A singular solution of the capillary equation

1975 ◽  
Vol 29 (2) ◽  
pp. 143-148 ◽  
Author(s):  
Paul Concus ◽  
Robert Finn
Keyword(s):  
Singular Solution ◽  
Capillary Equation

A singular solution of the capillary equation

1975 ◽  
Vol 29 (2) ◽  
pp. 149-159 ◽  
Author(s):  
Paul Concus ◽  
Robert Finn
Keyword(s):  
Singular Solution ◽  
Capillary Equation

scholarly journals IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion

1895 ◽  
Vol 186 ◽  
pp. 123-164 ◽  
Keyword(s):  
Singular Solution ◽  
Physical State ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Viscous Fluids ◽  
Linear Functions ◽  
Incompressible Viscous Fluids ◽  
Theoretical Results

1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St. Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a co-efficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption.

Recording the Aloes at Chelsea - a singular solution to a difficult problem

1996 ◽  
Vol 50 (1) ◽  
pp. 47-57
Keyword(s):  
Eighteenth Century ◽  
Royal Society ◽  
Singular Solution ◽  
Difficult Problem ◽  
General Series ◽  
History Of

In the Royal Society archives there is a collection of drawings of Aloes and other plants, made by two of the great botanical artists of the eighteenth century - Georg Dionysius Ehret and Jacob van Huysum. Although the Manuscripts General Series Catalogue records this manuscript only as a ‘Volume of 35 botanical paintings by Georg Dionysius Ehret’ of unknown provenance, the manuscript catalogue of the Arundel and other manuscripts, said to be the work of Jonas Dryander (1748-1810), provides the first clue linking these drawings to the two artists, and to the important collection of Aloes growing at that time in the Society of Apothecaries Physic Garden at Chelsea'. The history of the commissioning of the drawings is told briefly in the Journal Books of the Royal Society, and in the Minutes of Council, but the significance of these lovely and important drawings has been almost completely overlooked.

scholarly journals NEW SOLUTION TO THE PROBLEM OF A CRACK IN AN ORTHOTROPIC PLATE UNDER TENSION

2021 ◽  
Vol 56 (6) ◽  
pp. 902-910
Author(s):  
V. V. Vasil’ev ◽  
S. A. Lurie ◽  
V. A. Salov
Keyword(s):  
Singular Solution ◽  
Orthotropic Plate ◽  
Scale Parameter ◽  
Theory Of Elasticity ◽  
Crack Edge ◽  
Small Scale ◽  
Carbon Fiber Reinforced Plastic ◽  
Fiber Reinforced Plastic ◽  
Reinforced Plastic ◽  
Classical Plane

Abstract— A classical plane problem of the theory of elasticity about a crack in a stretched orthotropic elastic unbounded plane is considered, which leads to a singular solution for stresses in the vicinity of the crack edge. The relations of the generalized theory of elasticity, including a small scale parameter, are given. The equations of the generalized theory are of a higher order than the equations of the classical theory and allow eliminating the singularity of the classical solution. The scale parameter is determined experimentally. The results obtained determine the effect of the crack length on the bearing capacity of the plate and are compared with the experimental results for plates made of fiberglass and carbon fiber reinforced plastic.

Classification of singular solutions of porous media equations with absorption

2005 ◽  
Vol 135 (3) ◽  
pp. 563-584 ◽  
Author(s):  
Xinfu Chen ◽  
Yuanwei Qi ◽  
Mingxin Wang
Keyword(s):  
Porous Media ◽  
Singular Solution ◽  
Singular Solutions ◽  
Constant Independent ◽  
Porous Media Equation ◽  
Porous Media Equations ◽  
Self Similar ◽  
Image Position ◽  
Very Singular Solution

We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in Rn × [0, ∞)\{(0, 0)}, and satisfy u(x, 0) = 0 for all x ≠ 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u(c), called the fundamental solution with initial mass c, which satisfies ∫Rnu(·, t) → c as t ↘ 0. Also, there exists a unique singular solution u = u∞, called the very singular solution, which satisfies ∫Rnu∞(·, t) → ∞ as t ↘ 0.In addition, any singular solution is either u∞ or u(c) for some finite positive c, u(c1) < u(c2) when c1 < c2, and u(c) ↗ u∞ as c ↗ ∞.Furthermore, u∞ is self-similar in the sense that u∞(x, t) = t−αw(|x| t−αβ) for α = 1/(q − 1), β = ½(q − m), and some smooth function w defined on [0, ∞), so that is a finite positive constant independent of t > 0.

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