Cauchy number

Cauchy number number Nc o5 II 1 hO K- = bulk modulus (Inertial/compressible) forces Compressible flow... 

Cauchy number C pV2 p inertial force compressibility force Compressible flow, hydraulic transients... 

Archimedes number Bingham number Bingham Reynolds number Blake number Bond number Capillary number Cauchy number Cavitation number Dean number Deborah number Drag coefficient Elasticity number Euler number Fanning friction factor Froude number Densometric Froude number Hedstrom number Hodgson number Mach number Newton number Ohnesorge number Peclet number Pipeline parameter... 

Pressure coefiident = 1/(Euler number) 3 2/(Mach number) = Cauchy number 4 2/Weber number... 

The above system of linear equations shows that a = P = s = 6 = e = 0 thus all the dimensionless parameters are independent of each other. This result is important since it allows us to multiply and divide the individual dimensionless parameters for a given set to generate recognizable parameters. For example, Hi is the Euler number, II3 is the Reynolds number, and II5 is the inverse of the Cauchy number. However, 112 and Il4 are not recognized, named dimensionless numbers, but because this group of dimensionless parameters forms a basis set, that is, independent of each other, we can multiply and divide them to obtain new dimensionless parameters. Our only constraint is our analysis must produce five dimensionless parameters. For example, multiplying 112 by 113 gives us... 

In some cases, when the model and prototype use the same fluid, it may be difficult to meet the Reynolds number criterion. For gas processes, the corresponding velocities in the model can induce compression effects in which case the Cauchy number becomes a scaling parameter. For liquids, where compression is negligible, maintaining high fluid velocity requires excessive power. In this case, using a fluid of lower kinematic viscosity in the model, while meeting the Reynolds number criterion, provides scalable information [4]. 

Cauchy number A dimensionless number, Ca, it is a measure of the ratio of the inertial to elastic forces in the compressible flow of fluids. It is the product of the density, p, and square of the velocity, v, to the bulk modulus of the fluid, E ... 

These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

As. vi in Figure 6.15(a) is swept clockwise around the contour, it encircles two zeros and one pole. From Cauchy s theorem given in equation (6.46), the number of clockwise encirclements of the origin in Figure 6.15(b) is... 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

Thus the average cost per share for John is the arithmetic mean ofpi,p2, , p, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only ifpi = p2 = = Pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices Pi are distinct. One can also give a proof based on the Cauchy-Schwarz inequality. To this end, define the vectors... 

The Cauchy moments are derived and implemented for the approximate triples model CC3 with the proper N scaling (where N denotes the number of basis functions). The Cauchy moments are calculated for the Ne, Ar, and Kr atoms using the hierarchy of the coupled-cluster models CCS, CC2, CCSD, CC3 and a large correlation-consistent basis sets augmented with diffuse functions. A detailed investigation of the one- and A-electron errors shows that the CC3 results have the accuracy comparable to the experimental results. 

The Cauchy moments of Ne at the CCSD/q-aug-cc-pV5Z level were found in Ref. [4] to be converged within 1 % compared to the basis-set limit result. We have calculated the Cauchy moments also for the X—6 cardinal number. From the results in Table 1 it appears that the Cauchy moments at this level are significantly less than 1 % from the basis-set limit result. 

Table 2. The Cauchy moments S(k) for Ne calculated at various levels of the CC hierarchy using the d-aug-cc-pV6Z basis-set (all electrons correlated). All numbers in a.u.

Summarizing this subsection, the group-theoretic techniques allow us to obtain three maps S3 —>. S 2 whose velocity vectors are mutually orthogonal, and with the same linking number. Next, we have to build the Cauchy data of the electromagnetic knots based on these maps. 

The average force per unit area is Af,-/AS. This quantity attains a limiting nonzero value as AS approaches zero at point P (Cauchy s stress principle). This limiting quantity is called the stress vector, or traction vector T. But T depends on the orientation of the area element, that is, the direction of the surface defined by normal n. Thus it would appear that there are an infinite number of unrelated ways of expressing the state of stress at point P. 

Random error — The difference between an observed value and the mean that would result from an infinite number of measurements of the same sample carried out under repeatability conditions. It is also named indeterminate error and reflects the - precision of the measurement [i]. It causes data to be scattered according to a certain probability distribution that can be symmetric or skewed around the mean value or the median of a measurement. Some of the several probability distributions are the normal (or Gaussian) distribution, logarithmic normal distribution, Cauchy (or Lorentz) distribution, and Voigt distribution. Voigt distribution is... 

Finally, substitute the numbers into Eq. 10.53 to obtain 75.2 GPa (1 GPa = 10 Pa). This result is noticeably larger than the experimental value (percent error = 88.2 percent). The difference implies that a pairwise potential is not entirely adequate for describing NaCI, even though the Cauchy relation holds. 

L /polymer extensibility smectic-layer compressive modulus E E, Finger strain tensor B , Cauchy strain tensor yriso/r, capillary number characteristic ratio, defined by R )q — Ccotib translational diffusivity-------------------------... 

Theorem A.4 (Cauchy s Residue Theorem) If f z) is analytic in a simply connected domain D, except at a finite number of singular points zi,...,Zk and if C is a simple positively oriented (counterclockwise) closed contour that lies in D, then... 

Theorem A.5 (Evaluation of the Cauchy Principal Value of an Integral) If a function f z) is analytic in a simply connected main D, except at a finite number of singular points zi,..., 2jt, iff x) = P x)/Q x) where P(x) and Q(x) are polynomials, Q x) has no zeros, and the degree ofP x) is at least two less than the degree of Q(x), and ifC is a simple positively oriented (counterclockwise) closed contour that lies in D, then... 

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