Maxwell fluid models

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. 

An alternative attempt to account for the initial rapid entry of the ceU into micropipettes involved the application of a Maxwell fluid model with a constant cortical tension. Dong et al. [1988] used this model to analyze both the shape recovery of neutrophils following small, complete deformations in pipettes and the small-deformation aspiration of neutrophils into pipettes. Another study by Dong et al. [1991], they used a finite-element, numerical approach and a Maxwell model with constant cortical tension to describe... 

Using the data obtained for various q values, the frequency dependence of the surface tension could be described by a Maxwell fluid model (Ferry 1980) with a single exponential relaxation process. For this model the storage (G ((u)) and loss (G"([Pg.353]

On the other hand, the Maxwell fluid model explains the response of complex fluids to an oscillatory shear rate. The frequency-dependent behavior of this model, displayed into linear responses to applied shear rates has been found to be applicable to a variety of complex fluid systems. Although the linear viscoelasticity is useful for understanding the relationship between the microstructure and the rheological properties of complex fluids, it is important to bear in mind that the linear viscoelasticity theory is only valid when the total deformation is quite small. Therefore, its ability to distinguish complex fluids with similar micro- and nanostructure or molecular structures (e.g. linear or branched polymer topology) is limited. However, complex fluids with similar linear viscoelastic properties may show different non-linear viscoelastic properties [31]. 

Relaxation of the simple Maxwell fluid model will decrease the initial level of stress caused by the elasticity of the spring element by viscous elongation of the dashpot until all stress has been completely relieved. The solution of the first order linear differential equation in the case of = o = constant describes the time-dependent relaxation of stress for the Maxwell model as... 

To complete step 3, a Maxwell fluid model is assumed and E and T are determined using the following method. A constant strain-rate process is assumed, where the strain rate is estimated using the final radius at the contact time with the mold. Then different values of (P, t) and P, t) are used to estimate E and T, taking into account the concepts of initial elastic response and retarded viscous response. 

To complete part (3) of the problem, assume a Maxwell fluid model and determine E and t) using the following method ... 

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. 

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. 

In Chapter 1 it was pointed out that the Maxwell fluid is a very simple model of the first order effects observed with viscoelastic liquids. The equation of a Maxwell fluid is... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Together with Eq. 3.3-17, Eq. 3.3-16 is the White-Metzner constitutive equation, which has been used frequently as a nonlinear viscoelastic model. Of course, for small deformations, X(i) = dx/dt, and the single Maxwell fluid equation (Eq. 3.3-9) is obtained. 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

Vilchis et al. [81] presented a new idea to achieve better control of the particle size distribution by the synthesis in situ of a water-soluble copolymer of acrylic acid-styrene as suspension stabilizer without additional inorganic phosphate. Publications describe increasing the particle formation by using a physical (population balance, Maxwell fluid, power law viscosity, compartment mixing) modeling approach [22,60,98,105]. 

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. 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

Maxwell-Stefan (dusty gas) approach by taking the membrane to be the additional component in the mixture. When the model is extended to account for thermodynamic nonidealities (what may be considered to be a dusty fluid model) almost all membrane separation processes can be modeled systematically. Put another way, the Maxwell-Stefan approach is the most promising candidate for developing a generalized theory of separation processes (Lee et al., 1977 Krishna, 1987). 

In flow situations where the elastic properties play a role, viscoelastic fluid models are generally needed. Such models may be linear (e.g., Voigt, Maxwell) or nonlinear (e.g., Oldroyd). In general they are quite complex and will not be treated in this chapter. For further details, interested readers are referred to the textbooks by Bird et al. [6] and Barnes et al. [25],... 

Kaloni used Oldroyd model, Schtimmer a fourth order fluid model, while Wissler a nonhnear Maxwell model Employing the perturbation method, the authors observed that the inclusion of second-order perturbation terms (which bring in the non-Newtonian effects) predicted velocity profiles with superimposed secondary circulation patterns. 

As can be seen, the Maxwell-Weichert model possesses many relaxation times. For real materials we postulate the existence of a continuous spectrum of relaxation times (A,). A spectrum-skewed toward lower times would be characteristic of a viscoelastic fluid, whereas a spectrum skewed toward longer times would be characteristic of a viscoelastic solid. For a real system containing crosslinks the spectrum would be skewed heavily toward very long or infinite relaxation times. In generalizing, A may thus he allowed to range from zero to infinity. The concept that a continuous distribution of relaxation times should be required to represent the behavior of real systems would seem to follow naturally from the fact that real polymeric systems also exhibit distrihutions in conformational size, molecular weight, and distance between crosslinks. 

A schematic diagram of the unit cell for a vapor-Uquid-porous catalyst system is shown in Fig. 9.9. Each cell is modeled essentially using the NEQ model for heterogeneous systems described above. The bulk fluid phases are assumed to be completely mixed. Mass-transfer resistances are located in films near the vapor-liquid and liquid-solid interfaces, and the Maxwell-Stefan equations are used for calculation of the mass-transfer rates through each film. Thermodynamic equilibrium is assumed only at the vapor-liquid interface. Mass transfer inside the porous catalyst may be described with the dusty fluid model described above. 

Eq. (2), follows from the constitutive model for a Maxwell fluid, viz.,... 

The paradox of Maxwell s model. A popular representation of models in rheology mimics the equivalent electrical circuits with dipolar components. The elastic component is naturally symbolized by a spring and the viscous component by a damper or dashpot (a piston filled with a viscous fluid able to circulate). The viscoelastic relaxation is thus represented with these two components mounted in series, as shown in Figure 11.12a and is known as Maxwell s model (Oswald 2005). (In this representation, the customary notation is used for facilitating comparison with the literature.)... 

The structure is essentially that of a Maxwell fluid if Equation 9.23 is multiplied by Y, but with a relaxation time equal to that decreases with increasing stress.) Values of s are typically of order 0.02 or less, and at this level the shear viscosity and normal stress differences are insensitive to e. The parameter f arises in the network model as a slip coefficient that reflects the motion of the network relative to the continuum. 

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