Taylor’s equation

Taylor s Theory. To explain the behavior of in Figures 3 and 4 we borrow from G. I. Taylor s theory of continuous movements, one of the pioneering works in the representation of turbulence. Casting Taylor s equation over into our notation and restricting the theory to an exponentially-declining correlation function, as seems appropriate from Figure 5, yields the following for the dependence of... 

The Taylor-Aris result can be shown in a somewhat simpler mathematical way by starting with the complete convective diffusion equation (Eq. 4.6.7), including the axial diffusion term. The procedure is essentially the same as Taylor s. Equation (4.6.7) is integrated over the tube cross section, since what is of interest is the average concentration, and the radial concentration distribution is given by Eq. (4.6.21). The replacement of dddx by dctdx is still made. The analysis follows through as before. In addition to requiring we must... 

Replacing with a, Eq. (30) can be substituted into Eq. (33), producing Taylor s equation ... 

The main drawback of Taylor s equation is that it only predicts an increase in viscosity with addition of conq)onent 2 even if the viscosity of component 2 is lower... 

Figure 3. Viscosity as a function of PAI-1 concentration at 330 at two shear rates Conq)arison of observed viscosity and viscosity predicted by Taylor s equation rj = LCP, = PAI -1).

Based on a quasi-static assumption, we adopt Taylor s equation for the drainage rate of a film between two spheres of volumes x and x under the action of a constant force by replacing the constant force by a time-dependent F t) ... 

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. 

This weighting procedure for the linearized Arrhenius equation depends upon the validity of Eq. (6-7) for estimating the variance of y = In k. It will be recalled that this equation is an approximation, achieved by truncating a Taylor s series expansion at the linear term. With poor precision in the data this approximation may not be acceptable. A better estimate may be obtained by truncating after the quadratic term the result is... 

An equation is said to show analytic behavior if a Taylor s series expansion about a point in the solution set of the equation converges in the neighborhood of the point. 

Equation (11) represents the first iteration of the RWP idea, but it is not the most efficient. It also represents the first appearance of the damping procedure as used in Mandelshtam and Taylor s Chebyshev iteration [4]. 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

For the following relationships the sign = means approximately equal to, when X is small. These equations are derived by using a Taylor s series (see Series Summation and Identities ). 

In the preceding F = fc(r, r), H = tc(r, vt)G = k(vt, v) and the normalization constant C is fixed by equating the volume integral of n to unity. For further tractability, Sano and Mozumder expand (r v) in a Taylor s series and retain the first two terms only. The validity of this procedure can be established a posteriori in a given situation. At first, the authors obtain equations for the time derivatives of the expectation values and the correlations of dynamical variables. Then, for convenience of closure and computer calculation, these are transformed into a set of six equations, which are solved numerically. The first of these computes lapse time through the relation... 

We will not present here how to derive the first Pontryagin s equation for the probability Q(t, x0) or P(f,x0). The interested reader can see it in Ref. 19 or in Refs. 15 and 18. We only mention that the first Pontryagin s equation may be obtained either via transformation of the backward Kolmogorov equation (2.7) or by simple decomposition of the probability P(t, xq) into Taylor expansion in the vicinity of xo at different moments t and t + t, some transformations and limiting transition to r — 0 [18]. 

This is a non-linear relationship. Linearising using Taylor s series, as given in equation 7.24, Volume 3 ... 

The first approximation of Kasai and Oosawa (1969) has been improved by the efforts of two research groups. Zeeberg et al. (1980) employed an elegant approach with difference equations to the equilibrium exchange problem, and, through the use of a Taylor s series approximation as well as Stirling s approximation, obtained the following solution ... 

For the dilute solutions for which the osmotic coefficient is most useful, the namral logarithm in Equation (19.52) can be expanded in a Taylor s series, and terms of higher powers can be neglected. The result is... 

Tracer Determination of Longitudinal Dispersion Coefficient in Rivers. Tracers are generally used to determine longitudinal dispersion coefficient in rivers. Some distance is required, however, before the lateral turbulent diffusion is balanced by longitudinal convection, simitar to Taylor s (1953) analysis of dispersion in a laminar flow. This transport balancing distance, x is given by the equation... 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

Using the methods of Taylor s analysis as the basis of machine computation, Lutzky has computed "Spherical Taylor Wave for several high explosives (TNT, Pentolite, 65/35 60/40-Cyclotols, TNEtB, RDX, Tetryl and NGu) with the equation of state ... 

The calculation of AH° and AS° values from the pK-temperature data in each solvent mixture was performed by the nonempirical method of Clarke and Glew (26) as simplified by Bolton (27). In this method the thermodynamic parameters are considered to be continuous, well-behaved functions of temperature, and their values are expressed as perturbations of their values at some reference temperature 0 by a Taylor s series expansion. The basic equation is ... 

In Taylor s analysis (1953), 19546) an expression is derived for the variation of C with p when dCmldx is constant. Equation (6) of Taylor (19546) may be written in the notation of this paper... 

The case of turbulent flow and diffusion which was treated in Taylor s second paper (Taylor 1954a) clearly comes under the general case. If i/j and x are functions of p = (172 + 2)1/2 only and a is the radius of the tube equation (9) becomes... 

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