Influence of inclined electric field on decay of a liquid jet during heat treatment and surfacing

Introduction

The Kelvin–Helmholtz instability occurs in various research fields, ranging from terrestrial magnetohydrodynamics [1] and turbulent liquid mixing [2] to processes such as coating deposition by electrical explosion [3] and astrophysical phenomena like the solar wind [4]. This instability serves as a powerful trigger that disrupts the stability of systems involving the mixing of two or more liquids with different properties. Some of the most notable applications of this instability include acoustic modes in air-consuming systems such as boilers, jet engines, and gas turbines [5]. Another significant application of this instability is the breakup of a liquid jet into droplets in an electric field [6; 7]. This phenomenon underlies the operation of precision devices that are integral to various technological processes, such as electric arc welding and the production of ultrafine refractory powders [6; 7]. In [8], the influence of an inclined electric field on the Kelvin–Helmholtz instability of two ideal dielectric liquids was studied, revealing the conditions under which the liquid transitions to a stable mode under a horizontal electric field. It was also demonstrated that the vertical component of the electric field exerts a destabilizing effect. Another promising application of this instability is the accelerated cooling of rolled products. In [9], it was shown that by controlling the velocity of the air – water system, it is possible to achieve the formation of droplets in the nanometer range. Upon impacting rolled products, these droplets induce a thermoelastic wave, thereby increasing their impact toughness.

The origin and development of the Kelvin–Helmholtz instability of two viscous liquids were investigated in [10; 11]. A key feature of these studies is the use of the viscous potential approximation. This approach assumes the absence of shear stress components of the stress tensor at the interface, while the viscosity of the liquid is taken into account only in the condition of continuity of normal stresses at the interface [10]. In the general case, the situation is complicated by the need to determine the velocity profile of the fluid, and as shown in [11; 12], there is no analytical solution for the stability of a flow with a complex velocity profile. However, at high wavenumbers (short wavelengths), this approximation is justified, as demonstrated in.

The interaction of the Kelvin–Helmholtz instability with ultrasonic vibrations in the case of a problem with planar geometry under the viscous potential flow approximation was studied in [14; 15]. The interaction of the Kelvin–Helmholtz instability with ultrasonic vibrations in a planar geometry under the viscous potential approximation was studied in [14; 15]. It was assumed that the effect of acoustic vibrations is equivalent to an effective oscillating gravitational field. It was found that acoustic influence shifts the maximum growth rate toward higher wavenumbers [15], and a stability region was identified between weak acoustic excitation and parametric resonance.

The objective of this study was to investigate the combined influence of an inclined electric field and a transverse acoustic field on the stability of the planar surface of an electrically conductive liquid using the example of the iron – argon and air – water systems under the viscous potential flow approximation. The simultaneous application of these two factors enables the creation of micro- and nanodroplet flow patterns, which is crucial for the development of new technologies in welding, surfacing, and accelerated cooling of rolled products.

Problem statement

Consider the instability of the planar interface between two viscous electrically conductive fluids. The first fluid is characterized by density ρ₁, kinematic viscosity ν₁, electrical conductivity σ₁, and dielectric permittivity ε₁ (Fig. 1). It occupies the region (–∞ ≤ x ≤ +∞ and –h₁ ≤ z ≤ 0) and moves with a horizontal velocity U₁. The second fluid occupies the region (–∞ ≤ x ≤ +∞ and 0 ≤ z ≤ h₂) and is characterized by density ρ₂, kinematic viscosity ν₂, electrical conductivity σ₂, and dielectric permittivity ε₂. The horizontal velocity of the second fluid is U₂. The variable ξ represents the deviation of the interface from its equilibrium position (Fig. 1).

The flow velocities of the first and second fluids are much lower than the speed of sound; therefore, they can be considered incompressible. The kinematics of the interface motion is described by the function F(x, z, t) = z – ξ(x, t). Thus \(\vec n = \frac{{\nabla F}}{{\left| {\nabla F} \right|}} = \frac{{ - \frac{{\partial \xi }}{{\partial x}}{{\vec e}_x} + {{\vec e}_z}}}{{\sqrt {1 + {{\left( {\frac{{\partial \xi }}{{\partial x}}} \right)}^2}} }}\) the normal vector is defined as (where \(\nabla \) is the gradient operator. In the linear approximation, the normal vector \(\vec n =  - \frac{{\partial \xi }}{{\partial x}}{\vec e_x} + {\vec e_z}\). takes the form. The system under consideration is placed in an external transverse acoustic field and an inclined electric field relative to the planar interface of the fluid (Fig. 1). The electric field vector, taking into account the disturbances of the interface, is given by \(\vec E = {E_{0x}}{\vec e_x} - {E_{0z}}{\vec e_z} - \nabla \psi \) (where ψ is the disturbance of the electric potential, and E_0x and E_0z are the normal and tangential components of the unperturbed electric field, respectively). The inclination angle of the electric field (Fig. 1) relative to the unperturbed surface β is defined as \({\rm{arctg}}\left( {\frac{{{E_{0z}}}}{{{E_{0x}}}}} \right)\).

The fundamental linearized equations of the viscous potential flow model for relatively small disturbances, taking into account the electric field, according to [10; 15], are given as:

where Φ is represents the velocity potential disturbance.

When deriving the first and second equations of system (1), the terms associated with the electric field were neglected, which is valid for electrically conductive liquids [16 ‒ 18], in the absence of a volumetric charge. Therefore, the electric field component will only be considered in the boundary conditions at the liquid–gas interface. At the boundaries h₂ and h₁, we impose conditions ensuring the absence of disturbances in the flow velocity and the electric field:

The boundary conditions for the disturbances of the flow potential of the liquid at the interface, taking into account the electric field, are given by:

where U_i is the velocity of the i-th fluid; \({p_i} =  - {\rho _i}\left( {\frac{{\partial {\Phi _i}}}{{\partial t}} + {U_i}\frac{{\partial {\Phi _i}}}{{\partial x}}} \right)\) is the pressure disturbance in the i-th fluid; p_si is the disturbance of the pressure of the acoustic field; i = 1, 2 is the fluid index; γ is the surface tension; E_in and E_iτ are the normal and tangential components of the field, respectively.

The boundary conditions for the electric field at the interface are defined as [19]:

Substituting the above values of the normal vector and the electric field into equation (4) and subsequently linearizing, taking into account that at the interface of two conductors σ₁E_01z = σ₂E_02z, and E_01x = E_02x, leads to the following:

The solution to equations (1) will be sought in the form:

Substituting the third, fourth, and fifth equations of (6) into equation (5) leads to the following system of equations for the constants A₃ and A₄:

The solution to system (7) after transformation takes the form:

Then, the disturbances of the electric potential will take the following form:

To derive the disturbances of the flow potential, we substitute the first, second, and fifth equations of system (6) into the kinematic boundary conditions (3). As a result, we obtain:

The contribution of the acoustic field to the pressure is determined in the same way as in [14; 15]: p_si = ρ_ig_effξ₀exp(ωt + ikx) (where g_eff = g – ΩU_acos(Ωt) is the effective acceleration; Ω and Ua are the frequency and amplitude of the acoustic excitation. Substituting (9) and (10) into the dynamic boundary condition (3) and performing subsequent transformations, taking into account that at z = 0: σ₁E_01z = σ₂E_02z, E_01x = E_02x, leads to the following dispersion equation:

To analyze equation (11) while considering the influence of weak acoustic fields, we adopt the approach used in [14; 15]. According to this method:

where f  is a time-dependent function and represents the sum of a “slow” disturbance component A₁(t) = f₁exp(ωt), and a “fast” disturbance component, A₂(t) = f₂cos(Ωt), which corresponds to acoustic vibrations. Here Ω is the frequency of the acoustic influence [14; 15]. Substituting this sum into equation (7) and discarding the cosine and sine terms, considering that f₂ = –C_acA₁/(a₀Ω₂) [14], we obtain:

The solution of equation (7) takes the form:

The second root of equation (8) has no physical significance and is therefore not considered. The growth rate of disturbances at the liquid interface is determined as α = Re(ω₁). Consequently, we obtain:

The data for calculations using equation (9) are presented in the Table.

Research results and discussion

Air-water system

In Fig. 2, a, the dependencies of the growth rate of disturbances at the air – water interface on the wavenumber in the absence of an acoustic field under the influence of electric fields are shown. The velocity difference between the horizontal layers was 15 m/s. This function has only one maximum, regardless of the presence of an electric field (curves 1 – 3). A tangential electric field (β = 0) with a strength of approximately 3·10⁶ V/m stabilizes the Kelvin–Helmholtz instability, which is expressed in a decrease in α_m and a shift of the maximum mode km toward lower values (curve 2). A normal electric field (β = π/2) of the same strength, on the contrary, enhances this instability (curve 3), which is consistent with widely accepted concepts [8; 19]. At inclination angles of the electric field vector β π/6, π/4, π/3 (Fig. 2, b), an increase in k_m is observed from 59,170 m^–1 (λ_m = 106.19 μm) at β = π/6 to 119,709 m^–1 (λ_m = 52.48 μm) at β = π/3. The analysis of neutral curves (Fig. 2, c) showed that the presence of a vertical component of the electric field significantly narrows the range of relative velocity differences between the fluids, within which capillary forces and the tangential electric field suppress the Kelvin–Helmholtz instability. It should be noted that a similar effect was observed in [8] for the case of inviscid dielectric liquids.

Let us consider the combined influence of weak acoustic and inclined electric fields on the Kelvin–Helmholtz instability. Fig. 3, a shows the dependencies of the disturbance growth rate at an amplitude value of the acoustic oscillation velocity U_a = 5 m/s. These dependencies indicate that acoustic vibrations suppress the Kelvin–Helmholtz instability (curves 1 – 3), with the tangential electric field amplifying this effect (curve 2), while the normal electric field, on the contrary, weakens it. A weakening of the suppression effect is also observed at electric field inclination angles β π/6, π/4, π/3 (Fig. 3, b), which is confirmed by the analysis of neutral curves (Fig. 3, c). Notably, similar suppression phenomena of the Kelvin–Helmholtz instability were identified in [14]. However, in that study, the application of acoustic fields resulted in a rightward shift of k_m , while the maximum growth rate increased, indicating an enhancement of the Kelvin–Helmholtz instability. The discrepancy between the results of [14] and the present study may be explained by the fact that in [14], the ratio \(\frac{{C_{ac}^2}}{{2{a_0}{\Omega ^2}}}\) in equation (8) was taken with a negative sign, whereas in this study, it was taken with a positive sign.

Argon-iron system

Now, let us consider a different situation that arises in surfacing or welding processes. In these processes, when the electrode melts in an argon shielding gas environment, a jet of liquid metal forms, which, under certain conditions, breaks up into droplets [20]. The mode of material transfer determines the quality of the formed coating, making the development of methods for controlling this process an important task. In this system, the mutual flow of gas and liquid gives rise to the Kelvin–Helmholtz instability [20]. Fig. 4, a presents dispersion curves, showing that they exhibit two maxima. According to [13], the first maximum is hydrodynamic, while the second is viscosity-induced (curve 1).

The application of a tangential electric field with a strength of 3·10⁷ V/m leads to the near-complete suppression of the hydrodynamic maximum (curve 2), whereas in a normal electric field, this effect is weaker (curve 3). As results of [19] show, the change in the sign of the influence of the transverse electric field on capillary instability is associated with the ratio of the specific electrical conductivities σ = σ₂/σ₁ and the dielectric permittivities of the fluids ε = ε₂/ε₁. If ε > σ, the electric field has a stabilizing effect; otherwise, when (ε < σ) the electric field has a destabilizing effect [19]. In the case under consideration (see Table), for the argon–iron system σ ~ 0.00133, and ε ~ 0.0002, which should indicate a destabilizing effect of the normal field. However, this is not observed (Fig. 4, a). This discrepancy with capillary instability can be explained by the fact that, in this case, the relative velocity of the flowing layers becomes significant, altering the condition for the occurrence of Kelvin–Helmholtz instability maxima [21]. A further increase in the specific electrical conductivity ratio σ beyond 0.012 leads to the restoration of the hydrodynamic maximum and the complete suppression of the viscosity-induced maximum. When σ ≥ 0.015, the effect changes from stabilizing to destabilizing (Fig. 4, b). The results of studying the effect of inclined electric fields (Fig. 4, c) showed that these fields enhance the Kelvin–Helmholtz instability regardless of the values of σ and ε. In this case, the maximum growth rate is viscosity-induced. At field strengths E_20x < 3·10⁷ V/m, a weakly pronounced hydrodynamic maximum is observed. The neutral stability curves indicate that, as in the air – water system, a reduction in the stability region of interface disturbances is observed (Fig. 4, d).

The combined effect of acoustic vibrations with a velocity amplitude of 10 m/s and an electric field with a strength of 3·10⁷ /V/m at σ ~ 0.00133, and ε ~ 0.0002, on the contrary, leads to the complete suppression of the viscosity-induced instability maximum (Fig. 5, a, curve 1) and a sharp reduction in the maximum growth rate α_m of disturbances of hydrodynamic origin. The application of a horizontal electric field significantly weakens this effect (Fig. 5, a, curve 2), while in a vertical field, this effect, on the contrary, is amplified (Fig. 5, a, curve 3), despite the fact that at E_20x = 0 and E_20z = 3·10⁷ V/m, the value of k_m is greater than E_20x = 3·10⁷ V/m and E_20z = 0. A change in the sign of the effect to destabilizing also occurs at σ ≥ 0.015 (Fig. 5, b). The application of an inclined electric field, as in the absence of sound, enhances the Kelvin–Helmholtz instability, but α_m is somewhat lower (Fig. 5, c). The neutral curves (Fig. 5, d) show an increase in the range of fluid velocities in which capillary forces and tangential fields suppress the Kelvin–Helmholtz instability under the influence of an acoustic field.

Conclusions

It has been established that the combined application of inclined electric fields and acoustic vibrations in the air – water system allows the formation of liquid droplets in the range of 10 to 100 μm, even at low gas flow velocities. This opens up new prospects for the development of accelerated cooling technologies for rolled products to achieve high hardness and impact toughness.

Установлено, что совместное применение наклонных электрических полей и акустических колебаний в системе воздух – вода позволяет получать капли жидкости в диапазоне от 10 до 100 мкм даже при малых скоростях движения газа. Это открывает новые перспективы для разработки технологий ускоренного охлаждения проката, для достижения его высокой твердости и ударной вязкости.

For the argon–iron system, it has been shown that a vertical electric field at σ ~ 10^–3 and ε ~ 10^–4 practically suppresses the hydrodynamic maximum. An increase in the specific electrical conductivity ratio σ beyond 0.012, on the contrary, leads to the restoration of the hydrodynamic maximum and the complete suppression of the viscosity-induced one. At σ ≥ 0.015, the effect changes from stabilizing to destabilizing, regardless of the number of maxima in the dependence of the growth rate on the wavelength. The application of a tangential electric field completely suppresses the Kelvin–Helmholtz instability.

The combined effect of acoustic vibrations and a normal electric field leads to the complete suppression of the viscosity-induced instability maximum and a sharp reduction in the maximum growth rate of disturbances of hydrodynamic origin, while in a horizontal electric field, this effect is significantly weaker.