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

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

Journal of Non-Newtonian Fluid Mechanics

Volume 2, Issue 3, May 1977, Pages 255-270

https://doi.org/10.1016/0377-0257(77)80003-7 Get rights and content

Abstract

A continuum theory of viscoelasticity is developed which allows non-affine deformation, defined in an appropriate manner. The constitutive equation is a generalization of that obtained from molecular theory with the addition of one scalar parameter which becomes important for large deformations. The theory is applied to simple shear flows, the scalar parameter being determined to match certain experimental data. The theory shows good agreement with all data examined. The paper concludes with the development of a general non-affine thermodynamic theory.

References (10)

G. Astarita et al.
The dissipative mechanism in flowing polymers: theory and experiments
J. Non-Newtonian Fluid Mech.
(1976)
A.S. Lodge
(1974)
R.B. Bird et al.
Co-rotational rheological models and the Goddard expansion
AIChE J.
(1974)
C. Truesdell et al.
The Non-linear Field Theories of Mechanics

There are more references available in the full text version of this article.

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We consider shear wave propagation in soft viscoelastic solids of rate type. Based on objective stress rates, the constitutive model accounts for finite strain, incompressibility, as well as stress- and strain-rate viscoelasticity. The theory generalises the standard linear solid model to three-dimensional volume-preserving motions of large amplitude in a physically-consistent way. The nonlinear equations governing shear motion take the form of a one-dimensional hyperbolic system with relaxation. For specific objective rates of Cauchy stress (lower- and upper-convected derivatives), we study the propagation of acceleration waves and shock waves. Then we show that both smooth and discontinuous travelling wave solutions can be obtained analytically. We observe that the amplitude and velocity of steady shocks are very sensitive to variations of the stress relaxation time. Furthermore, the existence of steady shocks is conditional. Extension of these results to the case of multiple relaxation mechanisms and of the Jaumann stress rate is attempted. The analysis of simple shearing motions is more involved in these cases.
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Interplay of wall slip and shear banding in wormlike micelle solutions
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Understanding the behaviour of temperature dependent viscosity thin film flows on a vertically upward moving heated belt is crucial for improving predictive models in various engineering applications, such as in coating and lubrication processes. This paper presents the heat transfer analysis of temperature-dependent viscosity fluid flow on a vertical upward moving heated belt. The formulated coupled system of nonlinear ordinary differential equations is solved using the Adomian decomposition method. The study investigates the impact of flow controlling parameters: Stokes number $S_{t}$ , Brinkmann number $B r$ , variable viscosity parameter $β$ , Weissenberg number $W e$ , effective viscosity $ϕ$ and slip parameter $e$ is observed on velocity, temperature, and stationary points. It is perceived that the temperature of the fluid decreases with the increase of $S_{t}$ , $B r$ , $β$ , and $ϕ$ , while it increases with the increase of $W e$ and $e$ . Positions of stationary points relocate towards the heated belt surface by the increment of $S_{t}$ , $B r$ , $β$ , and $ϕ$ . On the other hand, their positions relocate away from the heated belt surface by the increment of $W e$ and $e$ . The analysis also highlights the influence of heat generated by viscous dissipation and heat transported by thermal diffusion on the velocity, temperature, and stationary points. Moreover, the flow variables discussed for the temperature dependent viscosity Johnson–Segalman fluid are also discussed for the temperature dependent viscosity Newtonian fluid, and an analogy between both is provided.
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¹: Supported by the Mathematics Research Center, the University of Wisconsin, Madison.

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

Abstract

J. Non-Newtonian Fluid Mech.

Co-rotational rheological models and the Goddard expansion

AIChE J.

The Non-linear Field Theories of Mechanics

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