Maxwell fluids components

In [91], Chen studies the long wave asymptotics of the concentric Poiseuille flow of two upper-converted Maxwell fluids under axisymmetric perturbations. He concludes that stability can generally be achieved by increasing the volume of the more elastic component , in agreement with the lubrication effect. The corresponding short wave asymptotic study of the same flow is done in [92]. In the case of coextrusion flows a study for arbitrary wavelengths and low Reynolds numbers is done in [93]. 

Plot the strain-time behavior for the following systems Newtonian fluid, Hoo-kean solid, non-Newtonian fluid, a Maxwell fluid (strain is made up of an elastic viscous component), 7 = 7e + 7v... 

The weak van der Waals potential between He atoms and the bosonic nature of He also are basic for understanding why He is the only bulk superfluid below Tc = 2.18 if at 0.05 bar. The rare isotope He, a fermion, on the other hand, only becomes superfluid at a three orders of magnitude lower temperature. There are many well known macroscopic manifestations of superfluidity such as (i) flow without resistance (ii) a vanishing viscosity (iii) the ability to creep out of vessels against the forces of gravity (iv) the fountain effect which is driven by a type of Maxwell demon which separates the superfluid from the normal fluid components and (v) an enormous thermal conductivity which is 30 times greater than that of copper. Table 7.1 compares some properties of liquid argon (also a cry-omatrix) with those of helium in the normal and in the superfluid state. 

Table 9.3. Components of the Maxwell fluid. Equation 9.16, for two-dimensional flows...

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. 

Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

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. 

Note first that if the fluid is at a state of equilibrium with no flow, then the time derivative d is equal to zero, and the velocity gradient Vv is also zero. This implies from the above equations that = G8. Hence cr, i = <7 2 = ct t, = G at equilibrium, and aj = 0, for i j. Thus, although the diagonal stress components are not zero at equilibrium, they are all equal to each other, and the nondiagonal components are all equal to zero. Hence, the stress tensor is isotropic, but nonzero at equilibrium. (If one redefines the stress tensor as H = a — G8, then S " = 0 at equilibrium. The upper-convected Maxwell equation can then be rewritten in terms of Z .)... 

Kaczmarski et al. used a similar model for the calculation of the band profiles of the enantiomers of 1-indanol on a chiral phase in HPLC [29,57]. These authors ignored the external mass transfer and assumed that local equilibrium takes place for each component between the pore surface and the stagnant fluid phase in the macropores (infinite fast kinetics of adsorption-desorption). They also assumed that surface diffusion contribution is much faster than pore diffusion and neglected pore diffusion entirely. Instead of the single file Maxwell-Stefan diffusion, these authors used the generalized Maxwell-Stefan diffusion (see Chapter 5).The calculation (see below) requires first the selection of equations to calculate the surface molecular flux [29,57,58],... 

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). 

Conversely, the correct approach to formulate the diffusion of a single component in a zeolite membrane is to use the Maxwell-Stefan (M-S) framework for diffusion in a non ideal binary fluid mixture made up of species 1 and 2, where 1 and 2 stand for the gas and the zeolitic material, respectively. In the M-S theory, it is recognized that to effect relative motions between the species 1 and 2... 

Another model referred to in the literature as a diffusion model [50] is similar in nature to the BFM, but is derived by assuming the membrane can be modelled as a dust component (at rest) present in the fluid mixture. The equations governing species transport are developed from the Stefan-Maxwell equations with the membrane as one of the mixture species. The resulting equation for species i is identical to Eq. (4.4) [50], thus the BFM and this diffusion model are equivalent. 

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.)... 

These results imply that the extension of equilibrium theories to nonequiUbrium states is not always valid in a straightforward way. Particularly, the diffusion tensor is proportional to the components of the pressure tensor or equivalently to the velocity gradient Vvq, which implies that the amplitude of the noise in the dynamics of the tagged particle is not simply thermal as in equilibrium since the diffusion tensor cannot be characterized entirely by the thermodynamic temperature. In similar manner, Eq. (5) does not depend on the irreversible heat flux. This is an anomaly of the Maxwell potential, for other potentials there will be an additional contribution to the drift vector that would depend on the any temperature gradient in the fluid. 

The governing equations for MHD have two components classical fluid dynamics and electromagnetics. The former includes mass continuity equation and N-S equation. The latter includes Maxwell s equation, current continuity equation, and constitutive equations. For an incompressible electrolyte solution of density, p, and viscosity, p, the continuity and N-S equations are, respectively, described as... 

The Maxwell-Boltzmann velocity distribution of the three-dimensional molecular velocity vector for a system at equilibrium with zero fluid velocity is equal to the product of each of the three independent normal distributions of the three independent velocity components ... 

The Giesekus-model is a generalization of a Maxwell-Oldroyd B fluid on the basis on a half-empirical structural analysis. The model is able to describe some flow phenomena more accurately than the Maxwell-model [4, 5]. The viscous component is defined as ... 

---