Hyperbolic expansion

Hyperbolic expansion The expansion of a fluid according to the law pV = Hypothalamus The temperature control center at the base of the brain, which regulates body temperature. Hypothermia The physiological state resulting when the deep core body temperature drops below 35 C. It results in vasoconstriction and shivering in an attempt to conserve body heat. 

Equation 26 was derived directly and contains no mathematical approximations. A similar equation was derived by Healy et al. (30). If the hyperbolic functions are approximated by a linear expansion, Equation 26 can be approximated... 

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. 

Unfortunately our initial and boundary data are incompatible and the hyperbolic Equation (102) has a discontinuous solution. Since the asymptotic expansion for c involves derivatives of c. Equation (102) does not suit our needs. As before, we proceed by following an idea from Rubinstein and Mauri (1986) and suppose that... 

Hence, we arrive at the conclusion that only in the limit a - 0 the Hookean body is the ideal energy-elastic one (r = 0) and the uniform deformation of a real system is accompanied by thermal effects. Equation (19) shows also that the dependence of the parameter q (as well as to) on strain is a hyperbolic one and a, the phenomenological coefficient of thermal expansion in the unstrained state, is determined solely by the heat to work and the internal energy to work ratios. From Eqs. (17) and (18), we derive the internal energy of Hookean body... 

The integrals I0 and /2 arise from a Fourier expansion for hyperbolic cosines of trigonometric argument [45]. With this potential, Parker calculated both the average number of collisions to establish rotational equilibrium, Zf, and that required to establish vibrational equilibrium, Z , as functions of temperature. One finds... 

If we carefully observe the expression of the characteristic function of the residence time distribution for the evolution of a liquid element (q>(s,Hd), relation (4.201)), we can notice that it is difficult to compute the expressions of the derivatives cp (0, Hd) and (p"(0, H ). Using the expansion of the hyperbolic sine and cosine respectively as multiplication series, we obtain the following simplified expression for the characteristic function ... 

Mathematically this happens because the first nonlinear term in the Taylor expansion series of exp x) is x 2 while for the hyperbolic function sinh(x) it is X /6. 

For low potentials several tens of mV for z = 1. the hyperbolic tangent may be replaced by the first, linear, term of its series expansion [A2.9), to give... 

At moderate pressures the water line AB is almost vertical and very close to the p axis its slope to the right is due to the excess of the expansion of the liquid by increase of temperature over the reduction due to the increase of pressure. The steam line CD is approximately hyperbolic (the vapour curves CE, DF starting from it are more nearly hyperbolic). According to Rankine CD may be represented by the equations ... 

During an experiment the sample is cold, ca 20K (= 14 cm ). The lowest internal vibrations are typically about 300 cm and the hyperbolic sine function will, except at the very lowest energies, have an argument greater than ten. The argument of the Bessel function is, therefore, less than 10 and it can be safely represented by the first term of its power series expansion. Where, for an arbitrary argument x ... 

The above relationship for electrostatic component of disjoining pressure has the following meaning the first term, in agreement with eq. (III.20), represents the osmotic pressure of ions in the center of a gap, while the second term is the osmotic pressure in the bulk of dispersion medium. One may thus say that the electrostatic component of disjoining pressure equals to the difference in osmotic pressure between the gap and the bulk, that forces the dispersion medium to flow into the gap between surfaces causing a disjoining action. For small values of (p(h/2) the expansion of hyperbolic cosine into series as cosh (y) 1 + A (y)2 readily yields eq. (V1I.21). 

In the experiments we always deal with film having thickness much less than the wavelength of the observable surface waves (see Section III.A). This means that in the experiments the long-wavelength limit (Iwl) is observed. Since the Iwl corresponds with the conditions Kh<. and w/i bending mode can be found from (5.24) and (5.25) by series expansion of the hyperbolic functions. The result for the squeezing mode is ... 

Expansion of the hyperbolic terms of (5.25) gives for the Iwl of the bending mode ... 

In the limit ajh -> 0, a straightforward expansion of the hyperbolic functions for large values of a yields... 

This accords with the original result of Lorentz (LI 1), obtained by employing a first reflection. In the opposite case, where the sphere is very near the wall, i.e., ajH- oo (where H = h — a), one finds from (131) that a. - 2Hla. Upon expansion of all the hyperbolic functions in (130) for small a, one obtains... 

The series expansions for the hyperbolic functions are similar to Eqs. (4.55) and (4.56), except that all terms have plus signs ... 

The analytical terms are computed through reducing them as such to contain the hyperbolic functions and then appl5ang the approximations of type (2.45). With this recipe we firstly evaluate for the fluctuation width the expansion of the term (Putz, 2009) ... 

Hyperbolic and Inverse Hyperbolic Table of expansion of certain functions into power series... 

Now, for small values of the argument, the hyperbolic cotangent is given by the expansion... 

An approach which couples flexibility to a fairly simple analytical formulation is that suggested by Murrell and coworkers decomposing the potential energy in two and three body terms. These terms are then represented using polynomial expansions in physical coordinates damped by exponential or hyperbolic tangent factors. We have suggested... 

Fig. 28. Volume dependencies of hyperfine field (T->0), Neel temperature and isomer shift (T = 300 K) of EuAlj. Isomer shift The dashed line gives the initial slope of dS/d In K The solid line is the hyperbolic fit discussed in text. Hyperfine field and Neel temperature The solid line is a power expansion fit. The value of at highest pressure was not included in the fit, to emphasize its weak increase. [Taken from Gleissner (1992).]...

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