Viscoelastic incompressible

I.M. Rutkevich, Some general properties of the equations of viscoelastic incompressible fluid mechanics, J. Appl. Math. Mech. (PMM), 33 (1969) 30-39. 

For the sake of simplicity we retain here the simpler form of the emulsion model proposed by Palierne [2]. For a dispersion of viscoelastic incompressible inclusions in a viscoelastic impressible... 

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

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

In the absence of body force, the dimensionless form of the governing model equations for two-dimensional steady-state incompressible creeping flow of a viscoelastic fluid are written as... 

The constant Tr is called the Trouton ratio10 and has a value of 3 in this experiment with an incompressible fluid in the linear viscoelastic limit. The elongational behaviour of fluids is probably the most significant of the non-shear parameters, because many complex fluids in practical applications are forced to extend and deform. Studying this parameter is an area of great interest for theoreticians and experimentalists. 

Note 3 Deformations and flows used in conventional measurements of properties of viscoelastic liquids and solids are usually interpreted assuming incompressibility. 

Note 3 The Finger strain tensor for a homogeneous orthogonal deformation or flow of incompressible, viscoelastic liquid or solid is... 

Equation relating stress and deformation in an incompressible viscoelastic liquid or solid. Note 1 A possible general form of constitutive equation when there is no dependence of stress on amount of strain is... 

This section summarizes results of the phenomenological theory of viscoelasticity as they apply to homogeneous polymer liquids. The theory of incompressible simple fluids (76, 77) is based on a very general set of ideas about the nature of mechanical response. According to this theory the flow-induced stress at any point in a substance at time t depends only on the deformations experienced by material in an arbitrarily small neighborhood of that point in all times prior to t. The relationship between stress at the current time and deformation history is the constitutive equation for the substance. 

For the solution of this problem, the momentum and continuity equations for the steady-state flow of an incompressible viscoelastic fluid are given by... 

Let us find the resistance force acting on a spherical particle of radius a which moves slowly with velocity u in an incompressible viscoelastic fluid. It means that the Reynolds number of the problem is small, the convective terms are negligibly small, and the equations of fluid motion are... 

In rubber and viscoelastic fluids, these two quantities are sufficient since, when incompressibility is taken into account, I3 = 1. 

The equations for a perturbation u of a steady solution u, of an incompressible viscoelastic fluid can be written as an abstract equation in a Hilbert space X,... 

In the small deformation approximation, it is assumed that the deformations undergone by the material are small, at least in the recent past. Approximations of different orders can be developed. The approximation of first order for an incompressible fluid is given by Boltzmann s equation of linear viscoelasticity,... 

To study the viscoelastic case it is convenient to assume a constant Poisson ratio. This hypothesis is essentially correct in most cases. In fact, if the material is incompressible, one has v = 1/2. Taking the input as the applied force instead of the distribution p r), Eq. (16.177) suggests that the time dependence of the displacement for a viscoelastic materials is given by... 

In formal rheology, relations between these three tensors are formulated and analyzed. Only for the two extremes of viscoelastic behaviour are such relations simple. For purely elastic materials there is a relation between the stress tensor and the strain tensor it contains the elasticity modulus and the Poisson ratio, accounting for the extent to which extension in one direction is accompamied by concomitant compression in the other two. For purely viscous fluids there is a relation between the stress tensor and the strain rate tensor. As extension in one direction is concomitant with (viscous) compression in the other two, in this case only one viscosity is required. For incompressible Newton fluids eventually an expression with only one viscosity results, see (1.6.1.131. 

In seal and O-ring design, three considerations are typically examined thermal mechanical, viscoelastic, and incompressibility [14], Thermal mechanical is basically the material changes that we have been discussing under load. Incompressibility is exhibited when a material has an isochoric (zero volume) change under pressure. If... 

Since the linear viscoelasticity of a material is described with a material function G(t), any experiment which gives full information on G(t) is sufficient it is not necessary to give the stresses corresponding to various strain histories. We will restrict the discussion to incompressible isotropic materials. In this case, different types of deformation such as elongation and shear give equivalent information in the range of linear viscoelasticity. Several types of experiments measure relaxation modulus, creep compliance, complex modulus etc which are equivalent to the relaxation modulus (1). 

The adsorption kinetics are slow in a surfactant concentration range around CAC and become fast again close to the CMC of the pure surfactant, where the surface layers with and without polymer become similar. Below CAC, they behave as incompressible layers. When subjected to small compression-expansion cycles, the layers exhibit a viscoelastic response, similar with the different polymers. Appreciable differences are seen only when the compression is more important (decrease of the surface area by a factor up to five) the layers with xanthan still behave as insoluble layers (even above CAC), whereas those with PAMPS appear as partially soluble. 

It is known that incompressible newtonian fluids at constant temperature can be characterized by two material constants the density p and the viscosity T. The characterization of a purely viscous nonnewtonian fluid using the power law model (or any of the so-called generalized newtonian models) is relatively straightforward. However, the experimental description of an incompressible viscoelastic nonnewtonian fluid is more complicated. Although the density can be measured, the appropriate expression for r poses considerable difficulty. Furthermore there is some uncertainty as to what other properties need to be measured. In general, for viscoelastic fluids it is known that the viscosity is not constant but depends on shear rate, that the normal stress differences are finite and depend on shear rate, and that the stress may also depend on the preshear history. To characterize a nonnewtonian fluid, it is necessary to measure the material functions (apparent viscosity, normal stress differences, etc.) in a relatively simple or standard flow. Standard flow patterns used in characterizing nonnewtonian fluids are the simple shear flow and shear-free flow. 

It is possible to model the deformation of film bubbles with a system of dimensionless equations that is derived according to the following assumptions [13] steady-state and axisymmetrical flow (z-axis) of an incompressible fluid thin and flat film external forces on the bubble are neglected Newtonian, pseudoplastic, or viscoelastic fluids and linear temperature profiles between die exit and freezeline position. The system of dimensionless fundamental equations can be represented, irrespective of the rheological constitutive equation used, as shown in the following equations ... 

It has been shown that for an incompressible linear viscoelastic liquid there is an interrelation between G and G" through the frequency relaxation spectrum, H(X) [Utracki, 2004] ... 

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