomena. In this method, the microscopic state of the liquid is described by variables that are local expectation values of microscopic features. The hypothesis of local thermodynamic equilibrium is extended to include information on the microscopic state, i.e., the energy of the liquid is assumed to depend on the entropy, specific volume, and microscopic variables. For compatibility with classical transport phenomena, the microscopic variables are taken to be extensive variables (per unit mass or volume), which obey convection-diffusion-generation equations. Restrictions on the constitutive laws of the diffusive fluxes and generation terms are derived by separating dissipation by transport (caused by gradients in the derivatives of the energy with respect to the state variables) and by relaxation (caused by non-equilibrated microscopic processes like polymer chain stretching and orientation), and by applying isotropy. When applied to unentangled, isothermal, non-diffusing polymer solutions, the equations developed according to the new method recover those developed by the Generalized Bracket [J. Non-Newtonian Fluid Mech. 23 (1987) Rheol. 38 (1994) 769]. Minor differences with published results obtained by the Generalized Bracket are found in the equations describing flow coupled to heat and mass transfer in polymer solutions. The new method is applied to entangled polymer solutions and melts in the general case where the rate of generation of entanglements depends nonlinearly on the rate of strain. A link is drawn between the mesoscopic transport equations of entanglements and conformation and the microscopic equation describing the configurational distribution of polymer segment stretch and orientation. Constraints are derived on the generation terms in the transport equations of entanglements and conformation, and the formula for the elastic stress is generalized to account for reversible formation and destruction of entanglements. A simplified version of the transport equation of conformation is presented which includes many previously published constitutive models, separates flow-induced polymer stretching and orientation, yet is simple enough to be useful for developing large-scale computer codes for modeling coupled fluid flow and transport phenomena in two-and three-dimensional domains with complex shapes and free surfaces.

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A model for viscoelastic fluid behavior which allows non-affine deformation

Cited by 442 publications

References 4 publications

Nonlinear Dynamics of an Interface between Shear Bands

Nonlinear Dynamics of an Interface between Shear Bands

Oscillations of a solid sphere falling through a wormlike micellar fluid

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

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