Effects of Asymmetric Gas Distribution on the Instability of a Plane Power-Law Liquid Jet

As a kind of non-Newtonian fluid with special rheological features, the study of the breakup of power-law liquid jets has drawn more interest due to its extensive engineering applications. This paper investigated the effect of gas media confinement and asymmetry on the instability of power-law plane jets by linear instability analysis. The gas asymmetric conditions mainly result from unequal gas media thickness and aerodynamic forces on both sides of a liquid jet. The results show a limited gas space will strengthen the interaction between gas and liquid and destabilize the power-law liquid jet. Power-law fluid is easier to disintegrate into droplets in asymmetric gas medium than that in the symmetric case. The aerodynamic asymmetry destabilizes para-sinuous mode, whereas stabilizes para-varicose mode. For a large Weber number, the aerodynamic asymmetry plays a more significant role on jet instability compared with boundary asymmetry. The para-sinuous mode is always responsible for the jet breakup in the asymmetric gas media. With a larger gas density or higher liquid velocity, the aerodynamic asymmetry could dramatically promote liquid disintegration. Finally, the influence of two asymmetry distributions on the unstable range was analyzed and the critical curves were obtained to distinguish unstable regimes and stable regimes.


Introduction
The disintegration of plane liquid jets into small droplets plays a significant role on combustion efficiency in numerous applications, such as gas turbines and rocket engines. As a result, a large number of investigations were carried out on the breakup of plane liquid jets, especially for inviscid fluids and Newtonian fluids [1][2][3][4][5][6][7].
However, non-Newtonian fluids are also extensively involved in engineering applications, such coal water slurry [8], gelled propellant [9], and sunflower oil. According to the rheological feature, there exist a lot of types of non-Newtonian fluids, such as viscoelastic fluid, power-law fluid and Binham fluid. The breakup mechanism of plane non-Newtonian liquid jets was investigated by Liu [10], Brenn [11], Yang [12,13] and Thompson [14] in details. Liu et al. [10] analyzed the instability of a viscoelastic sheet with two-dimensional disturbances. The results showed that viscoelastic liquid sheets had a higher growth rate than Newtonian liquid sheets both for symmetric and antisymmetric disturbances, resulted from the liquid elasticity. Brenn et al. [11] pointed out that two-dimensional disturbances dominated the instability of viscoelastic plane sheets both for symmetric and antisymmetric disturbances. Yang et al. [12] investigated the instability of planar viscoelastic liquid sheets with two gas streams which were subject to unequal velocities. It was found that larger Before disturbed, the mean flow field of liquid phase is expressed as follows at −h ≤ y ≤ h: The mean flow fields of right and left gas phases are separately at h ≤ y ≤ b1* and −b2* ≤ y ≤ h: where, j = 1 and 2 represent right and left gas media, respectively.
After a small disturbance is introduced, the gas-liquid interface will leave the balance position and the flow fields of both phases become: For incompressible liquid phase, the equations governing the conservation of mass and momentum in the Cartesian coordinate are expressed: 0 l ∇ ⋅ = U (5) ( ) l l l l t ρ ∂ + ⋅∇ = −∇ +∇⋅ ∂ U U P T (6) where, Ul is the velocity vector of liquid phase and Pl is the liquid pressure.
The power-law model is used to describe the rheological property of liquid phase: where T is the extra stress tensor of liquid phase, γ the strain-rate second order symmetric tensor, n the power law exponent, and K the viscosity coefficient. Before disturbed, the mean flow field of liquid phase is expressed as follows at −h ≤ y ≤ h: The mean flow fields of right and left gas phases are separately at h ≤ y ≤ b 1 * and −b 2 * ≤ y ≤ h: where, j = 1 and 2 represent right and left gas media, respectively. After a small disturbance is introduced, the gas-liquid interface will leave the balance position and the flow fields of both phases become: For incompressible liquid phase, the equations governing the conservation of mass and momentum in the Cartesian coordinate are expressed: where, U l is the velocity vector of liquid phase and P l is the liquid pressure.
The power-law model is used to describe the rheological property of liquid phase: where T is the extra stress tensor of liquid phase, . γ = ∇U + (∇U) T the strain-rate second order symmetric tensor, n the power law exponent, and K the viscosity coefficient.
The non-linear feature of liquid state shows the greatest discrepancy between Newtonian fluids and power-law fluids. It is very difficult to solve the governing equations due to the strong nonlinear viscosity term because of the power-law model, and we have to do some approximation. Besides, in the actual flow condition of a plane liquid jet, the scale of the disturbance and the jet velocity in the y-axis is very small. So the shear stress τ xy and normal stress τ yy in the y-axis are neglected as reported by Xia et al. [20], Deng et al. [21] and Chang et al. [22]. Then the momentum equations of liquid phase become: In order to keep the nonlinear feature, a coefficient g, characterizing the unit change of liquid velocity along x-axis, is introduced to linearize Equation (10). Then, Equation (10) is expanded according to Binomial theorem.
Substitute Equations (1) and (11) into Equation (5) and Equations (8)~(9) yields the governing equations of the liquid phase upon linearization: It is reasonable to neglect the viscous effect for two gas media, and then the disturbed governing equations are rewritten as: The disturbed parameters are given in the normal mode as following: The system should satisfy the kinematic and dynamic boundary conditions at the gas-liquid interfaces. At y = ±h + η j , the linearized kinematic boundaries are shown as: The dynamic conditions, after linearization, are expressed at y = ±h + η j as: Meanwhile, the additional boundary conditions at y = b 1 * and y = −b 2 * are: After a serial of derivation, the characteristic equation between the complex growth rate s and the wave number k is obtained: where, l = [1 + 2n(2g) n−1 Kk 2 (s + iku l ) −1 ] −0.5 k Choose h, u l , h/u l , ρ l and ρ l u l 2 as the reference scales of length, velocity, time, density and pressure, respectively. The dimensionless dispersion relation characterizing the instability of a plane power-law liquid jet into two asymmetric bounded gas media is given as: where, S = sh/u l is the dimensionless complex frequency, and its real part S r indicates the dimensionless growth rate of the disturbance. The other dimensionless parameters are Weber number We = ρ l u l 2 h/σ, the generalized Reynolds number Re = ρ l u l 2−n h n /K of liquid phase, the gas-liquid density ratio Q j = ρ gj /ρ l , the gas thickness B j = b j */h on each side of the jet, the wave number m = kh, the parameters G = gh/u l and L = lh In the present study, by adopting the temporal instability analysis, the dispersion equation Equation (23) of plane power-law liquid jets can be numerically solved, and the complex solutions of S can be calculated corresponding to each wavenumber m by specifying jet parameters including We, Re, Q j , G, B j and n. After solving Equation (23), a set of feature parameters, both for para-varicose mode disturbance and para-sinuous mode disturbance, can be obtained to indicate the instability characteristics of power-law liquid jets under different flow conditions. If Re → ∞, n = 1, Q 1 = Q 2 = Q and B 1 = B 2 = B → ∞, Equation (23) is identical to that of a plane inviscid liquid jet obtained by Squire [1].
If Q 1 = Q 2 = Q and B 1 = B 2 = B, Equation (23) becomes the dispersion relation characterizing the instability of a plane power-law liquid jet into symmetric bounded gas medium: When setting B → ∞, Equation (24) is changed into the case of a power-law jet in an unbounded gas media, which is deduced by Xia et al. [20] as:

Results and Discussion
For every unstable curve between the growth rate S r and the wave number m, there are three main unstable parameters, i.e., the dimensionless maximum growth rate S r,max , the dimensionless dominant wave number m dom , and the cutoff wave number m c . Under a given flow condition, the three parameters are respectively characterizing the unstable degree, the primary breakup scale and the unstable range of liquid jets. In practice, the parameter G is gained by the comparison between the Energies 2018, 11, 1854 6 of 14 theoretical prediction and the experiment results. In the current analysis, G is fixed at 2.5 × 10 −6 to investigate the effect of two asymmetric gas media.
The breakup length of the plane power-law liquid jet is defined as the distance from the nozzle outlet to the point where ligament formation occurs along the symmetry axes of the plane [18] According to the linear instability theory, the breakup length L b of the jet is estimated by Heislbetz et al. [23] which can be expressed by: where, s r, max represents the maximum growth rate gained by Equation (22), u l the jet velocity, η 0 the initial disturbance amplitude, η b the disturbance amplitude where breakup occurs and ln(η b /η 0 ) a parameter related with the amplitude of interface waves. In the Equation (27) the initial disturbance wave amplitude is determined by the jet geometry and the flow conditions, and the liquid velocity ul is given for different cases. However, η b cannot be calculated by the linear analysis. Therefore, ln(η b /η 0 ) cannot be obtained by the above analysis and is considered as an empirical constant. Researchers have made great efforts on this parameter, such as Deng [21], Weber [24], Dombrowski [25], Kroesser [26] and Kim [27]. Weber [24] pointed out that this parameter could be fixed at the value of 12 in their study range. Thus ln(η b /η 0 ) = 12 has been widely adopted [25]. Kroesser [26] gained a value of 11 for Newtonian liquid sheets. Nevertheless, with the development of technology, some investigators reported that the term of ln(η b /η 0 ) is not a universal value and has to be determined experimentally in each case, such as Xia et al. [20] and Kim [27]. Besides, compared with the Newtonian liquid jet, the liquid viscosity of power-law fluids will vary with the liquid velocity, which may cause the parameter ln(η b /η 0 ) to change.
It is found that the breakup length of theoretical results and experimental results are always in the same order, and their changing trends with jet velocity are also consistent when ln(η b /η 0 ) = 11 or 12. This show that the current analysis results are reliable although there are some discrepancies mainly resulted from the nonlinear feature of power-law fluids and some simplification of viscosity terms. If ln(η b /η 0 ) is considered as a correction coefficient of L b and varies with u l , the prediction curve 3 is obtained. It also demonstrates that, for power-law fluids, ln(η b /η 0 ) varies with the jet velocity. This may be mainly resulted from the shear-thinning feature.
Energies 2018, 11, x FOR PEER REVIEW 7 of 15 mainly resulted from the nonlinear feature of power-law fluids and some simplification of viscosity terms. If ln(ηb/η0) is considered as a correction coefficient of Lb and varies with ul, the prediction curve 3 is obtained. It also demonstrates that, for power-law fluids, ln(ηb/η0) varies with the jet velocity. This may be mainly resulted from the shear-thinning feature.  Figure 3 shows the effect of gas confinement on the instability of a plane power-law liquid jet in low-speed and high-speed cases.  Figure 3 shows the effect of gas confinement on the instability of a plane power-law liquid jet in low-speed and high-speed cases. Figure 2. Comparison between Deng's experiment data and the prediction curves in terms of breakup length Lb as a function of jet velocity ul. Lines 1-3 represent ln(ηb/η0) = 12, 11, and variable ln(ηb/η0). Figure 3 shows the effect of gas confinement on the instability of a plane power-law liquid jet in low-speed and high-speed cases. The parameters are fixed at Re = 50, Q 1 = Q 2 = 0.001, n = 0.8. The upper row is for low-speed jet (We = 100) case, and the lower row for high-speed jet (We = 2000) one. It should note that in order to ensure the existence of the gas media between the liquid phase and the solid wall, the distances b 1 * and b 2 * from gas boundaries of both sides to the x-axis must be larger than the half-thickness of the liquid h, i.e., B 1 > 1 and B 2 > 1. From Figure 3, it is found that, with the increase of B 1 and B 2 , the dimensionless maximum growth rate is decreased both for two jet cases and for two disturbances. The unstable range will be increased with a confined gas environment for both disturbances. This shows that a limited gas medium will promote the jet instability. The reasons are chiefly as follows. When Q 1 = Q 2 and B 1 = B 2 , Equation (23) reduces to Equation (24), and the gas media on both sides of the liquid jet will become symmetrical. Furthermore, setting B going to infinite, the gas boundary will disappear and Equation (25) is gained. Compared with Equation (25), it is found that the second term of Equation (24) is the counterpart of Equation (25) multiplied by an additional term coth[m(B-1)]. Note that the double cotangent function coth(x) is always larger than one when the independent variable x is larger than zero. When x tends to infinite, coth(x) tends to unit. This reveals that a limited gas medium will strengthen the relative motion between gas-liquid phases so as to promote the jet instability.

Effect of Gas Boundary Asymmetry
The physical mechanism is that the relative motion between gas molecules will become more intense in a limited gas space. This will cause more gas molecules to collide with gas-liquid interfaces at unit time so as to promote the jet instability. However, with the increase of the Weber number, the effect of gas confinement is becoming less and less. The reason is that, with the increase of liquid velocity, the apparent viscosity of liquid phase will be reduced resulting from the shear-thinning feature and the aerodynamic force is enlarged constantly. Accordingly, in high-speed jet case, the interface instability is mainly controlled by the gas-liquid relative motion. In contrary, the gas confinement has a more noticeable effect on the jet instability in a relative low-speed jet case.
In order to investigate the asymmetric degree between two gas gaps on both sides of the liquid jet, a parameters B c is defined as following, called as the boundary asymmetric degree: Clearly, with a limited gas environment, i.e., B 1 + B 2 = constant, B c < 1 represents that the gas thickness of left side is larger; otherwise B c > 1 represents that the right side is larger. When B c = 1, two gas thickness are the same. Figure 4 shows the dimensionless growth rate S r versus the wave number m under different We and the degree of gas boundary asymmetry. The low (We = 100) and high (We = 2000) Weber number cases denote low-speed and high-speed cases, respectively. The other parameters are fixed at Re = 50, Q 1 = Q 2 = 0.001, B 1 + B 2 = 100, n = 0.8. It is found that the liquid jet in unequal gaps is more unstable. In detail, with the asymmetric degree B c increasing, the dimensionless maximum growth rate of two modes S r for both low and high speed cases are amplified constantly.  The unstable range of para-sinuous mode is increased while it remains nearly constant for the para-varicose mode. This indicates that the asymmetric gas medium has an unstable effect on the jet instability. The para-sinuous mode always prevails over para-varicose mode under different asymmetric boundary conditions, implying that the gas boundary asymmetry is beneficial for the disintegration of power-law fluids.  The unstable range of para-sinuous mode is increased while it remains nearly constant for the para-varicose mode. This indicates that the asymmetric gas medium has an unstable effect on the jet instability. The para-sinuous mode always prevails over para-varicose mode under different asymmetric boundary conditions, implying that the gas boundary asymmetry is beneficial for the disintegration of power-law fluids.

Effect of the Aerodynamic Asymmetry
In this section, two parameters Q a and W are defined as follows to characterize the asymmetric degree of aerodynamic forces and the total perturbation energy of gas media on both sides of the jet respectively: Given that the gas media keeps still and the liquid velocity does not change, the two parameters are simplified to Q a = ρ g1 /ρ g2 and W = (ρ g1 + ρ g2 )u l 2 . Therefore, when the sum of two gas density ratios Q 1 + Q 2 is fixed, the total perturbation energy remains the same. In addition, Q a > 1 represents the perturbation energy in the right side of the liquid jet is larger than the left side, while Q a < 1 means the right side is weaker. Accordingly, the perturbation energy on both sides become equal at the condition of Q a = 1. In other words, the aerodynamic forces are distributed symmetrically. However, no matter the disturbance energy on the right or left side is stronger, the jet instability holds the line as long as the relative asymmetric degree between two gas densities Q d = |Q 1 − Q 2 | is fixed. As pointed out by Chang [22], the gas-liquid interaction force will become the leading factor to destabilize the liquid jet in high-speed cases. Figure 5 shows the effect of the aerodynamic asymmetry Q a , on the jet instability under a high speed flow condition. For Re = 50, We = 1850, n = 0.8, B 1 = B 2 →∞, it is found that with Q a increasing from 1 to 100 or decreasing from 1 to 1/100, the dimensionless maximum growth rate S r, max , the dimensionless dominant wave number m dom and the unstable range of para-sinuous mode all increase. On the contrary, these three unstable parameters of para-varicose mode all decrease for increasing or decreasing Q a from 1. When the total disturbance energy of two gas media on both sides of the liquid jet is equal (Q a = 1), the S r, max of para-varicose mode reaches the peak, while the counterpart of para-sinuous mode decreases to the minimum. This reveals that the aerodynamic asymmetry will destabilize para-sinuous mode, while damper para-varicose mode. As pointed out by Chang [22], the gas-liquid interaction force will become the leading factor to destabilize the liquid jet in high-speed cases. Figure 5 shows the effect of the aerodynamic asymmetry Qa, on the jet instability under a high speed flow condition. For Re = 50, We = 1850, n = 0.8, B1 = B2→∞, it is found that with Qa increasing from 1 to 100 or decreasing from 1 to 1/100, the dimensionless maximum growth rate Sr, max, the dimensionless dominant wave number mdom and the unstable range of para-sinuous mode all increase. On the contrary, these three unstable parameters of para-varicose mode all decrease for increasing or decreasing Qa from 1. When the total disturbance energy of two gas media on both sides of the liquid jet is equal (Qa = 1), the Sr, max of para-varicose mode reaches the peak, while the counterpart of para-sinuous mode decreases to the minimum. This reveals that the aerodynamic asymmetry will destabilize para-sinuous mode, while damper para-varicose mode. The reason is that when the total disturbance energy remains unchanged, one of the two aerodynamic forces will be enlarged while the other will be reduced with the change of Qa. Furthermore, two forms of interface waves have relationship with the aerodynamic forces on both sides. Para-sinuous mode is related with the larger one while the smaller one is associated with paravaricose mode. Similar results were reported by Yang et al. [5]. Therefore, as the asymmetric degree of two aerodynamic forces increases, the opposite tendency of both disturbance modes is observed.
Besides, the maximum growth rate of para-sinuous mode is larger than that of para-varicose The reason is that when the total disturbance energy remains unchanged, one of the two aerodynamic forces will be enlarged while the other will be reduced with the change of Q a . Furthermore, two forms of interface waves have relationship with the aerodynamic forces on both sides. Para-sinuous mode is related with the larger one while the smaller one is associated with para-varicose mode. Similar results were reported by Yang et al. [5]. Therefore, as the asymmetric degree of two aerodynamic forces increases, the opposite tendency of both disturbance modes is observed.
Besides, the maximum growth rate of para-sinuous mode is larger than that of para-varicose mode. This indicates that para-sinuous mode will dominate the jet breakup process and the aerodynamic asymmetry could effectively promote the breakup of power-law fluids. It is notable to see that the increasing rate of S r, max decreases with an increase of Q a until approaching a certain limit in Figure 5.
Considering that para-sinuous mode prevails over para-varicose mode in the jet breakup, Figure 6 will only show the maximum growth rate S r, max (a) and dominant wave number m dom (b) with para-sinuous mode for Re = 50, We = 1850, n = 0.8 and B 1 = B 2 → ∞, to reveal the effect of total perturbation energy W on the jet instability. As the total disturbance energy of two gas media increases, Sr, max and mdom all increase significant accordingly. This indicates that with larger total gas disturbance energy, the aerodynamic asymmetry enables power-law fluids to break quickly smaller droplets. In other words, the aerodynamic asymmetry could dramatically accelerate the liquid breakup process in a high density gas media. Besides, compared with the boundary asymmetry, the aerodynamic asymmetry is much effective to trigger shorter interface waves. In practice applications, high-speed jets can be employed to achieve better mixing and combustion. As a result, the dominant interface waves are located at the short wave range. The aerodynamic asymmetry could be more effective to improve the atomization quality of liquid fuels. Figure 7 presents the effect of the aerodynamic asymmetry on the jet instability at different Weber numbers, while other parameters are held at Re = 50, n = 0.8 and B1 = B2→∞. This gives more information about the plane jet of power-law fluids in asymmetric gas media. It is found that with a fixed Weber number, both the dimensionless maximum growth rate and the dimensionless dominant wave number increase with the increase of Qa. Specifically, this increasing trend becomes more significant with the increase of the Weber number constantly. This indicates that in the high-speed jet, the aerodynamic asymmetry could promote the breakup of power-law fluids more easily and also decrease the breakup scale. Thus, larger surface tensions or lower liquid velocities enable the plane jet of power-law fluids to become more stable in both asymmetric and symmetric cases. As the total disturbance energy of two gas media increases, S r, max and m dom all increase significant accordingly. This indicates that with larger total gas disturbance energy, the aerodynamic asymmetry enables power-law fluids to break quickly smaller droplets. In other words, the aerodynamic asymmetry could dramatically accelerate the liquid breakup process in a high density gas media. Besides, compared with the boundary asymmetry, the aerodynamic asymmetry is much effective to trigger shorter interface waves. In practice applications, high-speed jets can be employed to achieve better mixing and combustion. As a result, the dominant interface waves are located at the short wave range. The aerodynamic asymmetry could be more effective to improve the atomization quality of liquid fuels. Figure 7 presents the effect of the aerodynamic asymmetry on the jet instability at different Weber numbers, while other parameters are held at Re = 50, n = 0.8 and B 1 = B 2 →∞. This gives more information about the plane jet of power-law fluids in asymmetric gas media. It is found that with a fixed Weber number, both the dimensionless maximum growth rate and the dimensionless dominant wave number increase with the increase of Q a . Specifically, this increasing trend becomes more significant with the increase of the Weber number constantly. This indicates that in the high-speed jet, the aerodynamic asymmetry could promote the breakup of power-law fluids more easily and also decrease the breakup scale. Thus, larger surface tensions or lower liquid velocities enable the plane jet of power-law fluids to become more stable in both asymmetric and symmetric cases. fixed Weber number, both the dimensionless maximum growth rate and the dimensionless dominant wave number increase with the increase of Qa. Specifically, this increasing trend becomes more significant with the increase of the Weber number constantly. This indicates that in the high-speed jet, the aerodynamic asymmetry could promote the breakup of power-law fluids more easily and also decrease the breakup scale. Thus, larger surface tensions or lower liquid velocities enable the plane jet of power-law fluids to become more stable in both asymmetric and symmetric cases.

Effect of Gas Asymmetric Distribution on the Unstable Range
The effect of the boundary asymmetric degree B c on unstable range at Q 1 = Q 2 = 0.001, n = 0.8, and B 1 + B 2 = 1000 is presented in Figure 8. B = 1 represents the liquid fuel injecting into the symmetrical gas medium, then the cut-off wave numbers of para-sinuous and para-varicose modes are equivalent (m 1 = m 2 ). There exists a critical curve to distinguish the unstable regime with the stable regime. In Figure 8a, as B c is increased from 1 to 500, the unstable range of para-sinuous mode is enlarged while para-varicose mode remains nearly constant as shown in Figure 8b. Additionally, with the increase of the Weber number, the difference among the cut-off numbers in terms of para-sinuous mode becomes smaller and smaller in Figure 8a. This indicates that the boundary asymmetry could enlarge the disturbance in a power-law plane jet but it plays a more significant role on low Weber number cases.

Effect of Gas Asymmetric Distribution on the Unstable Range
The effect of the boundary asymmetric degree Bc on unstable range at Q1 = Q2 = 0.001, n = 0.8, and B1 + B2 = 1000 is presented in Figure 8. B = 1 represents the liquid fuel injecting into the symmetrical gas medium, then the cut-off wave numbers of para-sinuous and para-varicose modes are equivalent (m1 = m2). There exists a critical curve to distinguish the unstable regime with the stable regime. In Figure 8a, as Bc is increased from 1 to 500, the unstable range of para-sinuous mode is enlarged while para-varicose mode remains nearly constant as shown in Figure 8b. Additionally, with the increase of the Weber number, the difference among the cut-off numbers in terms of parasinuous mode becomes smaller and smaller in Figure 8a. This indicates that the boundary asymmetry could enlarge the disturbance in a power-law plane jet but it plays a more significant role on low Weber number cases. In order to explore the effect of the aerodynamic asymmetric degree Qa on the unstable range of power-law plane jets, Qa is set to increase from 1 to 1000. The results are shown in Figure 9, and the other parameters are Q1 + Q2 = 0.002, n = 0.8, and B1 = B2 → ∞. If Qa = 1, two gas media are symmetrical. It can be seen from Figure 9a,b that as the Weber number is increased from 100 to 10000, the cut-off wave number of both disturbances (m1, m2) all increase. Higher Qa leads to a larger unstable area of para-sinuous mode, but a smaller unstable area of para-varicose mode instead. It is noted that parasinuous mode relates with the liquid breakup in this study. Therefore, an asymmetric gas environment will enhance the interface instability and result in short breakup lengths. In order to explore the effect of the aerodynamic asymmetric degree Q a on the unstable range of power-law plane jets, Q a is set to increase from 1 to 1000. The results are shown in Figure 9, and the other parameters are Q 1 + Q 2 = 0.002, n = 0.8, and B 1 = B 2 → ∞. If Q a = 1, two gas media are symmetrical. It can be seen from Figure 9a,b that as the Weber number is increased from 100 to 10,000, the cut-off wave number of both disturbances (m 1 , m 2 ) all increase. Higher Q a leads to a larger unstable area of para-sinuous mode, but a smaller unstable area of para-varicose mode instead. It is noted that para-sinuous mode relates with the liquid breakup in this study. Therefore, an asymmetric gas environment will enhance the interface instability and result in short breakup lengths. Figure 8. Effect of the boundary asymmetry on the unstable range of power-law plane jets: (a) parasinuous mode (solid line); (b) para-varicose mode (dash line).
In order to explore the effect of the aerodynamic asymmetric degree Qa on the unstable range of power-law plane jets, Qa is set to increase from 1 to 1000. The results are shown in Figure 9, and the other parameters are Q1 + Q2 = 0.002, n = 0.8, and B1 = B2 → ∞. If Qa = 1, two gas media are symmetrical. It can be seen from Figure 9a,b that as the Weber number is increased from 100 to 10000, the cut-off wave number of both disturbances (m1, m2) all increase. Higher Qa leads to a larger unstable area of para-sinuous mode, but a smaller unstable area of para-varicose mode instead. It is noted that parasinuous mode relates with the liquid breakup in this study. Therefore, an asymmetric gas environment will enhance the interface instability and result in short breakup lengths.
(a) (b) Figure 9. Effect of the aerodynamic asymmetry on the unstable range of power-law plane jets: (a) para-sinuous mode (solid line); (b) para-varicose mode (dash line).

Conclusions
The instability characteristics of power-law fluids on a plane jet are investigated by a linear instability analysis in the presence of gas asymmetric distributions, including the boundary asymmetry and the aerodynamic asymmetry. It is found that a confined space will enhance the perturbation of gas medium on the jet instability. Under two kinds of gas asymmetric conditions, para-sinuous mode always dominates the liquid breakup. Also, the gas asymmetric distribution could accelerate the disintegration of power-law fluids. With a large Weber number, the aerodynamic asymmetry is more useful to promote the liquid breakup than the boundary asymmetry. For a larger gas density and a higher liquid velocity, the promotion of the aerodynamic asymmetry will become more significant. In addition, the effects of two asymmetric distributions on the unstable range of power-law plane jets regarding to para-sinuous and para-varicose disturbances are analyzed and the critical curves are obtained to characterized unstable and stable regimes. These conclusions are beneficial for better understanding of the breakup characteristic of power-law fluids in confined asymmetric gases.