In this paper, a Bondi-Hoyle accretion onto the rotating black hole in Einstein-Gauss Bonnet grav... more In this paper, a Bondi-Hoyle accretion onto the rotating black hole in Einstein-Gauss Bonnet gravity is studied. By injecting the gas from upstream region of the computational domain, we have found occurrence of the stable shock cones in the downstream region. The dynamical structures and oscillation properties of these shock cones strongly depend on the black hole spin parameter a and Gauss-Bonnet coupling constant α. It is found that the various values of α can lead the different amounts of matter to pile up close to the black hole horizon, higher α causes bigger oscillation amplitude in the mass accretion rate, and the required time to reach the steady state is getting smaller with the increasing in α. Moreover, increasing α in the negative direction causes a decrease in the shock opening angle and this angle slightly increases with the increasing α in the positive direction. We found that the negative values of Gauss-Bonnet coupling constant are more favored to have interesting physical outcomes such as accretion rate and oscillation. In addition, the higher the black hole rotation parameter a emerges the higher the accretion rate. It is also confirmed that, for α → 0, the black hole solution in EGB gravity converges to Kerr in general relativity. Furthermore, Gauss-Bonnet coupling constant could be used to constrain the size of observed shadow of M 87 * radius for various values of black hole rotation parameter.
In this paper, the numerical investigation of a Bondi-Hoyle accretion around a non-rotating black... more In this paper, the numerical investigation of a Bondi-Hoyle accretion around a non-rotating black hole in a novel four dimensional Einstein-Gauss-Bonnet gravity is investigated by solving the general relativistic hydrodynamical equations using the high resolution shock capturing scheme. For this purpose, the accreated matter from the wind-accreating X-ray binaries falls towards the black hole from the far upstream side of the domain, supersonically. We study the effects of Gauss-Bonnet coupling constant α in 4D EGB gravity on the accreated matter and shock cones created in the downstream region in detail. The required time having the shock cone in downstream region is getting smaller for α > 0 while it is increasing for α < 0. It is found that increases in α leads violent oscillations inside the shock cone and increases the accretion efficiency. The violent oscillations would cause increase in the energy flux, temperature, and spectrum of X-rays. So the quasi-periodic oscillations (QPOs) are naturally produced inside the shock cone when −5 ≤ α ≤ 0.8. It is also confirmed that EGB black hole solution converges to the Schwarzschild one in general relativity when α → 0. Besides, the negative coupling constants also give reasonable physical solutions and increase of α in negative directions suppresses the possible oscillation observed in the shock cone.
Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2014
Donen ve donmeyen karadelikler etrafindaki disklerin dinamik yapilarinin modellenmesi ve disklerd... more Donen ve donmeyen karadelikler etrafindaki disklerin dinamik yapilarinin modellenmesi ve disklerde olusan sok dalgalarinin ozelliklerinin ortaya cikarilmasi, genel rolativistlik hidrodinamik denklemlerin yuksek cozunurluge sahip numerik yontemler kullanilarak cozulmesi ile elde edilmistir. Diskler uzerinde duragan sok dalgalarinin olustugu gorulmus, bu dalgalar kuyruk sok dalgasi olarak adlandirilmistir. Olusan sok dalgalari karadeligin yuzeyi diye adlandirabilecegimiz olay ufku ile birlesmis ve duran iki sok dalgasi arasinda kalan bolge “kovuk” olarak adlandirilmistir. Bu sok dalgalarinin ve kovugun ozellikleri, diskin adyabatik indeksine, acisal momentumuna, basincina ve merkezde bulunan donen karadeligin spinine bagli oldugu bulunmustur. Diskin salinim ozelliklerini tanimlayan modlarin, bu tur sok dalgalari tarafindan yakalandigi goz onune alinirsa, diskin ve karadeligin bu parametreleri disk tarafindan yayilan X-isinlarinin ozelliklerini belirlemede onemli rol oynar. Diskin sali...
We present the numerical study of dynamical instability of a pressure-supported relativistic toru... more We present the numerical study of dynamical instability of a pressure-supported relativistic torus, rotating around the black hole with a constant specific angular momentum on a fixed space-time background, in case of perturbation by a matter coming from the outer boundary. Two dimensional general relativistic hydrodynamical equations are solved at equatorial plane using the HRSCS to study the effect of perturbation on the stable systems. We have found that the perturbed torus creates an instability which causes the gas falling into the black hole in a certain dynamical time. All the models indicate an oscillating torus with certain frequency around their instant equilibrium. The dynamic of accreted torus varies with the size of initial stable torus, black hole spin and other variables, such as Mach number, sound speed, initial radius of the torus etc., but not their instability. The precessing torus not only effects the gravitational radiation, but also generates it. On the other h...
In this paper, a numerical study of the dynamic of the non-self-gravitating, unmagnetized, non-ax... more In this paper, a numerical study of the dynamic of the non-self-gravitating, unmagnetized, non-axisymmetric, and rotating the torus around the non-rotating black hole is presented. We investigate the instability of the rotating torus subject to perturbations presented by increasing or decreasing the angular velocity of the stable torus. We have done, for the first time, an extensive analysis of the torus dynamic response to the perturbation of the angular velocity of the stable torus. We show how the high, moderate, and low values of the perturbations affect the torus dynamic and help us to understand the properties of the instability and quasi-periodic oscillation (QPO). Our numerical simulations indicate the presence of Papaloizou–Pringle instability (PPI) with global m = 1 mode and QPOs for the moderate and lower values of the perturbations on the angular velocity of the stable torus. Furthermore, with the lower values of the perturbations, the torus can lead to a wiggling initia...
The nonlinear dynamics and propagation of ion acoustic waves in a relativistic and ideal plasmas,... more The nonlinear dynamics and propagation of ion acoustic waves in a relativistic and ideal plasmas, which have the pressure variation of electrons and ions and degenerate electrons, are investigated using the analytic solution of KdV type equations performed applying (G′/G)-expansion and (G′/G,1/G)-expansion methods. The effects of various parameters, such as phase velocity of the ion acoustic wave, the ratio of ion temperature to electron temperature, normalized speed of light, electron and ion streaming velocities, arbitrary and integration constants, on the soliton dynamics are studied. We have found that dim and hump solitons and their amplitudes, widths and dynamics strongly depend on these plasma parameters and integration constants. The source term μ plays also a vital role in the formation of the solitons. Moreover, it is also found that the observed solitary wave solution can be excited from hump soliton to dip soliton. This dramatic change of the solitons can occur due to th...
In this paper, perturbations of an accretion disk by a star orbiting around a massive black hole ... more In this paper, perturbations of an accretion disk by a star orbiting around a massive black hole are studied. We report on a numerical experiment, which has been carried out by using a parallel-machine code originally developed by Ref.[1]. An initially steady state accretion disk near a non-rotating (Schwarzschild) black hole interacts with a ``star&amp;amp;amp;amp;#39;&amp;amp;amp;amp;#39;, modeled as an initially circular
We investigate the dynamics and morphology of jets propagating into the interstellar medium using... more We investigate the dynamics and morphology of jets propagating into the interstellar medium using 2D special relativistic hydrodynamics code. The calculations are performed assuming axisymmetric geometry and trace a long propagation of jets. The jets are assumed to be higher density than the ambient gas. The relativistic astrophysical jet is modeled for ultrarelativistic flow case. The dynamics of jet flowing is then determined by the ambient parameters such as densities, and velocities of the jets and the momentum impulse applied to the computational surface. We obtain solutions for the jet structure, propagation of jet during the time evolution, and variation in the Mach number, pressure and Lorentz factor on the computational domain at a fixed time.
The shocked wave created on the accretion disk after different physical phenomena (accretion flow... more The shocked wave created on the accretion disk after different physical phenomena (accretion flows with pressure gradients, star-disk interaction etc.) may be responsible observed Quasi Periodic Oscillations (QPOs) in $X-$ray binaries. We present the set of characteristics frequencies associated with accretion disk around the rotating and non-rotating black holes for one particle case. These persistent frequencies are results of the rotating pattern in an accretion disk. We compare the frequency's from two different numerical results for fluid flow around the non-rotating black hole with one particle case. The numerical results are taken from our papers Refs.\refcite{Donmez2} and \refcite{Donmez3} using fully general relativistic hydrodynamical code with non-selfgravitating disk. While the first numerical result has a relativistic tori around the black hole, the second one includes one-armed spiral shock wave produced from star-disk interaction. Some physical modes presented in the QPOs can be excited in numerical simulation of relativistic tori and spiral waves on the accretion disk. The results of these different dynamical structures on the accretion disk responsible for QPOs are discussed in detail.
How to cite Complete issue More information about this article Journal's homepage in redalyc.org ... more How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative
Stellar black hole candidates exhibit a variety of time-varying X-ray emissions from chaotic vari... more Stellar black hole candidates exhibit a variety of time-varying X-ray emissions from chaotic variations to quasi-periodic oscillations. The origins of these emissions are subjects of active study, and some of them are expected to result from hydrodynamic phenomena in the accretion disks. We have developed a numerical code for simulating fluid flow around a black hole in full general relativity.
The general relativistic hydrodynamical simulation of the spherical-shell accretion onto the stab... more The general relativistic hydrodynamical simulation of the spherical-shell accretion onto the stable torus around non-rotating and rotating black holes isotropically falling from a finite distance are constructed for the first time. This type of accretion might be used to explain the dynamics of the torus. The accreted matter sonically, supersonically or highly supersonically interacts with a torus and forms a newly developed dynamical structure. This spherical-shell changes the angular momentum of the torus and mediates torus instabilities which cause the termination of the torus. The impact of the rest-mass density of the perturbation is also studied which found that the high density perturbation destroys the torus in a few dynamical times. It is also found that the dumping time of the matter is much larger for the torus around a rotating black hole. On the other hand, the Papaloizou–Pringle instability from the spherical-shell accretion appears to be much more softer than the form...
In the present study, we perform the numerical simulation of a relativistic thin accretion disk a... more In the present study, we perform the numerical simulation of a relativistic thin accretion disk around the nonrotating and rapidly rotating black holes using the general relativistic hydrodynamic code with Kerr in Kerr–Schild coordinate that describes the central rotating black hole. Since the high energy X-rays are produced close to the event horizon resulting the black hole–disk interaction, this interaction should be modeled in the relativistic region. We have set up two different initial conditions depending on the values of thermodynamical variables around the black hole. In the first setup, the computational domain is filled with constant parameters without injecting gas from the outer boundary. In the second, the computational domain is filled with the matter which is then injected from the outer boundary. The matter is assumed to be at rest far from the black hole. Both cases are modeled over a wide range of initial parameters such as the black hole angular momentum, adiabat...
Monthly Notices of the Royal Astronomical Society, 2012
It is known from recent numerical calculations that Bondi-Hoyle accretion creates a shock cone be... more It is known from recent numerical calculations that Bondi-Hoyle accretion creates a shock cone behind compact objects. This type of accretion leads to instabilities, which can explain certain astrophysical phenomena. In this paper, our main goal is to find the flip-flop behaviour of the shock cone in the relativistic region. In order to do so we have modelled the dynamics of a shock cone around non-rotating and rotating black holes at the equatorial plane in 2D. The effects of the various parameters on the shock cones and instabilities, such as the asymptotic velocity, sound speed, Mach number and adiabatic index, are studied. We have determined the mass accretion rate, shock opening angle, shock cone oscillation, quasi-periodic oscillations (QPOs), and growth rate of instabilities to reveal the disc properties and its radiation. We have discovered, for the first time, flip-flop instabilities around a black hole in the relativistic region by solving the general relativistic hydrodynamical equations. The flip-flop instabilities are found for sound speeds C s,∞ < 0.2 with moderate Mach numbers (∼M = 3 and M = 4 for C s,∞ = 0.1 or M = 7 and M = 8 for C s,∞ = 0.05). Our calculation clearly confirms that the shock cone should be detached from the black hole in the Bondi-Hoyle accretion flow with ≥ 2 for non-rotating and rotating black holes. Results reveal that the flip-flopping shock cone not only creates a torque effect on the black hole but also produces continuous X-ray flares with a certain frequency. Furthermore, QPOs originate inside the shock cone and are stronger in regions that have a radius of a few gravitational radii away from the centre owing to the flip-flop oscillation. Finally, our results are compared with the results of numerical and theoretical calculations in Newtonian hydrodynamics, and it is found that they are in good agreement.
In this paper, a Bondi-Hoyle accretion onto the rotating black hole in Einstein-Gauss Bonnet grav... more In this paper, a Bondi-Hoyle accretion onto the rotating black hole in Einstein-Gauss Bonnet gravity is studied. By injecting the gas from upstream region of the computational domain, we have found occurrence of the stable shock cones in the downstream region. The dynamical structures and oscillation properties of these shock cones strongly depend on the black hole spin parameter a and Gauss-Bonnet coupling constant α. It is found that the various values of α can lead the different amounts of matter to pile up close to the black hole horizon, higher α causes bigger oscillation amplitude in the mass accretion rate, and the required time to reach the steady state is getting smaller with the increasing in α. Moreover, increasing α in the negative direction causes a decrease in the shock opening angle and this angle slightly increases with the increasing α in the positive direction. We found that the negative values of Gauss-Bonnet coupling constant are more favored to have interesting physical outcomes such as accretion rate and oscillation. In addition, the higher the black hole rotation parameter a emerges the higher the accretion rate. It is also confirmed that, for α → 0, the black hole solution in EGB gravity converges to Kerr in general relativity. Furthermore, Gauss-Bonnet coupling constant could be used to constrain the size of observed shadow of M 87 * radius for various values of black hole rotation parameter.
In this paper, the numerical investigation of a Bondi-Hoyle accretion around a non-rotating black... more In this paper, the numerical investigation of a Bondi-Hoyle accretion around a non-rotating black hole in a novel four dimensional Einstein-Gauss-Bonnet gravity is investigated by solving the general relativistic hydrodynamical equations using the high resolution shock capturing scheme. For this purpose, the accreated matter from the wind-accreating X-ray binaries falls towards the black hole from the far upstream side of the domain, supersonically. We study the effects of Gauss-Bonnet coupling constant α in 4D EGB gravity on the accreated matter and shock cones created in the downstream region in detail. The required time having the shock cone in downstream region is getting smaller for α > 0 while it is increasing for α < 0. It is found that increases in α leads violent oscillations inside the shock cone and increases the accretion efficiency. The violent oscillations would cause increase in the energy flux, temperature, and spectrum of X-rays. So the quasi-periodic oscillations (QPOs) are naturally produced inside the shock cone when −5 ≤ α ≤ 0.8. It is also confirmed that EGB black hole solution converges to the Schwarzschild one in general relativity when α → 0. Besides, the negative coupling constants also give reasonable physical solutions and increase of α in negative directions suppresses the possible oscillation observed in the shock cone.
Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2014
Donen ve donmeyen karadelikler etrafindaki disklerin dinamik yapilarinin modellenmesi ve disklerd... more Donen ve donmeyen karadelikler etrafindaki disklerin dinamik yapilarinin modellenmesi ve disklerde olusan sok dalgalarinin ozelliklerinin ortaya cikarilmasi, genel rolativistlik hidrodinamik denklemlerin yuksek cozunurluge sahip numerik yontemler kullanilarak cozulmesi ile elde edilmistir. Diskler uzerinde duragan sok dalgalarinin olustugu gorulmus, bu dalgalar kuyruk sok dalgasi olarak adlandirilmistir. Olusan sok dalgalari karadeligin yuzeyi diye adlandirabilecegimiz olay ufku ile birlesmis ve duran iki sok dalgasi arasinda kalan bolge “kovuk” olarak adlandirilmistir. Bu sok dalgalarinin ve kovugun ozellikleri, diskin adyabatik indeksine, acisal momentumuna, basincina ve merkezde bulunan donen karadeligin spinine bagli oldugu bulunmustur. Diskin salinim ozelliklerini tanimlayan modlarin, bu tur sok dalgalari tarafindan yakalandigi goz onune alinirsa, diskin ve karadeligin bu parametreleri disk tarafindan yayilan X-isinlarinin ozelliklerini belirlemede onemli rol oynar. Diskin sali...
We present the numerical study of dynamical instability of a pressure-supported relativistic toru... more We present the numerical study of dynamical instability of a pressure-supported relativistic torus, rotating around the black hole with a constant specific angular momentum on a fixed space-time background, in case of perturbation by a matter coming from the outer boundary. Two dimensional general relativistic hydrodynamical equations are solved at equatorial plane using the HRSCS to study the effect of perturbation on the stable systems. We have found that the perturbed torus creates an instability which causes the gas falling into the black hole in a certain dynamical time. All the models indicate an oscillating torus with certain frequency around their instant equilibrium. The dynamic of accreted torus varies with the size of initial stable torus, black hole spin and other variables, such as Mach number, sound speed, initial radius of the torus etc., but not their instability. The precessing torus not only effects the gravitational radiation, but also generates it. On the other h...
In this paper, a numerical study of the dynamic of the non-self-gravitating, unmagnetized, non-ax... more In this paper, a numerical study of the dynamic of the non-self-gravitating, unmagnetized, non-axisymmetric, and rotating the torus around the non-rotating black hole is presented. We investigate the instability of the rotating torus subject to perturbations presented by increasing or decreasing the angular velocity of the stable torus. We have done, for the first time, an extensive analysis of the torus dynamic response to the perturbation of the angular velocity of the stable torus. We show how the high, moderate, and low values of the perturbations affect the torus dynamic and help us to understand the properties of the instability and quasi-periodic oscillation (QPO). Our numerical simulations indicate the presence of Papaloizou–Pringle instability (PPI) with global m = 1 mode and QPOs for the moderate and lower values of the perturbations on the angular velocity of the stable torus. Furthermore, with the lower values of the perturbations, the torus can lead to a wiggling initia...
The nonlinear dynamics and propagation of ion acoustic waves in a relativistic and ideal plasmas,... more The nonlinear dynamics and propagation of ion acoustic waves in a relativistic and ideal plasmas, which have the pressure variation of electrons and ions and degenerate electrons, are investigated using the analytic solution of KdV type equations performed applying (G′/G)-expansion and (G′/G,1/G)-expansion methods. The effects of various parameters, such as phase velocity of the ion acoustic wave, the ratio of ion temperature to electron temperature, normalized speed of light, electron and ion streaming velocities, arbitrary and integration constants, on the soliton dynamics are studied. We have found that dim and hump solitons and their amplitudes, widths and dynamics strongly depend on these plasma parameters and integration constants. The source term μ plays also a vital role in the formation of the solitons. Moreover, it is also found that the observed solitary wave solution can be excited from hump soliton to dip soliton. This dramatic change of the solitons can occur due to th...
In this paper, perturbations of an accretion disk by a star orbiting around a massive black hole ... more In this paper, perturbations of an accretion disk by a star orbiting around a massive black hole are studied. We report on a numerical experiment, which has been carried out by using a parallel-machine code originally developed by Ref.[1]. An initially steady state accretion disk near a non-rotating (Schwarzschild) black hole interacts with a ``star&amp;amp;amp;amp;#39;&amp;amp;amp;amp;#39;, modeled as an initially circular
We investigate the dynamics and morphology of jets propagating into the interstellar medium using... more We investigate the dynamics and morphology of jets propagating into the interstellar medium using 2D special relativistic hydrodynamics code. The calculations are performed assuming axisymmetric geometry and trace a long propagation of jets. The jets are assumed to be higher density than the ambient gas. The relativistic astrophysical jet is modeled for ultrarelativistic flow case. The dynamics of jet flowing is then determined by the ambient parameters such as densities, and velocities of the jets and the momentum impulse applied to the computational surface. We obtain solutions for the jet structure, propagation of jet during the time evolution, and variation in the Mach number, pressure and Lorentz factor on the computational domain at a fixed time.
The shocked wave created on the accretion disk after different physical phenomena (accretion flow... more The shocked wave created on the accretion disk after different physical phenomena (accretion flows with pressure gradients, star-disk interaction etc.) may be responsible observed Quasi Periodic Oscillations (QPOs) in $X-$ray binaries. We present the set of characteristics frequencies associated with accretion disk around the rotating and non-rotating black holes for one particle case. These persistent frequencies are results of the rotating pattern in an accretion disk. We compare the frequency's from two different numerical results for fluid flow around the non-rotating black hole with one particle case. The numerical results are taken from our papers Refs.\refcite{Donmez2} and \refcite{Donmez3} using fully general relativistic hydrodynamical code with non-selfgravitating disk. While the first numerical result has a relativistic tori around the black hole, the second one includes one-armed spiral shock wave produced from star-disk interaction. Some physical modes presented in the QPOs can be excited in numerical simulation of relativistic tori and spiral waves on the accretion disk. The results of these different dynamical structures on the accretion disk responsible for QPOs are discussed in detail.
How to cite Complete issue More information about this article Journal's homepage in redalyc.org ... more How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative
Stellar black hole candidates exhibit a variety of time-varying X-ray emissions from chaotic vari... more Stellar black hole candidates exhibit a variety of time-varying X-ray emissions from chaotic variations to quasi-periodic oscillations. The origins of these emissions are subjects of active study, and some of them are expected to result from hydrodynamic phenomena in the accretion disks. We have developed a numerical code for simulating fluid flow around a black hole in full general relativity.
The general relativistic hydrodynamical simulation of the spherical-shell accretion onto the stab... more The general relativistic hydrodynamical simulation of the spherical-shell accretion onto the stable torus around non-rotating and rotating black holes isotropically falling from a finite distance are constructed for the first time. This type of accretion might be used to explain the dynamics of the torus. The accreted matter sonically, supersonically or highly supersonically interacts with a torus and forms a newly developed dynamical structure. This spherical-shell changes the angular momentum of the torus and mediates torus instabilities which cause the termination of the torus. The impact of the rest-mass density of the perturbation is also studied which found that the high density perturbation destroys the torus in a few dynamical times. It is also found that the dumping time of the matter is much larger for the torus around a rotating black hole. On the other hand, the Papaloizou–Pringle instability from the spherical-shell accretion appears to be much more softer than the form...
In the present study, we perform the numerical simulation of a relativistic thin accretion disk a... more In the present study, we perform the numerical simulation of a relativistic thin accretion disk around the nonrotating and rapidly rotating black holes using the general relativistic hydrodynamic code with Kerr in Kerr–Schild coordinate that describes the central rotating black hole. Since the high energy X-rays are produced close to the event horizon resulting the black hole–disk interaction, this interaction should be modeled in the relativistic region. We have set up two different initial conditions depending on the values of thermodynamical variables around the black hole. In the first setup, the computational domain is filled with constant parameters without injecting gas from the outer boundary. In the second, the computational domain is filled with the matter which is then injected from the outer boundary. The matter is assumed to be at rest far from the black hole. Both cases are modeled over a wide range of initial parameters such as the black hole angular momentum, adiabat...
Monthly Notices of the Royal Astronomical Society, 2012
It is known from recent numerical calculations that Bondi-Hoyle accretion creates a shock cone be... more It is known from recent numerical calculations that Bondi-Hoyle accretion creates a shock cone behind compact objects. This type of accretion leads to instabilities, which can explain certain astrophysical phenomena. In this paper, our main goal is to find the flip-flop behaviour of the shock cone in the relativistic region. In order to do so we have modelled the dynamics of a shock cone around non-rotating and rotating black holes at the equatorial plane in 2D. The effects of the various parameters on the shock cones and instabilities, such as the asymptotic velocity, sound speed, Mach number and adiabatic index, are studied. We have determined the mass accretion rate, shock opening angle, shock cone oscillation, quasi-periodic oscillations (QPOs), and growth rate of instabilities to reveal the disc properties and its radiation. We have discovered, for the first time, flip-flop instabilities around a black hole in the relativistic region by solving the general relativistic hydrodynamical equations. The flip-flop instabilities are found for sound speeds C s,∞ < 0.2 with moderate Mach numbers (∼M = 3 and M = 4 for C s,∞ = 0.1 or M = 7 and M = 8 for C s,∞ = 0.05). Our calculation clearly confirms that the shock cone should be detached from the black hole in the Bondi-Hoyle accretion flow with ≥ 2 for non-rotating and rotating black holes. Results reveal that the flip-flopping shock cone not only creates a torque effect on the black hole but also produces continuous X-ray flares with a certain frequency. Furthermore, QPOs originate inside the shock cone and are stronger in regions that have a radius of a few gravitational radii away from the centre owing to the flip-flop oscillation. Finally, our results are compared with the results of numerical and theoretical calculations in Newtonian hydrodynamics, and it is found that they are in good agreement.
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