Absence Of Viscosity Research Articles

Effects of surface tension on the collapse time of an empty bubble

The collapse of an empty spherical bubble in an ideal liquid, in the absence of viscosity and surface tension, was studied by Lord Rayleigh. Using energy conservation, he derived an exact expression for the total collapse time as a function of the initial radius of the bubble, the density of the liquid, and the far-field pressure. In the present work, we extend Rayleigh's expression to include surface tension effects. Results are found to depend on a dimensionless parameter ϵ that measures the ratio between the work done by surface tension and that done by pressure during the collapse. This parameter is small for large bubbles but can be of order unity or larger for bubbles of small radius and, eventually, small pressure. We show that the ratio between the collapse time in the presence of surface tension and Rayleigh's collapse time is proportional to a definite integral that is a smooth, monotonically decreasing function of ϵ. This function can be easily bounded analytically for any value of ϵ, yielding a simple and accurate approximation for the collapse time that, for all practical purposes, provides a complete analytical solution to the problem at hand. We finally extend results to the case of a hyperspherical collapsing empty bubble.

Study of motion stability of a viscoelastic rod

Stability of non-conservatively loaded elastic and inelastic bodies - a classic section of deformable solid mechanics that has been of interest for many years. In this paper, we study the motion stability of a free rod subjected to a constant tracking force on one of its ends. The defining ratio of the rod material is the Kelvin-Voigt model. The solution is presented in the form of an expansion in terms of beam functions. The number of terms of this expansion is substantiated. The values of the critical load in the presence and absence of viscosity are determined. The given analytical results are confirmed by numerical calculations.

Formation and structure of circumplanetary disks and envelopes during the late stages of giant planet formation

Giant planets are expected to form within circumstellar disks, which shape their formation history and the local environment. Here, we consider the formation and structure of circumplanetary disks that arise during the late stages of giant planet formation. During this phase, when most of the final mass is accumulated, incoming material enters the Hill sphere and falls toward the planet. In the absence of torques, the falling parcels of gas conserve their specific angular momentum and collect into a circumplanetary disk. Generalizing previous work, we consider a range of possible geometries for the flow entering the sphere of influence of the planet. Specifically, we consider five geometric patterns for the inward flow, ranging from concentration toward the rotational poles of the system to isotropic flow to concentration along the equatorial plane. For each case, we derive analytic descriptions for the density field of the infall region, the disk surface density in the absence of viscosity, and steady-state solutions for viscous disks. These results, in turn, specify the luminosity contributions of the planet, the circumplanetary disk, and the envelope. These power sources, in conjunction with the surrounding material, collectively determine the observational appearance of the forming planet. We conclude with an approximate determination of these radiative signatures.

On the Prevention of Vibrations in the Problem of the Time-Optimal Control of a System with Two Degrees of Freedom

We study a mechanical system with two degrees of freedom, consisting of two absolutely rigid bodies (material points) connected to each other by a weightless rectilinear viscoelastic rod that can be stretched or compressed. The bodies can move translationally along a fixed straight line. A control force limited in absolute value is applied to one of them, whose vector is directed along the rod. A continuous piecewise-linear control in time, which transfers the system from one equilibrium position to another in a length of time close to the minimum possible time, is constructed. In the absence of viscosity, with the constructed quasi time-optimal control, unwanted vibrations of the bodies are not excited either during the transition process or when it ends. In contrast to the time-optimal relay control, the constructed continuous control is also robust with respect to the uncertainty of the design parameters.

Superfluid Drain Vortex

Drain vortices are among the most common vortices observed in everyday life, yet their physics is complex due to the competition of vorticity’s transport and diffusion, and the presence of viscous layers and a free surface. Recently, it has become possible to study experimentally drain vortices in liquid helium II, a quantum fluid whose physics is characterised by the absence of viscosity and the quantisation of the circulation in the superfluid component. Using the Gross–Pitaevskii equation, we make a simple model of the problem which captures the essential physics ingredients, showing that the drain vortex of a pure superfluid consists of a bundle of vortex lines which, in the presence of a radial drain, twist, thus strengthening the axial flow into the drain.

Aerodynamic reduced-order Volterra model of an ornithopter under high-amplitude flapping

The unsteady aerodynamics of flapping low-aspect-ratio ellipsoidal-wings in ornithopters is analyzed and modeled by the use of three dimensional Computational Fluid Dynamics (CFD) simulations. The range of interest is high amplitude, moderate frequency flapping, and low to moderate angles of attack at Reynolds around 105, where autonomous ornithopters like GRIFFIN are able to perform complex maneuvers such as perching. The results obtained show that the Leading Edge Vortex is produced above a certain Strouhal and angle of attack at downstroke. The frequency response of the aerodynamic loads are compared with the classical analytical models, observing that analytical models based on absence of viscosity and small perturbations are not appropriate for the range of interest. Through the 3D CFD aerodynamic loads database, a finite memory Volterra model is identified in order to predict the characteristics of forces and moments produced by the flapping wing. A good agreement with the 3D CFD simulations has been found by considering a reduced-order model depending on the effective angle of attack of the surrogate airfoil located at 70% of the semi-span at three-quarters chord on the airfoil, in agreement with the literature. Finally, a methodology for validation with a high-accuracy Motion Capture System and without the need of wind tunnel is proposed. As a result the proposed model provides better estimates than classical analytical ones.

Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain mathbb {T}times mathbb {R}. In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their L^2 norm grows as t^{1/2} and this confirms previous observations in the physics literature. On the contrary, the solenoidal component of the velocity field experiences inviscid damping, namely it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order nu ^{-1/6} (with nu ^{-1} being proportional to the Reynolds number) on a time-scale nu ^{-1/3}, after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible flow, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

Heating of inhomogeneous electron flow in the hydrodynamic regime

We study the electron temperature profiles for an inhomogeneous electron flow in the hydrodynamic regime. We assume that the inhomogeneity is due to a weakly nonuniform distribution of the momentum relaxation time within a spherically constricted area. We show that the temperature profile dramatically depends on the drive strength and the viscosity of the electron liquid. In the absence of viscosity, a Landauer-dipole-like temperature distribution, asymmetrically deformed along the current by the inelastic electron-phonon scattering, emerges around the inhomogeneity. We find that both the Landauer-dipole temperature profile and its asymmetry in the direction of the driving electric field exist in all dimensionalities and are, therefore, universal features of inhomogeneous hydrodynamic electron flow. We further demonstrate that the electron viscosity suppresses the thermal Landauer dipole and leads to the appearance of a ``hot spot'' exactly at the center of the constriction. We also calculate the phonon temperature distribution, which can be directly measured in experiments on thermal nanoimaging.

A novel model of multi-temperatures theory in generalized thermo-viscoelasticity

In this paper, a new relation of multi-temperatures was presented. This new connection to the problem of the general thermo-viscoelastic material in the context of Lord-Shulman (L-S) theory was applied. The analytical expressions for physical quantities are obtained using normal mode analysis in the physical domain. These expressions are numerically calculated and graphically illustrated for a given material. The findings were predicted and tested in the presence and absence of viscosity by various models of the theory of one-temperature (1-Temperature), classical two-temperature (2-Temperature) and hyperbolic two-temperature (Hyper-Temperature).

Area waves on a slender vortex revisited

This paper considers the classic problem of the dynamics of axisymmetric waves on a rectilinear vortex, in the absence of viscosity. The waves alter the axial pressure distribution and thus generate axial flows which depend on the radial distribution of vorticity. To simplify this problem, models have been introduced which average over the cross-section and eliminate the radial dependence. One approach, pioneered by Lundgren and Ashurst (1989 J. Fluid Mech. 200 283–307), averages the momentum equation. Another averaging method, due to Leonard (1994 Phys. Fluids 6 765–77), focuses on the vorticity equation. The present paper takes a fresh look at the derivation of these two distinct models, which we refer to as the momentum wave model and vorticity wave model respectively, using the tools of differential geometry to develop a hybrid Eulerian–Lagrangian approach. We compare these models with area waves in the asymptotic limit of a slender vortex, with radial structure retained. Numerical calculations are presented to show the differences between waves in the full slender vortex system and those in the momentum and vorticity wave models. We also discuss modification of the vorticity wave model to allow an external irrotational flow, and simulations are presented where a vortex is subjected to uniform axial stretching. Our approach can also be developed to model more complicated configurations, such as occur during vortex collisions.

Rayleigh-Plateau Dissipative Instability

The instability of a freely falling jet of liquid in air taking into account the viscosity of the contacting media is considered. In neglecting the viscosities of both media, instability was studied by Rayleigh and Plateau. They showed that instability develops as a result of the action of surface forces, and is expressed in a change in the cylindrical shape of the boundary of a freely falling jet of liquid with air into a sequence of spherical drops. In subsequent works, by phenomenological consideration of viscosity by means of the Ohnesorge number, it is shown that the viscosity of each of the contacting media affects the nature of the instability. However, this method of taking viscosity into account is not entirely correct, because does not take into account the specificity of the boundary conditions existing at the interface. It is proposed to use percolation boundary conditions, the validity of which is proved by the example of the exact determination of the threshold velocity of occurrence of Kelvin-Helmholtz instability. A dispersion equation of the Rayleigh-Plateau problem with percolation boundary conditions that describes the instability taking into account the viscosity of both media is obtained. The dissipative nature of the development of such instabilities is substantiated. The growth rates of instabilities are determined in cases when: the jet and medium have a low viscosity (ideal fluids); the jet is characterized by high viscosity, and the environment is small; the jet and the environment are highly viscous. It is shown that the theoretical model of droplet decay of the jet in the absence of viscosity of both media is quite good, in quantitative terms, consistent with experimental results. The maximum increment is equal γmaxKGγ≈0.32, against the Rayleigh-Plateau increment γmaxKGγ≈0.34, for disturbances with the same wave number XMAX≈0.37. It was also shown that for viscous jets and a weakly viscous environment, the instability increment describes the experimental results with a rather high degree of accuracy. Numerical calculations show that for jets of comparable viscosity, the instability increment decreases with increasing viscosity of the environment. If the viscosity of the environment is constant, then the increment of instability will be greater where the viscosity of the stream is higher. It is shown that the results of theoretical calculations are in good agreement with the available experimental data.

Contribution of the chiral vortical effect to the evolution of the hypermagnetic field and the matter-antimatter asymmetry in the early Universe

In this paper, we study the contribution of the chiral vortical effect, in addition to that of the chiral magnetic effect, to the evolution of the hypermagnetic field and the matter-antimatter asymmetry in the symmetric phase of the early Universe in the temperature range 100GeV < T < 10TeV. We choose a fully helical Chern-Simons wave configuration for the velocity and the hypermagnetic vector potential fields. The latter makes the plasma force-free in the absence of viscosity. We show that the most pronounced effect of the chiral vorticity is the production and initial growth of the hypermagnetic field. In particular, we show that in the presence of a non-zero matter asymmetry, the hypermagnetic field can grow from zero initial value only in the presence of a non-zero vorticity field. Moreover, we show that larger initial growths not only result in larger maximum values of the hypermagnetic field, but also cause the saturation of the hypermagnetic field and the conversion of the lepton-baryon asymmetry to occur more quickly, i.e., at a higher temperature. We show that the damping of the vorticity due to the presence of viscosity, which typically occurs extremely rapidly, does not significantly affect the evolution.

Inviscid diffusion of vorticity in low-temperature superfluid helium

We numerically study the spatial spreading of quantized vortex lines in low temperature liquid helium. The vortex lines, initially concentrated in a small region, diffuse into the surrounding vortex-free helium, a situation which is typical of many experiments. We find that this spreading, which occurs in the absence of viscosity, emerges from the interactions between the vortex lines, and can be interpreted as a diffusion process with effective coefficient equal to approximately 0.5$\kappa$ where $\kappa$ is the quantum of circulation.

Dissipation-Induced Instabilities in Magnetized Flows

We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. A hydrodynamically stable flow can be destabilized by the magnetic field both in an ideal and a viscous and resistive system giving rise to the azimuthal magnetorotational instability. A special solution to the equations of ideal magnetohydrodynamics characterized by the constant total pressure, the fluid velocity parallel to the direction of the magnetic field, and by the magnetic and kinetic energies that are finite and equal—the Chandrasekhar equipartition solution—is marginally stable in the absence of viscosity and resistivity. Performing a local stability analysis, we find the conditions under which the azimuthal magnetorotational instability can be interpreted as a dissipation-induced instability of the Chandrasekhar equipartition solution.

Generation of dispersive shock waves in nonextensive plasmas

In this research we use a generalized hydrodynamic model to numerically investigate the quasi-neutral expansion and compression of an electron–ion plasma with nonextensive electron velocity distribution into or from vacuum. We study the effects of kinematic viscosity and electron–ion collisions on the expansion profile and compare our results to the numerical solutions of the standard Korteweg – de Vries (KdV) equation. It is found that the quasi-neutrality assumption in the hydrodynamic approach in the absence of viscosity and collisions, which leads to elimination of Poisson’s equation, sets the weak dispersion limit and becomes equivalent to the standard weakly dispersive KdV model. In the weak dispersion limit our model, as well as the KdV with small dispersion effect, predicts that a pulse-like initial profile evolves into solitary wave train. We further show that in a plasma expansion different shock profiles, such as purely dispersive, diffusive–dispersive, and dissipative ones, with significantly different characters may form. Finally, the effect of electron nonextensivity on oscillatory shock waves shows that the expansion profile is affected by changes of q-parameter. Our numerical solution is in qualitative agreement with some distinguished experiments showing the possibility of dispersive shock wave formation in rarefied plasma during an expansion into vacuum.

A complex scalar form of the incompressible Navier–Stokes equations

ABSTRACT It is demonstrated that in Fourier domain the Navier–Stokes equations for an incompressible fluid can be reduced to a single complex scalar equation. An advantage of this equation is that any solution of this equation (approximate or exact) automatically represents real incompressible velocity field. An attempt was undertaken to check that in the absence of viscosity this equation represents a Hamiltonian system expressed in non-canonical variables. However, only one of the two necessary conditions was shown to hold; the question of fulfillment of the second necessary condition (the Jacobi identity) remains open. Using the new representation, it was demonstrated that in the inviscous case there are only two translationally-invariant, second-order integrals of motion which correspond to conservation of energy and helicity.

Dynamical survey of the dual power Zakharov–Kuznetsov–Burgers equation with external periodic perturbation

We investigate the dynamical behavior of the newly introduced dual power Zakharov–Kuznetsov–Burgers equation. Using bifurcation theory of planar dynamical systems, we study bifurcations of traveling wave solutions of the dual power Zakharov–Kuznetsov–Burgers equation in presence and absence of viscosity (μ) effect. In presence of an external periodic perturbation, we discuss the periodic and chaotic motions of the perturbed dual power Zakharov–Kuznetsov–Burgers equation by analyzing phase projection analysis, time series analysis, Poincaré section and sensitivity analysis. The effect of viscosity (μ) plays an important role in the transition from chaotic motion to periodic motion of the perturbed dual power Zakharov–Kuznetsov–Burgers equation.

Emergent universe model with dissipative effects

Emergent universe model is presented in general theory of relativity with isotropic fluid in addition to viscosity. We obtain cosmological solutions that permit emergent universe scenario in the presence of bulk viscosity that are described by either Eckart theory or Truncated Israel Stewart (TIS) theory. The stability of the solutions are also studied. In this case, the emergent universe (EU) model is analyzed with observational data. In the presence of viscosity, one obtains emergent universe scenario, which however is not permitted in the absence of viscosity. The EU model is compatible with cosmological observations.

Explicit Weierstrass traveling wave solutions and bifurcation analysis for dissipative Zakharov–Kuznetsov modified equal width equation

We consider the dissipative Zakharov–Kuznetsov modified equal width (ZK-MEW) equation and discuss the effect of the dissipation on the existence and nature of traveling wave solutions of the equation. We use Lyapunov function and dynamical system theory in order to show that when viscosity term is added to the ZK-MEW equation, yet there exists still bounded traveling wave solutions in certain regions. Subsequently, we obtain general solution of the ZK-MEW equation in the presence and absence of viscosity in terms of Weirstrass $$\wp $$ functions and Jacobi elliptic functions. We also derive a new form of kink-type solution by exploring a factorization method based on functional transformation and the Abel’s first order nonlinear equation. Finally, we use the phase plane analysis and examine the stability of the viscous waves.

A fully conservative mimetic discretization of the Navier–Stokes equations in cylindrical coordinates with associated singularity treatment

We present a finite difference discretization of the incompressible Navier–Stokes equations in cylindrical coordinates. This currently is, to the authors' knowledge, the only scheme available that is demonstrably capable of conserving mass, momentum and kinetic energy (in the absence of viscosity) on both uniform and non-uniform grids. Simultaneously, we treat the inherent discretization issues that arise due to the presence of the coordinate singularity at the polar axis. We demonstrate the validity of the conservation claims by performing a number of numerical experiments with the proposed scheme, and we show that it is second order accurate in space using the Method of Manufactured Solutions.