Temporary Evolution of the Amplitudes of Capillary Waves on the Surface of a Charged Jet, Moving Relative to a Material Dielectric Medium

A solution to the problem of calculating the temporary evolution of amplitudes of capillary waves of an arbitrary symmetry on the surface of a cylindrical jet of an ideal incompressible conductive liquid moving relative to an ideal incompressible dielectric medium is offered, taking into account multimodal initial conditions. Analytical expressions for the temporary evolution of the amplitudes of waves on a jet, amplitude values of hydrodynamic potentials of velocity fields on a jet and in a medium, and the electric potential of the field in the neighborhood of the jet. An assessment of the characteristic time of a separation of a drop from a jet is performed.


INTRODUCTION
The problem of the study of patterns of the disintegration of a jet into separate drops has been a subject of recurrent interest among researchers (see, e.g., [1][2][3][4][5][6][7]) for over 150 years. During this time, so many modes of jet disintegration have been found that the necessity of classifying the latter became urgent, and the physical reasons for drop separation were investigated [8][9][10][11]. The disintegration mode in particular was found to depend on the symmetry of the waves on the jet surface. Common patterns of excitation of the first two nonaxisymmetric modes of the capillary waves on the jet were found in [12,13], and the images of these modes were shown in [14]. As a result, the stability of the waves had to be studied with an arbitrary symmetry on the surface of a cylindrical jet, described by functions of a more general form relative to axisymmetric functions, namely, , where s is the frequency of the waves, t is the time, ϕ is the azimuthal number, k is the wave number, and m is the azimuthal number. The axisymmetric wave corresponds to the case where m = 0, m = 1, and m = 2, for bending and deformation waves.
PROBLEM SETTING Let the R-radius jet of an ideal incompressible liquid with ρ in density move at a constant rate of ( be an ort of coordinate z) relative to an incompressible dielectric medium with ρ ex density and dielectric permittivity ε d . On the surface of the jet, a charge with ε d surface density appears. The coefficient of the surface tension of the interface is χ. The problem will be solved in a cylindrical system of coordinates, where the OZ axis coincides with the symmetry axis of an undisturbed jet, and the coordinate origin moves at the rate of the jet.
The form of a free surface of the jet (at the interface) is described as follows: where ξ(ϕ, z, t) is the perturbation of the equilibrium cylindrical surface of a jet, which occurs due to the capillary wave motion, which is generated by the thermal motion of molecules of the liquid [15]. The amplitude of these waves is extremely small, and it is determined using the expression , where κ is the Boltzmann constant, and T is the absolute temperature [15].
The mathematic formulation of the problem looks like as follows by analogy with [16]: are the fields of rates of the flux of liquid and hydrodynamic pressures in the jet (j = in) and in the external medium (j = ex).
For the section of the jet, whose length is equal to the length of wave λ: (1) where is the ort of the radial coordinate.
Let us accept that at the initial moment of time, the perturbation of a cylindrical surface of the jet is presented as a superposition of a final spectrum of waves with the different wave k l and azimuthal numbers m l, j : (2) where Ω is the spectrum of the wave numbers and initially excited waves, Ξ is the spectrum of the azimuthal numbers for each of the fixed wave number k l , and ζ l,j is the amplitudes of the initially excited modes.
All the following calculations are performed using dimensionless variables, where R = ρ in = σ = 1.

SCALARIZATION OF THE PROBLEM
Within the framework of the potential flux of the liquid in a quasielectrostatic approximation, let us imagine the field of rates and vector of intensity as the gradients of hydrodynamic [17] and electric [18] potentials, correspondingly:  Accounting for these formulas, the equations of continuity of the rate and electric field pass into the Laplace equations: The Euler equations are easily integrated, and as a result, the expressions for the hydrodynamic pressures inside the jet and in the external medium can be derived: The natural conditions for the potentials on the axis and on eternity are transformed into the following form: The boundary conditions on the surface of the jet are also reformulated into conditions for the potentials ψ in , ψ ex , and Φ.
The kinematic conditions on the jet surface will take the following form: The absence of the tangential component of the vector of the electric field intensity on the jet surface will transform into the condition of its equipotentiality: The dynamic boundary condition of problem (1) remains unchanged, except that the pressure of the electric forces is written via the electric potential: The problem is solved using the asymptotic method of many time scales in linear approximation with respect to small parameter ε [19]. All sought functions will depend on different time scales that are determined using small parameter ε: T p = ε p t; (p = 0, 1, 2,..). The time derivative is calculated according to the following rule [19][20]: Let us present the sought functions as an expansion of the small parameter of the problem [20]: ,

r z T T r z T T r z T T z r zT T r z T T r r z T T z T T z T T
The upper index in the brackets points to the order of smallness of the relevant component.
In the form of similar expansions, we shall present the pressures that enter the boundary condition on the surface of the jet: (5) After having inserted expansions (3)-(5) into the equation system of boundary and initial conditions of the scalarized problem and having grouped together the terms for proper powers of the small parameter ε, let us divide the original problem into the boundary problems of the zero and the first orders of smallness.

PROBLEM OF THE ZERO ORDER OF SMALLNESS
This problem describes the unperturbed equilibrium state of the system: a cylindrical jet, which moves relative to the medium at a constant rate: In the system of coordinates that has been introduced, which move together with the jet, the jet itself remains motionless, and the external medium moves at a rate of . Obviously, the potential of the rate of the external medium in an equilibrium state will look like the following: The electric potential distribution near the cylindrical jet is determined by the following boundary problem of the zero order of smallness: which means that the intensity of the field of the cylindrical jet tends to zero upon the infinite removal from it: The integral condition implies the charge storage on the length of the jet capillary wave. From now on, for the sake of simplicity, to designate the sign of the partial derivative along a certain coordinate x, the symbol ∂ x will be used, not .
The solution of the above problem with account of the axis symmetry of the equilibrium state is easy to find: The dynamic boundary condition on the equilibrium surface of the jet: determines the difference between the pressures in the medium and in the jet in an equilibrium state:

PROBLEM OF THE FIRST ORDER
OF SMALLNESS The mathematical formulation of the problem of the first order of smallness is as follows: the invariability of the charge on the length of the capillary wave: is the condition of invariability of the jet volume on the length of the capillary wave: the initial conditions are as given below: Let us write the Laplace equations for the hydrodynamic and electric potentials, harmonic with respect to coordinates (ϕ, z) and satisfying the natural boundary conditions [21] in the following form: (6) where I m (x) и K m (x) are the modified Bessel functions of the first and second kind [21].
The wave perturbation of the equilibrium surface of the jet will be written similarly: Inserting this expression and the expression for in the kinematic boundary condition of the scalarized problem and making the coefficients equal to each other at similar exponents, we shall find the connection between coefficients B 1 and B 2 and amplitudes α 1 and α 2 : is the derivative of the Bessel function. From the second kinematic condition, using the expression for the perturbation of the jet surface and the expression for , we also find the connection between coefficients A 1 , A 2 and amplitudes α 1 and α 2 : where the following designation is used: The connection of coefficients F 1 and F 2 and amplitudes α 1 and α 2 is obtained after the transformation of equipotentiality condition with account of the expressions of the explicit form for the perturbation of the surface and electric potential Φ (1) : Upon the insertion on the perturbed surface of the expressions for the hydrodynamic potentials and along with the electric potential in the dynamic boundary condition, we obtain the following ratio: (10) Equations (7)-(9) form a linear algebraic system for coefficients A q , B q , and F q , solving which we get their expressions via α j : Inserting coefficients (11) in ratios (10), we obtain differential equation of the second order for amplitudes α q (T 0 , T 1 ): (12) Equation (12) is evolutionary and determines the dependence of α q (T 0 , T 1 ) on the time scale T 0 . By inserting the solution project into it in the form of harmonic function α q ∼ exp(iωT 0 ), we obtain the dispersion equation: (13) The solutions of Eq. (13) are the roots ω 1 and -ω 2 : Frequencies (14) can be either real valued or complex valued, depending on the sign of the expression under the radical. If this becomes negative, the frequencies can obtain imaginary parts, and the waves that correspond to them lose their stability. The imaginary part of the complex frequency determines the increment of the equipotential increase in the wave  where Aα q (T 1 ) and Bα q (T 1 ) the functions, which describe the dependences of amplitudes on the scale of time T 1 , are defined in the problem of the following order of smallness.
Using the solutions of (6) and the ratios between coefficients of (11) after certain transformations and redesignations of the functions, which determine the dependence of solutions on the time scale, let us reduce the potentials and surface perturbations to the following form: (15) where ω 1 and ω 2 are determined by expressions (14), the form of functions A(T 1 ), B(T 1 ) and conjugated to them functions is defined in the problem of the following order of smallness.
Note that the general solutions for the potentials and surface perturbations should be written as the superpositions of specific solutions (15) for all possible values of the wave number k and azimuthal number m, i.e., in the form of integral over k and sum over m.

ANALYSIS OF SOLUTIONS OF THE PROBLEM OF THE FIRST ORDER OF SMALLNESS
Analysis of solutions of the dispersion equation is carried out numerically. Figure 1 shows the dependences on the wave number k of the real-valued and imaginary parts of both frequencies ω 1 and ω 2 determined using (14) for the main axisymmetric mode (m = 0) of the jet surface capillary waves (at the media interface) at different values of the rate of the jet motion U. Figures 1-3 denote the capillary waves' frequencies, and Figs. 4-6 designate the increments in the regions of values of the physical parameters where the capillary waves lose their stability. According to (14), the analytic expressions of frequencies consist of two terms, i.e., always of a real-valued first term (at any values of the input magnitudes) and the second term, a radical, which, depending on the values of the physical parameters, can be both real-valued and imaginary. The condition for arising of the imaginary component of the frequency (of an increment) or realization of a wave instability is a zero crossing of the expression being under the radical. In Fig. 1, the first components of frequency are depicted almost as straight lines in a small (multiplier γ m (k) of an order of one thousandth of a unit) vicinity of the axis of abscissa. In Fig. 1a these lines lie in the region of the negative values of Reω, and in Fig. 1b they are in the region of positive Reω. These frequency branches owe their origin to the presence of the jet motion relative to material medium, and at a rate equal to zero they turn into zero. Upon different values of the jet rates (see Fig. 1), the relevant lines differ by the line thickness. These lines finish by the rapid increase in frequency upon the expression under the radical becoming positive.
As for the rest, Figs. 1a and 1b are identic (see Eq. (14)), and in Figs. 2 and 3 we shall be limited to depiction of only ω 1 .
From Fig. 1 it follows that at ε ex = 1, χ 0 = 0, ρ ex = 0.001 and at different speeds, long capillary waves are unstable, and with an increase in the jet speed the instability zone expands due to the shift of the righthand boundary. In this case, the increments increase as well, i.e., with an increase in the speed of the jet motion the instability develops faster. The instability mainly develops under the effect of the capillary forces [4], with the contribution of the aerodynamic forces being insignificant. Short capillary waves are stable. Figure 2 shows the same dependences as those in Fig. 1, but for the bending mode (m = 1) of the capillary waves. Qualitatively, the shape of the curves is the same as in Fig. 1; however, at the zero speed the bending mode remains to be stable. As with the axisymmetric mode, with increases in speed, the instability zone grows due to the shift of the right-hand boundary of the instability zone with a simultaneous increase in the increments' values. Figure 3 shows the dependences analogous to those presented above for the deformation mode (m = 2). Unlike the previous cases, for the deformation zone the long capillary waves are stable, i.e., small values of k is the zone of stability. The instability zone originates in the region of k ≈ 3. In this case, we deal with the aerodynamic instability [22]. Naturally, aerodynamic instability is not realized at the zero speed. At other respects, the effect of the motion speed of the jet is completely identic to the aforementioned situations of the modes with m = 0 and m = 1. Figure 4 shows the dependences on the surface density of the charge χ 0 of the real-valued and imaginary parts of frequency ω 1 for the axisymmetric mode (m = 0) of the capillary waves on the jet surface at various values of the wave number k.
It is seen in Fig. 4 that at k = 1, the wave is unstable throughout the entire region of the charge variation. The instability of shorter waves is realized at surpass-ing of a certain final value by the charge. Upon an increase in the wave number, the boundary of the instability realization shifts to the region of large values of the surface charge densities. Figures 5 and 6 show the analogous dependences for the bending (m = 1) and deformation (m = 2) modes. The dependences of frequency ω 1 on the charge parameter in the graphs presented are identical to those obtained earlier for the axisymmetric mode.
The calculations show that there is also a marked dependence of the real-valued and imaginary parts of the frequency on the external medium density.

MEETING THE INITIAL CONDITIONS
The solution for the surface perturbation must meet the initial conditions. Let us insert the solution for perturbation ξ (1) in the initial condition for the sur- After having inserted ξ (1) into the left-hand part of the initial condition for the surface perturbation, having multiplied the obtained equality by exp(im′ϕ) and exp(-ik′z) and integrating over z in the limits from -∞ to ∞, over ϕ from -π to π we obtain the following:   Using the properties of δ-function and δ-symbol of Kroneсker, we obtain the following equation: (17) Performing the analogous operations with expression (2) for the derivative from ξ (1) , let us write the second equation: (18) As a result of solution of systems (17) and (18) we find coefficients :

T T A T im
Using (19) and (15), we obtain the final form of solutions of the first order of smallness, meeting the initial conditions: Further, we obtain the following: Here, it is taken into account that although expressions (20) are determined in the form of sums over the wave and azimuthal numbers of the initial perturbation, it is still obvious that the characteristic time will be determined by the wave with a maximum increment. The difference in the initial amplitudes of separate waves can only play a part when the increments are equal.
When the wave frequency becomes complex, its amplitude begins to grow in accordance with the order of value, according to the following law: (22) Let us estimate the characteristic time of disintegration of the drop from the jet (the time of realization of instability of the zero azimuthal mode (m = 0)). It is noteworthy that the separation of the drop from the jet occurs due to realization of instability of axisymmetric (m = 0) mode (necking) of the capillary waves (since the instability of the bending mode leads just to bending, and the instability of the deformation mode results in deformation).
Expression (22) is written in a dimensionless form. For the accepted non-dimensionalizing, both the amplitude of the initial deformation ζ, and the current amplitude ξ, are non-dimensionalized by the division by the R radius of the jet, the increment and wave frequency are non-dimensionalized by division into a combination of , and time is nondimensionalized by division into the contrary combination . Thus, the value of the unit of time non-dimensionalizing will depend on the jet radius, density, and value of the coefficient of the surface tension.
As the estimations of the order of value show, the initial amplitude of the capillary waves on the jet is [15], or ∼0.1 nm. Then, for the jet of water with the radius R = 1 mm, we obtain ≈ 2 × 10 2 s -1 , and ≈ 5 × 10 -3 s. If the jet radius is decreased by ten times, the units of non-dimensionalizing of frequency and time will change for ∼ 10 4 s -1 , ∼ 10 -4 s.
At radius R of the jet, the characteristic time of the drop disintegration τ will be determined by the time, until the growing amplitude of the capillary wave on the jet surface becomes ≈R. tion of the laws of mode excitation with m = 2 are yet to be set up. The mode of the branching jets detected experimentally in [8], reproduced in [9,11], was interpreted as a boundary mode, which was realized at extremely high intensities and by no means connected with azimuthal mode of m = 2. Today, based on the devices of the type used in [8,9,11,14] it can be studied thoroughly.
To experimentally verify the laws of disintegration of the drop from the jet the equipment of the type described in [24][25][26] can be used.
In the general case, using the equipment described in [27], it is possible to perform the experimental study of the laws of jet stability at the initial excitation of a complex of modes or separate modes.

CONCLUSIONS
The problem of the calculation of capillary wave motion in a charged jet of ideal incompressible liquid, which moves at a constant speed relative to the ideal nonconductive material medium, when the initial deformation of the jet surface is determined with a whole set of waves with arbitrary wave and azimuthal numbers. Analytical expressions are found that meet the initial conditions to determine the temporal evolu-tion of hydrodynamic potentials in the jet and medium, as well as the electric potential in the jet vicinity. An analytical expression is obtained for the characteristic time of disintegration of the drop from the jet. The order of excitation of various azimuthal modes is studied.