Ellipticity parameter

The film shape for the elliptoid contact region with an ellipticity parameter k = 2.9 is shown in Fig. 11 [45], which gives all characteristic features of medium loaded point... 

In solving the elastohydrodynamic problem for "point" contact, the Reynolds equation is coupled with the expressions for the elastic deformation of the bounding surfaces and for the influence of pressure and temperature on the viscosity of the lubricant, as in the solution for "line" contact. However, a single traverse across the contact zone does not suffice as the integration path in the case of "point" contact, where the contact area is elliptical or circular instead of rectangular. This brings into play the ellipticity parameter, which is simply... 

Fig. 8 All MTM ( ) and pin-on-disk ( ) data at different concentrations of aqueous glycerol mixtures, plotted on a lubrication-tegime map, obtained from the Esfahanian-Hamrock-Dowson equations [28] for a circular contact (ellipticity parameter k = 1). The four different regimes in the dimensionless viscosity (gv) versus elastic (gD parameter plot are iso-viscous rigid (IR), iso-viscous elastic (IE), piezo-viscous rigid (VR), and piezo-viscous elastic (VE). All the values reported in this study lie in the iso-viscous elastic regime. It should be noted that while the equations in [28] apply to rolling contact, the model can also be used for sliding geometries at the low speeds used in our pin-on-disk experiments [35]...

Hamrock, B.J., and Dowson, D., Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part I — Theoretical Formulation, Part II — Ellipticity Parameter Results, Part III — Fully Flooded Results, Part IV — Starvation Results, ASME J. of Lubrication Technology, 98 and 99 (1976-77). 

Here t plays the role of a parameter. Thus, we can use the results on smoothness up to the boundary for solutions to elliptic equations of the form (3.139) (see Mikhailov, 1976). This yields (3.138). 

Proof. Using elliptic regularization and the penalty approach, we construct an auxiliary problem which approximates (5.6)-(5.9). Its solution will depend on two positive parameters a and d which are related to the elliptic regularization and to the penalty approach, respectively. We will obtain a solution a, u by passing to the limit as a, (5 —> 0. So, consider the following boundary value problem in fl... 

Proof. The idea of the proof is to use an elliptic regularization for the penalty equations approximating (5.139)-(5.143). Solutions of the auxiliary problem will depend on two positive parameters s, 5. The first parameter is responsible for the elliptic regularization and the second one characterizes... 

At least two parameters are tested by the same laboratory on many nominally similar samples. In both cases, the simplest outcome is a round patch in the Youden plot, see Fig. 2.1, of points that signifies just noise, no correlation. .. no participating laboratory (or sample or point in time) is exceptional. On the other hand, an elliptical patch, especially if the slope deviates from what could be expected, shows that some effects are at work that need further investigation. After just noise, the... 

Parameters of dynamically hot galaxies , i.e. various classes of ellipticals and the bulges of spirals, generally lie close to a Fundamental Plane in the 3-dimensional space of central velocity dispersion, effective surface brightness and effective radius or equivalent parameter combinations (Fig. 11.10). This is explained by a combination of three factors the Virial Theorem, some approximation to... 

Increasing a leads to the effective double-well potential shown earlier with two elliptic (stable) and one hyperbolic (unstable) fixed points. The elliptic fixed points become unstable for parameter values below... 

The mass flux in the spray scales with liquid metal flow rate. Gas pressure tends to narrow the spray whereas melt superheat tends to flatten the spray)3] By changing the process parameters and/or manipulating the configuration and/or motion of the spray, the mass distribution profile can be tailored to the desired shape. For example, a linear atomizer produces a relatively uniform mass distribution in the spray. The mass flux distribution in the spray generated with a linear atomizer has been proposed to follow the elliptical form of the Gaussian distribution)178]... 

The solution for model-based clustering is based on the Expectation Maximi-zation (EM) algorithm. It uses the likelihood function and iterates between the expectation step (where the group memberships are estimated) and the maximization step (where the parameters are estimated). As a result, each object receives a membership to each cluster like in fuzzy clustering. The overall cluster result can be evaluated by the value of the negative likelihood function which should be as small as possible. This allows judging which model for the clusters is best suited (spherical clusters, elliptical clusters) and which number of clusters, k, is most appropriate. 

Stationary (i.e. for dA/ dz = 0) localized solutions to Eq.(3.2) represent nonlinear modes in the planar waveguide and may be found in an analytical form via matching the partial solutions of Eq.(3.2) at the core/cladding boundary. The partial solutions are Jacobi elliptical function in the core and 2l rccosh — )E]/E in the cladding (the functional dependence similar to a fundamental soliton in a uniform nonlinear medium). Here is a parameter which depends on the boundary conditions. Contrary to the modes of a linear waveguide, the transverse profile of a nonlinear mode depends on the power in the mode. 

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