Non-dissipative Fluids Research Articles

Complexity factor for charged dissipative dynamical system

In this paper, we examine the complexity factor for a dynamical spherical system with dissipative charged anisotropic fluid. We evaluate the Einstein-Maxwell field equations and structure scalars using Bel’s approach which help to discuss the structure as well as evolution of a self-gravitating system. We measure the complexity factor for the pattern of evolution through the homologous condition and homogeneous expansion. We also analyze the stability of vanishing complexity condition for dissipative and non-dissipative fluids. It is found that the complexity as well as stability of the spherical system increases and decreases, respectively, under the effects of electromagnetic field.

Evolution of expansion-free spherically symmetric self-gravitating non-dissipative fluids and some analytical solutions

We consider the distribution of spherically symmetric self-gravitating non-dissipative (but anisotropic) fluids under the expansion-free condition which requires the existence of vacuum cavity within the fluid distribution. The Darmois junction condition is investigated for matching the spherically symmetric metric to an internal vacuum cavity (Minkowski space-time). We have studied some analytical models, total of three family of solutions out of which two satisfy the junction conditions over both the hypersurfaces. The models are investigated under some known dynamical assumptions which further provide analytical solution in each family.

An equivalent system of Einstein Equations

We provide the full set of equations governing the evolution of self-gravitating systems by using a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor. We apply this formalism to spherical case and found that: the only static solution with homogeneous energy density, under the physical reasonable considerations, is the Schwarzchild solution; the conditions under which a non-dissipative fluid is static and finally that shear-free and isotropic pressure conditions are equivalent, for non-dissipative fluids with homogeneous energy density.

Field theory for string fluids

We develop a field theory description of nondissipative string fluids and construct an explicit mapping between field theory degrees of freedom and hydrodynamic variables. The theory generalizes both a perfect particle fluid and pressureless string fluid to what we call a perfect string fluid. Ideal magnetohydrodynamics is shown to be an example of the perfect string fluid whose equations of motion can be obtained from a particular choice of the Lagrangian. The Lagrangian framework suggests a straightforward extension of the perfect string fluid to more general anisotropic fluids describing higher dimensional branes such as domain walls. Other modifications of the Lagrangian are discussed which may be useful in describing relativistic superfluids and fluids containing additional currents.

Entropy current formalism for supersymmetric theories

The recent developments in fluid/gravity correspondence give a new impulse to the study of fluid dynamics of supersymmetric theories. In that respect, the entropy current formalism requires some modifications in order to be adapted to supersymmetric theories and supergravities. We formulate a new entropy current in superspace with the properties: 1) it is conserved off-shell for non-dissipative fluids, 2) it is invariant under rigid supersymmetry transformations, 3) it is covariantly closed in local supersymmetric theories, 4) it reduces to its bosonic expression on space–time.

Electromagnetic effects on the inhomogeneity of planar symmetry

In this work, we aim to identify the effects of electromagnetic field on the energy density inhomogeneity in self-gravitating plane symmetric spacetime filled with imperfect matter in terms of dissipation and anisotropic pressure. We formulate the Einstein–Maxwell field equation, conservation laws, evolution equations for the Weyl tensor and the transport equation for diffusion approximation. Inhomogeneity factors are identified for some particular cases of non-dissipative and dissipative fluids. For non-dissipative case, we analyze the inhomogeneity factor for dust, isotropic and anisotropic matter distributions while dissipative matter distribution includes the inhomogeneity factor only for geodesic dust fluid. We conclude that electric charge increases the inhomogeneity in the energy density which is due to shear, anisotropy and dissipation.

Effective field theory for hydrodynamics: Wess-Zumino term and anomalies in two spacetime dimensions

We develop the formalism that incorporates quantum anomalies in the effective field theory of nondissipative fluids. We consider the effect of adding a Wess-Zumino-like term to the low-energy effective action to account for anomalies. In this paper we restrict to two spacetime dimensions. We find modifications to the constitutive relations for the current and the stress-energy tensor, and, more interestingly, half a new propagating mode (one-and-a-halfth sound): a left- or right-moving wave with propagation speed that goes to zero with the anomaly coefficient. Unlike for the chiral magnetic wave in four dimensions, this mode propagates even in the absence of external fields. We check our results against a more standard, purely hydrodynamical derivation. Unitarity of the effective field theory suggests an upper bound on the anomaly coefficient in hydrodynamics.

Comments on Hall transport from effective actions

We consider parity-odd transport in 2+1 dimensional charged fluids restricting attention to the class of non-dissipative fluids. We show that there is a two parameter family of such non-dissipative fluids which can be derived from an effective action, in contradistinction with a four parameter family that can be derived from an entropy current analysis. The effective action approach allows us to extract the adiabatic transport data, in particular the Hall viscosity and Hall conductivity amongst others, in terms of the thermodynamic functions that enter as 'coupling constants'. Curiously, we find that Hall viscosity is forced to vanish, whilst the Hall conductivity is generically a non-vanishing function of thermodynamic data determined in terms of the hydrodynamic couplings.

Non-dissipative hydrodynamics: effective actions versus entropy current

While conventional hydrodynamics incorporating dissipative effects is hard to derive from an action principle, it is nevertheless possible to construct classical actions when the dissipative terms are switched off. In this note we undertake a systematic exploration of such constructions from an effective field theory approach and argue for the existence of non-trivial second order non-dissipative hydrodynamics involving pure energy-momentum transport. We find these fluids to be characterized by five second-order transport coefficients based on the effective action (a three parameter family is Weyl invariant). On the other hand since all flows of such fluids are non-dissipative, they entail zero entropy production; one can therefore understand them using the entropy current formalism which has provided much insight into hydrodynamic transport. An analysis of the most general stress tensor with zero entropy production however turns out to give a seven parameter family of non-dissipative hydrodynamics (a four parameter sub-family being Weyl invariant). The non-dissipative fluids derived from the effective action approach are a special case of the fluid dynamics constrained by conservation of the entropy current. We speculate on the reasons for the mismatch and potential limitations of the effective action approach.

Effective field theory for hydrodynamics: Thermodynamics, and the derivative expansion

We consider the low-energy effective field theory describing the infrared dynamics of nondissipative fluids. We extend previous work to accommodate conserved charges, and we clarify the matching between field-theory variables and thermodynamical ones. We discuss the systematics of the derivative expansion, for which field theory offers a conceptually clear and technically neat scheme. As an example, we compute the correction to the sound-wave dispersion relation coming from a sample second-order term. This formalism forms the basis for a study of anomalies in hydrodynamics via effective field theory, which is initiated in a companion paper.

Nonlinear perturbations for dissipative and interacting relativistic fluids

We develop a covariant formalism for studying nonlinear perturbations of dissipative andinteracting relativistic fluids. We derive nonlinear evolution equations for various covectorsdefined as linear combinations of the spatial gradients of the local number of e-folds and ofsome scalar quantities characterizing the fluid, such as the energy density and theparticle number density. For interacting fluids we decompose perturbations intoadiabatic and entropy components and derive their coupled evolution equations,recovering and extending the results obtained in the context of the linear theory.For non-dissipative and non-interacting fluids, these evolution equations reduceto the conservation equations that we have obtained in recent works. We alsoillustrate geometrically the meaning of the covectors that we have introduced.

Interpretation of Ultrasonic Experiments on Finite-Amplitude Waves

An analysis was made for the purpose of relating higher-order elastic constants to the growth of harmonics in an initially sinusoidal wave. It is supposed that the face of a semiinfinite nondissipative elastic medium—for which the equation of one-dimensional motion is ü = (∂2u/∂x2)g(∂u/∂x)—is subjected to the sinusoidal displacement u = u0 cos(ωt−φ). The function g(∂u/∂x) is given as g(ξ) = (1/ρ0)[M2+M3ξ+M4ξ2+⋯+Miξi−2], where Mi depends on elastic constants up to and including order i. A solution, including the effects due to terms as high as M5 in the function g(ξ), is expressed as u = u0 Σn{Dn sin[n(ωt− Kx)]+En cos[n(ωt− Kx)]}+F(x), where the summation over n goes from 1 to ∞. Dn and En are expanded in powers of a Mach number M and a dimensionless distance X (M = ωu0/c0, X = ωx/c0, c02 = M2/ρ0). It is found that as u0 → 0, the limiting value of the ratio of the second-harmonic amplitude, H2 = u0(D22+E22)12, to u0M is given by −M3X/8M2, independently of M4, M5, etc. Thus, with M2 already known, M3 can be calculated from an experimental determination of this ratio. Similarly, it is found that the limiting value of the ratio of the third harmonic amplitude to u0M2 depends on elastic constants only up through order four. [Tutorial material is included to make the paper reasonably self-contained and to relate the present analysis to corresponding work for fluids. Careful derivations show that purely longitudinal waves in elastic solids and nondissipative fluids are governed by an equation of motion of precisely the same form. Hence, with appropriate choice of the parameters, an analysis of the one case applies equally well to the other.]

Propagation of Plane Sound Waves of Finite Amplitude in Nondissipative Fluids

An extensive theoretical treatment is presented of the problem of plane progressive sound waves (simple waves) produced by continuous, high-amplitude motion of a piston in a lossless, semi-infinite tube. The fluid in the tube is assumed to be nondissipative; liquids as well as perfect gases are considered. The analysis is given in both Eulerian and Lagrangian coordinates. Earnshaw's exact solution in parametric form is first given for arbitrary piston motion. Special attention is then given to the case of sinusoidal piston motion. Approximate but explicit power- and Fourier-series solutions for this case are derived from the exact solution. A low-amplitude nonlinear theory of simple waves is proposed. In this theory nonlinear effects are considered in a simple yet general manner. A brief analysis of shock formation is also given. It is shown that the generally accepted formula for the distance at which the shock forms when the piston motion is sinusoidal is rigorously correct only under very special circumstances.

Absorption and Dispersion of Forced Spherical and Cylindrical Sound Waves According to the Navier-Stokes Equations

In his classical paper on infinitesimal sound waves in perfect gases endowed with shear viscosity, bulk viscosity, and heat conduction, Kirchhoff [Ann. Phys. 136, 177–193 (1868)] considered not only plane waves but also a certain class of curved waves. He showed that the absorption and dispersion of these waves is determined by the solution of a biquadratic equation for the complex frequencies, but in discussing the behavior of particular waves he employed only one of the two pairs of roots, and this he linearized with respect to the driving frequency. The biquadratic itself, as extended by Langevin so as to hold for fluids obeying an arbitrary equation of state, has been solved exactly and generally by C. A. Truesdell [J. Rational Mech. Anal. 2, 693–762 (1953)]. In his comprehensive analysis and interpretation of the solutions, however, Truesdell considered only forced plane waves. In the present article I give a new version of Kirchhoff's approach to curved waves of expansion (an arbitrary motion may be regarded as the vector sum of an isochoric irrotational motion, a set of vorticity waves, and a set of expansion waves. Vorticity waves are not considered in this note. As has been remarked in varying degrees of generality by M. Lessen [J. Aero. Sci. 21, 849–850 (1954)]; Lagerstrom, Cole, and Trilling, [“Problems in the theory of viscous compressible fluids,” GALCIT (1949)]; S. Jarvis, Jr., J. Acoust. Soc. Am. 27, 70–73 (1954)]; and U. Yao-Tsu Wu [J. Math. Phys. 35, 13–27 (1956)] the propagation of linearized vorticity waves is independent of bunk viscosity and heat conduction) and show how Truesdell's results on plane waves may be adjusted so as to apply to the class of curved waves considered. Spherical and cylindrical waves are analyzed in some detail. Proper treatment of this problem requires dimensionless variables and hence some reference standard of length. Two alternatives are considered: (1) the unit of length is a linear dimension of a given oscillator, (2) the unit of length is the wavelength corresponding to the driving frequency according to the theory of nondissipative fluids. In the brief remarks of Kirchhoff, apparently only the first possibility was noticed. For the second class of spherical waves, the absorption coefficient per cm is shown always to be much less than that for the first class. There is a frequency-dependent factor which tends to increase the amplitude with the frequency. Similar analysis is carried out for cylindrical waves, which me shown to behave in a fashion intermediate between plane and spherical waves. In principle, the results are contained in Kirchhoff's paper. However, the analysis there, expecially as given in the expository work of Rayleigh [Theory of Sound (Dover Publications, New York, 1945), Vol. II, pp. 319–323] does not make clear how much depends on linearization with respect to frequency and how much does not. The purpose of Sec. 1 of this note is to show that Kirchhoff's method of reducing the analysis of certain curved waves to that of plane waves is valid exactly, at all frequencies.

Absorption and Dispersion of Forced Spherical and Cylindrical Waves According to the Navier-Stokes Equations

Absorption and dispersion measures for forced spherical and cylindrical infinitesimal waves are calculated exactly within the classical linearized theory of fluids endowed with heat conduction and with both shear and bulk viscosity. A proper treatment of this problem requires a dimensionless standard of length. Two alternatives are considered: (1) the unit of length is fixed as the radius of a given oscillator, (2) the unit of length is the wavelength corresponding to the driving frequency according to the theory of nondissipative fluids. The characteristic equation for a general class of curved waves is shown to be the same as that obtained by Kirchhoff and Langevin for plane waves. The results of Truesdel's exact analysis of this equation are adjusted so as to apply to spherical and cylindrical waves. For the second class of spherical waves, the absorption coefficient per cm is shown always to be less than that for the first class, which was calculated by Kirchhoff. There is a frequency dependent factor which tends to increase the amplitude with the frequency.