On the stability of a cylindrical jet moving in a material medium along an external electrostatic field

A dispersion relation is derived for capillary waves with arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a jet of an ideal incompressible dielectric liquid moving in an ideal incompressible dielectric medium along an external uniform electrostatic field. A tangential discontinuity in the velocity field on the jet surface is shown to cause Kelvin-Helmholtz periodical instability at the interface and destabilize axisymmetric, flexural, and flexural-deformational waves. Both the flexural and flexural-deformational instabilities have a threshold and are observed not at an arbitrarily small velocity of the jet but starting from a certain finite value. It is shown that the instability of waves generated by the tangential discontinuity of the velocity field is periodic only formally (from the pure mathematical point of view). Actually, the jet disintegrates within the time of instability development, which is shorter than the half-cycle of the wave.     

On the stability of capillary waves on the surface of a charged jet moving relative to the environment

A dispersion relation is derived for capillary waves with arbitrary symmetry (arbitrary azimuthal numbers) on the surface of a charged cylindrical jet of an ideal incompressible conducting liquid moving relative to an ideal incompressible dielectric medium. It is shown that a tangential discontinuity in the velocity field on the surface of the jet leads to periodic instability of waves (similar to the Kelvin-Helmholtz instability) at the interface and destabilizes both axisymmetric and flexural waves. The wavenumber range for unstable waves and the instability growth rate increase with the field strength and relative speed of motion, varying as the square of these parameters. In the case of the neutral jet, the flexural instability is of the threshold character and sets in starting from a certain finite value of the speed rather than at an arbitrary small speed.     

On the stability of capillary waves on the surface of a space-charged dielectric liquid jet moving in a material medium

We have derived and analyzed the dispersion equation for capillary waves with an arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a space-charged cylindrical jet of an ideal incompressible dielectric liquid moving relative to an ideal incompressible dielectric medium. It has been proved that the existence of a tangential jump of the velocity field on the jet surface leads to a periodic Kelvin–Helmholtz- type instability at the interface between the media and plays a destabilizing role. The wavenumber ranges of unstable waves and the instability increments depend on the squared velocity of the relative motion and increase with the velocity. With increasing volume charge density, the critical value of the velocity for the emergence of instability decreases. The reduction of the permittivity of the liquid in the jet or an increase in the permittivity of the medium narrows the regions of instability and leads to an increase in the increments. The wavenumber of the most unstable wave increases in accordance with a power law upon an increase in the volume charge density and velocity of the jet. The variations in the permittivities of the jet and the medium produce opposite effects on the wavenumber of the most unstable wave.     

On the capillary stability of a cylindrical dielectric liquid jet in a longitudinal electrostatic field

A dispersion relation is derived for capillary waves with arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a cylindrical jet of an ideal incompressible dielectric liquid subjected to an electrostatic field aligned with the symmetry axis of the jet. It is shown that only long axisymmetric waves can experience capillary instability in such a system. The wavenumber range into which unstable waves fall begins with a zero value, and its width depends on the permittivities of the liquid and ambient and on the electrostatic field strength squared. As the field strength grows, the wavenumber range for unstable waves rapidly narrows and the capillary instability growth rate, as well as the wavenumber of the wave with the greatest growth rate, decreases.     

Stability of capillary waves with arbitrary symmetry on the surface of a jet in a longitudinal electric field periodically varying with time

The Mathieu differential equation for the evolution of the amplitudes of arbitrarily symmetric capillary waves (with arbitrary azimuthal numbers) propagating over the surface of a incompressible dielectric cylindrical liquid jet is analyzed. The jet is placed in a time-periodic uniform electric field that is parallel to the symmetry axis of the jet unperturbed by the wave flow. It is found that the time-varying electric field pressure parametrically builds up both axisymmetric waves on the jet surface, flexural waves, and flexural deformation waves. At a fixed frequency of the external field, waves with different wavelengths and symmetries (different azimuthal numbers) may build up simultaneously in the main demultiplication resonance, as well as in secondary and tertiary resonances. The parametric buildup of flexural deformation waves has a threshold relative to the external field frequency: it takes place at the field frequency exceeding a certain value depending on the jet radius and physicochemical properties of the liquid.     

Asymptotic analysis of nonlinear nonaxisymmetric waves on the surface of a neutral dielectric jet in a longitudinal electrostatic field

The problem of calculation of finite-amplitude waves on the cylindrical surface of an ideal incompressible dielectric liquid jet in a uniform electrostatic field collinear to the unperturbed jet axis is solved using a second-order asymptotic analytic procedure in ratio of the wave amplitude to the jet radius. Nonlinear corrections to the jet profile, velocity field potential, and electrostatic potentials inside and outside the jet are of resonant nature. The degenerate resonant interaction between the wave determining the initial strain and the waves excited due to nonlinearity of the hydrodynamic equations can take place for waves with different symmetries (different azimuth numbers).     

Spontaneous disintegration of a noncylindrical jet emitted from the liquid surface unstable against self-charge

A dispersion relation for waves on the surface of a charged viscous incompressible conducting liquid jet with an arbitrary azimuthal number is derived. It is shown that the influence of deformation on the growth rate and wavenumber of the most unstable mode varies according to the sign of local deformation relative to the cylindrical jet (the locality is specified by the wavelength), azimuthal number, and electric charge per unit length of the jet. This circumstance should be taken into account to comprehend conditions of liquid spontaneous electrodispersion.     

On the Stability of the Surface of a Short Charged Jet Moving with Respect to an External Material Medium

The problem of the stability of capillary waves on the surface of a charged jet of an ideal incompressible electroconducting liquid, which moves with respect to a material dielectric medium, is considered. There is a tangential discontinuity of the velocity field on the interface between the media. Solutions to the problem in two idealized models have been compared, i.e., when the jet has a finite and infinite length. It has been shown that the instability increments and the wave numbers of the most unstable waves, computed in both models, are linearly related, and velocity of motion of the jet acts as a coefficient of proportionality.     

Nonlinear analytical asymptotic investigation of oscillations of nonaxisymmetric modes of a charged jet of an ideal liquid

Technical Physics - A jet of an ideal incompressible conducting liquid with a uniformly charged surface is considered. The jet moves at a constant velocity along the symmetry axis of its...     

Nonlinear oscillations of a charged jet of a conducting liquid under multimode initial deformation of its surface

The subject of consideration is a uniformly charged jet of an ideal incompressible conducting liquid moving with a constant velocity along the symmetry axis of an undisturbed cylindrical surface. An evolutionary expression for the jet shape is derived accurate to the second order of smallness in oscillation amplitude for the case when the initial deformation of the equilibrium surface is a superposition of a finite number of both axisymmetric and nonaxisymmetric modes. The flow velocity field in the jet and the electric field distribution near it are determined. The positions of internal nonlinear secondary combined three-mode resonances are found, which are typical of nonlinear corrections to the analytical expressions for the jet shape, flow velocity field potentials, and electrostatic field in the vicinity of the jet.     

Influence of charge relaxation on the capillary disintegration of a jet of a viscous dielectric liquid placed in an electrostatic field collinear with the axis of the jet

A dispersion relation is derived for capillary waves with an arbitrary symmetry on the surface of a charged jet of a finite-conductivity viscous liquid placed in an electric field collinear with the axis of the jet. Analytical calculations are carried out in an approximation that is linear in dimensionless wave amplitude. In the case of axisymmetric waves, the instability of which causes disintegration of the jet into drops, the finiteness of the potential equalization rate has a noticeable effect only for jets of poorly conducting liquids. The charge relaxation shows up in that “purely relaxation” periodic and aperiodic liquid flows arise. When the conductivity of the liquid declines, the instability growth rates for unstable waves increase and their spectrum extends toward short waves. A charge present on the surface of the jet enhances its instability. An increase in the charge surface diffusion coefficient variously influences the capillary and relaxation branches of the solution: the damping ratio increases in the former case and decreases in the latter. As the diffusion coefficient rises, relaxation flows may become unstable.     

Instability Increments for Waves with Different Symmetries on a Space-Charged Jet Moving Relative to a Material Medium

Technical Physics - We have studied the instability increments of capillary waves with an arbitrary symmetry (arbitrary azimuthal numbers m) on the surface of a space-charged cylindrical jet of an...     

Determination of the dimension of a cavity in water behind a fast-moving cylindrical body

The cavity formation behind cylindrical bodies with variously shaped noses moving in water at velocities from 400 to 2000 m/s is investigated by numerically simulating a 2D axisymmetric gasdynamic problem under the assumption of an ideal compressible liquid. With regard to calculation and experimental data, a simple model of the process is proposed that considers the inertial expansion of a spherical cavity in an ideal incompressible liquid. It is shown that the collapse of a cavity behind a body moving in a liquid with backpressure may give rise to cumulative phenomena with the formation of a fast liquid jet running in the wake of the body.     

Nonlinear corrections to the frequencies of nonaxisymmetric Modes of a space-charged dielectric liquid jet

An expression for the shape of a uniformly charged (over volume) jet of an ideal incompressible dielectric liquid as a function of time is derived with analytical asymptotic calculations of the third order of smallness in jet oscillation amplitude for the case when the initial deformation of the equilibrium surface is governed by one (in general nonaxisymmetric) mode. Analytical expressions for nonlinear corrections to the frequencies and positions of inner nonlinear degenerate three-and four-mode resonances are found.     

非线性不可压缩霍尔磁流体方程组的精确解

得到了不可压缩理想霍尔磁流体方程组的平面波精确解,这些平面波是斜传播的左旋圆极化波或右旋圆极化波,并且涨落速度和涨落磁场通过波数联系起来。讨论了这些平面波解的叠加性质。当波的传播方向平行时,任意两支圆极化平面波都可以叠加;而当波的传播方向不平行时,只有极化方向相同,波数相同,且各自的旋度与自身都是同方向(或都是反方向)的两支圆极化平面波才可以叠加。     

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得到了不可压缩理想霍尔磁流体方程组的平面波精确解,这些平面波是斜传播的左旋圆极化波或右旋圆极化波,并且涨落速度和涨落磁场通过波数联系起来。讨论了这些平面波解的叠加性质。当波的传播方向平行时,任意两支圆极化平面波都可以叠加;而当波的传播方向不平行时,只有极化方向相同,波数相同,且各自的旋度与自身都是同方向(或都是反方向)的两支圆极化平面波才可以叠加。     

Stability of a velocity field tangential discontinuity in a three-layer density-stratified liquid with a moving middle layer

A dispersion relation is analytically derived for gravitational waves in an ideal incompressible threelayer liquid with a free surface in the presence of a velocity field tangential discontinuity between the layers. The discontinuity results from the motion of the middle layer. The instability of the tangential discontinuity is shown to depend on the relative velocity of contacting layers, which, in turn, depends on the ratio of their densities. The closer the density ratio to unity, the lower the moving layer velocity causing instability. In the given case, instability involves internal waves arising at the second and third interfaces in accordance with the Kelvin–Helmholtz concept of instability development. Internal waves with wavelengths far exceeding the thickness of the middle layer are found to interact with each other. Surface waves only change their frequencies.     

Traveling Wave Solutions of the Incompressible Ideal Hall Magnetohydrodynamics

The solutions of incompressible ideal Hall magnetohydrodynamics are obtained by using the traveling wave method.It is shown that the velocity and magnetic field parallel to the wave vector can be arbitrary constants.The velocity and magnetic field perpendicular to the wave vector are both helical waves.Moreover,the amplitude of the velocity perpendicular to the wave vector is related to the wave number and the circular frequency.In addition,further studies indicate that,no matter whether the uniform ambient magnetic field exists or not,the forms of the travelling wave solutions do not change.     

Interaction of strongly nonlinear waves on the free surface of a dielectric liquid in a horizontal electric field

The nonlinear dynamics of the free surface of an ideal dielectric liquid with a large relative permittivity in a strong horizontal electric field has been considered. It has been demonstrated that the interaction between oppositely propagating solitary waves in arbitrary geometry is elastic: they conserve their energy and momentum. The interaction between waves has been numerically simulated with the use of conformal variables. It has been shown that the interaction deforms the waves; this effect is weak for waves with a relatively small amplitude: deformation for oppositely propagating waves with the identical shape is determined by the fourth power of their amplitude. At multiple collisions of strongly nonlinear waves, a tendency to the formation of singularities, i.e., points with a high energy density of the field, is observed.     

Radiative Instability of an Unbounded Jet Flow

The problem of the linear dynamics of disturbances of an unbounded jet flow with a piecewise linear velocity profile is considered. The stable disturbances of a flow in an incompressible medium are so-called flow waves localized near a vorticity jump. In this work, it is shown that the amplitudes of these waves slowly increase in a compressible medium due to acoustic radiation; i.e., an instability appears. The asymptotic solution to the problem for small Mach numbers is represented in terms of Airy functions. An analytic expression for the growth increment of the disturbances is obtained.