Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

On the Denseness of the Set of Unsolvable Cauchy Problems in the Set of All Cauchy Problems in the Case of an Infinite-Dimensional Banach Space

We prove the following statement: Theorem 1. Let E and be an arbitrary infinite-dimensional Banach space and a continuous mapping, respectively. Then, for every and > 0, there exists a continuous mapping such that

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

Effect of a Short Roughness Strip on a Turbulent Boundary Layer

The response of a turbulent boundary layer to a short roughness strip is investigated using laser Doppler velocimetry (LDV) and laser induced fluorescence (LIF). Skin friction coefficients are inferred from accurate near-wall measurements. There is an undershoot in , where is the undisturbed smooth wall skin friction coefficient, immediately after the strip. Downstream of the strip, overshoots before relaxing back to unity in an oscillatory manner. The roughness strip has a major effect on the turbulent stresses ; these quantities increase, relative to the undisturbed smooth wall, in the region between the two internal layers originating at the upstream and downstream edges of the strip. The increase in the ratio suggests a decrease in near-wall anisotropy. From the flow visualizations, it is inferred that streamwise vortical structures are weakened immediately downstream of the strip. Consistently, streamwise length scales are also reduced; direct support for this is provided by measured two-point velocity correlations.     

Global Attractors: Topology and Finite-Dimensional Dynamics

Many dissipative evolution equations possess a global attractor with finite Hausdorff dimension d. In this paper it is shown that there is an embedding X of into , with N=[2d+2], such that X is the global attractor of some finite-dimensional system on with trivial dynamics on X. This allows the construction of a discrete dynamical system on which reproduces the dynamics of the time T map on and has an attractor within an arbitrarily small neighborhood of X. If the Hausdorff dimension is replaced by the fractal dimension, a similar construction can be shown to hold good even if one restricts to orthogonal projections rather than arbitrary embeddings.     

On a Boundary-Value Problem of Antiperiodic Type for First-Order Nonlinear Functional Differential Equations of Non-Volterra Type

We establish unimprovable (in a certain sense) sufficient conditions for the solvability and unique solvability of the boundary-value problem

Geometric Analysis of One Singular Cauchy Problem

We consider the Cauchy problem , x(0) = 0, where a ₀₀₀ = 0, a ₀₀₁ = 0, and a ₀₀₂ = 0, and prove the existence of continuously differentiable solutions x(0,] with required asymptotic properties.     

The flow regime in the turbulent near wake of a hemisphere

The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols C _p pressure coefficient, - D diameter of hemisphere - n vortex shedding frequency - p pressure on model surface - p ₀ static pressure - Re Reynolds number, - St Strouhal number, - U, V, W local mean velocity components - mean freestream velocity inX direction - U ^* shear velocity, - u, v, w velocity fluctuations inX, Y andZ directions - X Cartesian coordinate in longitudinal direction - Y Cartesian coordinate in lateral direction - Z Cartesian coordinate in direction perpendicular to the wall - it* boundary layer displacement thickness, - diameter of model surface roughness - elevation angleI - O boundary layer momentum thickness, - _w wall shearing stress - dynamic viscosity of fluid - density of fluid - streamfunction - _x longitudinal component of vorticity, - _y lateral component of vorticity, - _z vertical component of vorticity, This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

Laminar source flow between parallel porous disks

An analysis is presented for laminar source flow between infinite parallel porous disks. The solution is in the form of a perturbation from the creeping flow solution. Expressions for the velocity, pressure, and shear stress are obtained and compared with the results based on the assumption of creeping flow.Nomenclature a half distance between disks - radial coordinate - r dimensionless radial coordinate, /a - axial coordinate - z dimensionless axial coordinate, /a - radial coordinate of a point in the flow - R dimensionless radial coordinate of a point in the flow, /a - velocity component in radial direction - u =a/, dimensionless velocity component in radial direction - velocity component in axial direction - v = a/}, dimensionless velocity component in axial direction - static pressure - p = (a ²/ ²), dimensionless static pressure - =p(r, z)–p(R, z), dimensionless pressure drop - V magnitude of suction or injection velocity - Q volumetric flow rate of the source - Re source Reynolds number, Q/4a - reduced Reynolds number, Re/r ² - critical Reynolds number - R _w wall Reynolds number, Va/ - viscosity - density - =/, kinematic viscosity - shear stress at upper disk - ₀ = (a ²/ ²), dimensionless shear stress at upper disk - shear stress ratio, ₀/( ₀)_inertialess - u = , dimensionless average radial velocity - u/u, ratio of radial velocity to average radial velocity - dimensionless stream function     

Transient Free Convection from a Horizontal Surface in a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q ₁ on the surface. Then, at the heat flux on the surface is suddenly changed to q ₂ and maintained at this value for . Firstly, solutions which are valid for small and large are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state ( ) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times .     

Response of a viscous annular liquid layer in zero-gravity to different axial excitations of the rigid boundary plates

A cylindrical annular liquid layer between two plates and around a rigid center-core consisting of incompressible and viscous liquid is subjected to different axial excitations, such as one-sided, counter-directional and double-sided unequal excitations. The response of the free liquid surface, the velocity- and pressure-distribution has been determined.

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Zusammenfassung Eine zylindrische Flüssigkeitsschicht bestehend aus inkompressibler und viskoser Flüssigkeit wurde verschiedenen harmonischen Anregungsformen ausgesetzt. Dabei wurden die Fälle einseitiger, doppelseitiger entgegengesetzter und ungleicher doppelseitiger Anregung mit Phase behandelt. Die Vergrößerungsfunktionen für die freie Flüssigkeitsoberfläche, für die Geschwindigkeits- und Druckverteilung wurden bestimmt.

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List of symbols a radius of liquid layer - b radius of inner cylindrical core - (a–b) thickness of layer - e _r , e , k unit vectors in the radial, angular and axial direction resp. - h length of layer - I _m , K _m modified Bessel functions of first and second kind and order m - diameter ratio - p pressure - q 2na/h - q* na/h - r, , z cylindrical coordinates - complex frequency - S sa ²/ - t time - u, w velocity components in the radial- and axial direction - ₀ excitation amplitude - abbreviation - surface tension parameter - surface tension - dynamic viscosity - kinematic viscosity - density of liquid - free liquid surface elevation - dimensionless time - _rz shear stress - reduced forcing frequency - forcing frequency - stream function - _mn natural frequency of non-viscous liquid     

Free Convection from a Vertical Plate in a Porous Medium Subjected to a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

An analysis is made of the transient free convection from a vertical flat plate which is embedded in a fluid-saturated porous medium. It is assumed that for time a steady state temperature or velocity has been obtained in the boundary-layer which occurs due to a uniform flux dissipation rate . Then at time the heat flux on the plate is suddenly changed to and maintained at this value for 0$$ " align="middle" border="0"> . An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. These effects cause a new steady boundary layer. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from to the steady state, firstly by means of a step-by-step method and then by a matching technique. The transition between the two distinct solution methods is always observed to occur very near to the turning point of the plate surface temperature, a time at which the fluid temperature is close to its steady state profile. The solution obtained using the step-by-step method shows excellent agreement with the small time analytical solution. Results are presented to illustrate the occurrence of transients from both small and large increases and decreases in the levels of existing energy inputs.     

Analysis of thermally activated formation and breakdown of Kear-Wilsdorf barriers in Ni<Subscript>3</Subscript>Ge single crystals of various orientations

The orientation dependence of the yield stress in Ni₃Ge single crystals has been examined both theoretically and experimentally. The positive temperature dependence of the yield stress in the low temperature region is attributed to formation of Kear-Wilsdorf barriers. The forces driving the formation and breakdown of barriers are calculated within the framework of the Hirsch scheme. A distinctive feature of the model proposed is that the barrier is considered on the screw component of the a/2[ 01](111) superdislocation in the primary octahedral plane. The major role in barrier formation belongs to anisotropy of elastic moduli, energy of antiphase boundaries in the octahedral plane, shear stresses in the cubic and octahedral planes, and friction-induced stress in the cubic plane. A comparison of predicted values of the driving force of barrier formation and breakdown with experimental values reveals their good agreement. An analysis of the orientation dependence of the driving force of barrier formation in the temperature range T = 77–293 K shows that the dependence (T) has an extremum for crystals deformed along the [ 39] crystallographic direction, which is confirmed experimentally.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 116–125, January–February, 2005.     

Strong Solutions for Generalized Newtonian Fluids

We consider the motion of a generalized Newtonian fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). We focus on the three dimensional case with periodic boundary conditions and extend the existence result for strong solutions for small times from \tfrac{5}{3}$$ " align="middle" border="0"> (see [16]) to \tfrac{7}{5}.$$ " align="middle" border="0"> Moreover, for we improve the regularity of the velocity field and show that for all 0.$$ " align="middle" border="0"> Within this class of regularity, we prove uniqueness for all \tfrac{7}{5}.$$ " align="middle" border="0"> We generalize these results to the case when p is space and time dependent and to the system governing the flow of electrorheological fluids as long as      

Eigenvalue Characterization of a System of Difference Equations

We consider the system of difference equations

On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale

Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.Nomenclature material system; Section 2.1. - porous body (example of a material system); Sections 2.1, 3.1, 4.1 - fluid body (example of a material system); Sections 2.1, 3.1, 4.1 - weighting function; Sections 2.1, 2.3 - ,h weighting function corresponding to spherical averaging regions of radius and boundary mollifying layer of thicknessh; Section 3.2 - Euclidean space; Section 2.1 - V space of all displacements between pairs of points in; Section 2.1 - mass density field corresponding to; (2.3)₁ - ^P , ^f mass density fields for , ; (4.1) - P momentum density field corresponding to; (2.3)₂ - v velocity field corresponding to; (2.4) - S _r (X) interior of sphere of radiusr with centre at pointx; (3.3) - boundary ofany region - region in which ^p > 0 with = ,h; (3.1) - subset of whose points lie at least+h from boundary of ; (3.4) - abbreviated versions of ; Section 3.2, Remark 4 - strict interior of ; (3.7) - analogues of for fluid system ; Section 3.2 - general version of corresponding to any choice of weighting function; (4.6) - interfacial region at scale; (3.8) - ₀ scale of nearest-neighbour separations in ; Section 3.2. Remark 1 - porosity field at scales ( ₁; ₂); (3.9) - pore space at scales ( ₁; ₂); (3.12)     

Effects of couple-stresses on the dispersion of a soluble matter in a pipe flow of blood

Summary An analysis of the effects of couple-stresses on the effective Taylor diffusion coefficient has been carried out with the help of two non-dimensional parameters based on the concentration of suspensions and , a constant associated with the couple-stresses. It is observed that the concentration distribution increases with increasing or The effective Taylor diffusion coefficient falls rapidly with increasing when is negative.

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Zusammenfassung Der Einfluß der Momentenspannungen auf den effektiven Taylorschen Diffusionskoeffizienten wird untersucht. Dabei treten zwei dimensionslose Parameter and auf: Der erste bezieht sich auf die Suspensionskonzentration, der zweite kennzeichnet die Momentenspannungen. Man findet, daß die Verteilungsgeschwindigkeit mit wachsendem oder zunimmt. Dagegen fällt der Taylorsche Diffusionskoeffizient bei wachsendem stark ab, wenn negativ ist.

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a Tube radius - C Concentration - C _i Body moment vector - C ₀ Concentration at the axis of the tube - C _m Mean concentration - D Molecular diffusion coefficient - d _ij Symmetric part of velocity gradient - F Function of and characterising effective Taylor diffusion coefficient - f _i Body force vector - H A function of and - K ₂ Integration constant - K ^* Effective Taylor diffusion coefficient - k Radius of gyration of a unit cuboid with its sides normal to the spatial axes - I _n Modified Bessel's function ofnth order - L Length of the tube over which the concentration is spread - M Function ofH and - M _ij Couple stress tensor - P Function of - p Fluid pressure - Q Volume rate of the transport of the solute across a section of the tube - r Radial distance from the axis of the tube - T _ij Stress tensor - t Time coordinate - T _ij ^A Antisymmetric part of the stress tensor - u Relative fluid velocity - Average velocity - v _i Velocity vector - Fluid velocity at any point of the tube - v ₀ ⁿ Velocity of Newtonian fluid at the axis of the tube - _i Vorticity vector - x Axial coordinate - x ₁ Relative axial coordinate - z Non-Dimensional radial coordinate - Density - _ij Symmetric part of the stress tensor - µ Viscosity of the fluid - µ _ij Deviatoric part ofM _ij - , Constants associated with couple-stress With 3 figures     

An Approximate Solution for Fluid Infiltration near a Circular Opening in Unsaturated Media

A regular perturbation technique is employed to approximate the solution for fluid infiltration from a circular opening into an unsaturated medium. Introducing two empirical constitutive relations and relating the permeability k and water content with pore fluid pressure p, a nonlinear diffusion equation in terms of pore pressure is established. After rearranging the nonlinear diffusion equation, a parameter perturbation on is performed and an approximate solution with an error of is obtained, which correlate to a condition in which = . This approximate solution is verified by a finite difference solution and compared also with a linear solution in which the diffusivity is constant. It is shown that the perturbation solution with terms up to and including first-order can give a reasonably accurate solution for the parameter range for p ₀ selected in this paper. The solution procedure provided in this paper also avoids the numerical problem normally encountered for a small time solution. The solution may also be used to overcome difficulties arising in solution procedure by the similarity transformation (Boltzmann), commonly conducted on diffusion equation, which cannot be applied for a finite wellbore problem.     

Momentum balance in highly distorted turbulent pipe flows

An in depth study into the development and decay of distorted turbulent pipe flows in incompressible flow has yielded a vast quantity of experimental data covering a wide range of initial conditions. Sufficient detail on the development of both mean flow and turbulence structure in these flows has been obtained to allow an implied radial static pressure distribution to be calculated. The static pressure distributions determined compare well both qualitatively and quantitatively with earlier experimental work. A strong correlation between static pressure coefficient Cp and axial turbulence intensity is demonstrated.List of symbols C _p static pressure coefficient = (pw-p)/1/2 - D pipe diameter - K turbulent kinetic energy - (r, , z) cylindrical polar co-ordinates. / 0 - R, y pipe radius, distance measured from the pipe wall - U, V axial and radial time mean velocity components - mean value of u - u, u/ , / - u, , w fluctuating velocity components - axial, radial turbulence intensity - turbulent shear stress - u friction velocity, (u ² = ₀/p) - ₀ wall shear stress - ^* boundary layer thickness A version of this paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984