Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Stability of a radially bounded rotating fluid heated from below

The stability of a bounded rotating cylinder of fluid heated from below is treated mathematically under the assumptions of stationary onset and axisymmetry. Critical Rayleigh numbers are computed by Galerkin's method as a function of the Taylor number and cylinder aspect ratio for Taylor numbers,10⁶. The constraining effect of the side walls is shown to decrease with either increasing or increasing radius/height ratios. For>10⁶, most cylinders, excluding extremely tall ones, will appear infinite in horizontal extent as far as stability characteristics are concerned. The form of the motion at onset is discussed in relation to previous work.     

Effect of suspended particles on the stability of plane poiseuille flow

The behavior of the neutral stability curves is investigated for various values of the particle relaxation time and mass concentration 0 100 and 0 f 0.1. It is shown that as increases from zero the flow is at first destabilized and then at >6 becomes stable, while at >40 the stabilizing effect of the dispersed phase grows weaker. It is found that there is a certain interval 10< <40 on which the flow is most stable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 46–53, January–February, 1986.     

Dynamic-mechanical properties of a bitumen-silica composite

Summary The dynamic-mechanical behaviour of bitumensilica composites is described by a linear biparabolic model. Its mathematical expression allows the calculation of the mean relaxation times () either at different temperatures and given filler contents or for diverse filler contents () at imposed temperatures. At fixed filler concentration and within restricted temperature domains, obeys Arrhenius' law. The activation energies are respectively close to 10 kcal/mole (creep) and 30 kcal/mole (glass-transition). varies exponentially with. The mathematical treatment of the expressions ofE , as a function of temperature and of, leads to a general equation relating the complex modulus to temperature, frequency and filler content. A unique master curve, accounting for the viscoelastic behaviour of the composites, in limited ranges, can thus be constructed.

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Zusammenfassung Das dynamisch-mechanische Verhalten von Bitumen-Siliziumdioxyd-Zusammensetzungen kann durch ein lineares biparabolisches Modell beschrieben werden. Sein mathematischer Ausdruck erlaubt die Ausrechnung der mittleren Relaxationszeiten () entweder für verschiedene Temperaturen bei gegebenem Füllstoffgehalt oder für unterschiedliche Siliziumdioxydmengen () bei bekannter Temperatur. Für einen bestimmten Füllstoffgehalt folgt in einem beschränkten Temperaturbereich dem Arrheniusschen Gesetz. Die Aktivierungsenergien betragen näherungsweise 10 kcal/Mol (Fließprozeß) bzw. 30 kcal/Mol (Glasübergang). ändert sich exponentiell mit. Die mathematische Umformung der Ausdrücke fürE und als Funktion der Temperatur und des Parameters ergibt eine allgemeine Gleichung, die den komplexen Modul mit der Temperatur, der Frequenz und dem Füllstoffgehalt verknüpft. Man kann eine einzige Masterkurve bilden, die das viskoelastische Verhalten der Zusammensetzungen zumindest in begrenzten Bereichen beschreibt.

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Résumé Le comportement mécanique dynamique des composites à base de bitume et de silice peut être décrit par un modèle biparabolique linéaire. L'expression mathématique permet le calcul des temps moyens () de relaxation d'une part aux différentes températures, à taux de charge donné, et d'autre part pour diverses valeurs des taux de charge (paramètre) à température imposée. A taux de charge donné, et pour des domaines de température restreints, suit la loi d'Arrhénius. Les énergies apparentes d'activation sont respectivement voisines de 10 kcal/mole (processus de fluage) et de 30 kcal/mole (passage à l'état vitreux). Avec, varie exponentiellement. L'évaluation mathématique deE , de en fonction deT et de conduit à une expression générale du module complexe en fonction de la température, de la fréquence et du taux de charge. On peut donc construire une courbe maitresse unique qui décrit entièrement, mais dans des domaines restreints, le comportement viscoélastique des composites.

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With 6 figures     

On the compactness of quasi-conforming element spaces and the convergence of quasi-conforming element method

In this paper, the compactness of quasi-conforming element spaces and the—convergence of quasi-conforming element method are discussed. The well-known Rellich compactness theorem is generalized to the sequences of quasi-conforming element spaces with certain properties, and the generalized Poincare inequality. The generalized Friedrichs inequality and the generalized inequality of Poincare-Friedrichs are proved true for them. The error estimates are also given. It is shown that the quasi-conforming element method is convergent if the quasi-conforming element spaces have the approximability and the strong continuity, and satisfy the rank condition of element and pass the test IPT. As practical examples, 6-parameter, 9-paramenter, 12-paramenter, 15-parameter, 18-parameter and 21-paramenter quasi-conforming elements are shown to be convergent, and their L²²()-errors are O(h), O(h), O(h ² ), O(h ² ), O(h ), and O(h ⁴ ) respectively.     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress     

Die swell of filled polymer melts

The Barus effect in polypropylene and polystyrene blended with a variety of fillers at various concentrations was investigated using a capillary extrusion rheometer. If the die swell is defined as the square of the ratio of the extrudate diameterd to the die diameterD, it is found to depend on the apparent shear stress _W . Below a certain value of _w the relation =B _B ^A applies. The die swell, _M , of a filled polymer depends on the type, size and volume fraction of the filler. In particular,A increases as the volume fraction increases and is largest for powders, smaller for flakes and smallest for fibres, whereasB shows the opposite trend but to a lesser extent.     

On properties of solutions of functional differential equations continuously differentiable on (0, +∞)

We establish the saddle-point property of the system of functional differential equations (t) = Ax(t) + Bx((t)) + C ((t)) + f (x(t), x((t))), (0) = 0.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 302–310, July–September, 2004.     

Viscoelastic properties of semidilute poly(methyl methacrylate) solutions

Viscoelastic properties were examined for semidilute solutions of poly(methyl methacrylate) (PMMA) and polystyrene (PS) in chlorinated biphenyl. The number of entanglement per molecule, N, was evaluated from the plateau modulus, G _N . Two time constants, _s and ₁, respectively, characterizing the glass-to-rubber transition and terminal flow regions, were evaluated from the complex modulus and the relaxation modulus. A time constant _k supposedly characterizing the shrink of an extended chain, was evaluated from the relaxation modulus at finite strains. The ratios ₁/ _s and _k / _s were determined solely by N for each polymer species. The ratio ₁/ _s was proportional to N ^4.5 and N ^3.5 for PMMA and PS solutions, respectively. The ratio _k / _s was approximately proportional to N ^2.0 in accord with the prediction of the tube model theory, for either of the polymers. However, the values for PMMA were about four times as large as those for PS. The result is contrary to the expectation from the tube model theory that the viscoelasticity of a polymeric system, with given molecular weight and concentration, is determined if two material constants _s and G _N are known.     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

Zusammenfassung Für die eingefrorene laminare Grenzschichtströmung eines teilweise dissoziierten binären Gemisches entlang einer stark gekühlten ebenen Platte wird eine analytische Näherungslösung angegeben. Danach läßt sich die Wandkonzentration als universelle Funktion der Damköhler-Zahl der Oberflächenreaktion angeben. Für das analytisch darstellbare Konzentrationsprofil stellt die Damköhler-Zahl den Formparameter dar. Die Wärmestromdichte an der Wand bestehend aus einem Wärmeleitungs- und einem Diffusionsanteil wird angegeben und diskutiert. Das Verhältnis beider Anteile läßt sich bei gegebenen Randbedingungen als Funktion der Damköhler-Zahl ausdrücken.

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An analytical approximation for the frozen laminar boundary layer flow of a binary mixture

An analytical approximation is derived for the frozen laminar boundary layer flow of a partially dissociated binary mixture along a strongly cooled flat plate. The concentration at the wall is shown to be a universal function of the Damkohler-number for the wall reaction. The Damkohlernumber also serves as a parameter of shape for the concentration profile which is presented in analytical form. The heat transfer at the wall depending on a conduction and a diffusion flux is derived and discussed. The ratio of these fluxes is expressed as a function of the Damkohler-number if the boundary conditions are known.

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Formelzeichen A Atom - A₂ Molekül - C Konstante in Gl. (20) - c₁=1/(2C) Konstante in Gl. (35) - c_p spezifische Wärme bei konstantem Druck - D binärer Diffusionskoeffizient - Ec=u ² /(2h_f) Eckert-Zahl - h spezifische Enthalpie - h_t=h+u²/2 totale spezifische Enthalpie - h _A ⁰ spezifische Dissoziationsenthalpie - K_w Reaktionsgeschwindigkeitskonstante der heterogenen Wandreaktion - 1= /( ) Champman-Rubesin-Parameter - Le=Pr/Sc Lewis-Zahl - M Molmasse - p statischer Druck - Pr= c_pf/ Prandtl-Zahl - q_w Wärmestromdichte an der Wand - q_cw, q_dw Wärmeleitungsbzw. Diffusionsanteil der Wärmestromdichte an der Wand - universelle Gaskonstante - R=/(2M_a) individuelle Gaskonstante der molekularen Komponente - Re_x= u x/ Reynolds-Zahl - Sc=/( D) Schmidt-Zahl - T absolute Temperatur - T_d=h _A ⁰ /R charakteristische Dissoziationstemperatur - u, v x- und y-Komponenten der Geschwindigkeit - U=u/u normierte x-Komponente der Geschwindigkeit - x, y Koordinaten parallel und senkrecht zur Platte Griechische Symbole - =A/ Dissoziationsgrad - Grenzschichtdicke - ₂ Impulsverlustdicke - Damköhler-Zahl der Oberflächenreaktion - =T/T normierte Temperatur - =y/ normierter Wandabstand - Wärmeleitfähigkeit - dynamische Viskosität - , ^* Ähnlichkeitskoordinaten - Dichte - Schubspannung Indizes A auf ein Atom bezogen - M auf ein Molekül bezogen - f auf den eingefrorenen Zustand bezogen - w auf die Wand bezogen - auf den Außenrand der Grenzschicht bezogen     

Analysis of melt spinning transients in Lagrangian coordinates

Transients in melt spinning of isothermal power law and Newtonian fluids were found to be governed by an extremely simple partial differential equation ² ( ^1/n )/() = 0 in Lagrangian coordinates where is the cross-sectional area,n the power law exponent, the time and the the time at which a fluid molecule constituting the spinline left the spinneret. The general integral ^1/n =f() +g () of the above governing equation containing two arbitrary functions represents physically attainable spinline transients. Hitherto unknown analytical transient solutions of the above governing equation were obtained for the response of isothermal constant tension spinlines to a stepwise change in tension, spinneret hole area, extrusion speed or extrusion viscosity and for the starting transient in gravitational spinning. Linearized perturbation solutions and the stability limit of the spinline derived from the above new found nonlinear solutions were in agreement with previous findings and the above nonlinear response of the spinline to a step increase in the spinneret hole area was found to be equivalent to Orowan's tandem cylinder model of dent growth in filament stretching.     

Displacement type unified equation for bending, buckling and vibration of conical shells and its applications

By using Donnell's simplication and starting from the displacement type equations of conical shells, and introducing a displacement functionU(s,,) (In the limit case, it will be reduced to cylindrical shell displacement function introduced by V. S. Vlasov) and a generalized loadq,(s,,),the equations of conical shells are changed into an eighth—order solvable partial differential equation about the displacement functionU(s,,). As a special case, the general bending problem of conical shells on Winkler foundation has been studied. Detailed numerical results and boundary coefficients for edge unit loads are obtained.The project supported by the National Natural Science Foundation of China.     

Image processing of hydrogen bubble flow visualization for determination of turbulence statistics and bursting characteristics

A technique is described which employs automated image processing of hydrogen-bubble flow visualization pictures to establish local, instantaneous velocity profile information. Hydrogen bubble flow visualization sequences are recorded using a high-speed video system and then digitized, stored, and evaluated by a VAX 11/780 computer. Employing special smoothing and gradient detection algorithms, individual bubble-lines are computer identified, which allows local velocity profiles to be constructed using time-of-flight techniques. It is demonstrated how this techniques may be used to 1) determine local velocity behavior as a function of position and time, 2) evaluate time-averaged turbulence properties, and 3) correlate probe-type turbulent burst detection techniques with the corresponding visualization data.List of symbols Re Reynolds number based on momentum thickness, u / - t ⁺ nondimensional time tu ² / - T VITA variance averaging time period - u shear velocity = - u local instantaneous streamwise velocity,x-direction - u local fluctuating streamwise velocity,x-direction - u ⁺ nondimensional streamwise velocity, /u - local normal velocity,y-direction - w local spanwise velocity,z-direction - x ⁺ nondimensional coordinate in streamwise direction xu /v - y ⁺ nondimensional coordinate normal to wall, yu /v Greek momentum thickness, - kinematic viscosity - _w wall shear stress This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Quantitative analysis of flow visualizations in an unsteady channel flow

Quantitative results concerning the modulation of the ejection and bursting frequency in an unsteady channel flow obtained by flow visualizations are presented and compared with probe measurements. The frequency of the imposed velocity oscillations f covers a large range going from the quasi steady limit to the time mean bursting frequency in the corresponding steady flow. The imposed amplitudes of the velocity oscillations are 13% and 20% of the centerline velocity. The bursting process is identified by the intermittent lift up of the dye injected at the wall. Qualitative analysis of the flow visualizations show that the ejection activity at a given phase of the oscillation cycle is repetitive from one cycle to the other. The modulation amplitude of the ejection frequency f _e is sensitive to the imposed frequency. At low imposed frequency f _e is modulated as the wall shear stress, but the inner scaling does not hold when f ⁺ is high. Here, (+) corresponds to the quantities normalized with the inner variables, i.e. the friction velocity u and the viscosity . The grouping of the ejections into bursts show the coexistence of two categories of events which react differently to the forcing. The groups of ejections (Multiple Ejection Bursts) are governed by the modulation of the wall shear stress in the whole imposed frequency range. The solitary ejections (or the Single Ejection Bursts) have modulation amplitudes and phases which differ significantly from those of in the intermediate and high imposed frequency range. There is a good agreement between the flow visualization data and the probe measurements.     

Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers

Solution of a non-homogeneous Fredholm integral equation of the second kind [1], which forms the basis for the evaluation of the constriction resistance of an isothermal circular spot on a half-space covered with a surface layer of different material, is considered for the case when the ratio, , of layer thickness to spot radius is larger than unity. The kernel of the integral equation is expanded into an infinite series in ascending odd-powers of (1/) and an approximate kernel accurate to (^–(2M+1)) is derived therefrom by terminating the series after an arbitrary but finite number of terms, M. The approximate kernel is rearranged into a degenerate form and the integral equation with this approximate kernel is reduced to a system of M linear equations. An explicit analytical solution is obtained for a four-term approximation of the kernel and the resulting constriction resistance is shown to be accurate to (^–9). Solutions of lower orders of accuracy with respect to (1/) are deduced from the four-term solution. The analytical approximations are compared with very accurate numerical solutions and it is shown that the (^–9)-approximation predicts the constriction resistance exceedingly well for any 1 over a four orders of magnitude variation of layer-to-substrate conductivity ratio for both conducting and insulating layers. It is further shown that, for all practical purposes, an (^–3)-approximation gives results of adequate accuracy for > 2.     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

Translatory accelerating motion of a circular disk in a viscous fluid

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.Nomenclature a radius of the circular disk - b one half of the thickness of the circular disk - C dimensionless form of C ₁ - C ₁ magnitude of the constant force - D fluid drag force - f(t) externally applied force - F() dimensionaless form of applied force - F ₀ initial value of F - g gravitational acceleration - H() Heaviside step function - k magnitude of impulsive force - K dimensionless form of k - M a dimensionless parameter equals to (1+37#x03C0;_s/4_f) - S displacement of disk - t time - t ₁ time of application of impulsive force - u velocity of the disk - V dimensionless velocity - V ₀ initial velocity of V - V _t terminal velocity - parameter in (13) - parameter in (13) - (t) Dirac delta function - ratio of b/a - () function given in (5) - dynamical viscosity of the fluid - kinematic viscosity of the fluid - _f fluid density - _s mass density of the circular disk - dimensionless time - _i dimensionless form of t _i - dummy variable - dummy variable     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].