Deformation of free boundaries in plane viscous creep

Summary and Final Comments The deformation of a cylindrical free boundary lying in a thick infinite viscous body loaded by uniform stresses at infinity has been investigated. It is shown that when the free boundary can be initially conformally mapped onto a circle by a polynomial of degreen, it deforms so that its configuration remains within the family of curves described by a polynomial mapping of exactly degreen. A method of obtaining the exact motion of boundaries in this class is formulated and an example illustrating the calculations is offered; the example shows that a small irregularity on a free boundary may grow when compression is applied at infinity.The fact that polynomial free boundaries remain polynomial curves during deformation is one aspect of the kinematic reversibility (or memory) of linear viscous deformation, which greatly facilitates analysis of their motion. The deformation of completely arbitrary free boundaries is more difficult to obtain. In plane elasticity one often seeks approximate solutions to a problem by using a polynomia approximation to the exact mapping of a boundary.Muskhelishvili [3] has shown that if one chooses a sequence {_n} of mappings which converges onto the true mapping of the boundary , the sequences of stress functions {_n} and {_n} generated by {_n} converge onto the stress functions and associated with . In the viscous case Muskhelishvili's theorem always applies for computation of incipient motion; but, (21) and (26) show that when finite motion is considered it is not always adequate to replace the true mapping of the surface with a polynomial which is initially a close approximation. The general conditions under which one may expect the magnitude of higher order terms in the mapping of the free surface to diminish during deformation, so that a polynomial approximation to a mapping improves with time, have not been worked out in detail; however, it seems sufficient at this writing to require that the hydrostatic component of stress at infinity be tensile (P ₁+P₂>0). In the case that the higher order coefficients diminish, Equations (14), (15) and (18) indicate that the free boundary will tend to become an ellipse.

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Zusammenfassung Es wird die Verformung einer zylindrischen freien Oberfläche in einem unendlichen viskosen Körper untersucht, der sich unter dem Einfluss eines gleichförmigen Spannungszustandes im Unendlichen in einem ebenen Fliesszustand befindet. Es wird gezeigt, dass dann, wenn die freie Oberfläche ursprünglich durch ein Polynomn-ten Grades konform auf einen Kreis abgebildet werden kann, das gleiche auch für die deformierte Oberfläche gilt. Für diese Problemklasse wird eine Methode zur Ermittlung der exakten Oberflächenbewegung angegeben und an einem Beispiel illustriert. Dabei zeigt sich, dass eine kleine Unregelmässigkeit in der freien Oberfläche unter Druck im Unendlichen anwachsen kann.

     

Chain Lengths in Certain Random Directed Graphs

We study the random directed graph with vertex set {1, …, n} in which the directed edges (i, j) occur independently with probability c_n/n for i<j and probability zero for i ? j. Let M_n (resp., L_n) denote the length of the longest path (resp., longest path starting from vertex 1). When c_n is bounded away from 0 and ∞ as n→∞, the asymptotic behavior of M_n was analyzed in previous work of the author and J. E. Cohen. Here, all restrictions on c_n are eliminated and the asymptotic behavior of L_n is also obtained. In particular, if c_n/ln(n)→∞ while c_n/n→0, then both M_n/c_n and L_n/c_n are shown to converge in probability to the constant e.     

Effects of polymer–fiber interactions on rheology and flow behavior of suspensions of semi-flexible fibers in polymeric liquids

Rheological properties of suspensions of fibers in polymeric fluids are influenced by fiber–polymer interactions. In this paper, we investigate this influence from both experimental and modeling standpoints. In the experimental part of this investigation, we have changed the fiber–polymer interactions by treating the surface of the fibers. The resulting effects are observed using scanning electron microscopy and dynamic mechanical analysis techniques and quantified from the measurements of the viscosity in the start-up of shear flows and dynamic tests in the linear viscoelastic range region. The results are interpreted with the help of a mesoscopic rheological model developed for suspensions of fibers in viscoelastic fluids.     

A straightforward anionic coupling for the synthesis of ortho-bromobiaryls

Non-catalyzed anionic coupling of aryllithiums with 1,2-dibromobenzene gives straightforward access to valuable ortho-bromobiaryls.