Pub Date : 2019-01-01DOI: 10.26565/2304-6201-2019-43-05
V. Y. Kylynnyk, D. Kriutchenko, Y. Naumenko
Vibrations of an ideal incompressible fluid in shells of revolution have been considered. The shells of revolution under consideration include cylindrical and conical parts. It is assumed that the shell is subjected to vertical and horizontal excitations. The liquid in the shells is supposed to be an ideal and incompressible one. The fluid flow is the irrotational. Therefore the velocity potential that satisfies the Laplace equation exists. The non-penetration conditions are applied to the wetted surfaces of the shell and the kinematic and dynamic conditions on the free surface have been considered. The liquid pressure as the function of the velocity potential is defined using the Bernoulli equation. The problem of determining the fluid pressure is reduced to solving a singular integral equation. The numerical solution of the equation has been obtained by the method of discrete singularities. The method of simulating the free and forced oscillations of the fluid in the shells of revolution has been developed.
{"title":"Liquid oscillation in a cylindrical-conical shell under the action of vertical and horizontal excitation","authors":"V. Y. Kylynnyk, D. Kriutchenko, Y. Naumenko","doi":"10.26565/2304-6201-2019-43-05","DOIUrl":"https://doi.org/10.26565/2304-6201-2019-43-05","url":null,"abstract":"Vibrations of an ideal incompressible fluid in shells of revolution have been considered. The shells of revolution under consideration include cylindrical and conical parts. It is assumed that the shell is subjected to vertical and horizontal excitations. The liquid in the shells is supposed to be an ideal and incompressible one. The fluid flow is the irrotational. Therefore the velocity potential that satisfies the Laplace equation exists. The non-penetration conditions are applied to the wetted surfaces of the shell and the kinematic and dynamic conditions on the free surface have been considered. The liquid pressure as the function of the velocity potential is defined using the Bernoulli equation. The problem of determining the fluid pressure is reduced to solving a singular integral equation. The numerical solution of the equation has been obtained by the method of discrete singularities. The method of simulating the free and forced oscillations of the fluid in the shells of revolution has been developed.","PeriodicalId":33695,"journal":{"name":"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Matematichne modeliuvannia informatsiini tekhnologiyi avtomatizovani sistemi upravlinnia","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69003293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.26565/2304-6201-2019-44-05
Free vibrations of liquid in a rigid prismatic tank with vertical cross partitions are considered. These partitions divide the tank into four compartments. The partitions make it possible to reduce the amplitude of liquid sloshing in the tank under suddenly applied external loads due to earthquakes, terrorist attacks, emergencies, etc. It is assumed that the fluid is perfect and incompressible, and its motion is vortex-free. Under these conditions, there is a velocity potential that satisfies the Laplace equation. A non-leak condition is applied on the sides, bottom and partitions of the tank. On a free surface, kinematic and dynamic conditions are set. The kinematic condition is that the points of fluid that are on the free surface at the initial moment will remain on that surface for the entire subsequent motion. The dynamic condition is the equality of the fluid pressure on the free surface to the atmospheric pressure. An analytical solution of the boundary value problem for the Laplace equation is obtained for the case of the tank with a square bottom. The free surface oscillations have been found to be symmetrical. It should be noted that the oscillation patterns in each compartment are the same. The frequencies of free oscillations of the fluid in the tank with the cross partitions are increased in comparison with similar frequencies of oscillations of the prismatic tank without partitions. The frequencies obtained and the modes of natural oscillations of the fluid free surface allow us to solve the boundary value problem in case of sudden external loads. In this case, the velocity potential and the function describing the behaviour of the free surface are represented as the series according to the modes of natural fluctuations of the fluid free surface. Therefore it is possible to prevent the unwanted resonant frequencies at exploitation and transportation by designing prismatic tanks in a particular way.
{"title":"Liquid vibration modeling in prismatic tanks with quarter baffles","authors":"","doi":"10.26565/2304-6201-2019-44-05","DOIUrl":"https://doi.org/10.26565/2304-6201-2019-44-05","url":null,"abstract":"Free vibrations of liquid in a rigid prismatic tank with vertical cross partitions are considered. These partitions divide the tank into four compartments. The partitions make it possible to reduce the amplitude of liquid sloshing in the tank under suddenly applied external loads due to earthquakes, terrorist attacks, emergencies, etc. It is assumed that the fluid is perfect and incompressible, and its motion is vortex-free. Under these conditions, there is a velocity potential that satisfies the Laplace equation. A non-leak condition is applied on the sides, bottom and partitions of the tank. On a free surface, kinematic and dynamic conditions are set. The kinematic condition is that the points of fluid that are on the free surface at the initial moment will remain on that surface for the entire subsequent motion. The dynamic condition is the equality of the fluid pressure on the free surface to the atmospheric pressure. An analytical solution of the boundary value problem for the Laplace equation is obtained for the case of the tank with a square bottom. The free surface oscillations have been found to be symmetrical. It should be noted that the oscillation patterns in each compartment are the same. The frequencies of free oscillations of the fluid in the tank with the cross partitions are increased in comparison with similar frequencies of oscillations of the prismatic tank without partitions. The frequencies obtained and the modes of natural oscillations of the fluid free surface allow us to solve the boundary value problem in case of sudden external loads. In this case, the velocity potential and the function describing the behaviour of the free surface are represented as the series according to the modes of natural fluctuations of the fluid free surface. Therefore it is possible to prevent the unwanted resonant frequencies at exploitation and transportation by designing prismatic tanks in a particular way.","PeriodicalId":33695,"journal":{"name":"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Matematichne modeliuvannia informatsiini tekhnologiyi avtomatizovani sistemi upravlinnia","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69003447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.26565/2304-6201-2019-44-07
The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.
{"title":"The method for calculating singular integrals in problems of axially symmetric Stokes flows","authors":"","doi":"10.26565/2304-6201-2019-44-07","DOIUrl":"https://doi.org/10.26565/2304-6201-2019-44-07","url":null,"abstract":"The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.","PeriodicalId":33695,"journal":{"name":"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Matematichne modeliuvannia informatsiini tekhnologiyi avtomatizovani sistemi upravlinnia","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69003497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.26565/2304-6201-2019-43-04
The flow in the section of the Seversky Donets river in Kharkiv region is studied based on long-term measurements of the channel profile in a 10 cm increments. The geometry of the studied channel section on the Earth's surface has been determined by hydrological maps, and the cross-section profiles have been reconstructed by splines from the measurement results. The analysis of the results has revealed the profile variability in accordance with the change in the bottom sediments and the current year rainfall. A mathematical model describing the flow parameters in dependence on the slope and profile of the river channel has been developed. The model allows calculating flow velocities, dynamic pressure and viscous friction, predicting the evolution of coastal channel lines, the presence of stagnant zones with slow circulation, and predicting the dynamics of bottom drifts and channel overgrowing. Based on the three-dimensional flow of water in the channel with given geometry, numerical calculations by the finite element method are carried out. The flow rates are calculated and the presence of separated stagnant zones with slow circulation in which the channel overgrowth and water quality deterioration could be amplified is shown. Calculations of dynamic pressure and viscous friction shows the presence of areas with increased pressure which, in time, can ruin the riverbanks, contribute to the formation of bottom sediments, and increase the area of stagnant zones with slow circulation. Since there is a system of positive feedbacks in the river ecosystem, the resulting deterioration in circulation and water quality cannot be stopped naturally and require special engineering and hydrogeological measures. The developed model allows planning various specific measures to prevent river erosion and overgrowth, to improve circulation and water quality by introducing the changes into the original geometric model as well as quantifying the changes caused by hydrodynamic factors that affect the evolution of the river system.
{"title":"Mathematical modeling and forecasting the dynamics of a segment of the river bed of Seversky Donets river","authors":"","doi":"10.26565/2304-6201-2019-43-04","DOIUrl":"https://doi.org/10.26565/2304-6201-2019-43-04","url":null,"abstract":"The flow in the section of the Seversky Donets river in Kharkiv region is studied based on long-term measurements of the channel profile in a 10 cm increments. The geometry of the studied channel section on the Earth's surface has been determined by hydrological maps, and the cross-section profiles have been reconstructed by splines from the measurement results. The analysis of the results has revealed the profile variability in accordance with the change in the bottom sediments and the current year rainfall. A mathematical model describing the flow parameters in dependence on the slope and profile of the river channel has been developed. The model allows calculating flow velocities, dynamic pressure and viscous friction, predicting the evolution of coastal channel lines, the presence of stagnant zones with slow circulation, and predicting the dynamics of bottom drifts and channel overgrowing. Based on the three-dimensional flow of water in the channel with given geometry, numerical calculations by the finite element method are carried out. The flow rates are calculated and the presence of separated stagnant zones with slow circulation in which the channel overgrowth and water quality deterioration could be amplified is shown. Calculations of dynamic pressure and viscous friction shows the presence of areas with increased pressure which, in time, can ruin the riverbanks, contribute to the formation of bottom sediments, and increase the area of stagnant zones with slow circulation. Since there is a system of positive feedbacks in the river ecosystem, the resulting deterioration in circulation and water quality cannot be stopped naturally and require special engineering and hydrogeological measures. The developed model allows planning various specific measures to prevent river erosion and overgrowth, to improve circulation and water quality by introducing the changes into the original geometric model as well as quantifying the changes caused by hydrodynamic factors that affect the evolution of the river system.","PeriodicalId":33695,"journal":{"name":"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Matematichne modeliuvannia informatsiini tekhnologiyi avtomatizovani sistemi upravlinnia","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69003285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.26565/2304-6201-2019-44-10
A schedule ensuring the exactly minimal total tardiness can be found by the respective integer linear programming problem with infinities. In real computations, the infinity which shows that the respective states are either forbidden or impossible is substituted with a sufficiently great positive integer. An open question is whether the substitute can be selected so that the computation time would be decreased. The goal is to ascertain how the increment of the infinity substitute in the respective model influences the computation time of exact schedules. If the influence appears to be significant, then a recommendation on selecting the infinity substitute is to be stated in order to decrease the computation time. A pattern of generating instances of the job scheduling problem is provided. The instances of the job scheduling problem are generated so that schedules which can be obtained trivially, without the exact model, are excluded. Nine versions of the infinity substitute have been proposed. The increment of the infinity substitute in the model of total tardiness exact minimization rendered to solving an integer linear programming problem involving the branch-and-bound approach may have bad influence on the computation time of exact schedules. At least, the greater value of the infinity substitute cannot produce an optimal schedule faster in tight-tardy progressive 1-machine scheduling by idling-free preemptions of equal-length jobs. Roughly the best value of the infinity substitute is the maximal value taken over all the finite triple-indexed weights in the model and increased then by 1. The influence of the “max” infinity substitution is extremely significant. Compared to the case when the infinity is substituted with a sufficiently great integer, the “max” infinity substitution saves up to 50 % of the computation time. This saves hours and even days or months when up to 8 jobs of a few equal processing periods are scheduled for a few thousands of cycles or longer. Therefore, it is strongly recommended always to select the infinity substitute as less as possible in order to decrease the computation time.
{"title":"Infinity substitute in exactly minimizing total tardiness in tight-tardy progressive 1-machine scheduling by idling-free preemptions of equal-length jobs","authors":"","doi":"10.26565/2304-6201-2019-44-10","DOIUrl":"https://doi.org/10.26565/2304-6201-2019-44-10","url":null,"abstract":"A schedule ensuring the exactly minimal total tardiness can be found by the respective integer linear programming problem with infinities. In real computations, the infinity which shows that the respective states are either forbidden or impossible is substituted with a sufficiently great positive integer. An open question is whether the substitute can be selected so that the computation time would be decreased. The goal is to ascertain how the increment of the infinity substitute in the respective model influences the computation time of exact schedules. If the influence appears to be significant, then a recommendation on selecting the infinity substitute is to be stated in order to decrease the computation time. A pattern of generating instances of the job scheduling problem is provided. The instances of the job scheduling problem are generated so that schedules which can be obtained trivially, without the exact model, are excluded. Nine versions of the infinity substitute have been proposed. The increment of the infinity substitute in the model of total tardiness exact minimization rendered to solving an integer linear programming problem involving the branch-and-bound approach may have bad influence on the computation time of exact schedules. At least, the greater value of the infinity substitute cannot produce an optimal schedule faster in tight-tardy progressive 1-machine scheduling by idling-free preemptions of equal-length jobs. Roughly the best value of the infinity substitute is the maximal value taken over all the finite triple-indexed weights in the model and increased then by 1. The influence of the “max” infinity substitution is extremely significant. Compared to the case when the infinity is substituted with a sufficiently great integer, the “max” infinity substitution saves up to 50 % of the computation time. This saves hours and even days or months when up to 8 jobs of a few equal processing periods are scheduled for a few thousands of cycles or longer. Therefore, it is strongly recommended always to select the infinity substitute as less as possible in order to decrease the computation time.","PeriodicalId":33695,"journal":{"name":"Visnik Kharkivs''kogo natsional''nogo universitetu imeni VN Karazina Seriia Matematichne modeliuvannia informatsiini tekhnologiyi avtomatizovani sistemi upravlinnia","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69003534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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