Knotted periodic orbits in suspensions of Smale's horseshoe: Torus knots and bifurcation sequences

We construct a suspension of Smale's horseshoe diffeomorphism of the two-dimensional disc as a flow in an orientable three manifold. Such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the Duffing equation. From this suspension we construct a knot-hòlder or template—a branched two-manifold with a semiflow—in such a way that the periodic orbits are isotopic to those in the full three-dimensional flow. We discuss some of the families of knotted periodic orbits carried by this template. In particular we obtain theorems of existence, uniqueness and non-existence for families of torus knots. We relate these families to resonant Hamiltonian bifurcations which occur as horseshoes are created in a one-parameter family of area preserving maps, and we also relate them to bifurcations of families of one-dimensional quadratic like maps which can be studied by kneading theory. Thus, using knot theory, kneading theory and Hamiltonian bifurcation theory, we are able to connect a countable subsequence of one-dimensional bifurcations with a subsequence of area-preserving bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary. One implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area-preserving family: there are no universal routes to chaos!     

Effect of angular inertia of lubricant in MHD hydrostatic thrust bearings

Summary Circumferential motion of a conducting lubricant in a hydrostatic thrust bearing is caused either by the angular motion of a rotating disk or by the interaction of a radial electric field and an axial magnetic field. Under the assumption that the fluid inertia due to radial motion is negligibly small in comparison with that due to angular motion, it is found analytically that the rotor causes an increase in flow rate and a decrease in load capacity, while both are increased by the application of an electric field in the presence of an axial magnetic field. The critical angular speed of the rotor at which the bearing can no longer support any load is obtained, and the possibility of flow separation in the lubricant is discussed.Nomenclature a recess radius - b outside disk radius - B ₀ magnetic induction of uniform axial magnetic field - E ₀ radial electric field at r=a - E _r radial electric field - h half of lubricant film thickness - M Hartmann number = (B ₀ ² h ²/)^1/2 - P pressure - P ₀ pressure at r=a - P _e pressure at r=b - Q volume flow rate of lubricant - Q ₀ flow rate of a nonrotating bearing without magnetic field - r radial coordinate - r _s position of flow separation on stationary disk - u, v fluid velocity components in radial and circumferential directions, respectively - W load carrying capacity of bearing - W ₀ load capacity of a nonrotating bearing without magnetic field - z axial coordinate - coefficient of viscosity - _e magnetic permeability - fluid density - electrical conductivity - electric potential - angular speed of rotating disk - _c critical rotor speed at which W=0     

Generalized Rouse model for a dilute solution of clustered polymers

A theory analogue to tha of Rouse is given, to describe the rheological behavior of dilute solutions consisting of clusters of crosslinked polymers. The frequency-dependent behavior of the dynamic moduli of these fluids differs substantially from that of the well-known Rouse-like fluid (GG^1/2). In our case the storage modulus becomes proportional to ^3/2, while the loss modulus is proportional to . The loss modulus dominates the dynamic behavior for frequencies smaller than the largest normal frequency of the clusters.     

Unsteady flow of polymer melts: Polypropylene

Summary The unsteady flow effect has been studied experimentally in cylindrical capillaries for polypropylene melts with melt indices differing over an approximately 30-fold range. The entrance losses were negligibly small. The flow curves obtained on capillaries of different diameters, coincided, indicating the absence of any considerable near-wall slippage; hence the attainment of unsteady flow is not necessarily due to or accompanied by wall slippage.Two critical regimes can be distinguished distinctly in the flow of polypropylene melts, the first corresponding to the appearance of mattness on the extrudate surface, and the second to the appearance of defects such as a regular spiral with a constant longitudinal pitch. No unsteady flow of the melt fracture type was observed.The shear stresses in the first critical flow regime increase by 40 per cent when the temperature is raised from 180 to 240 °C. Within the same temperature range the stresses corresponding to attainment of the second critical flow regime change by 25 per cent.The critical flow parameters of polypropylene melts grow with increasing capillary length-to-diameter ratio, this effect not being damped even with big capillary lengths.The elastic deformations corrrsponding to attainment of the first critical flow regime of polypropylene melts_e2.7, while that corresponding to the second critical regime _e 3.3 with considerably changed critical stress values.

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Zusammenfassung Der Effekt des nicht-stationären Fließens wurde experimentell in zylindrischen Kapillaren für Polypropylen-Schmelzen mit Schmelz-Indizes zwischen 1 und 30 untersucht. Die Eintrittsverluste waren vernachlässigbar klein. Die Fließkurven, die bei Kapillaren verschiedenen Durchmessers ermittelt wurden, stimmten überein und zeigten das Fehlen irgendwelcher Gleitung in der Nähe der Wand. Demnach ist das Entstehen eines nicht-stationären Fließens nicht notwendigerweise abhängig bzw. begleitet von einer Wandgleitung.Zwei kritische Bereiche können eindeutig beim Fließen von Polypropylen-Schmelzen unterschieden werden: der erste im Zusammenhang mit dem Auftreten einer matten Extrudat-Oberfläche und der zweite mit dem Auftreten von Fehlern wie einer regelmäßigen Spirale mit konstanter Steigung. Ein nicht-stationäres Fließen vom Schmelzbruchtyp wurde nicht beobachtet.Die Schubspannung in dem ersten kritischen Fließbereich wächst bei einem Temperaturanstieg von 180 auf 240 °C um 40%. Im gleichen Temperaturbereich wachsen die Spannungen bei Erreichen des zweiten kritischen Fließgebietes um 25%.Die kritischen Fließparameter von Polypropylen-Schmelzen wachsen mit steigendem Verhältnis Kapillarlänge zu Durchmesser, wobei dieser Effekt auch durch große Kapillarlängen nicht gedämpft wird. Die elastischen Deformationen, die bei Erreichen des ersten kritischen Fließgebietes von Polypropylen-Schmelzen auftreten, betragen _e 2,7, während für das zweite kritische Gebiet bei deutlich geänderten kritischen Spannungswerten sich Deformationen von _e 3,3 ergeben.

     

Stabilization of bubble-liquid processes by an electric field

The stability of bubble-liquid and sedimentation processes in the presence of an electric field is investigated. To describe such processes in the case of polarizable particles or bubbles in a dielectric liquid in the presence of an electric field, the multivelocity model of a multiphase mixture interacting with an electric field [1] has been used. Gogosov, Naletova, and Shaposhnikova [2] have shown that a transition can take place to a regime of uniform ascent or sinking of the particles in the field of a planar capacitor. In the present paper, a criterion of stability of such a regime in an electric field is derived. It is shown that if the electric field is sufficiently strong (stronger than a certain critical E_cr) the regime is stable. The stability arises because in a mixture of polarizable phases in the presence of an electric field the difference between the permittivities results in a force F which acts on the disperse phase. In the one-dimensional case, under certain conditions, this force is proportional to the gradient of the volume concentration of the particles and has the opposite direction (F = -² ). Thus, this force, like the pressure gradient in a gas, tends to smooth perturbations of the density, velocity, and other parameters. It is found that if the electric field is absent or sufficiently weak the processes of uniform ascent and sinking are unstable with respect to small perturbations. A formula is derived for calculating the thickness of a liquid layer in which weak perturbations do not succeed in increasing their amplitude appreciably (the case when the flow parameters are such that the motion is unstable with respect to small perturbations). The thicknesses of this layer for different mixtures are given. It is shown that a magnetic field has a stabilizing influence on the considered processes in a magnetic liquid or in magnetizable particles in an ordinary liquid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 5–12, July–August, 1982.     

Particle mixing and diffusion in the turbulent wake of cylinder arrays

The collection of 10 DOP droplets from an aerosol flow around an array of five circular cylinders in a turbulent air stream was measured. Axial and angled alignments of the target array, normal to the flow, were examined. Cylinder Reynolds numbers ranged from 1,300 to 5,100, and effective Stokes numbers from 0.50 to 1.58. A continuous line source turbulent diffusion model was used with the centerline velocity and concentration distributions to derive the eddy diffusion coefficients for momentum and particle concentration in the near wake of the cyinders. Angled collection results showed an optimal target alignment at the edge of the momentum wake of the lead cylinder. Further examination with an LDA system suggests the peak in collection is due to a particle velocity surplus in the wake.List of symbols A _k projected area of cylinder k - A _t cross sectional area of flow - b wake width - C particle concentration - C ₁ concentration deficit - C _D.n drag coefficient of nth cylinder - d diameter - vortex shedding frequency - L _c length of cylinder - n cylinder rank - Q strength of line source - R _L Lagrangian time autocorrelation coefficient - Re Reynolds number (U d/) - T _L Lagrangian integral time scale - U x-directed velocity - U ₁ velocity deficit - u fluctuating axial velocity component - v fluctuating lateral velocity component - V drop volume - x intercylinder spacing - y lateral displacement - mixing length proportionality constant - A measured droplet diameter - eddy diffusivity - ^* corrected collection efficiency - contact angle of droplet on surface - viscosity of air - density - time - non-Stokesian drag correction factor - c cylinder - g gas - M momentum - p particles - theoretical (Stokes efficiency) - aneth freestream value - 1, n cylinder rank 1 or n     

Jet characteristics in confined swirling flow

Jets in confined swirling flow are investigated in a facility where the swirling flow in the tube is produced by a vane-type swirler. The jet is located centrally in the swirler, and the diameter ratio of the tube to the jet is 14. Both the jet and the swirling flow are fully turbulent. Results show that the confined jet is highly dissipative in nature. Consequently, the flow in the tube does not resemble a free jet with axial pressure gradient. The presence of swirl increases the rate of dissipation and the jet decays even faster. A fairly isotropic turbulence field is observed in the confined swirling flow. However, the introduction of the jet does not significantly affect this behavior and near isotropy of the turbulence field is again observed at 30 jet diameters downstream.     

Hydrodynamic models as applied to the investigation of magnetic plasma stability

In the present paper magnetohydrodynamic models are employed to investigate the stability of an inhomogeneous magnetic plasma with respect to perturbations in which the electric field may be regarded as a potential field (rot E 0). A hydrodynamic model, actually an extension of the well-known Chew-Goldberg er-Low model [1], is used to investigate motions transverse to a strong magnetic field in a collisionless plasma. The total viscous stress tensor is given; this includes, together with magnetic viscosity, the so-called inertial viscosity.Ordinary two-fluid hydrodynamics is used in the case of strong collisions=. It is shown that the collisional viscosity leads to flute-type instability in the case when, collisions being neglected, the flute mode is stabilized by a finite Larmor radius. A treatment is also given of the case when epithermal high-frequency oscillations (not leading immediately to anomalous diffusion) cause instability in the low-frequency (drift) oscillations in a manner similar to the collisional electron viscosity, leading to anomalous diffusion.Notation f particle distribution function - E electric field component - H₀ magnetic field - density - V particle velocity - e charge - m, M electron and ion mass - _i, _e ion and electron cyclotron frequencies - viscous stress tensor - P pressure - r_i Larmor radius - P pressure tensor - t time - frequency - T temperature - collision frequency - collision time - j current density - _i, _e ion and electron drift frequencies - k_x, k_y, k_z wave-vector components - n₀ particle density - g acceleration due to gravity. The authors are grateful to A. A. Galeev for valuable discussion.     

Existence of fast traveling waves for some parabolic equations: A dynamical systems approach

We study semilinear elliptic equationsu + cu _x =f(u,u) and ² u + cu _x =f(u,u, ² u) in infinite cylinders (x,y) × ⁿ⁺¹ using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limitc, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips.     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

Experimental analysis of scaling laws in low Re λgrid-generated turbulence

An experimental investigation of scaling laws in turbulence generated by a biplane grid for low Reynolds numbers (Re ) is presented. The present study covers a wide range of flow field conditions (from Re 2.5 to Re 36) that have not been analyzed from this point of view. It is shown that following the Kolmogorov theory a scaling range can not be observed since the separation between the energy production scales and the dissipative scales is too short. On the other hand, an extended form of scaling, the Extended Self-Similarity (ESS), permits the identification of a range of scales characterized by the same scaling exponent much wider than the one previously examined. Thus the scaling laws for the first six moments of the velocity structure function are accurately calculated by means of the ESS and an anomalous scaling with respect to the Kolmogorov theory is observed for Re down to the order of 10. As a matter of fact the scaling exponents are in good agreement with the ones that were determined at higher Re by previous experimental and numerical investigations. For Re 6 a regularization of the scaling exponents is observed as an effect of the dissipation. We also present an analysis of the universality properties of the ESS form of scaling by means of the form function and an analysis of the sensitivity of the scaling range to the Re .     

New concepts in relative permeabilities at high capillary numbers for surfactant flooding

Flooding oil reservoirs with surfactant solutions can increase the amount of oil that can be recovered. Macroscopic modelling of the process requires relative permeabilities to be functions of saturation and capillary number. With only limited experimental data, relative permeabilities have usually been assumed to be linear functions of saturation at high capillary numbers. The experimental data is reviewed, some of which suggest that this assumption is not necessarily correct. The basis for the assumption is therefore reviewed and it is concluded that the linear model corresponds to microscopically segregated flow in the porous medium. Based on new but equally plausible complementary assumptions about the flow pattern, a mixed flow model is derived. These models are then shown to be limiting cases of a droplet model which represents the mixing scale within the porous medium and gives a physical basis for interpolating between the models. The models are based on physical concepts of flow in a porous medium and so the approach described here represents a significant improvement in the understanding of high capillary number flow. This is shown by the fact that fewer parameters are needed to describe experimental data.Notation A total cross-sectional area assigned to capillary bundle - A ⁽ⁱ⁾ physical cross-sectional area of tube i - c ⁽ⁱ⁾ ordered configurational label for droplets in tube i - c configuration label for tube i (order not considered) - D defined by Equation (26) - E(...) expectation value with respect to the trinomial distribution - S _r ⁽⁾ fractional flow of phase - k absolute permeability - k _r relative permeability of phase - k _r ⁰ endpoint relative permeability of phase - L capillary tube length in bundle model - m ⁽ⁱ⁾ number of droplets of phase a occupying tube i - n exponent for phase a in Equation (2) - N number of droplets in bundle model - N _c capillary number - p pressure - p(c') probability of configuration c - Q ⁽ⁱ⁾ total volume flow rate in tube i - S saturation of phase - S flowing saturation of phase - S _r residual saturation of phase - S _r ⁽⁾ saturations when fractional flow of phase is 1 in the case of varying residual saturations for three-phase flow ( ) - t _c residence time for droplet configuration c - v ⁽ⁱ⁾ total fluid velocity in bundle tube i - , phase label - p pressure differential across capillary bundle - ⁽ⁱ⁾ tube conductivity defined by Equation (7) - viscosity of phase - interfacial tension - gradient operator - ... average over tube droplet configurations     

Flow visualization and analysis of aerodynamically detuned supersonic rotors

The effect of aerodynamic detuning on the supersonic steady and unsteady blade passage flow field is experimentally investigated on a free surface water table by means of color Schlieren and shadowgraph flow visualization techniques. Two aerodynamic detuning mechanisms are considered: (1) alternate circumferential spacing of adjacent airfoils; (2) the replacement of alternate airfoils with splitters. The steady flow visualization demonstrates the significant effect of aerodynamic detuning on the passage flow field and, in particular, the shock wave-airfoil surface intersection locations. The unsteady flow visualization studies show the importance of the interblade phase angle. A mathematical model is also described and utilized to demonstrate the enhanced aeroelastic stability associated with the altered cascade passage shock wave structure due to these aerodynamic detuning mechanisms.List of symbols a dimensionless perturbation sonic velocity - C airfoil chord - I _x mass moment of inertia - k reduced frequency, k = C/2 - K spring constant - M _R dimensionless unsteady aerodynamic moment - M Mach number - P _split split splitter circumferential spacing - P _start splitter leading edge location - R reference full chord airfoil - R _s reference splitter - Sp _chord splitter chord length - u dimensionless perturbation chordwise velocity - v dimensionless perturbation normal velocity - amplitude of oscillation - interblade phase angle - level of aerodynamic detuning - undamped natural torsional frequency - ₀ reference frequency - flutter frequency     

On the boundary conditions at the macroscopic level

We study the problem of the boundary conditions specified at the boundary of a porous domain in order to solve the macroscopic transfer equations obtained by means of the volume-averaging method. The analysis is limited to the case of conductive transport but the method can be extended to other cases. A numerical study enables us to illustrate the theoretical results in the case of a model porous medium. Roman Letters _sf interfacial area of the s-f interface contained within the macroscopic system m² - A _sf interfacial area of the s-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - d _s , d _f microscopic characteristic length m - g vector that maps to _s, m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l REV characteristic length, m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for the f-phase at the f-s interface - n _e outwardly directed unit normal vector at the dividing surface - T ^* macroscopic temperature field obtained by solving the macroscopic equation (3), K - V averaging volume, m³ - V _s , V _f volume of the considered phase within the averaging volume, m³ - volume of the macroscopic system, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² Greek Letters _s , _f volume fraction - ratio of thermal conductivities - _s , _f thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding to T ^* , K - spatial deviation temperature K - error on the temperature due to the macroscopic boundary conditions, K - spatial average - ^s , ^f intrinsic phase average     

Convexity conditions for strain-dependent energy functions for membranes

The forms that the convexity, polyconvexity, and rank-one convexity inequalities take when the strain energy is required to be a function of the strain G are studied. It is shown in particular that W(G) must be an increasing function of G, in the sense that W(G)W(G) if G – G is non-negative definite. Relatively simple sufficient conditions in terms of G alone are given. Necessary and sufficient conditions in terms of G alone are found to be rather complex.     

Time averaged temperature distribution in a cylindrical resistor with alternating current

The non-uniform heat generation in a cylindrical resistor crossed by an alternating electric current is considered. The time averaged and dimensionless temperature distribution in the resistor is analytically evaluated. Two dimensionless functions are reported in tables which allow one to determine the time averaged temperature field for arbitrarily chosen values of the physical properties and of the radius of the resistor, of the electric current frequency, of the Biot number and of either the power generated per unit length or the effective electric current.

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Zeitliche Temperaturverteilung in einem zylinderförmigen Wechselstromwiderstand

Zusammenfassung Es wird ungleichförmige Wärmeerzeugung in einem mit Wechselstrom belasteten Widerstand unterstellt, woraus sich die darin einstellende, zeitlich gemittelte, dimensionslose Temperaturverteilung analytisch berechnen läßt. Zwei tabellierte dimensionslose Funktionen gestatten die Bestimmung dieser Temperaturverteilung für beliebige Werte der Stoff- und Feldparameter, des Widerstandhalbmessers, der elektrischen Frequenz, der Biot-Zahl, sowie der erzeugten Leistung pro Längeneinheit oder des effektiven Stroms.

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Nomenclature A intregration constant introduced in Eq. (15) - Bi Biot numberhr ₀/ - c speed of light in empty space - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E amplitude of the electric field oscillations - electric permittivity - f function ofs and defined in Eq. (22) - function of defined in Eq. (45) - g function ofs and defined in Eq. (34) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=–1 - I electric current - I _eff effective electric currentI _eff=I/2 - Im imaginary part of a complex number - J current density - J _n Bessel function of first kind and ordern - thermal conductivity - magnetic permeability - ₀ magnetic permeability of free space - q _g power generated per unit volume - time average of the power generated per unit volume - Q time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - mass density - s dimensionless radial coordinates=r/r ₀ - s,s integration variables - electric conductivity - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - dimensionless temperature defined in Eq. (27) - x position vector - x arbitrary real variable - x integration variable - Y ₀ Bessel function of second kind and order 0 - z axial coordinate - z unit vector parallel to the axis of the cylinder - angular frequency - dimensionless parameter =r₀ - · modulus of a complex number - equal by definition     

Numerical investigation of interactions between shock waves and thermal inhomogeneities after “multipoint” energy emission

This paper discusses a possible mechanism for the various anomalous phenomena induced by spatially inhomogeneous energy emission. Two gas-dynamical models of multipoint energy emission in an ideal compressible gas have been numerically investigated using a second-order difference scheme. It is shown that qualitatively different scenarios are possible depending on the initial parameters.     

MHD-Effects Due to the Hypersonic Motion of a Cylindrical Body in the Magnetosphere of a Planet for Mutually Perpendicular Orientations of the Generatrix and Magnetic Field

MHD-effects associated with the movement of a weakly conducting cylindrical body in a planetary atmosphere are analyzed numerically. The longitudinal axis of the body is perpendicular to the direction of motion and the field lines of the planetary magnetic field. In the calculations we used data similar to those for the collision of the Shoemaker-Levy 9 comet with Jupiter. The simulation is developed for two atmospheric levels: H100 km (magnetic pressure number R _H1) and H280 km (R _H1). It is shown that the maximum strength of the induced magnetic field is 2–3 orders of magnitude greater than the planetary magnetic field. For small R _H, analytical expressions giving a fairly reliable estimate for the field strength in the shock-compressed space ahead of the body are obtained.     

Heat and mass transfer in laminar flow between parallel porous plates

Summary The problem of heat transfer in a two-dimensional porous channel has been discussed by Terrill [6] for small suction at the walls. In [6] the heat transfer problem of a discontinuous change in wall temperature was solved. In the present paper the solution of Terrill for small suction at the walls is revised and the whole problem is extended to the cases of large suction and large injection at the walls. It is found that, for all values of the Reynolds number R, the limiting Nusselt number Nu increases with increasing R.Nomenclature stream function - 2h channel width - x, y distances measured parallel and perpendicular to the channel walls respectively - U velocity of fluid at x=0 - V constant velocity of fluid at the wall - =y/h nondimensional distance perpendicular to the channel walls - f() function defined in equation (1) - coefficient of kinematic viscosity - R=Vh/ suction Reynolds number - density of fluid - C _p specific heat at constant pressure - K thermal conductivity - T temperature - x=x ₀ position where temperature of walls changes - T ₀, T ₁ temperature of walls for x<x ₀, x>x ₀ respectively - = (T – T ₁)/T ₀ – T ₁) nondimensional temperature - =x/h nondimensional distance along channel - R ^* = Uh/v channel Reynolds number - Pr = C _p/K Prandtl number - _n eigenvalues - B _n() eigenfunctions - B _n ⁽⁰⁾ , () eigenfunctions for R=0 - B ₀ ⁽ⁱ⁾ , B ₀ ⁽ⁱⁱ⁾ ... change in eigenfunctions when R0 and small - K _n constants given by equation (13) - h heat transfer coefficient - Nu Nusselt number - _m mean temperature - C _n constants given by equation (18) - perturbation parameter - B _0i () perturbation approximations to B ₀() - Q = B ₀/ ₀ derivative of eigenfunction with respect to eigenvalue - z nondimensional distance perpendicular to the channel walls - F(z) function defined by (54)     

Shell vibrations at high-frequency

The free vibration is called high-frequency when the frequency parameter is limited by the inequalities >max{R ₂ ^–2 (s)} and O(h ⁰). In this case there is only one boundary layer type of solution in the neighbourhood of any edge which is not sufficient to satisfy the two non-tangential boundary conditions to be dropped by the membrane equations at the edge, and is called non-complete.An asymptotic approach is presented in this paper, by means of which we find that there are two types of principal modes to be operative over the whole range of the shell surface, when the shell vibrates axisymmetrically at high frequency. One of the principal modes is a membrane type (¦u¦¦w¦, and the index of variation is zero) and the other is a quasi-transverse one with quick variation (¦u¦¦w¦, and the index of variation is equal to 1/2). Correspondingly, the set of frequency parameters can also be divided into two subsets, one of which corresponds to the membrane modes as their eigenvectors, while the other subset corresponds to the quasi-transverse modes with quick variation as their eigenvectors.