Spherical Flow Research Articles

Two-temperature advective transonic accretion flows around black holes

We present here unique transonic two-temperature accretion solutions in presence of radiation processes. Unlike one-temperature flow, the set of equations governing two-temperature flow is not complete, since for the latter, the number of equations is less than the number of flow variables. Consequently, a large number of transonic solutions of the equations of motion exist, for a given set of constants of motion. We invoke the second law of thermodynamics and identified the solution with the maximum entropy as the physically possible solution. In this paper, we compare spherical and rotating accretion two-temperature flows. We also show that the luminosity, as well as radiative efficiency of rotating flows, is higher by an order of magnitude, as compared to spherical flows.

Interaction of a Singular Surface with a Characteristic Shock in a Relaxing Gas with Dust Particles

Abstract A system of hyperbolic differential equations outlining one-dimensional planar, cylindrical symmetric and spherical symmetric flow of a relaxing gas with dust particles is considered. Singular surface theory used to study different aspects of wave propagation and its culmination to the steepened form. The evolutionary behavior of the characteristic shock is studied. A particular solution of the governing system of equations is used to discuss the steepened wave form, characteristic shock and their interaction. The results of the interaction between the steepened wave front and the characteristic shock using the general theory of wave interaction are discussed. Also, the influence of relaxation and dust parameters on the steepened wave front, the formation of a characteristic shock, reflected and transmitted waves after interaction and a jump in shock acceleration are investigated.

Spherical accretion flows onto modified Schwarzschild black hole

A steady state spherical accretion process of perfect fluid onto modified Schwarzschild black hole is studied in this paper. Assuming various forms of infalling fluid we find their solutions at critical points and determine critical velocity for each case. For few fluid types, solutions are presented in closed form by presenting the curves in phase diagram. Moreover, we also discuss the effect of the Rindler parameter on the mass accretion rate of underlying spacetime. It is seen that the small values of it yield the maximum accretion rate. The results we obtain here may be helpful to understand the physical mechanism of accretion onto the modified version of Schwarzschild black hole.

Asymptotically flat vacuum solution in modified theory of Einstein\u2019s gravity

A number of recent observations have suggested that the Einstein’s theory of general relativity may not be the ultimate theory of gravity. The f(R) gravity model with R being the scalar curvature turns out to be one of the best bet to surpass the general relativity which explains a number of phenomena where Einstein’s theory of gravity fails. In the f(R) gravity, behaviour of the spacetime is modified as compared to that of given by the Einstein’s theory of general relativity. This theory has already been explored for understanding various compact objects such as neutron stars, white dwarfs etc. and also describing evolution of the universe. Although researchers have already found the vacuum spacetime solutions for the f(R) gravity, yet there is a caveat that the metric does have some diverging terms and hence these solutions are not asymptotically flat. We show that it is possible to have asymptotically flat spherically symmetric vacuum solution for the f(R) gravity, which is different from the Schwarzschild solution. We use this solution for explaining various bound orbits around the black hole and eventually, as an immediate application, in the spherical accretion flow around it.

Radiative shocks in spherical accretion

ABSTRACT In order to explore various aspects of radiative shocks, we examine standing radiative shock waves in spherical accretion flows onto a central gravitating body under the equilibrium diffusion approximation. In contrast to the usual one-dimensional shock, in radiative shocks a radiative precursor appears in the pre-shock region before the shock front, due to the radiative diffusion effect. Furthermore, in spherical flows around a central object the gravitational potential varies in this radiative precursor, and a curvature effect also exists. We first formulate such radiative shocks in spherical flows, derive the overall jump conditions, and solve the structure of the radiative precursor for both the gas and radiation pressure dominated cases. Since the jump conditions contain the coordinates of both ends of the radiative precursor, we must obtain both the solution and the endpoints of the precursor simultaneously. We find that the gravitational effect is not significant, although it cannot be ignored. The curvature effect exerts a strong influence on the structure and width of the precursor. The precursor starting point x1 normalized by the shock radius is roughly expressed by $x_1={\cal M}_1^{1/7}$ for a radiation pressure dominated shock, while $x_1=1.21^{({\cal M}_1-1)}$ for a gas pressure one, where ${\cal M}_1$ is the pre-shock Mach number.

Определение времени восстановления давления при нестационарной радиально-сферической фильтрации вязкопластичной нефти в однородном пласте

The paper solves a non-steady-state hydrodynamic problem concerning a spherical flow of viscoplastic oil in a homogeneous formation. Pressure recovery in the formation occurs in two stages after well operation stops instantaneously. At the first stage the perturbation region appears at the well, expands and eventually reaches the external boundary, and keeps expanding during the second stage until steady-state pressure is reached in every formation region. We solved the differential equations for specific initial and boundary conditions, assuming pressure distribution laws over the formation and respective integral relations, and derived equations to determine pressure recovery time. At each stage pressure in the formation is a function of two variables, that is, a spatial coordinate and time. We assume that the first variable equals zero. We derived an expression for recovering the bottomhole pressure in the wellbore. We used the transient flow method to study the wells, plotted the pressure recovery curve (PRC) and determined formation parameters.

Method of characteristics for transient, spherical flows

• Spherically-symmetric unsteady flow is solved numerically by the method of characteristics. • Close agreement is demonstrated with an analytical solution for a single pulse propagating spherically. • Interaction of waves propagating internally in a duct with acoustic waves radiating from the duct's exit. • Flows containing both uni-axial and spherically-symmetric domains are calculated with a highly efficient use of resources. • Two-way coupling is demonstrated to be a significant advantage over conventional one-way coupling. The equations of motion for spherically-symmetric, unsteady, inviscid, compressible flow are expressed in a form that enables accurate numerical solutions to be obtained using a one-dimensional formulation based on the method of characteristics. Unlike the corresponding equations for uniaxial unsteady flows, the spherically-symmetric equations necessarily include terms involving the reciprocal of the radius and, close to the radial origin, numerical integration is unreliable. Nevertheless, good accuracy is obtained over a wide range of radii, including regions inside the range where the pressure and acceleration are approximately in phase. The range of validity of the method is assessed by comparison with an analytical solution for a single pulse and the method is then used to predict the radiation of acoustic waves from the exit of a duct in which a pulse is propagating internally. A method of obtaining efficient solutions of flows containing both uniaxial and spherically-symmetric domains is then obtained. Interfaces between the domains are solved in a manner that ensures continuity of pressure and flowrate. The use of spherically-symmetric assumptions limits the range of three-dimensional domains that can be approximated, but the combination of the two forms of one-dimensional analysis makes highly efficient use of resources. Furthermore, the two-way coupling is a significant advantage over two-step methods that use independent solutions of the internal, uniaxial domain to provide prescribed boundary conditions for solutions of the external domain.

Análise do rebaixamento da derivada logarítmica para interpretação de teste de bombeamento de taxa constante em aquíferos de substrato inclinado

Constant-rate pumping tests (CRPT) performed in aquifers having an inclined substratum (IS) cannot be interpreted using Theis-like models, due to an increasing thickness that is beyond conventional hydraulic assumptions. Using an empirical process based on numerical modelling, this study submits original tools for the detection of IS aquifers and the interpretation of their hydrodynamic responses to CRPT, using derivative [ds/dlog(t)] and flow dimension (n) analyses. It is shown that IS aquifers produce a drawdown log-derivative signal composed of two radial regimes and one spherical flow regime (n-sequence = 2–2–3). Comprehensive sensitivity analyses make it possible to constrain relationships between, on one hand, characteristic derivative responses, and on the other hand, hydraulic conditions such as pumping rate, distance from the well to the corner, substratum inclination, and aquifer properties, including anisotropic hydraulic conductivity. This study contributes to widening the array of tools available for the interpretation of pumping tests, by implementing a novel conceptual model for a specific type of nonuniform aquifer that has remained unaddressed; further, it provides another interpretation of the spherical flow regime, which has been interpreted in the petroleum literature to reflect partially penetrating/completing wells. Finally, a field application of the submitted interpretative tools to a CRPT into an IS aquifer is presented.

EVALUATING THE EFFECT WELL INCLINATION AND FORMATION ANISOTROPY ON CONING RATE IN DEVIATED WELLS

In oil reservoirs with bottom water drive, coning is a fundamental problem during oil production. Horizontal and deviated wells are often used to reduce the effect of water coning on total oil production. To avoid the premature breakthrough of water, the production of oil should be maximised through effective monitoring of the critical coning rate, and many researchers have developed models to determine the critical coning rate. However, very few studies have been conducted to evaluate the effect of good inclination and formation anisotropy on the critical coning rate. Therefore, we have incorporated well deviation and formation anisotropy into a water coning model to investigate their effects on the critical coning rate. Previous researches have been limited mainly to the calculation of water critical coning rate in the vertical well. In this study, a combination of radial and spherical flows in deviated wells has been adopted to develop a new model for analysing coning in non-vertical wells. The well inclination considered ranged from 20o to 80o, horizontal permeability (kh ) from 100 mD to 200 mD and the vertical permeability (kv ) was varied from 2 mD to 10 mD. Results showed that highly-deviated wells and formations with a high degree of reservoir heterogeneity (kv/kh) result in high critical coning rate and therefore less susceptible to early water breakthrough. This shows that more allowance to increase the production rate is possible for horizontal wells and anisotropic formations than in vertical and homogeneous reservoirs.
 Keywords: Water coning, deviated wells, reservoir heterogeneity, anisotropy permeability, critical coning rate, critical parameters.

Design Of An Optofluidic Device For The Measurement Of The Elastic Modulus Of Deformable Particles

Suspensions carrying deformable inclusions are ubiquitous in nature and applications. Hence, high-throughput characterization of the mechanical properties of soft particles is of great interest. Recently, a non-invasive optofluidic technique has been developed for the measurement of the interfacial tension between two immiscible liquids [8]. We have adapted such technique to the case of soft solid beads, thus designing a non-invasive optofluidic device for the measurement of the mechanical properties of deformable particles from real-time optical imaging of their deformation. The device consists of a cylindrical microfluidic channel with a cross-section reduction in which we make initially spherical soft beads flow suspended in a Newtonian carrier. By imaging the deformation of a particle in real time while it goes through the constriction, it is possible to get a measure of its elastic modulus through a theoretically derived-correlation. We provide both experimental and numerical validation of our device.

The observation of permeation grouting method as soil improvement technique with different grout flow models

This study concluded the results of a research on the features of cement based permeation grout, based on some important grout parameters, such as the rheological properties (yield stress and viscosity), coefficient of permeability to grout (kG) and the inject ability of cement grout (N and Nc assessment), which govern the performance of cement based permeation grouting in porous media. Due to the limited knowledge of these important grout parameters and other influencing factors (filtration pressure, rate and time of injection and the grout volume) used in the field work, the application of cement based permeation grouting is still largely a trial and error process in the current practice, especially in the local construction industry. It is seen possible to use simple formulas in order to select the injection parameters and to evaluate their inter-relationship, as well as to optimize injection spacing and times with respect to injection source dimensions and in-situ permeability. The validity of spherical and cylindrical flow model was not verified by any past research works covered in the literature review. Therefore, a theoretical investigation including grout flow models and significant grout parameters for the design of permeation grouting was conducted in this study. This two grout flow models were applied for three grout mixes prepared for w/c=0.75, w/c=1.00 and w/c=1.25 in this study. The relations between injection times, radius, pump pressure and flow rate for both flow models were investigated and the results were presented. Furthermore, in order to investigate these two flow model, some rheological properties of the grout mixes, particle size distribution of the cement used in this study and some geotechnical properties of the sand used in this work were defined and presented.

General relativistic two-temperature accretion solutions for spherical flows around black holes

Matter falling onto black holes is hot, fully ionized and has to be necessarily transonic. Since the electrons are responsible for radiative cooling via processes like synchrotron, bremsstrahlung and inverse-Compton, the electron gas and proton gas are supposed to settle into two separate temperature distribution. But the problem with two-temperature flow is that there is one more variable than the number of equations. Accretion flow in its simplest form is radial, which has two constants of motion, while the flow variables are the radial bulk three-velocity, electron and proton temperatures. Therefore, unlike single temperature flow, in the two-temperature regime, there are multiple transonic solutions, nonunique for any given set of constants of motion with a large variation in sonic points. We invoked the second law of thermodynamics to find a possible way to break the degeneracy, by showing only one of the solutions among all possible, has maximum entropy and therefore is the correct solution. By considering these correct solutions, we showed that the accretion efficiency increase with the increase in the mass accretion rate. We showed that radial flow onto super-massive black hole can radiate with efficiency more than 10%, if the accretion rate is more than 60% of the Eddington accretion rate, but accretion onto stellar-mass black hole achieve the same efficiency, when it is close to the Eddington limit. We also showed that dissipative heat quantitatively affects the two temperature solution. In the presence of explicit heat processes, the Coulomb coupling is weak.

Entropy Generation Analysis of Spherical and Non-Spherical Ag-Water Nanofluids in a Porous Medium with Magnetic and Porous Dissipation

The analysis of the entropy generation in spherical and non-spherical Ag-water nanofluids flow and heat transfer over a stretching cylinder is carried out in the presence of a viscous dissipation, magnetic field and porous medium. The Hamilton-Crosser nanofluid model for effective thermal conductivity is used to calculate the velocity and temperature profile. The modeled governing partial differential equations are transformed by a similarity transformation into self-similar ordinary differential equations and are solved numerically using the shooting method. The numerical solutions obtained for the velocity and temperature profile are used to calculate entropy generation number. Finally, changes in velocity, temperature, entropy generation number with physical flow parameters are presented graphically and discussed physically in detail. The results show that the entropy generation in the boundary layer flow of nanofluid-containing spherical nanoparticles is lower. It is also found that the flat stretching sheet causes less entropy generation compared to stretching cylinder.

The study on calculating flow pressure for tight oil well

This paper presents a new method to calculate flow pressure for tight oil well with partial penetration fracture. The solution to the diffusivity equation can be obtained using dimensionless transformation, Laplace transformation, Fourier cosine transformation and other mathematical methods. The calculation results of relative simple system are in good agreement with Kuchuk and Brighan’s analytical solution, suggesting that this method in this paper is accurate. The pressure dynamic flow curve can be divided into four stages: transient flow, first linear flow, first radial flow, and boundary dominated flow. The effect of parameters including fracture orientation, fracture length, the degree of penetration, reservoir boundary anisotropy, reservoir boundary conditions and reservoir width on the pressure and pressure derivative was fully investigated in detail. The fracture length mainly affects the early linear flow, and permeability anisotropy mainly affects the mid-radial flow, and the degree of penetration in the reservoir and fracture orientation mainly affect the late spherical flow, and the boundary conditions and reservoir border width mainly affect the control flow. The method can be used to determine the optimal degree of opening shot, vertical permeability and other useful parameters.

Rotating accretion flows in D dimensions: Sonic points, critical points, and photon spheres

We give the formulation and the general analysis of the rotational accretion problem on $D$-dimensional spherical spacetime and investigate sonic points and critical points. First, we construct the simple two-dimensional rotating accretion flow model in general $D$-dimensional static spherically symmetric spacetime and formulate the problem. The flow forms a two-dimensional disk lying on the equatorial plane and the disk is assumed to be geometrically thin and has uniform distribution in the polar angle directions. Analyzing the critical point of the problem, we give the conditions for the critical point and its classification explicitly and show the coincidence with the sonic point for generic equation of state (EOS). Next, adopting the EOS of ideal photon gas to the analysis, we reveal that there always exists a correspondence between the sonic points and the photon spheres of the spacetime. Our main result is that the sonic point of the rotating accretion flow of ideal photon gas must be on (one of) the unstable photon sphere(s) of the spacetime in arbitrary spacetime dimensions. This paper extends this correspondence for spherical flows shown in the authors' previous work to rotating accretion disks.

Asymptotic scaling laws and semi-similarity solutions for a finite-source spherical blast wave

A spherical blast wave generated by a sudden release of a sphere of compressed gas is an important model problem to understand blast phenomena such as volcanic eruptions and explosive detonations. The resulting explosion flow physics, such as the instability at the gas contact discontinuity and the interaction between the shock wave and the gas contact, are dictated by the initial pressure and sound-speed ratios between the compressed gas and the ambience. Since the initial pressure and sound-speed ratios vary over a wide range in practical applications, it is of interest to investigate the scaling laws and similarity solutions for the spherical symmetric explosion flow. In the present study, numerical simulation of a spherical blast wave is performed. A long-term length scale that incorporates the initial charge radius and the initial pressure ratio is introduced. The trajectories of the main shock normalized by the long-term length scale for a wide range of parameters collapse after a short transition time, indicating an asymptotic similarity solution exists for the far field in the long term. With the assistance of this similarity solution, the full evolution of the main shock can be obtained semi-analytically. For near-field features, i.e. the gas contact and the secondary shock wave, only semi-similarity solutions are observed, which depend on the initial sound-speed ratio but not the initial pressure ratio. The gas contact and the secondary shock share the same scaling relations. Asymptotic analysis is performed to obtain the short-term dynamics of the gas contact, including the gas contact acceleration and the Atwood number, which are the key parameters determining the Rayleigh–Taylor instability development at the gas contact. The asymptotic contact radius as$t\rightarrow \infty$is also obtained, which is found to be well represented by the long-term length scale and thus only depends on the initial pressure ratio. A simple model of an oscillating bubble is employed to explain the scaling relation of the asymptotic gas contact radius.

A new numerical well testing approach: Application to characterization of complex fault structures

Abstract Fluid flow in hydrocarbon reservoirs and consequently optimum scenario for hydrocarbon production, is heavily influenced by reservoir heterogeneities. Faults are one of the most common types of heterogeneity found in reservoirs. Leaky faults, baffles (limited extent faults) and complex multiple fault geometries are among the most complicated and important types of faults that are difficult to characterize. Leaky faults, unlike the sealing faults, are in partial communication with other portions of the reservoir. Because of faults' effect on reservoir connectivity and possible infill drilling plan for accessing all parts of the reservoirs, possible communication across the fault must be precisely modeled. In order to detect the effect of a fault on communication within the reservoir, we need to analyze dynamic data. There are a few analytical methods for modelling partially communicating faults, however, these methods may not be accurate enough and may be limited in application, especially in complex situations. Numerical methods (i.e. finite difference or finite element) are also not computationally economical when a large number of grid blocks are simulated. In the current work, the Fast Marching Method (FMM) is applied to effectively mimic fluid flow in the heterogeneous areas, such as complex faults. It is shown that FMM can capture the effect of different fault configurations on the bottom hole pressure and is also able to capture different linear, radial and spherical flows.

Reflection of a Rarefaction Wave from the Center of Symmetry: Theoretical Analysis of the Flow Features and Calculation by the Method of Characteristics

For an unsteady spherical rarefaction wave in an ideal (inviscid and non-heat-conducting) gas, the features of the flow near the reflection of the first characteristic from the center of symmetry are investigated. Computations performed by the method of characteristics on nearly uniform grids usually used in such problems reveal sawtooth irregularities in the parameter distributions near the reflection point, whereas similar irregularities in the cylindrical and plane cases are absent. The amplitudes of the irregularities and the sizes of the domains where they are observed remain nearly unchanged when the number of points of the characteristic grid is increased by many times. Away from the reflection point in both time and space, the numerical solution completely “forgets” about the irregularities. This finding explains why these irregularities were ignored or overlooked earlier, but the nature of this phenomenon remains an open question. The present study has established that the spherical rarefaction flow near the reflection point differs fundamentally in structure from its plane and cylindrical counterparts. In the spherical case, the rarefaction flow near the reflection point was found to be nearly conical (entirely conical in the linear approximation). Allowance for this feature in the method of characteristics led to continuous regular distributions of the parameters. The performed analysis and computations revealed that a spherical rarefaction wave is strengthened (cumulates) theoretically unlimitedly in a small neighborhood of the reflection point (center of symmetry) of the first characteristic. Moreover, the claim that a gradient catastrophe occurs in this neighborhood was found to be untenable.

Spherical accretion in giant elliptical galaxies: multitransonicity, shocks, and implications on AGN feedback

Isolated massive elliptical galaxies, or that are present at the center of cool-core clusters, are believed to be powered by hot gas accretion directly from their surrounding hot X-ray emitting gaseous medium. This leads to a giant Bondi-type spherical/quasi-spherical accretion flow onto their host SMBHs, with the accretion flow region extending well beyond the Bondi radius. In this work, we present a detailed study of Bondi-type spherical flow in the context of these massive ellipticals by incorporating the effect of entire gravitational potential of the host galaxy in the presence of cosmological constant $\Lambda$, considering a five-component galactic system (SMBH + stellar + dark matter + hot gas + $\Lambda$). The current work is an extension of Ghosh \& Banik (2015), who studied only the cosmological aspect of the problem. The galactic contribution to the potential renders the (adiabatic) spherical flow to become {\it multi-transonic} in nature, with the flow topology and flow structure significantly deviating from that of classical Bondi solution. More notably, corresponding to moderate to higher values of galactic mass-to-light ratios, we obtain Rankine-Hugoniot shocks in spherical wind flows. Galactic potential enhances the Bondi accretion rate. Our study reveals that there is a strict lower limit of ambient temperature below which no Bondi accretion can be triggered; which is as high as $\sim 9 \times 10^6 \, K$ for flows from hot ISM-phase, indicating that the hot phase tightly regulates the fueling of host nucleus. Our findings may have wider implications, particularly in the context of outflow/jet dynamics, and radio-AGN feedback, associated with these massive galaxies in the contemporary Universe.

One-dimensional turbulence modeling for cylindrical and spherical flows: model formulation and application

The one-dimensional turbulence (ODT) model resolves a full range of time and length scales and is computationally efficient. ODT has been applied to a wide range of complex multi-scale flows, such as turbulent combustion. Previous ODT comparisons to experimental data have focused mainly on planar flows. Applications to cylindrical flows, such as round jets, have been based on rough analogies, e.g., by exploiting the fortuitous consistency of the similarity scalings of temporally developing planar jets and spatially developing round jets. To obtain a more systematic treatment, a new formulation of the ODT model in cylindrical and spherical coordinates is presented here. The model is written in terms of a geometric factor so that planar, cylindrical, and spherical configurations are represented in the same way. Temporal and spatial versions of the model are presented. A Lagrangian finite-volume implementation is used with a dynamically adaptive mesh. The adaptive mesh facilitates the implementation of cylindrical and spherical versions of the triplet map, which is used to model turbulent advection (eddy events) in the one-dimensional flow coordinate. In cylindrical and spherical coordinates, geometric stretching of the three triplet map images occurs due to the radial dependence of volume, with the stretching being strongest near the centerline. Two triplet map variants, TMA and TMB, are presented. In TMA, the three map images have the same volume, but different radial segment lengths. In TMB, the three map images have the same radial segment lengths, but different segment volumes. Cylindrical results are presented for temporal pipe flow, a spatial nonreacting jet, and a spatial nonreacting jet flame. These results compare very well to direct numerical simulation for the pipe flow, and to experimental data for the jets. The nonreacting jet treatment overpredicts velocity fluctuations near the centerline, due to the geometric stretching of the triplet maps and its effect on the eddy event rate distribution. TMB performs better than TMA. A hybrid planar-TMB (PTMB) approach is also presented, which further improves the results. TMA, TMB, and PTMB are nearly identical in the pipe flow where the key dynamics occur near the wall away from the centerline. The jet flame illustrates effects of variable density and viscosity, including dilatational effects.