Uniform Circular Cross-section Research Articles

On the deflected configuration of a slender elastic rod subject to parallel terminal forces and moments

This paper investigates the three-dimensional configurations of a slender elastic rod of uniform circular cross-section subject to parallel terminal forces and moments. The nonlinear, equilibrium equations for the rod are established for a Cartesian coordinate system and solved analytically without linearization. Consequently, the results are applicable for large nonlinear elastic deformations.

Unsteady axial viscoelastic pipe flows

The main objective of this work is to examine in detail basic unsteady pipe flows and to investigate any new physical phenomena. We take the viscoelastic upper-convected Maxwell fluid as our non-Newtonian model and consider the flow of such a fluid in pipes of uniform circular cross-section in the following three cases: 1. (a) when the pressure gradient varies exponentially with time; 2. (b) when the pressure gradient is pulsating; 3. (c) a starting flow under a constant pressure gradient. In the first problem we looked separately at the pressure gradient rising exponentially with time and falling exponentially with time, i.e. the pressure gradient is proportional to e ± α 2 τ . The behaviour of the flow field depends to a large extent on β where β 2 = α 2(1 ± Hα 2) with H being the quotient of the Weissenberg and Reynolds numbers. In both cases for small | βη|, η being the radial distance from the axis, the velocity profiles are seen to be parabolic. However, for large | βη| the flows are vastly different. In the case of increasing pressure gradient the flow depicts boundary-layer characteristics while for decreasing pressure gradient the velocity depends on the wall distance. The case of a pulsating pressure gradient is investigated in the second problem. Here the pressure gradient is proportional to cos nτ. Again the flow depends to a large extent on a parameter β ( β 2 = i n − n 2 H). For small values of | βη| the velocity profile is parabolic. However, it is found that, unlike Newtonian fluids, the velocity distribution for the upper-convected Maxwell fluid is not in phase with the exciting pressure distribution. In the case of large | βη| the solution displays a boundary-layer characteristic and the phase of the motion far from the wall is shifted by half a period. The final problem examines a flow that is initially at rest and then set in motion by a constant pressure gradient. A closed form solution has been obtained with the aid of a Fourier-Bessel series. The variation of the velocity across the pipe has been sketched and comparison made with the classical solution.

On the Flexural-Torsional Behavior of a Straight Elastic Beam Subject to Terminal Moments

This paper discusses the large-displacement flexural-torsional behavior of a straight elastic beam with uniform circular cross-section subject to arbitrary terminal bending and twisting moments. The beam is assumed to be free from any kinematic constraints at both ends. The equilibrium equation is solved analytically with the full expression for curvature to obtain the deformed configuration in a three-dimensional Cartesian coordinate system. The results show the influence of the terminal moments on the beam’s deflected configuration.

Yarn Guide pipe of the Automatic Winder

The present paper studies the flow pattern in the yarn guide pipe with a uniform circular cross section provided with a long slot along the pipe axis. Some experiments and the numerical analysis of the flow are made for both the straight and the circular curved pipes.The results will give some useful knowledges for the improvement of the yarn guide apparatus from the view point of engineering practice.

Effective Properties of Fiber-Reinforced Materials

A material consisting of a matrix in which aligned fibers of uniform circular cross section are embedded is considered. A method is developed for computing the effective axial shear modulus and transverse thermal conductivity (which are mathematically analogous) for such materials. The method, an approximate technique based on pairwise interactions, introduces a new consistency concept uniquely appropriate for this problem, and is universal in the sense that it is readily and quickly applicable to any fiber arrangement. It overcomes an apparent in consistency in other interaction methods.

The optimization of modal sound attenuation in ducts, in the absence of mean flow

The optimization of modal, axial sound attenuation rates, in the sense defined by Cremer, is examined in detail for zero mean flow in the duct. Cremer's result, for the lowest order mode pair in the absence of flow, is generalized so that the optimum impedance and attenuation rate can now be obtained for any mode pair in ducts of uniform rectangular, circular or annular cross-section. One qualification to Cremer's result emerges in the derivation of the Green's function : at the optimum condition the mode pair degenerates into a single mode with the expected exponential attenuation rate but this is offset by an amplification rate which is directly proportional to the distance from the source. In spite of this effect the axial decay rate of the Green's function can reach a maximum for a wall impedance close to Cremer's optimum value. Computed results for the attenuation compare favourably with available experimental values. The computation method produces slightly more accurate values of the optimum wall impedance for maximum attenuation of the “least damped mode” than those previously obtained by Cremer and by Morse. The theoretical approach developed is suitable for extension to attenuation (and attenuation optimization) studies in ducts carrying mean flow.

Steady laminar flow with suction or injection and heat transfer through porous pipe

Following the method due to Bhatnagar (P. L.) [Jour. Ind. Inst. Sic., 1968, 1, 50, 1], we have discussed in this paper the problem of suction and injection and that of heat transfer for a viscous, incompressible fluid through a porous pipe of uniform circular cross-section, the wall of the pipe being maintained at constant temperature. The method utilises some important properties of differential equations and some transformations that enable the solution of the two-point boundary value and eigenvalue problems without using trial and error method. In fact, each integration provides us with a solution for a suction parameter and a Reynolds number without imposing the conditions of smallness on them. Investigations on non-Newtonian fluids and on other bounding geometries will be published elsewhere.

Elecromagnetic induction in a system of straight parallel nonmagnetic conductors of equal circualr cross-section

A method is described for calculating the alternating-current resistance of any member of a general system of straight parallel nonmagnetic conductors of equal uniform circular cross-section. The total current in each conductor is assumed to be known in magnitude and phase. The theory is a generalisation of that developed by Butterworth for two parallel conductors. The solution for two conductors carrying equal or opposite currents and the solution for three symmetrically arranged conductors carrying 3-phase balanced currents are derived from the general solution. These particular solutions agree with Butterworth's solution for two conductors and with Costello's solution for the 3-phase system. The present solutions have been evaluated numerically to high accuracy with the aid of a computer and the results are found to agree with available experimental results within the limits of experimental error. New solutions are derived for systems of four and six conductors and numerical results are compared with new experimental work, again yielding agreement within the limits of experimental error. The general solution may therefore be applied with some confidence to any system of parallel conductors.

Electromagnetic induction in a system of straight parallel nonmagnetic conductors of equal circular cross-section

A method is described for calculating the alternating-current resistance of any member of a general system of straight parallel nonmagnetic conductors of equal uniform circular cross-section. The total current in each conductor is assumed to be known in magnitude and phase. The theory is a generalisation of that developed by Butterworth for two parallel conductors. The solution for two conductors carrying equal or opposite currents and the solution for three symmetrically arranged conductors carrying 3-phase balanced currents are derived from the general solution. These particular solutions agree with Butterworth's solution for two conductors and with Costello's solution for the 3-phase system. The present solutions have been evaluated numerically to high accuracy with the aid of a computer and the results are found to agree with available experimental results within the limits of experimental error. New solutions are derived for systems of four and six conductors and numerical results are compared with new experimental work, again yielding agreement within the limits of experimental error. The general solution may therefore be applied with some confidence to any system of parallel conductors.

Some mathematical considerations relating to the sagging of bars of uniform circular cross section

In order to estimate creep parameters from sagging bar tests, a creep modulus F has been defined, by analogy to elastic moduli, such that F = σ n /(d ϵ/d t) where σ is the stress, ϵ is the strain, t is the time and n is a constant. A mathematical analysis has been made of this relation leading to a general form of bending moment equation. Expressions are derived for particular values of n which relate the sagging rates of the mid-points of end supported uranium bars to the creep modulus F. The expressions are used in experimental determinations of the parameter n from sagging bar tests carried out on uranium.

Rubber models of yarns and cords: the twisting of multi-filament models

The axial load and couple about the axis of rubber models of yarns containing 7, 19 and 127 rubber filaments of uniform circular cross section have been measured at various twists when the yarns are subjected to small (similar 5%) and large (66%) axial extensions. The results are in satisfactory agreement with the theoretical predictions at the high extension, but in general are less so at the lower extensions. The discrete filament theory, which is applicable to the two smaller models, shows that the couple induced into the yarns on twisting is mainly dependent on the tensile force along the filaments at finite yarn strains. The corrections to the load-torsion characteristics for the effects of transverse forces between filaments are shown to be of considerable importance particularly at high twists.