Theories Maxwell model

It is likely that most biomaterials possess non-linear elastic properties. However, in the absence of detailed measurements of the relevant properties it is not necessary to resort to complicated non-linear theories of viscoelasticity. A simple dashpot-and-spring Maxwell model of viscoelasticity will provide a good basis to consider the main features of the behaviour of the soft-solid walls of most biomaterials in the flow field of a typical bioprocess equipment. 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

MODELLING OF SIMULTANEOUS MASS AND HEAT TRANSFER WITH CHEMICAL REACTION USING THE MAXWELL-STEFAN THEORY—I. MODEL DEVELOPMENT AND ISOTHERMAL STUDY... 

Both these models find their basis in network theories. The stress, as a response to flow, is assiimed to find its origin in the existence of a temporary network of junctions that may be destroyed by both time and strain effects. Though the physics of time effects might be complex, it is supposed to be correctly described by a generalized Maxwell model. This enables the recovery of a representative discrete time spectrum which can be easily calculated from experiments in linear viscoelasticity. 

Many theoretical and empirical models have been proposed to predict the effective thermal conductivity of two phase mixtures. Comprehensive review articles have discussed the applicability of many of these models that appear to be more promising [34-36]. First, using potential theory. Maxwell [20] obtained a simple relationship for the conductivity of randomly distributed and non-interacting homogeneous spheres in a homogeneous medium. Maxwell model is good for low solid concentrations. Relative thermal conductivity enhancement (ratio of the effective thermal conductivity keffO nanofluid to base fluid kj) is. 

Impedance is the ratio of the voltage across a system to the current passing through the system. It measures the dielectric properties (permittivity and conductivity) of the system. The dielectric behavior of colloidal particles in suspension is generally described by Maxwell s mixture theory [26]. This relates the complex permittivity of the suspension to the complex permittivity of the particle, the suspending medium and the volume fraction. Based-on Maxwell s mixture theory, shelled-models have been widely used to model the dielectric properties of particles in suspension [35-40]. A single shelled spherical model is shown in Fig. la. 

We consider the orientational dynamics only and ignore the spatial coordinates of interacting rods (an analog of the Maxwell model of binary collisions in kinetic theory of gases, see e.g. [15]). Since the motor residence time on microtubules (about 10 sec) is much smaller than the characteristic time of pattern formation (10 min or more), we model molecular motor - microtubule inelastic interaction as an instantaneous colUsion in which two rods change... 

As we shall see in Chapter 13, the relaxation time is a function of the viscosity and modulus (G) of the polymer and, according to the Maxwell model, x = (t /G). The modulus will be much less tranpraature dependent than the viscosity, so we can write aj = (rir/TiJ, which demonstrates the equivalence of the empirical Equation 12.13 with that derived from the free-volume theory, shown in Equation 12.10 and Equation 12.12. 

The effective thermal conductivity of two-phase materials can be predicted by various models (empirical or theoretical). Using the potential theory, Maxwell and Eucken obtained a simple relationship for the conductivity of randomly distributed and non-interacting homogeneous spheres in a homogeneous medium ... 

Comparison of the forms of equations 58 to 61 with equations 21 to 23 of Chapter 9 and equations 23 and 24 of Chapter 3 shows that the time and frequency dependence correspond to a generalized Maxwell model as in the Rouse theory and its various modifications, but here the spring constants (or discrete contributions to the relaxation spectrum) are not necessarily all equal they are proportional to the concentrations of the various types of strands, v e. The molecular weight does not enter explicitly, but it may be expected that the higher the molecular weight the greater the concentrations of strands which find it difficult to leave the network and hence have large values of the time parameter... 

Coalescence in quiescent blends containing spherical dispersed droplets can be induced by interdroplet molecular forces, namely van der Waals forces and/ or Brownian motion. An approximate theory of the molecular forces and Brownian motion-driven coalescence was derived [107] which considers the interaction of a droplet only with its nearest neighbor. Furthermore, it considers the system as monodispersed, and also treats the Brownian motion in a very approximate manner. The theory was derived for Newtonian droplets in a Newtonian matrix, and for Newtonian droplets in a viscoelastic matrix described by the Maxwell model. Coalescence in viscoelastic matrix was shown to be more rapid than in the Newtonian matrix with the same viscosity. 

To resolve such problems, rigorous mass-transfer theory has been applied to a distillation stage in combination with the required heat transfer models (Krishna and Standart, 1979 Taylor and Krishna, 1993). Based on such theories, numerical models have been developed wherein correlations of mass-transfer and heat-transfer coefficients for the distillation device, of packed or plate type, are incorporated (Krishnamurthy and Taylor, 1985 Taylor etoL, 1994). For multicomponent systems, Maxwell-Stefan formalism (Section 3.1.5.1) provided a structural framework for such models. Such theories are known as a rate based approach for modding distillation where equilibrium between phases is nonexistent except at the vapor-liquid interfeice. 

Maxwell (1867) first proposed this equation for the viscosity of gases Despite his initial misapplication of a good theory, rhe-ologists have forgiven him and embrace eq. 3.2.18 as the Maxwell model. It is often represented as a series combination of springs, elastic elements, and dashpots, viscous ones as shown in Hgure 3.2.2. 

In the late 1800s, two scientists, Ludwig Boltzmann and James Maxwell, independently proposed a model to explain the properties of gases in terms of particles in motion. This model is now known as the kinetic-molecular theory. The model makes the following assumptions about the size, motion, and energy of gas particles. 

It is worth remarking that the development of both types of model, like so many other aspects of the kinetic theory of gases, relies heavily on ideas of Clerk Maxwell. Some of these were rediscovered by later workers, but there is remarkably little that was not anticipated, at least in outline, by Maxwell. 

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