The Kepler problem spinor regularization and quantization on the manifolds S2

Summary In a previous research we have shown that the KS-transformation, developed by Kustaanheimo and Stiefel for the regularization of the Kepler problem, may be interpreted as the correspondence which associates to each null 4-vector of the space of Minkowski a one-index spinor, defined up to a phase factor, and we have obtained a new form of the KS-transformation. In the present research we show that this formulation allows a straight derivation of the Hopf fibering of the sphere S³ (characterized by unit spinors) having the base space given by the section (sphere S²) of the light cone, and we show that the KS-transformation allows the quantization of the symplectic manifold S² in the sense of Souriau. The sphere S³ turns out to be a contact quantized manifold. The bilinear relation characteristic of the KS-theory and the column vectors of the KS-matrix are intimately related to the contact structure.

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Sommario In un precedente lavoro si è mostrato che la trasformazione KS, introdotta da Kustaanheimo e Stiefel per regolarizzare il problema di Keplero, è riconducibile alla ben nota corrispondenza fra vettori del cono isotropo dello spazio di Minkowski e spinori semplici, definiti a meno della fase, e si è pervenuti ad una nuova formulazione della KS. Nel presente lavoro si mostra come da tale formulazione scaturisca in modo naturale la fibrazione di Hopf della sfera S³ (caratterizzata dagli spinori unitari) avente quale base una sezione (sfera S²) del cono isotropo e si mette in luce come la trasformazione KS consenta di effettuare la quantizzazione della varietà simplettica S² nel senso di Souriau e di ottenere la sfera S³ quale varietà quantica di contatto. La relazione bilineare caratteristica della teoria KS ed i vettori colonna della matrice KS risultano intimamente legati alla struttura di contatto.

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Presented at the VI Congresso Nazionale dell'Associazione Italiana di Meccanica Teorica ed Applicata (AIMETA), Genova, October 1982. Work performed under the auspices of G.N.F.M. of the C.N.R. (Consiglio Nazionale delle Ricerche).     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.     

Thermal and diffusive relaxation of an evaporating drop with internal heat liberation

The problem of nonsteady-state evaporation or growth of a radiating drop with uniformly distributed internal heat sources is considered. The Reynolds R=ua/v 1 and Peclet P_D= ua/D 1 numbers are assumed to be small (a is the radius of the drop, u the velocity of its relative motion, andv, D, the coefficients of viscosity, diffusion and thermal diffusivity of the vapor-gas medium). This enables the convective transfer of vapor and heat to be neglected, and the concentration and temperature fields to be regarded as spherically symmetric [1]. In view of the fact that the density of saturated vapor is less than the density of liquid the convective flow caused by the change in radius of the drop is not taken into account [2]. It has already been shown [3,4], that for _r (, _r are the coefficients of molecular and radiative thermal conductivity) there exists a bounded region ryo (1/) /_r ( is the absorption coefficient for radiation in the gas), in which the effect of radiation on the temperature relaxation of the vapor-gas medium is negligible. If the conditiona (1/) /_r is satisfied, then the temperature at the outer boundary of this region will be practically the same as the temperature at infinity T=T. This means that terms in the energy equation connected with energy transferred by radiation can be neglected. It is assumed that the free path of molecules in the gas is less than the radius of the drop, and so concentration and temperature discontinuities close to the surface of the drop can be neglected [2].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 78–87, January–February, 1970.The authors are grateful to V. G. Levich for discussing the results of the paper.     

On mean rotations in finite deformations

Summary Firstly, the classical interpretation of the mean rotation tensor of Cauchy and Novozhilov is revised. Indeed, two distinctive and defective features of this interpretation reflect a severe restriction on the class of admissible deformations. Secondly, an alternative measure of mean rotation is introduced and its explicit aspect for pure rotations, pure strains and additively pure rotations is determined.

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Sommario In questa nota viene riesaminata l'interpretazione classica del tensore di rotazione media di Cauchy e Novozhilov. Preliminarmente, si mostra come due tratti distintivi di questa interpretazione ne limitino l'ampiezza e riflettano una severa restrizione della classe delle deformazioni ammissibili. Successivamente, si perviene ad una misura alternativa di rotazione media, il cui calcolo esplicito è condotto nei casi di rotazioni pure, deformazioni pure e rotazioni additivamente pure.

     

Instability of a plane axisymmetric flow with a shear discontinuity in the velocity profile

It is proposed to investigate the stability of a plane axisymmetric flow with an angular velocity profile (r) such that the angular velocity is constant when r < r_O – L and r > r_O + L but varies monotonically from ₁ to ₂ near the point r_O, the thickness of the transition zone being small L r_O, whereas the change in velocity is not small ¦₂ – ₁¦ ₂, ₁. Obviously, as L O short-wave disturbances with respect to the azimuthal coordinate (k=m/r_O 1/r_O) will be unstable with a growth rate-close to the Kelvin—Helmholtz growth rate. In the case L=O (i.e., for a profile with a shear-discontinuity) we find the instability growth rate _O and show that where the thickness of the discontinuity L is finite (but small) the growth rate does not differ from _O up to terms proportional to kL 1 and 1/m 1. Using this example it is possible to investigate the effect of rotation on the flow stability. It is important to note that stabilization (or destabilization) of the flow in question by rotation occurs only for three-dimensional or axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–114, January–February, 1985.     

A simplified flood routing model: variable prameter muskingum (VPM)

Summary Flood routing methods are numerical methods for estimating the movement of a flood wave along a channel reach, on the basis of the knowledge of the discharge hydrograph at the upstream end and of the hydraulic characteristics of the reach and, usually, in the hypothesis that no perturbation is coming from downstream (free boundary condition). The flood routing method wich is proposed is similar to the Muskingum one, but with variable and hydraulic parameters; it is able to estimate water levels too; is effective even if kinetic terms are not completely negligible; take advantage of the insignificance of the downstream condition and make it possible to obtain results starting upstream and proceeding downstream; for simplicity's sake, take advantage of the fact that the discharge loop of normal flood waves is quite small. Obtained results are much better that those obtainable from constant parameters methods and indeed, if the flood loop is less that 10%, very similar to those obtainable from more complex and time consuming models.

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Sommario I metodi di flood routing studiano la propagazione di un'onda di piena lungo un tratto di un corso d'acqua, assegnato l'andamento temporale della portata nella sezione di monte e le caratteristiche dell'alveo, e usualmente nell'ipotesi di assenza di perturbazioni provenienti da valle (condizione di valle passiva). Viene qui proposto un procedimento di flood routing, formalmente simile ad un Muskingum ma con i parametri variabili e calcolati per via idraulica; idoneo a stimare anche i livelli idrici; valido anche se i termini cinetici non sono del tutto trascurabili; che sfrutta l'irrilevanza della condizione di valle procedendo a cascata da monte a valle; che sfrutta, a vantaggio della semplicità, il fatto che per le normali onde di piena dei corsi d'acqua il cappio di portata è di dimensioni modeste. I risultati ottenuti sono molto migliori di quelli ottenibili con metodi a parametri costanti e, almeno per i casi in cui il cappio relativo è inferiore al 10%, paragonabili a quelli ottenuti con metodi molto più complessi ed onerosi.

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List of symbols x, t channel distance, starting upstream; time - z water surface height above datum - Q volumetric rate of discharge - P(x, z) steady rating curve - q=Q–P flood loop - g acceleration of gravity - A, B cross section wetted area and free surface width - I, S water surface slope and friction slope - c kinematic wave velocity - F Froude number - L,L ₀,L ₁,L ₂,L ₃ characteristic lengths of the channel - T=L/c characteristic time of the channel - D diffusion - p, l time and space steps - K, X Muskingum parameters - C ₁,C ₂,C ₃,C ₄ Muskingum coefficients - f _x=f/x,f _t=f/t etc. for the partial derivatives Paper presented at the First Italian Meeting of Computational Mechanics held in Milan, June 24–26, 1986.     

Some problems concerning nonstationary currents in dense plasma systems confined by walls

Nonstationary currents are examined in a dense magnetized plasma with 1, in which energy release and heat loss by thermal conduction and radiation are possible. Solutions are found in two limiting cases: ¦f¦ ¦ div (T)¦ and ¦f¦ ¦ div(T)¦ (f is the radiation intensity, is the coefficient of heat conduction, and T is the temperature). In the first case a solution was obtained of some problems of the cooling and heating of a plasma illustrated in part by the evolution in time of the temperature profile in the boundary layer. In the second case an isomorphic solution was found for an arbitrary dependence of the coefficient of heat conduction on the temperature, pressure, and magnetic field.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–8, January–February, 1972.The author is grateful to G. I. Budker for formulating the problem.     

The effect of background particulates on the delayed transition of a heated 9:1 ellipsoid

An experimental study was done to quantify the effects of a variety of background particulates on the delayed laminar-turbulent transition of a thermally stabilized boundary layer in water. A Laser-Doppler Velocimeter system was used to measure the location of boundary layer transition on a 50 mm diameter, 9:1 fineness ratio ellipsoid. The ellipsoid had a 0.15 m RMS surface finish. Boundary layer transition locations were determined for length Reynolds numbers ranging from 3.0 × 10⁶ to 7.5 × 106. The ellipsoid was tested in three different heating conditions in water seeded with particles of four distinct size ranges. For each level of boundary layer heating, measurements of transition were made for clean water and subsequently, water seeded with 12.5 m, 38.9 m, 85.5 m and 123.2 m particles, alternately. The three surface heating conditions tested were no heating, T = 10°C and T = 15°C where T is the difference between the inlet model heating water temperature, T _i, and free stream water temperature, T . The effects of particle concentration were studied for 85.5 m and 123.2 m particulates.The results of the study can be summarized as follows. The 12.5 m and 38.9 m particles has no measurable effect on transition for any of the test conditions. However, transition was significantly affected by the 85.5 m and 123.2 m particles. Above a length Reynolds number of 4 × 10⁶ the boundary layer transition location moved forward on the body due to the effect of the 85.5 m particles for all heating conditions. The largest percentage changes in transition location from clean water, were observed for 85.5 m particles seeded water.Transition measurements made with varied concentrations of background particulates indicated that the effect of the 85.5 m particles on the transition of the model reached a plateau between 2.65 particulates/ml concentration and 4.2 particles/ml. Measurements made with 123.3 m particles at concentrations up to 0.3 part/ml indicated no similar plateau.     

New results on the semidiscrete Boltzmann equation for a binary gas mixture

Summary A plane semidiscrete model of the Boltzmann equation for a binary gas mixture with molecular collisions ruled by the hard-spheres interaction potential is described. After establishing a model, a theorem demostrating the global existence of mild solutions of the initial-value problem is given and the propagation of unidimensional shock waves examined.

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Sommario Si propone un modello semidiscreto piano dell'equazione di Boltzmann per una miscela binaria con collisioni molecolari soggette al potenziale di interazione delle sfere rigide. Costruito il modello, si dà un teorema di esistenza globale di soluzioni generalizzate per il problema di Cauchy, e si analizza la propagazione di onde d'urto unidimensionali.

     

Stress-intensity factors for a semi-infinite plane crack with a wavy front

The stress-intensity factors for a semi-infinite plane crack with a wavy front are determined when the crack faces are subjected to normal and shearing tractions. The results are derived using asymptotic methods and are valid to O(²) where =A/1; A is the amplitude and is the wavelength of the wavy front. The normal and shearing tractions are in the form of line loads parallel to the crack front.The results are then used to evaluate, in a qualitative manner, the growth characteristics of a semi-infinite plance crack with a wavy front under combined mode loading. This provides a possible explanation of crack front segmentation observed experimentally.     

Gasdynamics of a plasma in a multiple-mirror magnetic trap with “point” mirrors

Equations are derived for the gasdynamics of a dense plasma confined by a multiple-mirror magnetic field. The limiting cases of large and small mean free paths have been analyzed earlier: ₀ and k, where is the length of an individual mirror machine, ₀ is the size of the mirror, and k is the mirror ratio. The present work is devoted to a study of the intermediate range of mean free paths ₀ k. It is shown that in this region of the parameters the process of expansion of the plasma has a diffusional nature, and the coefficients of transfer of the plasma along the magnetic field are calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 14–19, November–December, 1974.The authors thank D. D. Ryutov for the statement of the problem and interest in the work.     

Conditions of simulation of stagnation point heat transfer from a high-enthalpy flow

The problem of local simulation of stagnation point heat transfer to a blunt body is solved within the framework of boundary layer theory on the assumption that the simulation subsonic high-enthalpy flow is in equilibrium outside the boundary layer on the model, while the parameters of the natural flow are in equilibrium at the outer edge of the boundary layer on the body. The parameters of the simulating subsonic flow are expressed in terms of the total enthalpyH ₀, the stagnation point pressurep _w and the velocityV ₁ for the natural free-stream flow in the form of universal functions of the dimensionless modeling coefficients=R _m ^* /R _b ^* ( .<1),=V ₁/2H ₀ ( .<1) whereR _m ^* and R _b ^* are the effective radii of the model and the body at their stagnation points. Approximate conditions for modeling the heat transfer from a high-enthalpy (including hypersonic) flow to the stagnation point on a blunt body by means of hyposonic (M1) flows, corresponding to the case ²1, are obtained. The possibilities of complete local simulation of hypersonic nonequilibrium heat transfer to the stagnation point on a blunt body in the hyposonic dissociated air jets of a VGU-2 100-kilowatt induction plasma generator [4, 5] are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 172–180, January–February, 1993.     

Interaction of spheres in a viscoelastic fluid

Summary The interaction between two identical spheres of radiusa in a second-order fluid is studied, if the undisturbed flow is a general homogeneous flow. WithR the (instantaneous) distance between the sphere centers only the situationa/R 1 is considered. It turns out that it is not sufficient to know thea/R-term of the perturbation velocity, since certain contributions of the (a/R)²-terms are also needed. For two spheres sedimenting in a quiescent fluid a change of the relative position vector is predicted: the distance decreases and so does the orientation, i.e. the spheres tend to fall along their line of centers. If the motion of the individual sphere is restrained via a rigid connection (rigid dumbbell) this change of orientation implies that the dumbbell rotates until its axis is parallel to the direction of the applied force (stable orientation). In simple shear the first-order dumbbell (a/R-terms due to interaction) ultimately ends up in the plane normal to the gradient direction, independent of the rate of shear. This contrasts the behavior of a second-order dumbbell: if the symmetry axis lies in the plane of flow it will rotate around the vorticity axis at small rates of shear. Increasing the shear rate this dumbbell reaches a spinfree terminal state in which the angle between the symmetry axis and the flow direction is non-zero (although it is small). It is conjectured that for arbitrary initial orientations (not in the flow plane) the axis of the second-order dumbbell will not rotate in the Jeffrey orbits but rather show a systematic drift to become oriented parallel to the vorticity axis.

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Zusammenfassung Die Wechselwirkung zweier identischer Kugeln vom Radiusa in einer beliebigen homogenen Strömung einer Flüssigkeit zweiter Ordnung wird untersucht. MitR dem (augenblicklichen) Abstand der Kugelzentren beschränken wir uns auf die Situationa/R 1. Es zeigt sich, daß es nicht genügt, diea/R-Glieder der Störungsgeschwindigkeit zu kennen, da einige Beiträge der (a/R)²-Terme ebenso benötigt werden. Für zwei in einer ruhenden Flüssigkeit sedimentierende Kugeln wird eine Änderung der relativen Position vorausgesagt: der Abstand verkleinert sich, und das gleiche gilt für die Orientierung, d. h. die Kugeln streben die Situation, hintereinander zu fallen, an. Schränkt man die Bewegung der individuellen Kugeln durch eine starre Verbindung ein (starre Hantel), so zieht diese Orientierungsänderung eine Rotation nach sich, die die Hantelachse parallel zur Richtung der angreifenden Kraft ausrichtet (stabile Orientierung). Bei einer einfachen Scherung wandert unabhängig von der Schergeschwindigkeit die Achse einer Hantel erster Ordnung (die nur die Wechselwirkungsgliedera/R enthält) in der Ebene, deren Normale in Gradientenrichtung zeigt. Damit verhält sie sich völlig anders als eine Hantel zweiter Ordnung: Liegt bei letzterer die Symmetrieachse in der Strömungsebene, so rotiert diese bei kleinen Schergeschwindigkeiten um eine Achse senkrecht zur Strömungsebene. Bei einer Vergrößerung der Schergeschwindigkeit wird dagegen eine rotationsfreie Lage erreicht, bei der die Hantelachse unter einem kleinen Winkel zur Strömungsrichtung steht. Bei einer beliebigen Anfangsorientierung (außer in der Strömungsebene) schließen wir auf eine Wanderung der Achse einer Hantel zweiter Ordnung, bis diese parallel zur indifferenten Richtung steht.

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With 5 figures     

Motion of an evaporating drop

The quasi steady evaporation of a spherical drop with internal heat release is studied. Energy is transferred from the drop to the ambient vapor—gas medium by molecular heat conduction, convection, and radiation. The differences of the temperature and the concentration between the surface of the drop and the region far from it are assumed to be small. The Reynolds and Péclet numbers, determined, respectively, using the free stream velocity and the mass-average velocity of the vapor—gas medium on the surface of the drop, satisfy the conditions Re R, Pe P, R Re 1, P Pe 1, Re Pe 1. The aim of the paper is to investigate the influence of asymmetry of the heat flux on the drag of the evaporating drop and to establish the conditions of applicability of the model of spherically symmetric evaporation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–10, January–February, 1984.     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

The dynamic and steady shear properties of synovial fluid and of the components making up synovial fluid

Steady-shear and dynamic properties of a pooled sample of cattle synovial fluid have been measured using techniques developed for low viscosity fluids. The rheological properties of synovial fluid were found to exhibit typical viscoelastic behaviour and can be described by the Carreau type A rheological model. Typical model parameters for the fluid are given; these may be useful for the analysis of the complex flow problems of joint lubrication.The two major constituents, hyaluronic acid and proteins, have been successfully separated from the pooled sample of synovial fluid. The rheological properties of the hyaluronic acid and the recombined hyaluronic acid-protein solutions of both equal and half the concentration of the constituents found in the original synovial fluid have been measured. These properties, when compared to those of the original synovial fluid, show an undeniable contribution of proteins to the flow behaviour of synovial fluid in joints. The effect of protein was found to be more prominent in hyaluronic acid of half the normal concentration found in synovial fluid, thus providing a possible explanation for the differences in flow behaviour observed between synovial fluid from certain diseased joints compared to normal joint fluid.Nomenclature A Ratio of angular amplitude of torsion head to oscillation input signal - G Storage modulus - G Loss modulus - I Moment of inertia of upper platen — torsion head assembly - K Restoring constant of torsion bar - N ₁ First normal-stress difference - R Platen radius - S ⁽ⁱ⁾ Geometric factor in the dynamic property analysis - t ₁ Characteristic time parameter of the Carreau model - X, Y Carreau model parameters - Z () Reimann Zeta function of - Carreau model parameter - Shear rate - Apparent steady-shear viscosity - ^* Complex dynamic viscosity - Dynamic viscosity - Imaginary part of the complex dynamic viscosity - ₀ Zero-shear viscosity - ₀ Cone angle - Carreau model characteristic time - Density of fluid - Shear stress - Phase difference between torsion head and oscillation input signals - ₀ Zero-shear rate first normal-stress coefficient - Oscillatory frequency     

A Stefan-type free boundary problem in a thermal switch

Summary We study a mathematical model of thermal switch in a glass under the Joule heating. We prove the existence of solutions for the free boundary problem related to the model. This problem is a Stefan-type problem with a source depending on the free boundary and the latent heat depending on the history of the free boundary.

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Sommario Si studia un modello matematico che descrive un fenomeno di cambiamento di fase in un vetro attraversato da una corrente elettrica. In particolare si dimostra l'esistenza di soluzioni per il problema a frontiera libera connesso con il modello. Questo problema è del tipo di Stafan, con sorgente funzione della frontiera libera e con calore latente che dipende dalla storia della frontiera.

     

Effect of nose shape on hypersonic flow past slender blunt bodies at an angle of attack

We examine some characteristics of hypersonic flow past slender blunt bodies of revolution at a small angle of attack 1, where is the relative body thickness. It is shown that, within the framework of hypersonic theory, for a correct-consideration of the effect of the conditions in the transitional section between the nose and the lateral surface it is necessary, in the general case, to specify the circumferential distribution of the force effect for the nose and the mass of the gas. For small , the effect of the nose, just as in two-dimensional flows [1–4], shows up only through its drag coefficient c_x, for =0. On this basis, the similarity law [1–4] for flow past such bodies, with arbitrary form of the lateral surface and differing in the shape of the nose blunting, which is valid over the entire disturbed region, with the exception of a small vicinity of the nose, is extended to the case in question.The notation r₀ and L maximum nose radius and characteristic body length - V, M, and density, velocity, Mach number, and adiabatic exponent of the gas in the approaching stream - , V²i, and V²p density, enthalpy, and pressure - x, r, and coordinate system of the cylindrical body with its center at the transitional section between the nose and the side surface - Vu, Vv, and Vw corresponding velocity components     

Boundary kinetic effects in the problem of slow flow over an evaporating heat-conducting spherical particle

It is shown that for sufficiently large values of the thermal conductivity of the condensed phase' as compared with the thermal conductivity of the vapor (/' Kn) the effects associated with the presence of a Knudsen layer on the evaporating surface must be taken into account in order to obtain a solution of the problem of a spherical particle in a slow (Re, 1) continuum (Knudsen number Kn 1) flow of its own vapor. The drag is calculated for various types of boundary conditions on the particle surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 134–139, March–April, 1987.In conclusion the authors wish to thank V. S. Galkin and M. N. Kogan for useful discussions.     

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada