Viscous Flow Through Tubes of Multiply Connected Cross Sections

1959 ◽  
Vol 26 (4) ◽  
pp. 573-576
Author(s):  
F. A. Gaydon ◽  
H. Nuttall
Keyword(s):  
Lower Bounds ◽  
Cross Sections ◽  
Essential Feature ◽  
Ritz Method ◽  
Cylindrical Tube ◽  
Volume Flow ◽  
Upper And Lower Bounds ◽  
Internal Boundary ◽  
Relaxation Methods ◽  
Multiply Connected

Abstract A new method is proposed for estimating the volume flow of a viscous incompressible fluid through a cylindrical tube of multiply connected cross section. The method brackets the magnitude of the volume flow between upper and lower bounds. The essential feature of the method is that the calculation of both upper and lower bounds is based upon the same approximating function for the velocity distribution, thus avoiding the usual approach to a lower bound via the Rayleigh-Ritz method. For multiply connected cross sections of the form discussed, a Rayleigh-Ritz solution of sufficient accuracy becomes extremely laborious. Efforts to solve the problem by relaxation methods are also rendered difficult by the presence of high-velocity gradients in the vicinity of an internal boundary, particularly when this is a small circle. In contrast to these methods the one presented achieves the result with considerably less labor; moreover, the method is directly applicable to simply connected cross sections with many-sided boundaries.

Bounds for the Natural Frequencies of a Simply Supported beam Carrying a Uniformly Distributed Axial Load

1967 ◽  
Vol 9 (2) ◽  
pp. 149-156 ◽  
Author(s):  
G. Fauconneau ◽  
W. M. Laird
Keyword(s):  
Lower Bounds ◽  
Axial Load ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Distributed Load ◽  
Made In

Upper and lower bounds for the eigenvalues of uniform simply supported beams carrying uniformly distributed axial load and constant end load are obtained. The upper bounds were calculated by the Rayleigh-Ritz method, and the lower bounds by a method due to Bazley and Fox. Some results are given in terms of two loading parameters. In most cases the gap between the bounds over their average is less than 1 per cent, except for values of the loading parameters corresponding to the beam near buckling. The results are compared with the eigenvalues of the same beam carrying half of the distributed load lumped at each end. The errors made in the lumping process are very large when the distributed load and the end load are of opposite signs. The results also indicate that the Rayleigh-Ritz upper bounds computed with the eigenfunctions of the unloaded beam as co-ordinate functions are quite accurate.

Upper and Lower Bounds of Frequencies for Cantilever Bars of Variable Cross Sections

1966 ◽  
Vol 33 (4) ◽  
pp. 948-950 ◽  
Author(s):  
J. H. Gaines ◽  
Enrico Volterra
Keyword(s):  
Lower Bounds ◽  
Cross Sections ◽  
Transverse Shear ◽  
Natural Frequencies ◽  
Numerical Results ◽  
Upper And Lower Bounds ◽  
Variable Cross ◽  
Tabular Form ◽  
Rotatory Inertia ◽  
Transverse Vibrations

Upper and lower bounds of frequencies of transverse vibrations of cantilever bars of variable cross sections are presented, taking into account the effects of transverse shear and of rotatory inertia. Numerical results for the first four natural frequencies are presented in tabular form for different inertia characteristics of the bars.

Bounds for the torsional rigidity of shafts with arbitrary cross-sections containing cylindrically orthotropic fibres or coated fibres

2007 ◽  
Vol 463 (2088) ◽  
pp. 3291-3309 ◽  
Author(s):  
Tungyang Chen ◽  
Robert Lipton
Keyword(s):  
Cross Section ◽  
Lower Bounds ◽  
Cross Sections ◽  
Equilateral Triangle ◽  
Torsional Rigidity ◽  
Transverse Cross Section ◽  
Upper And Lower Bounds ◽  
Cylindrical Shaft ◽  
Cylindrically Orthotropic ◽  
Additional Constraints

In this paper we derive bounds for the torsional rigidity of a cylindrical shaft with arbitrary transverse cross-section containing a number of cylindrically orthotropic fibres or coated fibres. The exact upper and lower bounds depend on the constituent shear rigidities, the area fractions, the locations of the reinforcements as well as the geometric shape of the cross-sections. Specific bounds are derived for circular shafts, elliptical shafts and cross-sections of equilateral triangle. Simplified expressions are also deduced for reinforcements with isotropic constituents. We verify that when additional constraints between the constituent properties of the phases are fulfilled, the upper and lower bounds will coincide. In the latter case, the fibres or coated fibres become neutral under torsion and the bounds recover the previously known exact torsional rigidity.

Torsion of Cylindrical and Prismatic Bars in the Presence of Steady Creep

1958 ◽  
Vol 25 (2) ◽  
pp. 214-218
Author(s):  
S. A. Patel ◽  
B. Venkatraman ◽  
P. G. Hodge
Keyword(s):  
Closed Form ◽  
Lower Bounds ◽  
Cross Sections ◽  
Creep Behavior ◽  
Upper And Lower Bounds ◽  
Elastic Analysis ◽  
Steady Creep ◽  
Pure Torsion ◽  
Closed Form Solutions ◽  
Creep Problem

Abstract This paper is concerned with the steady creep behavior of cylindrical and prismatic bars in which the deformations are caused by pure torsion. The creep problem is first reduced to one in nonlinear elasticity by means of the elastic analog. The elastic analysis is then carried out by means of the principles of minimum energies. These principles yield upper and lower bounds on the angle of twist. Closed-form solutions also are presented for some cross sections.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

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