Spherical Container Research Articles

Steady rotation of a composite sphere in a concentric spherical cavity

The problem of steady rotation of a composite sphere located at the centre of a spherical container has been investigated. A composite particle referred to in this paper is a spherical solid core covered with a permeable spherical shell. The Brinkman’s model for the flow inside the composite sphere and the Stokes equation for the flow in the spherical container were used to study the motion. The torque experienced by the porous spherical particle in the presence of cavity is obtained. The wall correction factor is calculated. In the limiting cases, the analytical solution describing the torque for a porous sphere and for a solid sphere in an unbounded medium are obtained from the present analysis.

Creeping motion of a porous approximate sphere with an impermeable core in a spherical container

The creeping motion of a porous approximate sphere with an impermeable core at the instant it passes the center of a spherical container is discussed. The flow in the spherical container is governed by the Stokes equation. The flow inside the porous approximate sphere is governed by Brinkman’s equation. The boundary conditions used at the interface are stress jump condition for tangential stresses, continuity of the normal stresses and velocity components. The drag experienced by the porous approximate spherical particle is obtained. The wall correction factor is calculated. The variations of drag coefficient and wall correction factor are studied with respect to permeability, separation parameters, deformation parameters and stress jump coefficient.

Stokes flow between eccentric rotating spheres with slip regime

The steady axisymmetric flow problem of a viscous fluid contained between two eccentric spheres that rotate about an axis joining their centers with different angular velocities is considered. A linear slip of Basset-type boundary condition at both surfaces of the spherical particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in the spherical coordinate systems based on the inner solid particle and the spherical container. The boundary conditions on the particle’s surface and spherical container are satisfied by a collocation technique. Numerical results for the coupling coefficient acting on the particle are obtained with good convergence for various values of the ratio of particle-to-container radii, the relative distance between the centers of the particle and container, the slip coefficients and the relative angular velocity. In the limiting cases, the numerical values of the coupling coefficient for the solid sphere in concentric position with the container and when the particle is near the inner surface of the container are obtained, and the results are in good agreement with the available values in the literature. The variation of the coupling coefficient with respect the parameters considered are tabulated and displayed graphically.

Compressible inviscid instability of rapidly expanding spherical material interfaces

A high-order weighted essentially non-oscillatory scheme is employed to investigate the stability of a rapidly expanding material interface produced by a spherical shock tube. The flow structure is characterized by a forward moving primary shock, a backward moving secondary shock, and a spherical contact interface in-between. We consider herein the linear inviscid regime and focus on the development of the three-dimensional perturbations around the contact interface by solving a one-dimensional system of partial differential equations. Numerical simulations are performed to illustrate the effects of the contact interface's density discontinuity on the growth of the disturbances for various spherical wave numbers. In a spherical shock tube the instability is influenced by various mechanisms which include classical Rayleigh-Taylor (RT) effects, Bell-Plesset or geometry/curvature effects, the effects of impulsively accelerating the interface, and compressibility effects. Henceforth, the present instability will be referred to as non-classical RT instability to distinguish it from classical RT instability. For an extended intermediate time period, it can be shown that the small disturbances grow exponentially as in the classical RT instability. During this stage, the exponential growth rate increases with the spherical wave number, until it saturates for very large wave numbers due to the finite thickness limitation of the numerical representation of the contact interface. The results compare favorably with previous theoretical models; but indicate that in addition to compressibility, the space-time evolution of the contact interface's thickness plays a significant role. A parametric study is performed that varies the pressure and density ratios of the initial spherical container. The characteristics of the contact interface and the applicability of various instability theories is investigated for these regimes. Furthermore, varying the pressure and density ratios aids in understanding significance of compressibility effects on the instability at different operating conditions.

Transition of velocity distributions in collapsing self-gravitatingN-body systems

By means of N-body simulations, we study the evolution of gravity-dominated systems from an early relaxation to a collapse, focusing on the velocity distributions and thermodynamic properties. To simulate the dynamical evolution, we consider self-gravitating small N-body systems enclosed in a spherical container with adiabatic or semipermeable walls. It is demonstrated that in the early relaxation process, the velocity distribution is non-Gaussian and q-Gaussian, since the system is in quasiequilibrium states (here q is the Tsallis entropic parameter). Thereafter, the velocity distribution undergoes higher non-Gaussian distributions, especially when the core forms rapidly in the collapse process; i.e., q tends to be larger than that for the quasiequilibrium state, since the velocity distribution further deviates from Gaussian. However, after the core forms sufficiently, the velocity distribution gradually relaxes toward a Gaussian-like distribution. Accordingly, the velocity distribution evolves from a non-Gaussian distribution through a higher non-Gaussian distribution to a Gaussian-like distribution; i.e., the velocity distribution does not monotonically relax toward a Gaussian-like distribution in our collapse simulations. We clearly show such a transition of the velocity distribution, based not only on the Tsallis entropic parameter but also on the ratio of velocity moments. We also find that a negative specific heat occurs in a collapse process with mass and energy loss (such as the escape of stars from globular clusters), even if the velocity distribution is Gaussian-like.

Engineering Design of a Multiple-Use Spherical Explosion Containment Vessel Subjected to Internal Blast Loading From 25 kg TNT High Explosive

Explosion containment vessels have been widely used to contain the internal blast effects. In this paper, we present engineering design and experimental results of a real multiple-use spherical explosion containment vessel, which has been employed to withstand internal blast loading from 25 kg trinitrotoluene (TNT) high explosive in the past 10 years. It is shown that strain growth should be considered in the analysis and design of multiple-use explosion containment vessels, and the dynamic load factor (DLF) method with the consideration of strain growth may achieve a safe and proper design.

Time-Reversibility, Instability and Thermodynamics in N-body Systems Interacting with Long-Range Potentials

Long-range attractive potentials cause self-gravitating N-body systems to exhibit not only chaotic behavior but also peculiar features such as gravothermal catastrophe, negative specific heat and nonextensive statistical mechanics. Especially when its potential energy is significantly dominated, a system should gradually evolve from quasi-equilibrium states through collapses to core-halo states. In dynamical evolution, velocity distributions are generally expected to monotonically relax from non-Gaussian towards Gaussian (Maxwell–Boltzmann) distributions. To clarify the velocity relaxation, we numerically examine the long-term evolution of a self-gravitating N-body system enclosed in a spherical container with adiabatic walls. We found that the velocity distribution non-monotonically relaxes from a non-Gaussian distribution to a Gaussian-like distribution when a core forms rapidly through the collapse process.

Calculating energy shifts in terms of phase shifts

We clarify the relation in interacting systems, and in particular ultracold atomic gases, between energy shifts arising from interparticle interactions and scattering phase shifts. A principle result of this paper is an expression for the energy levels of an atom in a container of arbitrary shape in the presence of a short-range scatterer, generalizing the result of Lüscher to containers of shapes other than cubic. We show that, while the energy shifts for a spherical container are under many conditions proportional to the phase shift, those for a cubic container have a more complicated structure. The general relation is extended to problems of particles in traps with smoothly varying potentials, including, e.g., the interaction of a small neutral atom with a Rydberg atom. Finally, we discuss why, even though individual energy levels are very sensitive to boundary conditions, the energies of many-body systems are not.

Numerical investigations of unconstrained melting of nano-enhanced phase change material (NEPCM) inside a spherical container

This paper presents a numerical study of unconstrained melting of nano-enhanced phase change materials (NEPCM) inside a spherical container using RT27 and copper particles as base material and nano-particle, respectively. Numerical studies are performed for three different Stefan number and volume fraction of nano-particles with an initial sub-cooling of 6 °C. Transient numerical simulations are performed for axi-symmetric melting inside a sphere. The simulation results show that the nano-particles cause an increase in thermal conductivity of NEPCM compared to conventional PCM. The enhancement in thermal conductivity with a decrease in latent heat results in higher melting rate of NEPCM.

Importance of unsteady contributions to force and heating for particles in compressible flows. Part 2: Application to particle dispersal by blast waves

Particle dispersal by blast waves is an interesting phenomenon. A model problem, i.e., a sudden release of a compressed gas–particle mixture contained in a spherical container, is employed to investigate the fundamental physics of particle dispersal. The problem is simulated by the multiphase flow models proposed in Part 1 of this article that include unsteady contributions in momentum and energy exchange between gas and particles. At early times, when particles are accelerated in the expansion fan, unsteady force and heating contributions are much larger than the corresponding quasi-steady contributions. Consequently, neglecting unsteady contributions leads to significant errors in particle front location (the boundary of the particle cloud). The complex wave interactions in the flow have strong influence on the particle motion. As a result, the particle motion is a non-monotonic function of particle density or diameter and the evolution of particle concentration is non-uniform and unsteady.

Damage detection in elevated spherical containers partially filled with liquid

Damage detection through changes in the dynamic properties has received considerable attention in recent years. However, approaches in structures supporting tanks partially filled with liquid are scarce in the technical literature. In this paper, a numerical–experimental study of damage detection in coupled fluid–structure elevated spherical tank systems is presented. The main objective is to investigate the feasibility to detect structural damage in the support structure by monitoring changes in natural frequencies. The major difficulty arises due to the changes in natural frequencies when the liquid level varies. Thus, in order to gain insight into the dynamical behaviour of the spherical containers and distinguish between the frequency shift caused by container filling conditions or by structural damage, experimental free vibration tests with small vibration amplitudes on a scaled spherical tank model are performed. The dependency of the identified frequencies on the structural damage severity is studied assuming three increasing levels of damage in the support structure. The results indicate that it is possible to detect structural damage, with acceptable confidence, up to liquid filling level of 30%. Moreover, only the “associated structural frequency” reflects the structural damage with a perceptible drop. Next, a numerical model of a real spherical container that takes into account the coupling between fluid and structure is presented to demonstrate the usefulness and validity of the results.

Seismic reliability of spherical containers retrofitted by means of energy dissipation devices

For the purpose of enhancing the seismic reliability of structures, external energy dissipation systems may be advantageously used. In this paper, the reliabilities of a spherical storage tank in original and updated states are assessed by means of simulation. The effect of the energy dissipation system is numerically evaluated by performing a set of nonlinear dynamic analyses on a spherical tank based on a detailed model that takes the fluid–structure interaction into account. To obtain robust estimators of the reliability, an ensemble including acceleration records with markedly different characteristics is used in the simulation study. The full reliability analysis shows that, for a typical spherical container widely used in the petrochemical industries, a reduction of the limit state probability in the order of 85% may be expected by introducing additional energy dissipation systems.

Influence of the block hydrophilicity of AB2 miktoarm star copolymers on cluster formation in solutions

We investigated the formation of various micelle shapes of lipid-like amphiphilic AB(2) miktoarm star copolymers in a solution, by performing dissipative particle dynamics simulations. AB(2) miktoarm star copolymer molecules are modeled with coarse-grained structures that consist of a relatively hydrophilic head (A) group with a single arm and a hydrophobic tail (B) group with double arms. A decrease in the hydrophilicity of the head group leads to a reduction of the polymer-solvent contact area, causing cluster structure changes from spherical micelles to vesicles. Consequently, a spherical exterior with multi-lamellar or cylindrical phase interior structures forms under poor solvent conditions without the introduction of spherical hard-wall containers. Furthermore we observed that, for small head group lengths, vesicles were formed in much wider range of solvent-head interaction strength than for long head groups, indicating that molecules with short head group offer a superior vesicle forming property. A phase diagram, the structure and kinetics of the cluster formation, a density profile, and a detailed shape analysis are presented to discuss the molecular characteristics of potential candidates for drug carriers that require superior and versatile vesicle forming properties. We also show that, under certain solvent-hydrophilic head group interaction conditions, initially formed cylindrical micelles transform to bilayer fragments through redistribution of copolymers within the cluster.

The inaccuracy of heat transfer characteristics for non-insulated and insulated spherical containers neglecting the influence of heat radiation

In this investigation, the differences of heat transfer characteristics for insulated and non-insulated spherical containers between considering and neglecting the influence of heat radiation are studied by the simulations in some practical situations. It is found that the heat radiation effect cannot be ignored in conditions of low ambient convection heat coefficients (such ambient air) and high surface emissivities, especially for the non-insulated and thin insulated cases. In most practical situations when ambient temperature is different from surroundings temperature and the emissivity of insulation surface is different from that of metal wall surface, neglecting heat radiation will result in inaccurate insulation effect and heat transfer errors even with very thick insulation. However, the insulation effect considering heat radiation will only increase a very small amount after some dimensionless insulated thickness (such insulation thickness/radius ≧ 0.2 in this study), thus such dimensionless insulated thickness can be used as the optimum thickness in practical applications. Meanwhile, wrapping a material with low surface emissivity (such as aluminum foil) around the oxidized metal wall or insulation layer (always with high surface emissivity) can achieve very good insulated effect for the non-insulated or thin insulated containers.

On fluid flows in precessing spheres in the mantle frame of reference

We investigate, through both asymptotic and numerical analysis, precessionally driven flows of a homogeneous fluid confined in a spherical container that rotates rapidly with angular velocity Ω and precesses slowly with angular velocity Ωp about an axis that is fixed in space. The precessionally driven flows are primarily characterized by two dimensionless parameters: the Ekman number E providing the measure of relative importance between the viscous force and the Coriolis force, and the Poincaré number Po quantifying the strength of the Poincaré forcing. When E is small but fixed and |Po| is sufficiently small, we derive a time-dependent asymptotic solution for the weakly precessing flow that satisfies the nonslip boundary condition in the mantle frame of reference. No prior assumption about the spatial-temporal structure of the precessing flow is made in the asymptotic analysis. A solvability condition is derived to determine the spatial structure of the precessing flow, via a selection from a complete spectrum of spherical inertial modes in the mantle frame. The weakly precessing flow within the bulk of the fluid is characterized by an inertial wave moving retrogradely. Direct numerical simulation of the same problem in the same frame of reference shows a satisfactory agreement between the time-dependent asymptotic solution and the nonlinear numerical simulation for sufficiently small Poincaré numbers.

Thermodynamic properties of an evaporation process in self-gravitatingN-body systems

By means of N -body simulations, we consider self-gravitating open systems enclosed in a spherical container with semipermeable reflecting walls, in order to investigate the thermodynamics of the evaporation process in self-gravitating N -body systems (such as the escape of stars from globular clusters). To simulate the evaporation process, when the energy of a particle exceeds a certain threshold value, the particle passes through the semipermeable reflecting wall freely. We show that the thermodynamic properties of the evaporation process, such as the dependence of the temperature on energy, agree well with those of stellar polytropes, if the system is in an approximate virial equilibrium state. However, in a lower-energy region or for a rapid evaporation process, the thermodynamic properties deviate from those for the stellar polytrope. Nevertheless, we found that a negative specific heat occurs even in the lower-energy region or for a rapid evaporation process.

Effects of Insulated and Isothermal Baffles on Pseudosteady-State Natural Convection Inside Spherical Containers

The effects of insulated and isothermal thin baffles on pseudosteady-state natural convection within spherical containers were studied computationally. The computations are based on an iterative, finite-volume numerical procedure using primitive dependent variables. Natural convection effect is modeled via the Boussinesq approximation. Parametric studies were performed for a Prandtl number of 0.7. For Rayleigh numbers of 104, 105, 106, and 107, baffles with three lengths positioned at five different locations were investigated (120 cases). The fluid that is heated adjacent to the sphere rises replacing the colder fluid, which sinks downward through the stratified stable thermal layer. For high Ra number cases, the hot fluid at the bottom of the sphere is also observed to rise along the symmetry axis and encounter the sinking colder fluid, thus causing oscillations in the temperature and flow fields. Due to flow obstruction (blockage or confinement) effect of baffles and also because of the extra heating afforded by the isothermal baffle, multi-cell recirculating vortices are observed. This additional heat is directly linked to creation of another recirculating vortex next to the baffle. In effect, hot fluid is directed into the center of the sphere disrupting thermal stratified layers. For the majority of the baffles investigated, the Nusselt numbers were generally lower than the reference cases with no baffle. The extent of heat transfer modification depends on Ra, length, and location of the extended surface. With an insulated baffle, the lowest amount of absorbed heat corresponds to a baffle positioned horizontally. Placing a baffle near the top of the sphere for high Ra number cases can lead to heat transfer enhancement that is linked to disturbance of the thermal boundary layer. With isothermal baffles, heat transfer enhancement is achieved for a baffle placed near the bottom of the sphere due to interaction of the counterclockwise rotating vortex and the stratified layer. For some high Ra cases, strong fluctuations of the flow and thermal fields indicating departure from the pseudosteady-state were observed.

Nonequilibrium process of self-gravitating N-body systems and quasi-equilibrium structure using normalized q-expectation values for Tsallis' generalized entropy

To clarify the nonequilibrium processes of self-gravitating systems, we examine a system enclosed in a spherical container with reflecting walls, by N-body simulations. To simulate nonequilibrium processes, we consider loss of energy through the reflecting wall, i.e., a particle reflected at a non-adiabatic wall is cooled to mimic energy loss. We also consider quasi-equilibrium structures of stellar polytropes to compare with the nonequilibrium process, where the quasi-equilibrium structure is obtained from an extremum-state of Tsallis' entropy. Consequently, we numerically show that, with increasing cooling rates, the dependence of the temperature on energy, i.e., the ϵ-Ť curve, varies from that of microcanonical ensembles (or isothermal spheres) to a common curve. The common curve appearing in the nonequilibrium process agrees well with an ϵ-Ť curve for a quasi-equilibrium structure of the stellar polytrope, especially for the polytrope index n ∼ 5. In fact, for n > 5, the stellar polytrope within an adiabatic wall exhibits gravothermal instability [Taruya, Sakagami, Physica A, 322 (2003) 285]. The present study indicates that the stellar polytrope with n ∼ 5 likely plays an important role in quasi-attractors of the nonequilibrium process in self-gravitating systems with non-adiabatic walls.

Reconstruction of Aperture Functions during Full Fusion in Vesicular Exocytosis of Neurotransmitters

Individual vesicular exocytosis of adrenaline by dense core vesicles in chromaffin cells is considered here as a paradigm of many situations encountered in biology, nanosciences and drug delivery in which a spherical container releases in the external environment through gradual uncovering of its surface. A procedure for extracting the aperture (opening) function of a biological vesicle fusing with a cell membrane from the released molecular flux of neurotransmitter as monitored by amperometry has been devised based on semi-analytical expressions derived in a former work [C. Amatore, A. I. Oleinick, I. Svir, ChemPhysChem 2009, 10, DOI: 10.1002/cphc.200900646]. This precise analysis shows that in the absence of direct information about the radius of the vesicle or about the concentration of the adrenaline cation stored by the vesicle matrix, current spikes do not contain enough information to determine the maximum aperture angle. Yet, a statistical analysis establishes that this maximum aperture angle is most probably less than a few tens of degrees, which suggests that full fusion is a very improbable event.

Resonant frequencies in an elevated spherical container partially filled with water: FEM and measurement

In this paper, a numerical-experimental study of the overall dynamical response of elevated spherical tanks subjected to horizontal base motion is presented. The main objective is to gain insight in the physical response of this particular structural typology widely used in the petrochemical industry as liquefied petroleum gas (LPG) containers. In order to identify the natural frequencies of the modes that mainly contribute to the response, experimental free vibration tests on an elevated spherical tank model for different liquid levels were carried out. Next, a numerical model that takes into account the coupling between fluid and structure was developed and validated against the experimental results. A very good agreement between experimental and numerical results was obtained. The results obtained show the influence of liquid levels on natural frequencies and indicate that the sloshing has a significant effect on the dynamical characteristics of the analyzed system. In order to obtain a good representation of the overall dynamical behaviour of the system by means of a simplified lumped mass model, a minimum of three masses is suggested. Finally, appropriate names of these three masses are proposed in the present paper.