Class Of Exact Solutions Research Articles

Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface

A class of exact solutions is presented describing the time evolution of insoluble surfactant to a stagnant cap equilibrium on the surface of deep water in the Stokes flow regime at zero capillary number and infinite surface Péclet number. This is done by demonstrating, in a two-dimensional model setting, the relevance of the forced complex Burgers equation to this problem when a linear equation of state relates the surface tension to the surfactant concentration. A complex-variable version of the method of characteristics can then be deployed to find an implicit representation of the general solution. A special class of initial conditions is considered for which the associated solutions can be given explicitly. The new exact solutions, which include both spreading and compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads to a forced complex Burgers equation with linear damping. Generalized exact solutions to the latter equation at infinite surface Péclet number are also found and used to study how reaction effects destroy the surfactant cap equilibrium.

A Tolman-like Compact Model with Conformal Geometry.

In this investigation, we study a model of a charged anisotropic compact star by assuming a relationship between the metric functions arising from a conformal symmetry. This mechanism leads to a first-order differential equation containing pressure anisotropy and the electric field. Particular forms of the electric field intensity, combined with the Tolman VII metric, are used to solve the Einstein–Maxwell field equations. New classes of exact solutions generated are expressed in terms of elementary functions. For specific parameter values based on the physical requirements, it is shown that the model satisfies the causality, stability and energy conditions. Numerical values generated for masses, radii, central densities, surface redshifts and compactness factors are consistent with compact objects such as PSR J1614-2230 and SMC X-1.

Invariance properties of exact solutions of the radiative transfer equation

In this work, special invariance properties of a class of exact solutions of the radiative transfer equation (RTE) pertaining to a uniform Lambertian illumination of any non-absorbing homogeneous and inhomogeneous volume are presented and discussed. This class of solutions of the RTE traces a reference ground under which light propagation can be studied in a special simplified regime. Despite the difficulties to obtain general solutions of the radiative transfer equation, the condition of Lambertian illumination determines a unique regime of photon transport where quite easy and simple invariant solutions can be obtained in all generality for homogeneous and inhomogeneous geometries. These solutions are invariant both with respect to the geometry (size and shape of the volume) and with respect to the scattering properties, i.e. scattering coefficient, scattering function and homogeneity of the considered domain. Another evident advantage of these solutions is that they are exact solutions known with arbitrary precision and can thus be used as reference standard for photon migration studies.

Exact Solutions to the Navier–Stokes Equations with Couple Stresses

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

Magnetized Kerr\u2013Newman\u2013Taub-NUT spacetimes

We find a new class of exact solutions in the Einstein–Maxwell theory by employing the Ernst magnetization process to the Kerr–Newman–Taub-NUT spacetimes. We study the solutions and find that they are regular everywhere. We also find the quasilocal conserved quantities for the spacetimes, the corresponding Smarr formula and the first law of thermodynamics.

Anisotropic generalization of Buchdahl bound for specific stellar models

Anisotropy is one factor that appears to be significantly important in the studies of relativistic compact stars. In this paper, we make a generalization of the Buchdahl limit by incorporating an anisotropic effect for a selected class of exact solutions describing anisotropic stellar objects. In the isotropic case of a homogeneous distribution, we regain the Buchdahl limit 2M/R le 8/9. Our investigation shows a direct link between the maximum allowed compactness and pressure anisotropy vi-a-vis geometry of the associated 3-space.

On primordial black holes from rapid turns in two-field models

We study rapid-turn trajectories in a class of two-field cosmological models, whose scalar manifold is the Poincaré disk. Background solutions in multi-field inflation, with field-space trajectories exhibiting sharp turns, can seed primordial black hole (PBH) formation. We investigate a class of exact solutions with hidden symmetry and show that they exhibit the kind of transient rapid-turn period, needed to induce PBH generation. Furthermore, we relax the symmetry condition and find, in a certain regime, modified solutions with improved behavior of the Hubble η-parameter, which preserve the desired shape of the turning rate function. Interestingly, the modified solutions describe a brief ultra-slow roll phase, followed by long-term slow roll inflation. It is notable that slow roll occurs near the center (not near the boundary) of the Poincaré disk, unlike in the standard α-attractor case.

Generating new class of exact solutions to the Einstein–Maxwell system

For a static spherically symmetric charged distribution of matter, the Einstein–Maxwell system is solved. The solution is regular, well behaved and complies with all the requirements of a realistic stellar model. For some specific choices of the model parameters, the new class of solutions is shown to regain some of the previously reported realistic models of compact stars.

Magnetogasdynamic shock wave propagation using the method of group invariance in rotating medium with the flux of monochromatic radiation and azimuthal magnetic field

Abstract The propagation of a cylindrical shock wave in rotating medium with azimuthal magnetic field under the action of monochromatic radiation using a method of group invariance is investigated. To derive similarity solutions as well as exact solutions, the group invariance technique is used. All classes of the solutions depending on the absorption coefficient are discussed by considering absorption coefficient to be variable or constant. A similarity solution is obtained, when the absorption coefficient is assumed to be variable. Two cases of solutions with a power law shock path are obtained by the different choices of arbitrary constants involving in the infinitesimal generators of the Lie group of transformations. To obtain the similarity solution in the case of the power law shock path, the density, magnetic field, axial and azimuthal velocity components are assumed to be varying and obeying power laws in the undisturbed medium. It is observed that with increase in the values of Alfven Mach number, adiabatic exponent and rotational parameter, shock strength decreases. The effects of variation of magnetic field strength, adiabatic exponent, rotational parameter and initial magnetic field variation index on the flow variables and on shock waves are analyzed graphically. Also, all classes of exact solutions are obtained by considering a constant absorption coefficient.

New exact even-dimensional pure Lovelock interior metrics

While it is well known that Nth-order pure Lovelock spacetimes of dimension $$d = 2N + 1$$ do not admit bounded compact objects, this is not the case for the $$d = 2N + 2$$ case. In the $$N = 2$$ , pure Gauss–Bonnet gravity exact models for the Vaidya–Tikekar superdense star ansatz are investigated and new classes of metrics are discovered in terms of elementary functions. An astrophysical model of a stellar distribution that is consistent with the physical expectations extrapolated from four-dimensional gravity is exhibited. The isothermal sphere is re-considered and found to be inconsistent in pure Lovelock theory. Finally, we briefly consider the case of the third-order Lovelock polynomial term and several classes of exact solutions emerge.

Charged analogues of isotropic compact stars model with buchdahl metric in general relativity

In this work, we examine a spherically symmetric compact body with isotropic pressure profile, in this context we obtain a new class of exact solutions of Einstein's-Maxwell field equation for compact stars with uniform charged distributions on the basis of Pseudo-spheroidal space-time with a particular form of electric field intensity and the metric potential g_rr. Indicating these two parameters takes into account further examination to be done in deciding unknown constants and depicts the compact strange star candidates likes PSR J1614-2230, 4U 1608-52, SAX J1808.4-3658, 4U 1538-52, SMC X-1, Her X-1, and Cen X-3. By the isotropic Tolman-Oppenhimer-Volkoff(TOV)equation, We explore the equilibrium among hydrostatic, gravitational and electric forces. Then, we analyze the stability of the model through the adiabatic index(gamma) and velocity of sound (0 <dp/c^2dr< 1). we additionally talk about other physical features of this model like, for example, the pressure, redshift, density, energy conditions and mass-radius of the stars in detail and demonstrate that our results are satisfying all the basic prerequisites of a physically legitimate stellar model. We have seen the measurement of basic physical parameter such as pressure, density, energy and redshift are satisfied the reality condition. All the physical quantities such as density, pressure pressure-density ratio and speed of sound is monotonically decreasing. The outcomes acquired are valuable in exploring the strength of other compact objects like white dwarfs, gravastars and neutron stars. Finally, we have shown that the obtained solutions are compatible with observational data for compact objects.

A class of exact solutions with spatial acceleration for the description of viscous incompressible fluid flows in the field of mass forces

The article presents a new class of exact solutions to the system of Navier–Stokes equations, which allows one to take into account the nonlinear distribution of the pressure field and the influ-ence of external volumetric forces, as well as the possibility of horizontal fluid outflow/inflow when modeling its vertical motion. This class is a generalization of the Lin–Sidorov–Aristov class, which assumes the linear distribution of two projections of the fluid flow velocity vector along a part of the coordinates and the independence of the third projection of the velocity vector from these coor-dinates.

Thermodynamically stable asymptotically flat hairy black holes with a dilaton potential: the general case

We extend the analysis, initiated in [1], of the thermodynamic stability of four-dimensional asymptotically flat hairy black holes by considering a general class of exact solutions in Einstein-Maxwell-dilaton theory with a non-trivial dilaton potential. We find that, regardless of the values of the parameters of the theory, there always exists a sub-class of hairy black holes that are thermodynamically stable and have the extremal limit well defined. This generic feature that makes the equilibrium configurations locally stable should be related to the properties of the dilaton potential that is decaying towards the spatial infinity, but behaves as a box close to the horizon. We prove that these thermodynamically stable solutions are also dynamically stable under spherically symmetric perturbations.

Cylindrical spacetimes due to radial magnetic fields

We continue our previous study of cylindrically symmetric, static electrovacuum spacetimes generated by a magnetic field, involving optionally the cosmological constant, and investigate several classes of exact solutions. These spacetimes are due to magnetic fields that are perpendicular to the axis of symmetry.

Noether symmetry approach in energy-momentum squared gravity

In this paper, we investigate the newly developed f(R, T 2) theory (R is the Ricci scalar and T 2 = T α β T α β , T α β demonstrates the energy-momentum tensor) to explore some viable cosmological models. For this purpose, we use the Noether symmetry approach in the context of flat Friedmann-Robertson-Walker (FRW) universe. We solve the Noether equations of this modified theory for two types of models and obtain the symmetry generators as well as corresponding conserved quantities. We also evaluate exact solutions and investigate their physical behavior via different cosmological parameters. For the prospective models, the graphical behavior of these parameters indicate consistency with recent observations representing accelerated expansion of the Universe. In the first case, we take a special model of this theory and obtain new class of exact solutions with the help of conserved quantities. Secondly, we consider minimal and non-minimal coupling models of f(R, T 2) gravity. We conclude that conserved quantities are very useful to derive the exact solutions that are used to study the cosmic accelerated expansion.

Square\u2013torsion gravity, dark matter halos and the baryonic Tully\u2013Fisher relation

Square–torsion gravity is applied to the long standing dark matter problem. In this context the theory reduces to General Relativity complemented by a dark stress–energy tensor due to the torsion of spacetime and is studied under the simplifying assumption of spherical symmetry. The dark stress–energy tensor satisfies an anisotropic structure equation. In vacuum this is equivalent to a wave equation with sources. A natural class of exact solutions is found which explicitly perturbs any seed spacetime metric by a conformal factor. This leads to the concept of dark coating. Static solutions are then used to construct structures that model dark matter halos surrounding baryonic bodies. In the Newtonian régime the baryonic mass m_b and the flat rotation curve velocity v_f are related by the baryonic Tully–Fisher relationm_bpropto v_f^4, a hitherto purely empirical result. The example of a dark halo on the Schwarzschild geometry is made as a toy model for a galaxy. Qualitative an quantitative features of galactic rotation curves are recovered. In this setting, a boundary of staticity is found, called torsion sphere, placed between the photon sphere and the event horizon. The phenomenon of dark radiation is briefly exposed.

Exact Solutions of Incompressible Navier–Stokes Equations in the Case of Oil and Gas Industrial Problems

Classes of exact solutions corresponding to vortex and potential flows are presented within the framework of a hydrodynamic model describing flows of a viscous incompressible fluid. The study of exact solutions is a prerequisite for creating a core simulator, which is associated with modeling fluid dynamics in a porous medium and the response of the field to dynamic influences in order to increase oil recovery.

Electromagnetic and anisotropic extension of a plethora of well-known solutions describing relativistic compact objects

We demonstrate a technique to generate new class of exact solutions to the Einstein-Maxwell system describing a static spherically symmetric relativistic star with anisotropic matter distribution. An interesting feature of the new class of solutions is that one can easily switch off the electric and/or anisotropic effects in this formulation. Consequently, we show that a plethora of well known stellar solutions can be identified as sub-class of our class of solutions. We demonstrate that it is possible to express our class of solutions in a simple closed form so as to examine its physical viability for the studies of relativistic compact stars

On Modeling a Layered Viscous Conductive Fluid Flow in a Region Changing in Time

The flow of a viscous conductive incompressible fluid in a time-varying region is investigated. Based on the model of a layered fluid flow, the class of exact solutions of the equations of magnetohydrodynamics in the region moving in time is considered. We study the change in the structure of the fluid flow as a result of the volume effect by the magnetic field and the movement of the boundary of the flow region. The heat dissipation effect due to internal friction and Joule heating is considered. The presented results are relevant to the research on optimizing problems of controlling the dynamics of an incompressible fluid and creating the “digital field” using the domestic technology.

Relativistic anisotropic models of ultra-dense stellar objects under embedding class I

In this paper, we present a new class of exact solutions satisfying Einstein’s field and modified TOV-equations. The thermodynamic quantities of stellar matter like anisotropic pressures, baryon density, red-shift and velocity of sound have been investigated using the embedding class I methodology with the Karmarkar condition. The solutions satisfy the static stability criterion, energy conditions, stability factor, adiabatic index and causality condition. In addition to it, we perform complete graphical analysis of neutron stars in Vela $$X-1$$ and Her $$X-1$$ in the setting of the Karmarkar space-time.