Free Molecule Flow Through Slit and Annular Orifices in the Presence of Participating Bounding Walls

1969 ◽  
Vol 36 (4) ◽  
pp. 715-722
Author(s):  
E. M. Sparrow ◽  
H. S. Yu ◽  
T. S. Lundgren
Keyword(s):  
Integral Equations ◽  
Approximate Model ◽  
Geometrical Parameters ◽  
Free Molecule ◽  
Closed Form Solutions ◽  
Wide Range ◽  
Flow Through ◽  
Bounding Surfaces ◽  
Molecule Flow ◽  
Range Of Values

The effect of actively participating bounding surfaces on the free molecule flow through a slit or an annular orifice situated in a wall separating two regions of different pressure is analyzed. The flow through the slit or orifice depends on the distributions of the flux of mass leaving the bounding surfaces. These distributions are found by formulating and solving pairs of integral equations. In the case of the slit, the integral equations are formulated by employing kinetic theory methods, while for the annular orifice it was found advantageous to use the techniques of radiative transfer. In addition to exact solutions, closed-form solutions based on an approximate model are derived. Results are presented for a wide range of values of the relevant geometrical parameters.

Free Molecule Flow Through a Vacuum Seal

1970 ◽  
Vol 92 (3) ◽  
pp. 405-410
Author(s):  
H. S. Yu ◽  
E. M. Sparrow
Keyword(s):  
Mass Flow ◽  
Numerical Solutions ◽  
Rarefied Gas ◽  
Coupled System ◽  
Mass Fluxes ◽  
Wide Range ◽  
Flow Through ◽  
Molecule Flow ◽  
Range Of Values

An analysis is made of the rate of the mass flow through a vacuum seal separating two rarefied gas environments. The determination of the mass throughflow characteristics involves the formulation and solution of a coupled system of six integral equations. The formulation is performed using the methods of kinetic theory. Numerical solutions are carried out for a wide range of values of the seal geometrical parameter. Mass flow results evaluated from these solutions are presented graphically. In addition, representative distributions of the mass fluxes at the participating surfaces are given.

Orifice flow at high Knudsen numbers

1961 ◽  
Vol 10 (3) ◽  
pp. 371-384 ◽  
Author(s):  
Roddam Narasimha
Keyword(s):  
Flow Field ◽  
Three Dimensional ◽  
Mean Free Path ◽  
Free Molecule ◽  
First Order ◽  
Orifice Flow ◽  
Flow Through ◽  
The Mean ◽  
Molecule Flow ◽  
Inverse Knudsen Number

Several interesting features of the flow field in free-molecule flow through an orifice are discussed. An estimate is then made of the deviation of the mass flow $\dot{m}$ through the orifice from its limiting free-molecule value $\dot{m}$ for small departures from the limit. Using an iteration method proposed by Willis, it is shown that this deviation is of the first order in ε, the inverse Knudsen number, defined as the ratio of the radius of the hole to the mean free path in the gas at upstream infinity. An estimate of the coefficient is obtained making some reasonable assumptions about the three-dimensional nature of the flow, and the value so derived, giving $\dot{m}=\dot{m}(1+0.25\epsi)$, shows fair agreement with the measurements of Liepmann. It appears that ‘nearly’ free-molecular conditions prevail up to ε ∼ 1.0.

scholarly journals Centroid-Based Clustering with αβ-Divergences

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 196 ◽  
Author(s):  
Auxiliadora Sarmiento ◽  
Irene Fondón ◽  
Iván Durán-Díaz ◽  
Sergio Cruces
Keyword(s):  
Real Data ◽  
Practical Applications ◽  
Theoretical Contribution ◽  
Closed Form Solutions ◽  
Research Fields ◽  
Wide Range ◽  
The Family ◽  
Two Parameters ◽  
Range Of Values

Centroid-based clustering is a widely used technique within unsupervised learning algorithms in many research fields. The success of any centroid-based clustering relies on the choice of the similarity measure under use. In recent years, most studies focused on including several divergence measures in the traditional hard k-means algorithm. In this article, we consider the problem of centroid-based clustering using the family of α β -divergences, which is governed by two parameters, α and β . We propose a new iterative algorithm, α β -k-means, giving closed-form solutions for the computation of the sided centroids. The algorithm can be fine-tuned by means of this pair of values, yielding a wide range of the most frequently used divergences. Moreover, it is guaranteed to converge to local minima for a wide range of values of the pair ( α , β ). Our theoretical contribution has been validated by several experiments performed with synthetic and real data and exploring the ( α , β ) plane. The numerical results obtained confirm the quality of the algorithm and its suitability to be used in several practical applications.

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