Hollow Spherical Shell Research Articles

Electrochemical properties of spherical polyelectrolytes: II. Hollow sphere model for membranous vesicles

The nonlinearized Poisson-Boltzmann equation has been numerically solved to obtain the electrostatic potential inside and outside of hollow spherical shells suspended in an aqueous medium. The geometrical characteristics of the shells were chosen to correspond to those of the small (100 to 600 A radius) vesicles typically formed by sonication of phospholipids or membranes. The boundary conditions were specified using the following assumptions: (i) The shells are permeable to ions, permitting an electrochemical equilibrium to exist between the inner and outer solutions, (ii) Both inner and outer surfaces of the shell bear ionizable groups, but only the total (in plus out) degree of dissociation is given, (iii) The cell model for finite concentrations of vesicles was employed. The additional assumption of electroneutrality within the vesicles was used in most of the reported calculations. It is shown that for vesicles having radii in the range studied here, the computed results obtained with and without the electroneutrality constraint are nearly the same. The electrostatic potential was studied as a function of salt concentration, vesicle concentration, shell thickness, radius of shell, and surface area per ionizable group. The ionic distribution, activity coefficients, Δp K , and osmotic coefficient were obtained from the electrostatic potential. The degree of dissociation under all circumstances was found to be less on the inside than the outside. The average potential of the inside solution is larger in magnitude than that outside the shell at low salt and polyelectrolyte concentrations. At high salt concentration the potentials are similar inside and outside, whereas at high polyelectrolyte concentration, but low salt, the magnitude of the potential is larger outside than inside.

Radial oscillations of nonhomogeneous, thick-walled cylindrical and spherical shells subjected to finite deformations

The infinitesimal breathing motions of long cylindrical tubes and hollow spherical shells of arbitrary wall thickness subjected to a finite deformation field caused by uniform internal and/or external pressures are investigated. A neo-Hookean material with a material constant varying continuously along the radial direction is used. The shell is first subjected to finite static deformations and is then exposed to a secondary dynamic displacement field. Based on the theory of small deformations superposed on large deformations, closed form expressions are obtained for the frequency of small oscillations about the highly prestressed state. Frequency versus initial deformation parameter curves are given for several nohomogeneity functions and for various wall thicknesses.

Resonant oscillations of a hollow spherical shell containing liquid

Resonant oscillations of a hollow spherical shell containing liquid

Hollow hydrogen spheres for laser-fusion targets

Many laser fusion experiments require the use of hollow spherical shells of hydrogen (deuterium-tritium mixture) as targets. We have produced hollow particles of normal hydrogen in preparation for producing shells of the various hydrogen isotopes and controlled mixtures of these isotopes. The hollow spheres were produced by ultrasonically nucleating bubbles in superheated liquid drops of hydrogen. Evaporation into the bubble increased its size and produced the shells.

A note on the radial vibrations of an isotropic spherical shell

Results and a graphical appeciation for the study of the frequencies of radial vibration in a hollow spherical shell are given.

Bellman's new approach to the numerical solution of Fredholm integral equations with positive kernels

In this paper, using Bellman's new technique, we show how to solve numerically the integral equation of the source function of the radiation field in a homogeneous, isotropically scattering spherical medium or hollow spherical shell medium with no flux of radiation incident on the outer surface and with an isotropically emitting central point source. By Bellman's method, the integral equation for the source function is converted into a system of linear simultaneous, first order ordinary differential equations of the initial value type.

Large impact deformation response of spherical shells

A combination of experiment and simple theory is used to investigate the deformation response of complete, hollow spherical shells subjected at elevated temperature to high-velocity normal impact upon a planar rigid target. Temperature of the shells at impact is a few hundred degrees Fahrenheit below their melting point. A semi-inverse method of analysis is developed which is based upon energy conservation during the impact process. The method for prediction of dynamic behavior is essentially a static description of the deformation process. An assumed shell deformation pattern closely resembling the postimpact pattern obtained by test is used in formulating the external work and strain energy expressions. Consideration of one-dimension al stress wave propagation is used to develop the external work expression. Bending and stretching are included in the strain energy. A general expression for energy conservation is developed and subsequently considerably simplified by assuming the shell wall to behave as a rigid-linear strain hardening material. The average absolute difference between predictions of the analysis and test results is 8.6%.

Thermal explosion in the sphere and the hollow spherical shell heated at the inner face

The steady state heat conduction equation for reactants of spherical symmetry decomposing by a zero-order exothermic process has been converted by means of the fifth-power (quintic) approximation ▪ (where α and β are constants) into a form which is soluble analytically. The method of solution depends upon the physical model under examination: two models have been discussed, that of self ignition of a hollow spherical shell of reactant heated at the inner face and cooling to the surroundings from the outer face and the solid sphere immersed in a constant temperature bath. Approximate Frank-Kamenetsky criticality criteria for hollow spheres of varying thickness and hot face temperatures were derived analytically by a method involving the use of incomplete elliptic integrals of the first kind. The results were compared with numerical solutions of the equation wherein the exponential approxiamation had been made and found to be in good agreement. It has been shown how these results fit into the pattern established by those for the asymmetrically heated slab and the infinite hollow cylindrical shell heated at the inner face. For the first time an analytical criticality criterion (δ crit. = 3/4α 4β), critical centre temperature rise [θ mcrit. = (□2−1)(α/β)] and temperature distribution, given by ( α + β θ ) 2 = 1 ⋅ 5 α 2 [ ( 1 + z 2 ) − ( 1 − z 2 ) { 1 − ( 4 α 4 δ β / 3 ) } 1 2 ] [ δ β ( 1 − z 2 ) 2 α 4 + 3 z 2 ] have been determined for the sphere by a method in which the phenomenon of criticality emerges naturally from the analysis.

Response of an embedded spherical shell to a plane wave

The interaction of plane elastic waves with a thin hollow spherical shell surrounded by an elastic medium is analysed by assuming that each vibrating shell element causes plane waves to radiate into the surrounding medium. By using a modal approach together with the Laplace Transform method a closed form solution for the response of the dilatational mode is obtained and illustrated by a numerical example.

Contributions to the mathematical theory of organic form. II Asymmetric metabolism of cellular aggregates

The case is studied of a cellular aggregate forming a hollow spherical shell, the physical constants and the metabolic rate being non-uniform throughout the shell. This leads to asymmetries of concentration distributions, which are calculated here for some simple cases.