Single-fluid Case Research Articles

Effects of diamagnetic drift on nonlinear interaction between multi-helicity neoclassical tearing modes

A numerical study of the diamagnetic drift effect on the nonlinear interaction between multi-helicity neoclassical tearing modes (NTMs) is carried out using a set of four-field equations including two-fluid effects. The results show that, in contrast to the single-fluid case, 5/3 NTM cannot be completely suppressed by 3/2 NTM with diamagnetic drift flow. Both modes exhibit oscillation and coexist in the saturated phase. To better understand the effect of the diamagnetic drift flow on multiple-helicity NTMs, the influence of typical relevant parameters is investigated. It is found that the average saturated magnetic island width increases with increasing bootstrap current fraction f b but decreases with the ion skin depth δ. In addition, as the ratio of parallel to perpendicular transport coefficients χ ∥/χ ⊥ increases, the average saturated magnetic island widths of the 3/2 and 5/3 NTMs increase. The underlying mechanisms behind these observations are discussed in detail.

Investigating dark matter-admixed neutron stars with NITR equation of state in light of PSR J0952-0607

The fastest and heaviest pulsar, PSR J0952-0607, with a mass of M = 2.35±0.17 M ⊙, has recently been discovered in the disk of the Milky Way Galaxy. In response to this discovery, a new RMF model, 'NITR' has been developed. The NITR model's naturalness has been confirmed by assessing its validity for various finite nuclei and nuclear matter properties, including incompressibility, symmetry energy, and slope parameter values of 225.11, 31.69, and 43.86 MeV, respectively. These values satisfy the empirical/experimental limits currently available. The maximum mass and canonical radius of a neutron star (NS) calculated using the NITR model parameters are 2.355 M ⊙ and 13.13 km, respectively, which fall within the range of PSR J0952-0607 and the latest NICER limit. This study aims to test the consistency of the NITR model by applying it to various systems. As a result, its validity is extensively calibrated, and all the nuclear matter and NS properties of the NITR model are compared with two established models such as IOPB-I and FSUGarnet. In addition, the NITR model equation of state (EOS) is employed to obtain the properties of a dark matter admixed NS (DMANS) using two approaches (I) single-fluid and (II) two-fluid approaches. In both cases, the EOS becomes softer due to DM interactions, which reduces various macroscopic properties such as maximum mass, radius, tidal deformability, etc. The various observational data such as NICER and HESS are used to constrain the amount of DM in both cases. Moreover, we discuss the impact of dark matter (DM) on the nonradial f-mode frequency of the NS in a single fluid case only and try to constrain the amount of DM using different theoretical limits available in the literature.

Revised formalism for slowly rotating superfluid neutron stars in general relativity

We discuss slowly-rotating, general relativistic, superfluid neutron stars in the Hartle-Thorne formulation. The composition of the stars is described by a simple two-fluid model which accounts for superfluid neutrons and all other constituents. We apply a perturbed matching framework to derive a new formalism for slowly-rotating superfluid neutron stars, valid up to second-order perturbation theory, building on the original formulation reported by Andersson and Comer in 2001. The present study constitutes an extension of previous work in the single-fluid case where it was shown that the Hartle-Thorne formalism needs to be amended since it does not provide the correct results when the energy density does not vanish at the surface of the star. We discuss in detail the corrections that need to be applied to the original two-fluid formalism in order to account for non vanishing energy densities at the boundary. In the process, we also find a correction needed in the computation of the deformation of the stellar surface in the original two-fluid model in all cases (irrespective of the value of the energy density at the surface). The discrepancies found between the two formalisms are illustrated by building numerical stellar models, focusing on the comparison in the calculation of the stellar mass, the deformation of the star, and in the Kepler limit of rotation. In particular, using a toy-model equation of state for which the energy density does not vanish at the boundary of the star we demonstrate that the corrections to the formalism we find impact the structure of slowly-rotating superfluid neutron stars in a significant way.

Anisotropic two-fluid stellar objects in general relativity

We apply the $1+1+2$ covariant semitetrad approach to describe a general static and spherically symmetric relativistic stellar object which contains two fluids with anisotropic pressure. The corresponding Tolman-Oppenheimer-Volkoff equations are then obtained in covariant form for the anisotropic case. These equations are used to obtain new exact solutions using direct resolution and reconstruction techniques. Finally, we show that three of the generating theorems known for the single-fluid case can also be used to obtain two-fluid solutions from single-fluid ones.

Two-fluid stellar objects in general relativity: The covariant formulation

We apply the 1+1+2 covariant approach to describe a general static and spherically symmetric relativistic stellar object which contains two interacting fluids. We then use the 1+1+2 equations to derive the corresponding Tolman-Oppenheimer-Volkoff (TOV) equations in covariant form in the isotropic, non-interacting case. These equations are used to obtain new exact solutions by means of direct resolution and reconstruction techniques. Finally, we show that the generating theorem known for the single fluid case can also be used to obtain two-fluid solutions from single fluid ones.

Effect of imposed shear on the dynamics of a contaminated two-layer film flow down a slippery incline

The linear instability of a surfactant-laden two-layer falling film over an inclined slippery wall is analyzed under the influence of external shear, which is imposed on the top surface of the flow. The free surface of the flow and the interface among the fluids are contaminated by insoluble surfactants. Dynamics of the fluid layers are governed by the Navier–Stokes equations, and the surfactant transport equations regulate the motion of the insoluble surfactants at the interface and free surface. Instability mechanisms are compared by imposing the external shear along and opposite to the flow direction. A coupled Orr–Sommerfeld system of equations is derived using the perturbation technique and normal mode analysis. The eigenmodes corresponding to the Orr–Sommerfeld eigenvalue problem are obtained by employing the spectral collocation method. The numerical results imply that the stronger external shear destabilizes the interface mode instability. However, a stabilizing impact of the external shear on the surface mode is noticed if the shear is imposed in the flow direction, which is in contrast to the role of imposed external shear on the surface mode for a surfactant-laden single layer falling film. Furthermore, in the presence of strong imposed shear, the overall stabilization of the surface mode by wall velocity slip for the stratified two-fluid flow is also contrary to that of the single fluid case. The interface mode behaves differently in the two zones at the moderate Reynolds numbers, and higher external shear magnifies the interfacial instability in both zones. An opposite trend is observed in the case of surface instability. Moreover, the impression of shear mode on the primary instability is analyzed in the high Reynolds number regime with sufficiently low inclination angle. Under such configuration, dominance of the shear mode over the surface mode is observed due to the weaker impact of the gravitational force on the surface instability. The shear mode can also be stabilized by applying the external shear in the counter direction of the streamwise flow. Conclusively, the extra imposed shear on the stratified two-layer falling film plays an active role in the control of the attitude of the instabilities.

Scale-to-scale energy transfer rate in compressible two-fluid plasma turbulence.

We derive the exact relation for the energy transfer in three-dimensional compressible two-fluid plasma turbulence. In the long-time limit, we obtain an exact law which expresses the scale-to-scale average energy flux rate in terms of two point increments of the fluid variables of each species, electric and magnetic field and current density, and puts a strong constraint on the turbulent dynamics. The incompressible single fluid and two-fluid limits and the compressible single fluid limit are recovered under appropriate assumption. In the single fluid limits, analyses are done with and without neglecting the electron mass thereby making the exact relation suitable for a broader range of application. In the compressible two-fluid regime, the total energy flux rate, unlike the single fluid case, is found to be unaltered by the presence of a background magnetic field. The exact relation provides a way to test whether a range of scales in a plasma is inertial or dissipative and is essential to understand the nonlinear nature of both space and dilute astrophysical plasmas.

Numerical study of variable density turbulence interaction with a normal shock wave

Accurate numerical simulations of shock–turbulence interaction (STI) are conducted with a hybrid monotonicity-preserving–compact-finite-difference scheme for a detailed study of STI in variable density flows. Theoretical and numerical assessments of data confirm that all turbulence scales as well as the STI are well captured by the computational method. Linear interaction approximation (LIA) convergence tests conducted with the shock-capturing simulations exhibit a similar trend of converging to LIA predictions to shock-resolving direct numerical simulations (DNS). The effects of density variations on STI are studied by comparing the results corresponding to an upstream multi-fluid mixture with the single-fluid case. The results show that for the current parameter ranges, the turbulence amplification by the normal shock wave is much higher and the reduction in turbulence length scales is more significant when strong density variations exist. Turbulent mixing enhancement by the shock is also increased and stronger mixing asymmetry in the postshock region is observed when there is significant density variation. The turbulence structure is strongly modified by the shock wave, with a differential distribution of turbulent statistics in regions having different densities. The dominant mechanisms behind the variable density STI are identified by analysing the transport equations for the Reynolds stresses, vorticity, normalized mass flux and density specific volume covariance.

Shock structures of astrospheres

The interaction between a supersonic stellar wind and a (super-)sonic interstellar wind has recently been viewed with new interest. We here first give an overview of the modeling, which includes the heliosphere as an example of a special astrosphere. Then we concentrate on the shock structures of fluid models, especially of hydrodynamic (HD) models. More involved models taking into account radiation transfer and magnetic fields are briefly sketched. Even the relatively simple HD models show a rich shock structure, which might be observable in some objects. We employ a single fluid model to study these complex shock structures, and compare the results obtained including heating and cooling with results obtained without these effects. Furthermore, we show that in the hypersonic case valuable information of the shock structure can be obtained from the Rankine-Hugoniot equations. We solved the Euler equations for the single fluid case and also for a case including cooling and heating. We also discuss the analytical Rankine-Hugoniot relations and their relevance to observations. We show that the only obtainable length scale is the termination shock distance. Moreover, the so-called thin shell approximation is usually not valid. We present the shock structure in the model that includes heating and cooling, which differs remarkably from that of a single fluid scenario in the region of the shocked interstellar medium. We find that the heating and cooling is mainly important in this region and is negligible in the regions dominated by the stellar wind beyond an inner boundary.

An immersed boundary method for two-fluid mixtures

We present an immersed boundary method for interactions between elastic boundaries and mixtures of two fluids. Each fluid has its own velocity field and volume-fraction. A penalty method is used to enforce the condition that both fluidsʼ velocities agree with that of the elastic boundaries. The method is applied to several problems: Taylorʼs swimming sheet problem for a mixture of two viscous fluids, peristaltic pumping of a mixture of two viscous fluids, with and without immersed particles, and peristaltic pumping of a mixture of a viscous fluid and a viscoelastic fluid. The swimming sheet and peristalsis problems have received much attention recently in the context of a single viscoelastic fluid. Numerical results demonstrate that the method converges and show its capability to handle a number of flow problems of substantial current interest. They illustrate that for each of these problems, the relative motion between the two fluids changes the observed behaviors profoundly compared to the single fluid case.

Quasi-periodic oscillations in superfluid magnetars

We study the time-evolution of axisymmetric oscillations of superfluid magnetars with a poloidal magnetic field and an elastic crust, working in Newtonian gravity. Extending earlier models, we study the effects of composition gradients and entrainment on the magneto-elastic wave spectrum and on the potential identification of the observed quasi periodic oscillations (QPOs). We use two-fluid polytropic equations of state to construct our stellar models, which mimic realistic composition gradient configurations. The basic features of the axial axisymmetric spectrum of normal fluid stars are reproduced by our results and in addition we find several magneto-elastic waves with a mixed character. In the core, these oscillations mimic the shear mode pattern of the crust as a result of the strong dynamical coupling between these two regions. Incorporating the most recent entrainment configurations in our models, we find that they have a double effect on the spectrum: the magnetic oscillations of the core have a frequency enhancement, while the mixed magneto-elastic waves originating in the crust are moved towards the frequencies of the single-fluid case. The distribution of lower-frequency magneto-elastic oscillations for our models is qualitatively similar to the observed magnetar QPOs with $\nu < 155 $Hz. \emph{In particular, some of these QPOs could represent mixed magneto-elastic oscillations with frequencies not greatly different from the crustal modes of an unmagnetised star.} We find that many QPOs could even be accounted for using a model with a relatively weak polar field of $B_{p} \simeq 3\cdot 10^{14}$G, because of the superfluid enhancement of magnetic oscillations. Finally, we discuss the possible identification of 625 and 1837~Hz QPOs either with non-axisymmetric modes or with high-frequency axisymmetric QPOs excited by crustal mode overtones.

Molecular dynamics of fluids and droplets in patterned nanochannels

Molecular dynamics studies of chain molecule fluids in nanochannels reveal new kinds of stationary flows when the walls of the channel are patterned chemically. In the single-fluid case, the patterning is shown to give rise to spatial switching between Poiseuille-like and plug-flow behaviours. In the two-fluid case, such that the two immiscible fluids wet different patches of a chemically patterned channel, the velocity field of the individual components is modulated in a way that is phase shifted relative to the single-fluid case. When the channel is homogeneous and wetting to one fluid component but non-wetting to the other then the resulting coating of the channel walls provides dissipation and thus stabilises flow of the other component. Wettability and geometrical patterning of substrates gives rise to a lotus-like effect in a rolling motion of nanodroplets along the substrate.

An efficient numerical method for the two-fluid Stokes equations with a moving immersed boundary

We consider the immersed boundary problem in which the boundary separates two very viscous fluids with differing viscosities. The moving elastic boundary may exert a force on the local fluid. The model solution is obtained using the immersed interface method, which computes second-order accurate approximations by incorporating known jumps in the solution or its derivatives into a finite difference method. These jump conditions become coupled when the fluid viscosity has a jump across the boundary, and this coupling renders the application of the immersed interface method challenging. We present a method that first uses boundary integral equations to reduce the two-fluid Stokes problem to the single-fluid case, and then solves the single-fluid problem using the immersed interface method. Using this method, we assess, through two numerical examples, how the fluid dynamics are affected by differing viscosities in the two-fluid regions. We also propose an implicit algorithm and a fractional-step algorithm for advancing the boundary position. Because both algorithms make use of the integral form of the solution, neither one requires the solution of a large system of coupled nonlinear equations, as is traditionally the case. Numerical results suggest that, for sufficiently stiff problems, the fractional time-stepping algorithm is the most efficient, in the sense that it allows the largest time-interval between subsequent updates of global model solutions.

Nonlinear wave propagation in binary mixtures of Euler fluids

In the present paper, we study the propagation of acceleration and shock waves in a binary mixture of ideal Euler fluids, assuming that the difference between the atomic masses of the constituents is negligible. We evaluate the characteristic speeds, proving that they can be separated into two groups: one is related to the case of a single Euler fluid, provided that an average ratio of specific heats is introduced; the other is new and related to the propagation speed due to diffusion. We evaluate the critical time for sound acceleration waves and compare its value to that of a single fluid. We then study shock waves, showing that three types of shock waves appear: sonic and contact shocks, which have counterparts in the single fluid case, and the diffusive shock, which is peculiar to the mixture. We discuss the admissibility of the shock waves using the Lax-Liu conditions and the entropy growth criterion. It is proved that the sonic and the characteristic shock obey the same properties as in the single fluid case, while for the diffusive shock there exists a locally exceptional case that is determined by a particular value of the concentration of the constituents, for which the genuine nonlinearity is lost and no shocks are admissible. For other values of the unperturbed concentration, the diffusive shock is stable in a bounded interval of admissibility.

Two-Fluid Marangoni–Bénard Convection with a Deformable Interface

Two immiscible fluid layers that are subjected to a temperature gradient perpendicular to their interface exhibit a range of behaviors that is considerably richer than for the single-fluid case. We describe a numerical technique for calculating thermally driven flows in two fluid layers which uses a simple technique based on a Landau transformation to map the physical domain into a reference domain, enabling the unknown location of the deformable interface to be determined. The coupled system of nonlinear partial differential equations, comprising mapping, continuity, momentum, and energy equations and the appropriate boundary conditions, is solved using the finite-element method in two-dimensional domains. Numerical bifurcation techniques are used to investigate the multiplicity of the solution set. The case of heating from above is considered in some detail and the results of finite-element computations are compared with linear stability calculations performed on unbounded domains. The principal advantages of the finite-element approach are the ability to determine the effect of non-90° contact angles (when the conducting solution no longer exists and traditional linear stability approaches fail), the ability to determine the role of finite aspect ratio domains and the relative volume fractions of the two fluids, and the capability of calculating the nonlinear development of flows beyond the critical temperature gradient.

Studies on the Reservoir Performance of a Water Drive Field

From the mathematical point of view, water drive fields present us many interesting problems. There are several solutions for the diffusivity equation for unsteady-state flow of a compressible liquid in a porous media. However, all these solutions are for the single fluid case. In the solutions to be presented here, the writer tried to solve the equation under several assumptions, taking into account the existence of the aquifer around the oil reservoir and applying the Laplace transformation.

Effect of Dip on Five-Spot Sweep Pattern

Published in Petroleum Transactions, Volume 207, 1956, pages 111–117. Abstract Distortions caused by dip on five-spot sweep pattern were determined by visual experiments on a model composed of parallel plates of glass with holes drilled at the well positions. This model was designed to simulate a reservoir depleted of its primary oil and containing a gas phase. Prior to interference, it was found that the injected fluid took a nearly circular shape about the injection well for the cases studied. However, these circles moved downdip considerably under the influence of gravity. It was concluded that pattern flooding should be practical for many dipping reservoirs. A quantitative measure is given of how much the well position should be altered to allow for dip. Experimental determinations of injectivity showed this to be about the same in a dipping reservoir as in a horizontal one. Introduction This investigation was undertaken at the suggestion of the Pacific Coast area, Shell Oil Co., where a pilot water flood in a dipping reservoir was being carried out at the time this paper was in preparation. Prior to the undertaking of this flood there had been considerable speculation as to how gravity would distort a flood pattern in a dipping reservoir. The thought was expressed in some quarters that a five-spot flood would not be applicable to a dipping reservoir because the flood pattern would elongate in the direction of dip and would never form a "seal" around each five-spot. If the flood pattern elongated sufficiently, both the oil and the water might flow largely downdip rather than into the pattern producers. In view of the importance of the California water floods, it was deemed important to determine the effects of gravity on a five-spot pattern in a dipping reservoir. Previous work had also been carried out in this laboratory, but only for the single-fluid case. The present investigation extends these results to multifluid cases. Several sets of pictures of flood fronts are presented so that the process can be visualized. From these pictures, curves are obtained which show the correct drilling position for achieving equal times of oil-bank breakthrough. Several injectivity curves are also presented which may aid in prediction of injection rate.