1 Introduction

The electrically driven jet is an important and interesting problem in the area of physics, applied mathematics, and fluid mechanics. The mathematical model describing the fluid flow in this system traces back to 1969 in the work published by Melcher and Taylor^[1-2]. Even though this problem has been known for so long, most of its understanding lies at the linear stage of the flow. In 2001, there was a significant development in the theory of electrically driven jets in the work published by Hohman et al.^[3-4]. The work in Ref. [3] extended the currently available understanding of electrically driven jets which at that time relied on very idealistic fluid flows where a zero viscosity or infinite electrical conductivity was assumed. The work from Hohman et al.^[3] resulted in a model that can analyze more realistic flows over a range of wave numbers of practical relevance. Their work and efforts were concentrated at the linear temporal stage of problem. The efforts in this paper are towards the nonlinear stage of electrically driven jets^[5-10].

The governing equations we study here are applicable to various physical processes. Electrically driven jets have practical applications in the area of electrospraying^[11] and electrospinning^{[3-4, 9-10, 12-14]}. Electrospinning is a technology used to control and produce small fibers by the use of large electric fields. These fibers have applications that range from electrical circuits to bio-medical applications. Electrospinning has been known for many years, but due to the complexity in the mathematical modeling, this process continues to lack a mathematical formulation for prediction of nonlinear instability. This gives us the motivation to undertake this nonlinear problem.

This paper studies the nonlinear regime of electrically driven jets by considering nonlinear resonant three-wave interactions. The present study can be seen as an extension of the work done in Ref. [15] in which two-wave interactions for spatio-temporal instabilities were studied. For a more detailed introduction, we refer the readers to Ref. [15]. Other developments trace back to either space or time alone evolving instabilities^{[3-4, 16-17]}. The mathematical formulation at least as the main model equations (the system of four partial differential equations) remains the same, but the nonlinear three-wave interactions explored here provide an entire new study in the jet flow system.

In this investigation, in contrast with Ref. [15], we now consider instability waves of triad type (three-wave interactions) which resulted in higher level of instabilities in the jet flow system. We also include wave interactions of a single and multiple triad modes. A more detailed derivation for the fluid parameters is now presented. The linear instability modes explored here are completely different from those in Ref. [15], and the present modes are much closer to the neutral instability curve (the onset of instability). The nonlinear instability results we present are completely different from those in Ref. [15] since the linear instability modes are used to study the nonlinear wave interactions in the jet flow system. The nonlinear wave interaction formulation has also been revisited and improved to take into account different solutions of the jet flow system at their respective order expansions (first-and second-orders). We also include the nonlinear solution of the jet's electric field which extends the currently available results. All results and computations presented in this investigation are completely different from those in Ref. [15], but they rely on the formulation presented in Ref. [15], and we decide to leave the formulation here to have a self-contained paper.

The present study extends the linear^{[3-4, 16]} and nonlinear^{[15, 17]} understanding of the problem. We find, in particular, that by considering resonance triad modes, we are able to find new operating modes that cannot be detected by the linear stability theory alone. It is found that nonlinear instability provided interesting operating modes, in which the thickness of the jet flow is reduced and the electric field inside the jet is properly oriented. The nonlinear triad resonance investigation is able to provide better agreement with the physical problem of a jet that stretches and thins due to the use of large applied electric fields.

2 Governing equations for jet flow

The mathematical formulation for the electrically driven flows is given by a system of four nonlinear partial differential equations^[16],

(1)

where is the total derivative, u is the velocity vector, P is the pressure, E is the electric field vector, Φ is the electric potential, q is the free charge density, ρ is the fluid density, μ is the dynamic viscosity, K is the electric conductivity, and t is the time variable.

The above equations are further simplified by the axisymmetric assumptions on the jet flow as it was done in Refs. [3]-[4], [16], and [18] and non-dimensionalized using r₀ (radius of the crosssectional area of the nozzle orifice at z = 0), , t₀ = (ρr₀/γ)^1/2, r₀/t₀, and as scales for length, electric field, time, velocity, and surface charge, respectively. The system to be studied under the triad resonant instability approach has the following form^[16]:

(2)

(3)

(4)

(5)

where the independent variables are in terms of h(z, t), v(z, t), σ(z, t), and E(z, t) for the jet's radius, velocity, surface charge, and electric field, respectively. The conductivity K is assumed to be a function of z in the form ^[16], where K₀ is a constant of dimensional conductivity, and is a non-dimensional variable function. is the non-dimensional conductivity parameter. is the permittivity ratio constant of the fluids. is the non-dimensional viscosity parameter (it is also possible to use the Reynolds number but here we follow the original formulation due to Hohman et al.^[3]), and 1/χ is the local aspect ratio (χ is the ratio of the length scale and the radius of the electrically induced viscous jet)^{[3, 10, 17-19]}, which is assumed to be small. E_b is the externally applied electric field E_b(z) =Ω(1-δz), where δ is a small parameter which will be defined later. In the present work, we consider both cases of uniform and nonuniform axially dependent electric fields, which are known experimentally to be the cases if the direction of the jet is initially inclined perpendicular to the capacitor plates. Riahi^[16] found that for such a variable external field, the equilibrium state, that was related to the earlier one in Ref. [3] for their considered uniform base state, can be satisfied only if the conductivity of the fluid can be nonuniform, as is the case in a number of fluid flow cases.

3 Linearized jet flow dynamics

For linear stability in the jet flow, we consider the solution of Eqs. (2)-(5) to be a sum of an equilibrium solution (electro-static point) plus a small perturbation.

There are three basic states in which the linear stability analysis can be performed^[3]. Two of the basic states are associated with an applied field, but no surface charge. The other is associated with a surface charge, but with no applied electric field. These two cases are in fact perfectly cylindrical states and are not relevant for a thinning jet. Following Hohman et al.^[3], the basic state that we are interested in is the one that a jet can be at a distance away from its initial Taylor cone formation and is thin due to a tangential electric field, which is the main driving force. Hence, the gravity effect can be neglected as it is known to be rather small. Reneker et al.^[12] and Hohman et al.^[3] did not include the gravity effect since it is very minor on the jet instabilities that promote significant jet radius reduction. Based on the modeling conditions, the stability characteristics of a thinning jet can be locally approximated by a charged cylinder of constant radius. Hence, our basic state solution has a jet radius, velocity, and charge density of constant value and variable applied electric field.

(6)

(7)

(8)

(9)

where h₁, v₁, σ₁, and E₁ are assumed to be small. s and f are the growing disturbances for the spatial and temporal cases, respectively, k is the wave number, and ω is the real frequency. Here, both Ω and σ₀ are prescribed with constant values. σ₀ is the background free charge density, and Ω accounts for the strength of the applied electric field. We set to be a small parameter (δ1). The parameter δ can be zero if σ₀ = 0. For δ = 0 and δ≠0, the model represents situations in which the applied electric field is uniform or non-uniform. In the experimental setting, this relates to how good collector plate aligns in reference to the orifice. Using the approximation of the jet's aspect ratio χ in the limit of jet flows with small wave numbers as done in Ref. [3], we set 1/χ = 0.89k. Plugging Eqs. (6)-(9) into Eqs. (2)-(5) gives

(10)

(11)

(12)

(13)

Equations (10)-(13) can be written in a matrix-vector form. This system admits non-trivial solutions as long as the determinant of the coefficient matrix for the unknown quantities is equal to zero. The dispersion relation takes the form of a single nonlinear equation,

(14)

which can be expressed as D(s, f, w; k; ν^*; K^*; σ₀; Ω; β) = 0. After prescribing the particular fluid parameters, one obtains

(15)

We can decompose the lengthy expression into its real and complex components as follows:

(16)

Then, we set the two components in the above equation to be zero to formulate the solution of the dispersion relation,

(17)

(18)

Equations (17) and (18) are solved using Newton's method implemented in MATHEMATICA software.

We consider two cases of a water-glycerol mixture, one with lower viscosity (ν^* = 0.607 64) and the other with higher viscosity (ν^* = 1.01). Using these cases of viscosity leads to triad growth rates reasonably close to the neutral modes of instability, where the present weakly nonlinear approach can be applicable. More details about these values were given in Refs. [3] and [10] (see Section 6). We present the solutions of the problem of linear spatial and temporal instability under different strengths and types of externally applied electric field. These solutions at the linear stage are then used to understand the nonlinear evolution of the problem.

4 Nonlinear dynamics via resonant triad interactions

To study the nonlinear problem, we use the resonant theory of three-wave interactions^{[17, 20-24]}. Using h = 1 + h₁, v = 0 + v₁, σ = 1 + σ₁, and E = E_b + E₁ in Eqs. (2)-(5), we set up the problem with linear and quadratic terms on left-and right-hand side, respectively. After dropping the subscripts, we get

(19)

(20)

(21)

(22)

In order to capture the nonlinear wave interactions ^{[20, 22]} in the flow, we let the solution of the above equations be expressed as (h, v, σ, E)=(h₁, v₁, σ₁, E₁)+ ²(h₂, v₂, σ₂, E₂), which allows for the problem to be treated at the first-and second-orders. The expressions have the following forms:

(23)

(24)

where h_1n, v_1n, σ_1n, and E_1n are constants, and h_2n(z_s, t_s), v_2n(z_s, t_s), σ_2n(z_s, t_s), and E_2n(z_s, t_s) are functions of slowly varying spatio-temporal variables, wave numbers k_n, frequencies ω_n, small growth rates f_n, and s_n (f_n = , s_n = , n=1, 2, 3 with , and as the first-order quantities). Here, A_n(z_s, t_s) are the amplitude functions accounting for the nonlinearity presented in the problem. Using Eqs.(23)-(24), and , where z_s:= and t_s:= in Eqs.(19)-(22), gives a system at the second-order ,

(25)

(26)

(27)

(28)

The above system given by Eqs.(25)-(28) requires the solutions from Eq.(15) that satisfy the triad resonance conditions (k₃, ω₃)=(k₁, ω₁) + (k₂, ω₂). The nonlinear problem formulation studied here requires the amplitude values in Eq.(23) (h_1n, v_1n, σ_1n, E_1n) for n=1, 2, 3. Finding such constant amplitude values allows us to construct linear solutions (explained in detail in Section 6) based on the linear theory and to later study the problem in a triad resonance sense (based on the weakly nonlinear theory). Using Eq.(23) in Eqs.(19)-(22) leads to linear systems at the first-order O(),

(29)

(30)

(31)

(32)

Prescribing suitable values for h_1n for n=1, 2, 3 according to the weakly nonlinear theory ^{[20, 22]} and using triad resonant solutions, the above system is solvable. To set up the solvability of the nonlinear problem, we make use of the adjoint variables (h_1n^(a), v_1n^(a), σ_1n^(a), E_1n^(a)). Here, the inner product is used, (a, b)=ab, where a refers to the conjugate transpose of the vector a, and the overbar takes the meaning of the complex conjugate. Using the inner product of the linear system formed by the left-hand sides of Eqs.(25)-(28) with Eq.(24) against (h_1n^(a), v_1n^(a), σ_1n^(a), E_1n^(a)) gives the single equation (33). The left-hand sides of Eqs.(25)-(28), denoted by Φ₍₂₅₎, Φ₍₂₆₎, Φ₍₂₇₎, and Φ₍₂₈₎ can be computed by simply using the left-hand sides of Eqs.(29)-(32) with (h_2n, v_2n, σ_2n, E_2n) instead of (h_1n, v_1n, σ_1n, E_1n) for n=1, 2, 3. In other words, the notation (the left-hand side of Eq.(25)) represents the second-order O(²) quantities found on the left-hand side of Eq.(25) after the spatio-temporal type of disturbances are used from Eqs.(23)-(24).

(33)

The above equation is a direct result of the solvability in Eqs.(25)-(28). We further express Eq.(33) as φ₁h_1n +φ₂v_1n+φ₃σ_1n+φ₄E_1n=0 which leads to a system of equations that can be solved for the adjoint amplitude variables for n=1, 2, 3.

Using the constant amplitude values and Eqs.(23)-(24) in the nonlinear Eqs.(25)-(28), we set up the nonlinear resonant problem^{[22, 25]} for the case n=1. This gives a system at the second-order O(²),

(34)

(35)

(36)

(37)

where the overbar in the amplitude function A(z_s, t_s) denotes the complex conjugate, and the coefficients (b_i, c_i, d_i) for i=1 to 4 are lengthy and will not be given. The system in Eqs.(34)-(37) can be expressed as Lx = N with the adjoint problem formulation L^(a)x^(a) = 0. x represents the vector with the amplitude values (h_1n, v_1n, σ_1n, E_1n), L represents the coefficient matrix, and N represents the right-hand side vector. Pairing N against x^(a) gives (N, x^(a))=(Lx, x^(a))=(x, L^(a) x^(a))=(x, 0)=0. For n=1, this gives the solvability condition for the above system^[26]. This reduces to a nonlinear partial differential equation,

(38)

Considering the cases for n=2 and n=3, we complete the formulations for the conditions. The system governing the evolution of the amplitude functions A₁(z_s, t_s), A₂(z_s, t_s), and A₃(z_s, t_s) is given below,

(39)

(40)

(41)

where the coefficients (C_i, D_i, G_i) for i=1, 2, 3 are given in Appendix A, and f_n = , s_n = , n=1, 2, 3 with and are the first-order quantities.

5 Amplitude functions

The amplitude functions are solved numerically with standard finite differences that consist of first-order approximation in time and second-order approximation in space. This approach is often used in physics and optics problems. In this context, the problem reads

(42)

(43)

(44)

where

and A₁^t₀ and A₁^t_s represent the amplitude function A₁(z_s, t_s) at a previous time step t₀ and at the current time step t_s, respectively. Equations (42)-(44) are now ordinary differential equations, which can be solved by choosing the initial condition and boundary conditions that comply with the weakly nonlinear theory^[22]. We also make use of the Courant-Friedrichs-Lewy (CFL) condition for convergence in solving partial differential equations. Solving for these amplitude functions and using them in Eq.(23) give the nonlinear solutions in the jet flow system.

6 Jet flow linear and nonlinear evolution

In this paper, we extend the linear and nonlinear work previously reported in Refs.[15] and [17]-[18]. The spatial and temporal growth rates are detected over a larger range of values of the wave number k when compared with those results in Ref.[18]. This is attributed to the effectiveness of the algorithm used to solve the dispersion relation in Eq.(14). One can see how these new results in Figs. 1-2 and Figs. 6-7 are very distinct from those in Ref.[15]. For the experimental values used in the present study, we follow the work of Refs.[3] and [10]. Parameter values for different quantities used for modeling electrically driven jets are compared and verified against the theoretical framework^[3-4]. We use the following average values of quantities for viscosity and conductivity parameters for glycerol fluid^{[3-4, 10, 17-19]}. We set ν=0.001 49 m²/s, K=1 μs/m, r₀=0.0005m, γ=0.065 N/m, ρ=1200 kg/m³, and = 8.854× 10^-12 C²/(N·m²). We consider two cases of water-glycerol mixture one with ν=1.66217 × 10^-4m²/s as a relative higher viscosity case and the other with ν=1.0× 10^-4 m²/s as a lower viscosity case, and the rest quantities are the same as those in the case for glycerol.

We consider two types of fluids (mentioned above) for the jet flow with the parameters K^*=19.60, ν^* (higher viscosity water-glycerol) =1.01, ν^* (low viscosity water-glycerol) = 0.60764, and β=77. We also use σ₀ =0.0 (uniform =0.01. For the electric field, we consider several different magnitudes Ω =0.5, 1.0, 1.5, and 2.0. For the triad wave interactions, we consider two cases in which one case is composed of all positive growth rates. The multiple triad mode or mixed triad mode is a mode that uses both negative and positive growth rates to form a triad wave interaction.

The solutions of Eq.(14) are computed for both cases of the applied electric field (δ=0, δ ≠0). The solutions satisfying the triad resonance condition are used to study nonlinear evolution in the jet flow. These linear solutions uncover the Rayleigh instability^{[3-4, 27]} in the flow. At the nonlinear stage, we solve Eqs.(42)-(44) and use these amplitude functions A₁(z, t), A₂(z, t), and A₃(z, t) in Eq.(23) which satisfies Eqs.(19)-(22) in a triad sense (these are the nonlinear triad solutions). These solutions give the time and space evolution of an electrically driven jet in the context of axisymmetric imperfections. We label these results as the perturbation plots. We will make a comparison of the nonlinear vs. linear solutions. The linear solutions are also in the form of Eq.(23) except that the amplitude functions A_n(z, t) (n=1, 2, 3) are of constant value (amplitudes will not be modulated by nonlinear triad interactions). These constant values are obtained from the data used to solve the governing system in Eqs.(42)-(44). We use the initial configurations (in time and space) of the amplitude functions to set the constant amplitude values needed for the solutions in absence of wave interactions. Once these constant amplitudes are known, one can reconstruct linear solutions of similar type as in Eq.(23), but with constant amplitudes. These solutions are known as the solutions in absence of nonlinear effects or simply the linear solutions.

6.1 Nonlinear resonance structure

The smallness of the small parameters and nonlinear theories is also discussed. Many dispersion relations including the present one support resonances, but not all have the nearly-neutral resonances in weakly nonlinear cases. Nearly neutral resonances mean that over particular initial stage in time or axial variable (in the present study), the effect of the positive growth rates keeps the magnitude of the perturbation quantities O(). This means that products of f, t, s, and z for the particular time interval t and initial axial interval z, which enter the exponential parts for the expression of the linear perturbation quantities, need to be relatively small. In the present study and for those cases where s or f is not too small, we consider these initial time and initial axial intervals to be sufficiently small in order to keep the magnitude of the linear perturbation quantities of O(). About any relation that spatial and temporal growth rates of a spatio-temporal instability mode may have, it can be, in general, different from what has been known between the temporal growth rate of a temporal instability mode and the spatial growth rate of a spatial instability mode^{[22, 28]}. It is known^[27] that a local instability mode is called absolutely unstable if it enhances in time and spreads in both upstream and down-stream, while it is called convectively unstable if it sweeps away from its initial source. From our results and the generated data, we find in Figs. 3-5 and Figs. 8-10 that for given fixed values of real wave number and real frequency (fixed phase speed) of a triad mode, which is in fact a wave packet instability mode, spatio-temporal instability of such a mode can be swept away from the initial impulse source and spread in time and in the axial direction in both downstream and upstream so that it can be classified as both convectively and absolutely unstable mode^[27] as we described before. We now present the results of this study including two-dimensional and three-dimensional nonlinear solutions for the jet flow.

6.2 Lower viscosity flow

Consider the single triad mode for the lower viscosity jet. Figures 1 and 2 represent the combined temporal and spatial growth rate f and s versus the wave number k for the uniform applied electric field (σ₀ =0.0), respectively, and for different values of Ω. Figure 1 shows that the temporal growth rate f decreases as Ω increases which gives rise to the classical Rayleigh instability^{[3, 22, 24]}. Temporal growth rates are found to be always stable for the largest case of the applied field. Unstable or positive growth rates are found for smaller Ω and k < 0.4. Figure 2 illustrates that by increasing the magnitude of the electric field Ω, the instabilities are suppressed which exhibits the Rayleigh instability^[3-4]. The growth rate s increases for larger wave number values. Both temporal and spatial growth rates are slightly suppressed for the case of non-uniform applied field.

For the case of δ=0, we find several modes that satisfy the three-wave resonant conditions. Figures 3(a)-3(b) contain the three-dimensional solutions for the jet radius h₁(z, t) and the jet electric field E₁(z, t) along with their corresponding contour plots underneath in Figs. 3(c)-3(d). Darker regions in contour plots represent steep transitions with the decreasing magnitude, and lighter regions indicate the increasing magnitude. For these plots, the vertical axis describes t, and the horizontal axis is set to be z. It is important for the electric field E₁(z, t) (the mechanism in electrically driven jets) to increase and for the jet radius h₁(z, t) to decrease. Figures 3(a)-3(b) demonstrate important characteristics for both the electric field and the jet radius. The jet radius decreases for t>4, and the jet electric field increases on this same region. For this investigation, we compare perturbation plots accounting for nonlinear wave interaction (left-hand side figures) and in the absence of such interactions (right-hand side figures). For the spatial mode (fixing t), Fig. 3(f) illustrates that the jet radius (thin line) decreases for z>80, the velocity increases (dotted line), the surface charge (dashed line) stays neutral, and the electric field (thick line) increases. The nonlinear effects considered in Fig. 3(e) provide a faster jet radius reduction at much smaller z values. With regard to time evolution (fixing z), Fig. 3(h) shows how the linear instability is modified by changing the oscillatory evolution of the perturbed quantities into the ones shown in Fig. 3(g) in which both the electric field and the jet radius are significantly enhanced. The instability modes presented in Fig. 3 under the triad interaction significantly change the trajectories for perturbation quantities into ones of interest for applications.

For the case of δ ≠ 0, we include another triad mode shown in Fig. 4. From the nonlinear solutions of h₁(z, t) and E₁(z, t), it can be seen that the jet radius decreases, and the electric field increases without any domain restrictions. Selecting different cross-sections from Figs. 4(a)-4(b) leads to Figs. 4(e)-4(h). Perturbation quantities for space evolution undergo significant changes in amplitude and direction due to nonlinear interaction, and this actually is attained with a more reduced z-axis in Fig. 4(e) compared with Fig. 4(f). For the case of time evolving perturbations, the nonlinear interaction modifies an already thinning jet radius (see Fig. 4(h)) to a jet radius that also decreases but at a shorter t-axis and at a higher rate (see Fig. 4(g)).

Consider the multiple triad mode for lower viscosity water-glycerol mixture jet. For the lower viscosity water-glycerol fluid and σ₀=0.1, we consider multiple triad modes that included stable and unstable modes (positive and negative growth rates). Results for h₁(z, t) and E₁(z, t) using the multiple triad modes are shown in Fig. 5(a)-5(b). The jet radius undergoes reduction on the entire domain, while the electric field increases. Compared with the single triad mode as in Fig. 3, the solutions with multiple triads exhibit less restriction on their space and time domain. For the space evolution, triad interaction allows a more favorable mechanism specially in the jet radius from thickening to thinning jet (see Figs. 5(e)-5(f)), and for z>40, the jet is reduced at an increased rate. Similarly, for the time evolution(see Figs. 5(g)-5(h)), the nonlinear triad wave interactions modify the linear instability results and allow a more realistic mechanism in the jet flow system.

6.3 Higher viscosity flow

Consider the single triad mode for the higher viscosity water-glycerol jet. Figures 6 and 7 present the combined temporal growth rate f and the spatial growth rate s versus the wave number for the case of (σ₀=0.0) and for different values of Ω. f is found to be suppressed for increasing values of k, and the temporal growth rate f is found to be stabilized for k>0.2. It can be seen that the Rayleigh instability sets in for k < 0.4, and for other values of the wave number k, the conducting instability is present. For the spatial growth rates, the Rayleigh type of instability is detected, and the growth rates s increases for larger wave numbers.

We find another mode that satisfies the triad resonant condition for the case of the higher viscosity fluid and for δ=0. It can be seen from Figs. 8(a)-8(b) that the jet radius h₁(z, t) decreases and the electric field E₁(z, t) increases with no restriction on the time or space domain. For the space evolution, the perturbation plots dominate on their nonlinear interaction, as shown from the linear counterpart. This includes not affecting the dynamics of an already good solution and a reduction on the t space variable that indicates a high level of nonlinear wave interaction (see Figs. 8(g)-8(h)). On the space evolution, Figs. 8(e)-8(f) demonstrate that linear results are improved by the weakly nonlinear approach. The dominant effects on the glycerol fluid due to nonlinear interactions are noticeable when the nonlinear interaction reduces the jet radius dramatically for z>70.

For σ₀=0.1, we find oscillatory resonant modes that satisfy the triad conditions, and these modes significantly enhance the perturbation quantities (nonlinear solutions), but also restrict the space and time domains for which the solutions provide interest for application. The jet radius h₁(z, t) and the electric field E₁(z, t) are shown in Figs. 9(a)-9(b), and they both demonstrate favorable solution paths for which the jet radius decreases and the electric field increases. Combining their contour plots and three-dimensional plots information, we can consider solution paths for both time and space evolution with regard to the time and space instabilities. Space evolving instabilities are significantly enhanced with a considerable jet radius rate of reduction when the resonant triad modes are considered (see Figs. 9(e)-9(f)). For evolution in time, the resonant modes favor the realistic features expected in modeling of thinning jets. The linear results are modified by the triad interactions allowing a proper orientation of the electric field along with a reduction in jet radius (see Figs. 9(g)-9(h)). The effects of the triad resonant interactions become more dominant for the case of a non-uniform applied field than that of a uniform applied field as shown in Fig. 8.

Consider multiple triad modes for the higher viscosity water-glycerol mixture jet. For this fluid parameter and σ₀=0.1, we consider multiple triad modes that include stable and unstable modes (positive and negative growth rates). The nonlinear solutions for h₁(z, t) and E₁(z, t) using the multiple triad modes are shown in Figs. 10(a)-10(b). The jet radius undergoes reduction except for t>1.7, and the electric field in the jet finds the increasing paths for t < 2. Using the information about the electric field and jet radius as shown in Figs. 10(a)-10(d), we can provide time and space evolution for the perturbation plots in which the jet flow favors the results relevant for applications. For the case of space evolution, the linear results provide a mechanism of no interest for application as the jet radius oscillates and the electric field decreases for z>80 (see Fig. 10(f)). The nonlinear wave interactions of the multiple modes we considered modify those results. It can be seen in Fig. 10(e) that around z=60, the linear spatial stability predicted by linear theory saturates and the jet dynamics is changed to a jet radius that is decreasing at a high rate. For the time evolution, the linear and nonlinear solutions are very close in value. Their perturbation plots are about 10^-2 different, and this is attributed to the combination of stable and unstable temporal growth rates used giving an oscillatory behavior, as shown in Figs. 10(g)-10(h).

6.4 Triad vs. dyad resonance

Comparison between triad and dyad resonance is as follows. Comparing the present investigation results with those obtained in Refs.[15], [17], and [29], we find significant difference in the instabilities (amplitude, interaction, and evolution) when we consider triad nonlinear wave interactions to study viscous jets induced by electric fields. Studying nonlinear combined spatial and temporal instabilities via three-wave interactions provides a more natural and realistic mathematical modeling for the disturbances in the jet compared with the more restricted cases (only resonant temporal instability reported in Ref.[17] and linear spatio-temporal instability reported in Ref.[18]). We find under the triad resonance investigation, higher level of instabilities that lead to larger rates of increase in magnitude for the electric field E₁(z, t) and jet thickness h₁(z, t) to dramatically decrease at larger rates than those found under the dyad resonance investigation. This investigation demonstrates that under triad resonance, much more reduced z-axis (distance away from orifice) and t-axis (time evolution on jet) are required in order to achieve the proper nonlinear mechanism favored in applications in which a thinning jet is desired. We also find better agreement with the physical problem of a thinning jet under the triad wave interaction approach that we conducted on this investigation.

7 Concluding remarks and future work

The goal in this paper is to contribute in the understanding of electrically forced jets and to continue to expand and extend the understanding that is available at the linear stage and make contributions at the nonlinear stage of the problem. We conduct a nonlinear resonant study in the jet flow by considering spatio-temporal instabilities due to wave-like disturbances of axysymmetric type.

We find nonlinear properties in our investigation that allows a favorable change in the dynamics of the jet flow that changes from a thickening to a thinning jet radius with proper jet electric field orientation. Considering the electrically driven jet model in a nonlinear wave theory approach under the triad resonant settings provides a mechanism which is very different from that obtained from a classical linear stability theory point of view. This mechanism may be of high interest in practical applications (i.e., electrospinning) to structure the control and prediction of instabilities in the jets in order to produce higher quality fibers.

For future work, we consider it very important to conduct studies of electrically driven flows for the case of non-axisymmetric flows^[30]. This problem will be addressed in an entire new manuscript since it involves lengthy and complex formulation for a jet flow with a moving center line. This type of flow is associated with the so-called whipping instability which is directly linked to the applications of electrified jets. We plan to extend our work to the nonlinear regime of the non-axisymmetric flow with dyad and triad nonlinear wave interactions and fully time dependent numerical simulations.

Appendix A

The coefficients C_n, D_n, and G_n (n=1, 2, 3) in Eqs.(39)-(41) are

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)