Spherical Sinks Research Articles

The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensor technique.

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model.

Experimental Study on the Longitudinal Motion Performance of a Spherical Robot Rolling on Sandy Terrain

To provide the necessary theoretical models of sphere–soil interaction for the structural design, motion control, and simulation of spherical robots, this paper derives analytical expressions for traction force and driving torque when spherical robots slide and sink into sandy terrain, based on terramechanics and multibody dynamics. Furthermore, orthogonal experimental analysis identifies the load, joint angular acceleration, and maximum joint angular velocity of spherical robots as influencing factors, highlighting that the load significantly affects their longitudinal motion performance. Experimental results indicate that rolling friction and additional resistance on sandy terrain cannot be ignored. The corrected theoretical model effectively replicates the temporal variation of driving torque exerted by spherical robots on sandy terrain. Numerical computations and experimental analyses demonstrate that increasing the radius of the sphere shell, the load, and the slip ratio all lead to increased traction force and driving torque. However, traction force and driving torque begin to decrease once the slip ratio reaches approximately 0.5. Therefore, in the design of spherical robot structures and control laws, appropriate parameters such as load and slip ratio should be chosen based on the established sphere–soil interaction theoretical model to achieve high-quality longitudinal motion performance on sandy terrain.

Velocity Distribution Upstream of Circular and Square Intakes

In the present study, a practical method is developed to predict the radial velocity of the flow towards circular and square intakes. It is shown that one single spherical sink surface passing from the desired points that have radial velocity vectors of identical magnitude within the ambient fluid can be used for the prediction of the radial velocity. The specific imaginary center of the corresponding spherical sink surface should be located on the center-line axis downstream of the entrance of the intake. As opposed to the previous studies, the proposed method does not require the solution of equations in closed-integral form, and it can also be used for the orifice-intake not opening into the atmosphere (submerged outlet), and an intake with a pipe, or intake-pipe projecting into the ambient fluid. It is shown that velocity field can be predicted with the introduced method even in the case of circulation imposed or induced ambient fluid flow. The agreement between the theoretical results and the available experimental/numerical results was found to be good.

Accurate analytical calculation of the rate coefficient for the diffusion-controlled reactions due to hyperbolic diffusion.

Using an approach based on the diffusion analog of the Cattaneo-Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski's boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski's theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice's formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper bound of the exact value. Here, the obtained formula seems to be of great significance for bridging a known gap between an analytically estimated rate coefficient on the one hand and molecular dynamics simulations together with experimentally observed results for the short times regime on the other hand. A particular emphasis has been placed on the rigorous mathematical treatment and important properties of the relevant initial boundary value problems in parabolic and hyperbolic diffusion theories.

Flow Rate along the Length of the Swirling Vortex Axis at an Intake

In this study, the characteristics of the flow in the region of swirling vortex are examined. The potential flow model based on the summing infinite number of spherical sinks along the vortex core is introduced to predict the flow field and the flow rate along the vortex axis. The flow towards the swirling vortex core has considerable effects on the radial velocity distribution within the ambient fluid flow region near the intake. The agreement between available test data relating to the radial velocity and the method introduced in this study is found to be satisfactory.

Occurrence of a Swirling Vortex without an Intake

In this study, mechanisms that cause the occurrence of a swirling vortex without an intake are explained. There are more than one different reasons which may cause the occurrence of an air-core vortex or a swirling-vortex flow filament within an ambient fluid. An air-core vortex or a swirling vortex flow filament can also occur in an ambient fluid media without an intake provided that the coaxial subsurface depressions and thereby the imaginary subsurface spherical sink surface sectors are hydraulically developed by any means (such as rotation of the fluid mass of spherical sector-form, impinging or penetrating jet flow, continuously entering solid body or string etc.). Although studies related to occurrence of a swirling vortex in case of an intake can be found in available literature, there are currently no studies which explains the occurrence mechanisms of a swirling vortex without an intake in detail. In this study, it is showed that the causes for the occurrence of the air-core vortex or the swirling vortex filament can be different but the main mechanism of development of the air-core vortex or the swirling vortex filament is identical for all cases.

Critical submergence for a horizontal pipe intake

Incipient air entrainment due to a free-surface vortex at a horizontal pipe intake passing through a dead-end wall located in a channel is investigated. Available potential flow based methods fail to predict the critical submergence due to friction effects introduced by the dead-end wall. In this study, velocity at the critical spherical sink surface, which is an imaginary spherical surface having identical center and the discharge as the intake, is associated with the average approach flow velocity instead of assuming it as a constant as in the previous studies. This simpler method for determination of critical submergence is tested with available experimental data and the agreement is found to be good. Scale effects are also discussed. It is shown that for the similarity between the model and prototype horizontal intakes in terms of air-entrainment, there is no need to consider Froude, Reynolds, and Weber numbers similarities.

Allometric scaling and morphological variation in sinking rate of phytoplankton.

Since the first decades of the last century, several hypotheses have been proposed on the role of phytoplankton morphology in maintaining a favorable position in the water column. Here, by an extensive review of literature on sinking rate and cell volume, we firstly attempted to explore the dependency of sinking rate on morphological traits using the allometric scaling approach. We found that sinking rate tends to increase with increasing cell volume showing the allometric scaling exponent of 0.43, which is significantly different than the Stokes' law exponent of 0.66. The violation of the 2/3 power rule clearly indicates that cell shape changes as size increases. Both size and shape affect how phytoplankton sinking drives nutrient acquisition and losses to sinking. Interestingly, from an evolutionary perspective, simple and complex cylindrical shapes can get much larger than spherical and spheroidal shapes and sink at similar rates, but simple and complex cylindrical shapes cannot get small enough to sink slower than small spherical and spheroidal shapes. Cell shape complexity is a morphological attribute resulting from the combination of two or more simple geometric shapes. While the effect of size on sinking rate is well documented, this study deepens the knowledge on how cell shape or geometry affect sinking rates that still needs further consideration.

Heat transfer performance investigation of the spherical dimple heat sink and inclined teardrop dimple heat sink

The convective heat transfer is significant in the cooling channel, but the heat transfer enhancement in a cooling passage will be more useful or increases if a dimple is employed on their surfaces. In the case of the nozzle, heat exchanger, the fuel of nuclear reactor and heat sink, the application of the dimple is handy. In the present Investigation, a numerical and experimental investigation has been carried out to study a Nusselt number and the pressure loss performance of a spherical dimple heat sink and inclined teardrop dimple heat sink under different flow conditions. The study is being performed for a Reynolds number varying from 1800 to 7000. To get a reasonable comparison of the dimple heat sink, both the geometries have the same longitudinal spacing of 10 mm, the transverse spacing of 5 mm and dimple depth to dimple diameter ratio of 0.2. Dimple is formed on both the sides of the aluminum heat sink, both inclined teardrop dimple and spherical dimple heat sink exhibit the heat transfer augmentation of about 1.09 to 1.15 times and 1.04 to 1.08 times as compared to the flat heat sink. Inclined teardrop dimple heat sink shows the highest heat transfer of about 8.6% higher than the spherical dimple heat sink. Additionally, friction factor, thermal performance, and flow structure are obtained numerically by using Ansys Fluent software. Inclined teardrop dimple heat sink exhibits the increase of pressure loss of about 3.45% to 4.98% as corresponding to the spherical dimple heat sink.

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

The generalized method of separation of variables (GMSV) is applied to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (e.g., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We consider both exterior and interior problems and all most common boundary conditions: Dirichlet, Neumann, Robin, and conjugate one. Using the translational addition theorems for solid harmonics to switch between the local spherical coordinates, we obtain a semi-analytical expression of the Green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions. Although the numerical computation of the coefficients involves series truncation and solution of a system of linear algebraic equations, the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the GMSV particularly efficient, especially for exterior problems. The obtained Green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate, escape probability, harmonic measure, residence time, and mean first passage time, to name but a few. The relevant aspects of the numerical implementation and potential applications in chemical physics, heat transfer, electrostatics, and hydrodynamics are discussed.

Prediction of critical submergence in a two-layer-stratified fluid flow

This study examines the critical submergence of a pipe intake located in a two layer-stratified flow. The critical submergence for a vertically downward flowing pipe intake in a two-layer stratified fluid field is defined as the vertical distance of the interface of the fluid layers to the intake level for which the upper fluid just enters the intake. The available potential flow solution-based theory developed for the critical submergence of an intake located in a single fluid flow (with no stratification) is utilized to predict the critical submergence for an intake located in a two-layer-stratified flow. It is shown that whole boundary blockage effects may be disregarded, and an imaginary complete critical spherical sink surface (CSSS) can be used for the prediction of the critical submergence. The results of several available experimental studies indicate that the velocity at the imaginary complete CSSS is a constant for a given geometrical and flow conditions and it can be obtained by conducting a few experiments. Since this approach does not require computation of blockages caused by the flow boundaries, it is easy to use in practice. This theory can also be applied to the intakes located in general geometrical and flow conditions [for both two-layer-stratified flows and single layer flows (without stratification)]. Theoretical results obtained from this study match very well with the available experimental data.

Scale and flow boundary effects for air-entraining vortices

A challenge when designing water intake structures in rivers, channels and reservoirs is to reduce the entrainment of air through free air-core vortices. Entrained air can cause discharge reduction, vibration and cavitation in the water-conveying system. This study firstly investigates flow boundary effects at the critical submergence of an intake, which is the depth up to which air entrainment occurs. The friction effect of a boundary section close to the free surface dominates the boundary blockage effect and retards the occurrence of the air-entraining vortex. Potential flow solution methods overpredict the critical submergence of an intake under the influence of boundary friction, so a design criterion is presented to correct this. Furthermore, previous work shows that Froude similarity is not always reliable for predicting critical submergence at prototype intakes. In the second part of this study, kinematic similarity – the equality of the ratios of intake velocity to the velocity at the imaginary critical spherical sink surface between the model and prototype intakes – is proposed and validated against experimental results. In addition, a theoretical explanation of the equal intake velocity concept is given using kinematic similarity.

Modification on theory of sink strength: An Object Kinetic Monte Carlo study

Sink strength is a fundamental quantity in modeling the microstructure evolution of irradiated materials by the mean-field approaches. The analytical expressions for different sinks have been extensively studied. The Object Kinetic Monte Carlo (OKMC) simulations were subsequently performed to corroborate the expressions and to guide the development of analytical theories. Although a general agreement was found between the theory and simulation, there are still some discrepancies in the case of the small spherical sinks and the dislocation lines. In this work, OKMC simulations were performed to study the sink strength of spherical sinks and dislocation lines. Our results revealed the origins of discrepancies between the theory and simulation for small sinks, high volume fraction sinks and periodically distributed dislocation lines. The theoretical corrections were also proposed accordingly. These corrections can extend the capabilities of mean-field approaches to properly reproduce the defect evolution of irradiated materials, and provide a better insight into the experimental phenomena, such as the defect cluster nucleation, the irradiation-induced swelling and the blistering.

A spherical chaotic system

This paper proposes a spherical chaotic system by appropriately combining a celestial conservative system with a dissipative chaotic system. The spherical system spins around its axis chaotically while revolving along an elliptical orbit. The spherical system takes on rich dynamics: having spherical four-wing chaotic attractors, spherical ring-like chaotic attractors, spherical periodic orbits and spherical sink orbits, as the dissipative sub-system is perturbed through parameter variations.

Effects of Froude, Reynolds, and Weber numbers on an air-entraining vortex

The present study highlights the need for using both the kinematic and dynamic similarities in scaled models of the critical submergence of vertical vortices at intake structures. A semi-theoretical approach, which is based on the principle of flow continuity, is used in combination with published experimental data. The present study shows that only the kinematic similarity is needed to model the ratio of the critical submergence to the diameter of the intake. For intakes that have an identical ratio of the intake velocity to the velocity at the critical spherical sink surface (CSSS), the ratio of the critical submergence to the diameter of the intake is identical regardless of the flow and the geometrical conditions. The new finding is that dynamic similarity is not required for the prediction of the ratio of the critical submergence to the intake diameter, provided that the ratio of the intake velocity to the velocity at the CSSS is kept the same.

Nutrient shielding in clusters of cells

Cellular nutrient consumption is influenced by both the nutrient uptake kinetics of an individual cell and the cells' spatial arrangement. Large cell clusters or colonies have inhibited growth at the cluster's center due to the shielding of nutrients by the cells closer to the surface. We develop an effective medium theory that predicts a thickness ℓ of the outer shell of cells in the cluster that receives enough nutrient to grow. The cells are treated as partially absorbing identical spherical nutrient sinks, and we identify a dimensionless parameter ν that characterizes the absorption strength of each cell. The parameter ν can vary over many orders of magnitude among different cell types, ranging from bacteria and yeast to human tissue. The thickness ℓ decreases with increasing ν, increasing cell volume fraction φ, and decreasing ambient nutrient concentration ψ(∞). The theoretical results are compared with numerical simulations and experiments. In the latter studies, colonies of budding yeast, Saccharomyces cerevisiae, are grown on glucose media and imaged under a confocal microscope. We measure the growth inside the colonies via a fluorescent protein reporter and compare the experimental and theoretical results for the thickness ℓ.

Study on the critical submergence of surface vortices and the design of anti-vortex intakes

Surface vortex behavior in front of the tunnel intake was investigated in this paper. The critical submergence of vortex was discussed based on the concept of ‘critical spherical sink surface’ (CSSS). The vortex formation and evolution at the tunnel intake were analyzed based on the theory of CSSS considering the effect of circulation. A theory was proposed to explain the surface vortex. The theoretical development was verified by the physical model experiments of Xiluodu hydropower station. The radial velocity and vortex circulation were considered as the main factors that influence the formation and evolution of surface vortex. Finally, an anti-vortex intake configuration was proposed to weaken the air-core vortex in front of the tunnel intakes of the hydraulic structures.

Compared techniques for the critical submergence of an intake in water flow

Air-entraining vortex at intake is an important problem encountered in hydraulic engineering. Intake submergence depth could result in formation of the air-entraining free surface vortices. Unless the dangerous air entrainment is eliminated, air entraining causes mechanical damage, vibration in pipelines and loss of pump performance. The value of the intake's submergence when the vortex starts entraining air is known as ''critical submergence''. In this study, the critical submergence for a circular intake pipe in still-water and open channel flow for permeable and impermeable bottom was investigated. Experimental results were used to compare with critical spherical sink surface (CSSS), radial basis function based neural network (RBNN) and general linear model (GLM). The CSSS has the same center and discharge as the intake with the critical submergence. The GLM underlies most of the statistical analyses that are used in applied research. And the RBNN is one of the most used network models. The ranking of prediction on critical submergence is obtained as RBNN, GLM and CSSS, respectively.

An artificial neural network model for the prediction of critical submergence for intake in a stratified fluid medium

Density differences may occur because of temperature differentials, suspended sediments, dissolved salts or other chemicals. Most of the large surface reservoirs are stably stratified throughout most, or all, of the year. One of the means of assisting the management is to allow a selective withdrawal from the reservoir. And while an intake is used for withdrawal (from the lower layer), a maximum discharge is required not allowing the uptake of the upper layer fluids. The value of the intake's vertical distance from the upper layer elevation (submergence) when the upper layer fluids begin to be drawn into the intake is known as ‘critical submergence’. In this study, the critical submergence for a circular intake pipe in a stratified body (which has different layer thickness) is investigated. Experiments were conducted on a vertically flowing downward intake pipe in a still-water reservoir. Artificial neural network (ANN) models and formulas, which are found by the theoretical analysis of critical spherical sink surface (CSSS), are used for the analysis of experimental results. The CSSS has the same centre and discharge as the intake. The ANN model and CSSS results are compared with the experimental results.

Ground deformation in an area later damaged by an earthquake: monitoring the Avcilar district of Istanbul, Turkey, by satellite radar interferometry 1992-1999

SUMMARY The Avcilar district of Istanbul was severely damaged by the M 7.4 Izmit (Kocaeli) earthquake of 1999. The same area underwent ground subsidence before the earthquake, as revealed by geodetic monitoring. Analysis of 14 synthetic aperture radar images acquired by the ERS-1 and ERS-2 satellites between 1992 and 1999 by interferometry (InSAR) measures the rate of subsidence. Using the General Inversion for Phase Technique (GIPhT), we analyse a set of 12 interferometric pairs. The interferometric fringe patterns show a rounded triangular shape that we interpret as secular subsidence at a constant rate. The maximum subsidence rate of 6 mm per year occurs at a point located at latitude 40.98 ◦ N and longitude 28.71 ◦ E. A simple four-parameter elastic Mogi model, consisting of three infinitesimal spherical sinks at a depth of 2.4 ± 0.4 km deflating at 78 000 ± 16 000 cubic metres per year, describes subsidence signal to first order. The model also accounts for tropospheric effects by estimating a vertical phase gradient and an additive offset for each image acquisition epoch. The model fits the data with a cost of 0.18 cycles per datum for the 4644 phase measurements included in the inversion. This fit is significantly better than either the null hypothesis or the initial model with 95 per cent confidence for 32 free parameters. The association of ground deformation with earthquake damage may be interpreted in terms of weak, compressible material in shallow subsurface layers.