Fluids Maxwell element

We can get a first approximation of the physical nature of a material from its response time. For a Maxwell element, the relaxation time is the time required for the stress in a stress-strain experiment to decay to 1/e or 0.37 of its initial value. A material with a low relaxation time flows easily so it shows relatively rapid stress decay. Thus, whether a viscoelastic material behaves as a solid or fluid is indicated by its response time and the experimental timescale or observation time. This observation was first made by Marcus Reiner who defined the ratio of the material response time to the experimental timescale as the Deborah Number, Dn-Presumably the name was derived by Reiner from the Biblical quote in Judges 5, Song of Deborah, where it says The mountains flowed before the Lord. ... 

Maxwell element or model Model in which an ideal spring and dashpot are connected in series used to study the stress relaxation of polymers, modulus Stress per unit strain measure of the stiffness of a polymer, newtonian fluid Fluid whose viscosity is proportional to the applied viscosity gradient. 

A Maxwell element shows an instantaneous elastic deformation and thereafter unlimited flow. For polymers in the solid condition the latter is not realistic. For a fluid polymer it is more relevant moreover, the instantaneous elastic deformation is in accordance with the real behaviour when the stress is released a polymer fluid shows an instantaneous recoil. 

The models described so far provide a qualitative illustration of the viscoelastic behaviour of polymers. In that respect the Maxwell element is the most suited to represent fluid polymers the permanent flow predominates on the longer term, while the short-term response is elastic. The Kelvin-Voigt element, with an added spring and, if necessary, a dashpot, is better suited to describe the nature of a solid polymer. With later analysis of the creep of polymers, we shall, therefore, meet the Kelvin-Voigt model again in more detailed descriptions of the fluid state the Maxwell model is being used. 

The Maxwell element, which represents a fluid because it does not resist deformation under an applied stress, is defined as... 

A single Maxwell element is not realistic for characterizing a polymer as no transient response is shown in a creep test, i.e., the creep response is linear with time. A single Kelvin element is also not accurate as no instantaneous elastic response occurs in a creep test. A more realistic result for creep is obtained if a Kevin solid is combined with a Maxwell fluid to obtain the four-parameter fluid as in Fig. 3.22. 

The four-parameter fluid can also be evaluated in relaxation but typically, Maxwell elements in parallel are used for relaxation and Kelvin elements in series are used for creep. 

Note that this Four-Parameter Fluid model is composed of a Kelvin element (subscripts 1) and a Maxwell element (subscripts 0). Thus, the constitutive laws (differential equations) for the Kelvin and Maxwell elements need to be used in conjunction with the kinematic and equilibrium constraints of the system to provide the governing differential equation. Again, treating the time derivatives as differential operators will allow the simplest derivation of Eq. 5.12. The derivation is left as an exercise for the reader as well as the determination of the relations between the pi and q, coefficients and the spring moduli and damper viscosities (see problem 5.1). 

Variation of the relaxation modulus and creep compliance of a Maxwell model on linear-linear (left) and log-log (right) scales are shown in Figs. 7.15-7.16. Notice the rapid decay of the modulus as the time approaches the selected relaxation time and the flow at long times due to the fluid nature of the Maxwell model. The behavior of the modulus and compliance for a simple Maxwell element is similar to that for many polymers in the glassy and transition region. 

Since (j) and brq are symmetric, positive definite matrices (the case of a semipositive definite brq is omitted here for the sake of brevity this case corresponds to a viscoelastic fluid, as modeled by a free dash pot attached to Kelvin or Maxwell elements), they can be diagonalized simultaneously and (6.16) is expressible in the form... 

This molecular weight dependence of D has been seen experimentally in melts by Klein (1978) and in concentrated solutions by Legeretal. (1981). Reviews of diffusion behavior are available (Tirrell, 1984 Kausch and Tirrell, 1989). One can also deduce the molecular weight dependence of the viscosity by a nonrigorous but plausible argument. Suppose the entire fluid behaves as a simple viscoelastic solid (Maxwell element) then its relaxation time would be... 

As shown in Figure 15.4, the sudden application of stress to a Maxwell element causes an instantaneous stretching of the spring to an equilibrium value of xJG (or aJE if a tensile stress is applied), where Xg is the constant applied shear stress (or Og is the constant applied extensional stress). The dashpot extends linearly with time with a slope of xjx] (or Maxwell element is a fluid, because it will continue to deform as long as it is stressed. The creep response of a Maxwell element is therefore... 

It is useful as an aid to understanding viscoelastic phenomena to interpret this function in terms of the behavior of a mechanical assembly consisting of a linear (Hookean) spring and a linear dash-pot, connected in series as shown in Fig. 4.5. A dash-pot is an element in which the force is proportional to the rate of displacement and is thus analogous to a Newtonian fluid. This assembly was proposed by Maxwell as a model for the behavior of gases, and it is referred to as Maxwell element. 

Figure 4.10 Storage and loss moduli divided by Gg versus <0for a viscoelastic fluid as modeled by a single-Maxwell element.The slopes approach two and unity respectively as the frequency approaches zero on this double-logarithmic plot.

A simple way to illustrate the viscoelastic properties of materials subjected to small deformations is to evaluate the stress that results from combining a linear spring that obeys Hooke s law and a simple fluid that obeys Newton s law of viscosity. An example of such combination is the mathematical representation of the Maxwell element. Even though this model is inadequate for quantitative correlation of polymer properties, it illustrates the quahtative nature of real behavior. Furthermore, it can be generahzed by the concept of a distribution of relaxation times so that it becomes adequate for quantitative evaluation. Maxwell s element is a simple one combining one viscous parameter and one elastic parameter. Mechanically, it can be visualized as a Hookean spring and a Newtonian dashpot in series ... 

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. 

An example describing the application of this algorithm to the finite element modelling of free surface flow of a Maxwell fluid is given in Chapter 5. 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

The simplest model that can show the most important aspects of viscoelastic behaviour is the Maxwell fluid. A mechanical model of the Maxwell fluid is a viscous element (a piston sliding in a cylinder of oil) in series with an elastic element (a spring). The total extension of this mechanical model is the sum of the extensions of the two elements and the rate of extension is the sum of the two rates of extension. It is assumed that the same form of combination can be applied to the shearing of the Maxwell fluid. 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

A fluid composed of a single species is described by five fields the three components of the velocity, the mass density, and the temperature. This is a drastic reduction of the full description in terms of all the degrees of freedom of the particles. This reduction is possible by assuming the local thermodynamic equilibrium according to which the particles of each fluid element have a Maxwell-Boltzmann velocity distribution with local temperature, velocity, and density. This local equilibrium is reached on time scales longer than the intercollisional time. On shorter time scales, the degrees of freedom other than the five fields manifest themselves and the reduction is no longer possible. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

This equation is derived by integrating Eq.( 11-29) with boundary condition)/ = 0, T = To at r = 0. Although the model has some elastic character the viscous response dominates at all but short times. For this reason, the element is known as a Maxwell fluid. 

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