Viscoelasticity differential

Depending on the method of analysis, constitutive models of viscoelastic fluids can be formulated as differential or integral equations. 

The Oldroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950)... 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

In general, the utilization of integral models requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under general concUtions. 

In the following section representative examples of the development of finite element schemes for most commonly used differential and integral viscoelastic models are described. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The following type of differential equation is encountered in the text, for example, in the analysis of the models for viscoelastic behaviour ... 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

This is indeed a system of three second-order differential equations. The tensor elements Cyki may be complex-valued in case of viscoelasticity. Analysis shows that the propagation can be split into three orthogonally polarized planar waves propagating along a wave vector k. Those three waves may have different propagating celerities. Phase celerity and polarization ilj are connected through Christoffel equation ... 

Assuming that a number of NMR data sets (e.g., 2-D or 3-D maps of displacement vectors resulting from an external periodic excitation) from an object are acquired, the remaining difficulty is their reconstruction into viscoelastic parameters. As written in Section 2 the basic physical equation is a partial differential equation (PDE, Eq. (3)) relating the displacement vector to the density, the attenuation, Young s modulus and Poisson s ratio of the medium. The reconstruction problem is indeed two-fold ... 

Evidence for the formation of gels from aPS systems is obtained from simple mechanical, (1.4.5) viscoelastic, (7.8) thermodynamic (1.6) and spectroscopic ( ) techniques. Simple tube tilting, falling ball methods and differential scanning calorimetry have been used to determine the phase diagrams for a number of systems. Viscoelastic measurements on the aPS-carbon disulfide system show that the low frequency response indicative of a... 

Note 7 There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in Definition 5.2. (See, for example, [11].) Such definitions have certain advantages regarding their mathematical generality. However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models). 

Note 2 For specimens without mass, the linear-viscoelastic interpretation of the resulting deformations follows a differential equation of the same form as that for a uniaxial extensional forced oscillation, namely... 

In the Maxwell model for viscoelastic deformation, it is assumed that the total strain is equal to the elastic strain plus the viscous strain. This is expressed in the two following differential equations from Equations 14.2 and 14.3. 

Boggs combined, in a mathematically elegant approach, a constitutive equation, including normal stress terms with equations of motion to form differential equations similar to the Navier-Stokes equations. He found that viscoelasticity had a destabiliz-... 

The periodic response of a linear viscoelastic cooling tower to a prescribed recurring sequence of pressure fluctuations and earth accelerations are analyzed. An approximate analysis, based on the bending theory of shells, is presented. The problem is reduced to a double sequence of boundary-value problems of linear ordinary differential equations. 19 refs, cited. 

Viscoelasticity has already been introduced in Chapter 1, based on linear viscoelasticity. However, in polymer processing large deformations are imposed on the material, requiring the use of non-linear viscoelastic models. There are two types of general non-linear viscoelastic flow models the differential type and the integral type. 

Differential Viscoelastic Models. Differential models have traditionally been the choice for describing the viscoelastic behavior of polymers when simulating complex flow systems. Many differential viscoelastic models can be described by the general form... 

For the viscoelastic stress, we can use differential or integral constitutive models (see Chapter 2). For differential models we have the general form... 

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