Taylor expression

Although the dominant mixing mechanism of an immiscible liquid polymeric system appears to be stretching the dispersed phase into filament and then form droplets by filament breakup, individual small droplet may also break up at Ca 3> Ca. A detailed review of this mechanism is given by Janssen (34). The deformation of a spherical liquid droplet in a homogeneous flow held of another liquid was studied in the classic work of G. I. Taylor (35), who showed that for simple shear flow, a case in which interfacial tension dominates, the drop would deform into a spheroid with its major axis at an angle of 45° to the how, whereas for the viscosity-dominated case, it would deform into a spheroid with its major axis approaching the direction of how (36). Taylor expressed the deformation D as follows... 

It is not difScult to observe that the Taylor expression can be transposed as Eq. (5.3) where the index i has been extracted because it stays unchanged along the relation ... 

Further contributions to the subject were made by Taylor in 1938. Two important consequences of the non-linearity of the Navier-Stokes equations were identified First, the skewness of the probability distribution of the difference between the velocities at two points, and the existence of an interaction or modulation between components of turbulence having different length scales. Secondly, the Fourier transform of the correlation between two velocities is an energy spectrum function in the sense that it describes the distribution of kinetic energy over the various Fourier wave-number components of the turbulence [164]. Taylor expressed in mathematical form the relation between the correlation function and the ID spectrum function. 

To add effects of specific interactions, the Gordon-Taylor expression can be expanded into a virial expression, as in the Schneider equation, listed in Fig. 7.69 [30]. The variable W2c is the expansivity-corrected mass fraction of the Gordon-Taylor expression W2c = kW2 / (Wj + kWj). The Schneider equation can be fitted with help of the constants Kj and K2 to many polymer/polymer solutions, as is illustrated in Sect. 7.3.2. The parameter Kj depends mainly on differences in interaction energy between the binary contacts of the components A-A, B-B, and A-B, while Kj accounts for effects of the rearrangements in the neighborhood of the contacts. 

This concept benefits from the Taylor expression. Similar to the direct method given in Plate 3.1 and tabulated below, the Newton-Raphson as stated earlier can be viewed at for comparison A reference is made to Plate 3.2. Additional solution procedures... 

A Taylor expression to the first order of the relation P E is generally... 

Alternatively, expansion of equation (A2.5.1). equation (A2.5.2) or equation (A2.5.3) into Taylor series leads ultimately to series expressions for the densities of liquid and gas, / and p, in temis of their sum (called the diameter ) and their difference ... 

Evidently, this fomuila is not exact if fand vdo not connnute. However for short times it is a good approximation, as can be verified by comparing temis in Taylor series expansions of the middle and right-hand expressions in (A3,11,125). This approximation is intrinsically unitary, which means that scattering infomiation obtained from this calculation automatically conserves flux. 

Raman scattering has been discussed by many authors. As in the case of IR vibrational spectroscopy, the interaction is between the electromagnetic field and a dipole moment, however in this case the dipole moment is induced by the field itself The induced dipole is pj j = a E, where a is the polarizability. It can be expressed in a Taylor series expansion in coordinate isplacement... 

In Section III.D, we shall investigate when this happens. For the moment, imagine that we are at a point of degeneracy. To find out the topology of the adiabatic PES around this point, the diabatic potential matrix elements can be expressed by a hrst order Taylor expansion. 

III- slow growth expression can be derived from the thermodynamic perturbation expres sion (Equation (11.7)) if it is written as a Taylor series ... 

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Taylor, B. N. Kuyatt, G. E. Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, 1994. 

Although Eq. (10.50) is still plagued by remnants of the Taylor series expansion about the equilibrium point in the form of the factor (dn/dc2)o, we are now in a position to evaluate the latter quantity explicitly. Equation (8.87) gives an expression for the equilibrium osmotic pressure as a function of concentration n = RT(c2/M + Bc2 + ) Therefore... 

As for the change of dipole moment, the change of polarizability with vibrational displacement x can be expressed as a Taylor series... 

Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-49 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point x, is assigned the symbol Cj = c(Xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are ... 

The authors would like to thank Dr Mark Taylor, Dr Raquel Duarte-Davidson and the other participants who contributed to our understanding during a workshop on the ecological effects of sex hormone disrupters held at the Institute in January 1998. We also acknowledge the financial support provided by the UK Department of the Environment, Transport and the Regions for the work at lEH on endocrine disruption. However, the opinions expressed in this paper are those of the authors and do not necessarily represent those of any government department or agency. 

All fluid properties are functions of space and time, namely p(x, y, z, t), p(x, y, z, t), T(x, y, z, t), and u(x, y, z, t) for the density, pressure, temperature, and velocity vector, respectively. The element under consideration is so small that fluid properties at the faces can be expressed accurately by the first two terms of a Taylor series expansion. For example, the pressure at the E and W faces, which are both at a distance l/26x from the element center, is expressed as... 

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

Estr is the energy function for stretching a bond between two atom types A and B. In its simplest form, it is written as a Taylor expansion around a namral , or equilibrium , bond length Rq- Tenninating the expansion at second order gives the expression... 

More accurate difference expressions may be found by expanding the Taylor series. For example, f (x) to V(h) is given by forward difference by... 

The content of Eq. (3-81) is sometimes expressed in a somewhat different way by writing the (formal) Taylor series expansion of in the form... 

The conditions which must be satisfied at the plait point may be deduced as follows Expand by Taylor s theorem the expressions on the right of (9) and (10), omitting terms of higher orders than the second ... 

In the Taylor-Prandtl modification of the theory of heat transfer to a turbulent fluid, it was assumed that the heat passed directly from the turbulent fluid to the laminar sublayer and the existence of the buffer layer was neglected. It was therefore possible to apply the simple theory for the boundary layer in order to calculate the heat transfer. In most cases, the results so obtained are sufficiently accurate, but errors become significant when the relations are used to calculate heat transfer to liquids of high viscosities. A more accurate expression can be obtained if the temperature difference across the buffer layer is taken into account. The exact conditions in the buffer layer are difficult to define and any mathematical treatment of the problem involves a number of assumptions. However, the conditions close to the surface over which fluid is flowing can be calculated approximately using the universal velocity profile,(10)... 

Derive the Taylor-Prandtl modification of the Reynolds analogy between heat and momentum transfer and express it in a form in which it is applicable to pipe flow. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

The coupled cluster expression for frequency-dependent second hyperpolarizability, Eq. (30), can now be expanded in a Taylor series in its frequency arguments around its static limit as ... 

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