Research Article  Open Access
Mohammed Rizwan Sadiq Iqbal, "AirAided Shear on a Thin Film Subjected to a Transverse Magnetic Field of Constant Strength: Stability and Dynamics", International Scholarly Research Notices, vol. 2013, Article ID 748613, 24 pages, 2013. https://doi.org/10.1155/2013/748613
AirAided Shear on a Thin Film Subjected to a Transverse Magnetic Field of Constant Strength: Stability and Dynamics
Abstract
The effect of air shear on the hydromagnetic instability is studied through (i) linear stability, (ii) weakly nonlinear theory, (iii) sideband stability of the filtered wave, and (iv) numerical integration of the nonlinear equation. Additionally, a discussion on the equilibria of a truncated bimodal dynamical system is performed. While the linear and weakly nonlinear analyses demonstrate the stabilizing (destabilizing) tendency of the uphill (downhill) shear, the numerics confirm the stability predictions. They show that (a) the downhill shear destabilizes the flow, (b) the time taken for the amplitudes corresponding to the uphill shear to be dominated by the one corresponding to the zero shear increases with magnetic fields strength, and (c) among the uphill shearinduced flows, it takes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to large shear values when the magnetic field intensity increases. Simulations show that the streamwise and transverse velocities increase when the downhill shear acts in favor of inertial force to destabilize the flow mechanism. However, the uphill shear acts oppositely. It supports the hydrostatic pressure and magnetic field in enhancing films stability. Consequently, reduced constant flow rates and uniform velocities are observed.
1. Introduction
A nonlinear fourthorder degenerate parabolic differential equation of the form: where , , and are arbitrary continuous functions of the interfacial thickness , represents a scalar conservation law associated to the flow of a thin viscous layer on an incline under different conditions. The stability and dynamics of equations of the type of (1) is a subject of major interest [1–18] because of their robustness in regimes where viscosity dominates inertia [19]. Such studies have focused attention primarily on the isothermal and nonisothermal instability analysis, mainly for nonconducting fluids.
Since the investigation of Chandrasekar [20] on the stability of a flow between coaxial rotating cylinders in the presence of a magnetic field held in the axial direction, the laminar flow of an electrically conducting fluid under the presence of a magnetic field has been studied extensively. For instance, Stuart [21] has reported on the stability of a pressure flow between parallel plates under the application of a parallel magnetic field. Among other earlier investigations, Lock [22] examined the stability when the magnetic field is applied perpendicular to the flow direction and to the boundary planes. Hsieh [23] found that the magnetic field stabilizes the flow through Hartmann number when the electrically conducting fluid is exposed to a transverse magnetic field, provided that the surfacetension effects are negligible in a horizontal film. Ladikov [24] studied a problem in the presence of longitudinal and transverse magnetic fields. The author observed that the longitudinal magnetic field plays a stabilizing role and that the effect of instability at small wave numbers could be removed if the longitudinal magnetic field satisfies certain conditions. Lu and Sarma [25] investigated the transverse effects of the magnetic field in magnetohydrodynamic gravitycapillary waves. The flow of an electrically conducting fluid over a horizontal plane in the presence of tangential electric and magnetic fields was reported by Gordeev and Murzenko [26]. They found that the flow suffers from instability not due to the Reynolds number but due to the strength of the external electric field.
In applications such as magneticfieldcontrolled materialprocessing systems, aeronautics, plasma engineering, MEMS technology, and magnetorheological lubrication technologies, the hydromagnetic effects are important. Also, liquid metal film flows are used to protect the solid structures from thermonuclear plasma in magnetic confinement fusion reactors, and this application requires a better understanding of the instability mechanism arising in a thin magnetohydrodynamic flow over planar substrates [27]. Furthermore, the presence of an external magnetic field regulates the thickness of a coating film. It prevents any form of direct electrical or mechanical contact with the fluid thereby reducing the risk of contamination [28]. Renardy and Sun [29] pointed out that the magnetic fluid is effective in controlling the flow of ordinary fluids and in reducing hydraulic resistance. The reason is that the magnetic fluids can be easily controlled with external magnetic fields and that coating streamlined bodies with a layer of lessviscous magnetic fluid significantly reduces the shear stress in flow boundaries. Magnetohydrodynamic flow can be a viable option for transporting weakly conducting fluids in microscale systems such as flows inside a microchannel network of a labonachip device [30, 31]. In all of the above applications, considering the associated stability problem is important because it gives guidance in choosing the flow parameters for practical purposes. In this regard, in the past two decades, the emerging studies on the hydromagnetic effects have focused their attention on stability problems and on analyzing the flow characteristics [28, 29, 32–43].
The shearing effect of the surrounding air on the fluid in realistic situations induces stress tangentially on the interfacial surface. The hydrodynamic instability in thin films in the presence of an external air stream attracted attention in the mid 1960s, which led Craik [44] to conduct laboratory research and study theory. He found that the instability occurs regardless of the magnitude of the air stream when the film is sufficiently thin. Tuck and VandenBroeck [45] reported on the effect of air stream in industrial applications of thin films of infinite extent in coating technology. Sheintuch and Dukler [46] did phase plane and bifurcation analyses of thin wavy films subjected to shear from countercurrent gas flows and found satisfactory agreement of their results connected to the wave velocity along the flooding curve with the experimental results of Zabaras [47]. Although the experimental results related to the substrate thickness did not match exactly with the theoretical predictions, still the results gave qualitative information about the model. Thin liquid layer supported by steady airflow surfacetraction was reported by King and Tuck [48]. Their study models the surfacetractionsupported fluid drops observed on the windscreen of a moving car on a rainy day. Incorporating the superficial shear offered by the air, Pascal [49] studied a problem which models the mechanism of windaided spreading of oil on the sea and found quantitative information regarding the maximum upwind spread of the gravity current. Wilson and Duffy [50] studied the steady unidirectional flow of a thin rivulet on a vertical substrate subjected to a prescribed uniform longitudinal shear stress on the free surface. They categorized the possible flow patterns and found that the direction of the prescribed shear stress affects the velocity in the entire rivulet. The generation of roll waves on the free boundary of a nonNewtonian liquid was numerically assessed using a finite volume method by Pascal and D’Alessio [51], revealing the significant effect of air shear on the evolution of the flow. Their study showed that the instability criteria was conditional and depends on the directionally induced shear. Kalpathy et al. [52] investigated an idealized model suitable for lithographic printing by examining the shearinduced suppression of two stratified thin liquid films confined between parallel plates taking into account the van der Waals force. A film thickness equation for the liquidliquid interface was derived in their study using lubrication approximation. They found that the effect of shear affects the imaginary part of the growth rate, indicating the existence of traveling waves. Furthermore, they also observed a critical shear rate value beyond which the rupture mechanism could be suppressed. This study motivated Davis et al. [53] to consider the effect of unidirectional air shear on a single fluid layer. For a twodimensional ultrathin liquid, Davis et al. [53] showed that the rupture mechanism induced by the London van der Waals force could be suppressed when the magnitude of the wind shear exceeds a critical value, as observed by Kalpathy et al. [52]. Recently, Uma [18] measured numerically the profound effect of unidirectional wind stress on the stability of a condensate/evaporating powerlaw liquid flowing down an incline.
In the present investigation, the effects of downhill and uphill air shear on a thin falling film in the presence of a transverse magnetic field are studied. Such an investigation will illustrate the realistic influence of the natural environment acting upon the flows. Or, it may illustrate the need to control the flow mechanism through artificial techniques, which blow air when the hydromagnetic effects are considered. The outline of this paper is organized as follows. Section 1 presents the introduction. In Section 2, mathematical equations governing the physical problem are presented. Section 3 discusses the longwave Benneytype equation. In Section 4, linear stability analysis, weakly nonlinear stability analysis, and the instability arising due to sideband disturbances are analyzed. The equilibria of a truncated bimodal dynamical system is mathematically presented in Section 5. While the results in Section 6 discuss nonlinear simulations of the film thickness evolution, Section 7 highlights the main conclusions of the study and includes future perspectives.
2. Problem Description
A thin Newtonian liquid layer of an infinite extent falling freely over a plane under the influence of gravitational acceleration, , is considered. The flow is oriented towards the axis, and the plane makes an angle with the horizon. Properties of the fluid like density (), viscosity (), and surfacetension () are constants. The magnetic flux density is defined by the vector , where is the magnitude of the magnetic field imposed along the direction (Figure 1). It is assumed that there is no exchange of heat between the liquid and the surrounding air, but an air flow (either in the uphill or in the downhill direction) induces a constant stress of magnitude on the interface and moves tangentially along the surface. The axis is perpendicular to the planar substrate such that, at any instant of time , measures the film thickness.
The magnetohydrodynamic phenomena can be modeled by the following equations, which express the momentum and mass balance: The last term in (2) arises due to the contribution of Lorenz body force, based on Maxwell’s generalized electromagnetic field equations [28, 54, 55].
In a realistic situation corresponding to a threedimensional flow, the total current flow can be defined using Ohm’s law as follows: where , , , and represent the current density, electric field, velocity vector, and electrical conductivity, respectively. The Lorenz force acting on the liquid is defined as . In the rest of the analysis, a short circuited system corresponding to a twodimensional problem is considered by assuming that . This assumption simplifies the last term corresponding to the electromagnetic contribution in (2). In this case, the pondermotive force acting on the flow (the last term in (2)) has only one nonvanishing term in the direction; therefore, where and is the pressure.
Boundary conditions on the planar surface and the interface are added to complete the problem definition of (2)(3). On the solid substrate, the noslip and the nopenetration conditions are imposed, which read as The jump in the normal component of the surfacetraction across the interface is balanced by the capillary pressure (product of the mean surfacetension coefficient and the local curvature of the interface), which is expressed as where is the mean film curvature and is the pressure afforded by the surrounding air. The unit outward normal vector at any point on the free surface is , and represents the unit vector along the tangential direction at that point such that . The tangential component of the surfacetraction is influenced by the air stress and reads as The location of the interface can be tracked through the following kinematic condition:
In order to remove the units associated with the model (2)–(9) involving the physical variables, reference scales must be prescribed. In principle, one can nondimensionalize the system based on the nature of the problem by choosing one of the following scales [9, 16]: kinematic viscosity based scales, gravitational acceleration as the flow agent based scales, mean surfacetension based scales, and Marangoni effect as the flow agent based scales. However, for very thin falling films, the main characteristic time is the viscous one [7–9, 16]. An advantage of choosing the viscous scale is that either both the large and the small inclination angles could be considered by maintaining the sine of the angle and the Galileo number as separate entities [7, 16] or the Reynolds number as a product of Galileo number and sine of the inclination angle could be defined as a single entity [9]. Also, such a scale plays a neutral role while comparing the action of gravity and Marangoni effect in nonisothermal problems [16]. Choosing the viscous scale, the horizontal distance is scaled by , vertical distance by , streamwise velocity by , transverse velocity by , pressure by , time by , and, finally, the shear stress offered by the wind by . In addition, the slenderness parameter is considered small, and a gradient expansion of the dependent variables is done [7–9]. The horizontal length scale, , is chosen such that , where is a typical wavelength larger than the film thickness.
The dimensionless system is presented in Appendix A (the same symbols have been used to avoid new notations). The set of nondimensional parameters arising during nondimensionalization procedure are (the Reynolds number) [9], (the Galileo number), (the Hartmann number, which measures the relative importance of the drag force resulting from magnetic induction to the viscous force arising in the flow), (the surfacetension parameter), and (the shear stress parameter). The surfacetension parameter is usually large; therefore, it is rescaled as and set as in accordance to the waves observed in laboratory experiments. All of the other quantities are considered . The longwave equation is derived in the next step.
3. LongWave Equation
The dependent variables are asymptotically expanded in terms of the slenderness parameter to derive the longwave equation. Using the symbolic math toolbox available in MATLAB, the zeroth and the firstorder systems are solved. These solutions are then substituted in the kinematic condition to derive a Benneytype model accurate up to of the form (1) as The standard procedure for the derivation [7, 9, 14, 15, 28] is skipped here. The expressions for , , and are, the following: It should be remarked that, when , the above terms agree with the longwave equation derived by Tsai et al. [28] when the phasechange effects on the interface are neglected. The dimensionless parameters differ from the nondimensional set presented in Tsai et al. [28] because viscous scales are employed here. Although one can consider different scales, in principle, the structure of the evolution equation remains unchanged regardless of the dimensionless parameters appearing in the problem. Furthermore, the error in the second term associated with in Tsai et al. [28] is corrected here, which as per the convention followed in their paper should read as . Also, the case corresponding to can be recovered from (11)–(13) when . In this case, the functions in (11)–(13) read as which match with the evolution equation derived by Miladinova et al. [9] in the absence of Marangoni and air shearing effect. The effect of magnetic field and air shear affects the leading order solution through term in (11). This term contributes towards wave propagation and steepening mechanism. The effect of hydrostatic pressure is measured by the terms within the first flower bracket in in (12). The rest of the terms in affect the mean flow due to inertial, air shear and magnetic field contributions. The function corresponds to the mean surfacetension effect. Although the effect of cannot be properly judged based on its appearance in (11)–(13), it is obvious that in the absence of the magnetic field both and increase (decrease) when . The stability of the longwave model (10) subject to (11)–(13) is investigated next.
4. Stability Analysis
The Nusselt solution corresponding to the problem is
For a parallel shear flow, (10) subject to (11)–(13) admits normal mode solutions of the form , where is the unsteady part of the film thickness representing the disturbance component such that [56, 57]. Inserting in (10) and invoking a Taylor series expansion about , the unsteady nonlinear equation representing a slight perturbation to the free surface is obtained as where (a prime denotes the order of the derivative with respect to ) It should be remarked that, while expanding the Taylor series, there are two small parameters, namely, and whose orders of magnitude should be considered such that . When terms are retained and since , appears as a unique contribution of order in (16). This term, although present in the unsteady equation (16), does not contribute when a multiplescale analysis is done (refer to Section 4.2.1 and Appendix B), where equations only up to are considered while deriving a complex GinzburgLandautype equation [12, 14, 17, 56, 57]. In addition, such a term did not appear in earlier studies [12, 14, 17, 56, 57] because was a mathematical function of second degree in , whose higherorder derivatives are zero.
Equation (16) forms the starting point for the linear stability analysis and describes the behavior of finiteamplitude disturbances of the film. Such an equation predicts the evolution of timewise behavior of an initially sinusoidal disturbance given to the film. It is important to note that the constant film thickness approximation with longwave perturbations is a reasonable approximation only for certain segments of flow and implies that (16) is only locally valid.
4.1. Linear Stability
To assess the linear stability, the linear terms in (16) are considered. The unsteady part of the film thickness is decomposed as (a tilde denotes the complex conjugate) where is a complex disturbance amplitude independent of and . The complex eigenvalue is given by such that represents the streamwise wavenumber. The linear wave velocity and the linear growth rate (amplification rate) of the disturbance are , , respectively. Explicitly, they are found as
The disturbances grow (decay) when (). However, when , the curve and the positive branch of represent the neutral stability curves. Identifying the positive branch of as ( is the critical wavenumber), the wavenumber corresponding to the maximal growth rate is obtained from . This gives such that . The linear amplification of the most unstable mode is calculated from .
A parametric study considering the elements of the set is done in order to trace the neutral stability and linear amplification curves by assuming the slenderness parameter to be 0.1. Only those curves which are relevant in drawing an opinion are presented.
The influence of the magnetic field on the critical Reynolds number () varying as a function of the shear parameter is presented in Figure 2. For each , there is a below which the flow is stable. The critical Reynolds number decreases when the angle of inclination increases, and, therefore, the flow destabilizes. The stabilizing effect of the magnetic field is also seen when increases. There exists a certain such that the flow remains unstable beyond it.
(a)
(b)
Figures 3 and 4 display the neutral stability curves, which divide the and planes into regions of stable and unstable domains. On the other hand, Figure 5 shows the linear amplification curves. The shear stress offered by the air destabilizes the flow when it flows along the downhill direction () and increases the instability threshold compared to the case corresponding to (Figures 3 and 4). However, the flow mechanism is better stabilized when the applied shear stress offered by the air is in the uphill direction () than when (Figures 3 and 4). As seen from Figure 3, the portion of the axis corresponding to the unstable Reynolds numbers increases and extends towards the left when the angle of inclination increases, thereby reducing the stabilizing effect offered by the hydrostatic pressure at small inclination angles. From curves 2 and 3 corresponding to Figure 3, it is observed that the force of surfacetension stabilizes the flow mechanism. Figure 4 supports the information available from Figure 3. When the magnitude of the Hartmann number increases, the instability region decreases because the value of the critical wavenumber decreases. Comparing Figure 4(a) with Figure 4(c), it is also observed that the inertial force destabilizes the flow mechanism. The growth rate curves (Figure 5) agree with the results offered by the neutral stability curves (Figures 3 and 4).
(a)
(b)
(a)
(b)
(c)
(a)
(b)
The linearly increasing graphs of and the decreasing plots of with respect to and , respectively, are presented in Figure 6. The effect of inertia doubles the linear wave speed, . But when the magnitude of , increases, the linear wave speed decreases. The downhill effect of the shear stress on the interface makes the linear wave speed larger than the cases corresponding to and .
The linear stability results give only a firsthand information about the stability mechanism. The influence of airinduced shear on the stability of the flow under the application of a transverse magnetic field will be better understood only when the nonlinear effects are additionally considered. To analyze and illustrate the nonlinear effects on the stability threshold, a weakly nonlinear study is performed in the next step.
4.2. MultipleScale Analysis
4.2.1. Weakly Nonlinear Theory
In order to do multiplescale analysis, the following slow scales are introduced following Sadiq and Usha [14] (the justification for stretching the scales is provided in Lin [56] and in Krishna and Lin [57]): such that Here, is a small parameter independent of and measures the distance from criticality such that . The motivation behind such a study lies in deriving the complex GinzburgLandau equation (CGLE), which describes the evolution of amplitudes of unstable modes for any process exhibiting a Hopf bifurcation. Using such an analysis, it is possible to examine whether the nonlinear waves in the vicinity of criticality attain a finite height and remain stable or continue to grow in time and eventually become unstable. Refer to Appendix B for the detailed derivations of the threshold amplitude, , and the nonlinear wave speed, .
The threshold amplitude subdivides the flow domain according to the signs of and [14]. If and , the flow exhibits subcritical instability. However, when and , the Landau state is supercritically stable. If, on the other hand, and , the flow is subcritically stable. A blowup supercritical explosive state is observed when and .
Different regions of the instability threshold obtained through multiplescale analysis are illustrated in Figure 7. The subcritical unstable region is affected due to the variation of . For , such a region is larger than the ones corresponding to . In addition, the subcritical unstable and the stable regions increase when increases. The explosive state region (also called the nonsaturation zone) decreases when than when . The strip enclosing asterisks is the supercritical stable region where the flow, although linearly unstable, exhibits a finiteamplitude behavior and saturates as time progresses [8, 9, 14]. The bottom line of the strip is the curve which separates the supercritical stable region from the explosive region [56, 58, 59]. Within the strip , different possible shapes of the waves exist [7, 9, 14].
Tables 1 and 2 show the explosive and equilibration state values for different flow parameters. The wavenumber value corresponding to the explosive state increases when the inertial effects increase. This increases the unstable region (Table 1). Such a wavenumber decreases either when the hydromagnetic effect is increased or when the air shear is in the uphill direction. It is evident from Table 1 that the explosive state occurs at small values of and , which is also true for large values of when . Also, it is observed that either when the Hartmann number or the value of the uphill shear is increased, the critical wavenumber becomes zero (Table 2). Therefore, (Table 1).


The threshold amplitude profiles display an asymmetric structure (Figure 8), increasing up to a certain wavenumber (>) and then decreasing beyond it in the supercritical stable region. The induced amplitudes show larger peak amplitude than the cases corresponding to and . The peak amplitude value decreases when increases. The nonlinear wave speed curves represent a counterclockwise rotation of the mirror image of the alphabet when is small. The nonlinear speed decreases to a particular value in the vertical direction and thereafter traces a constant value beyond it as the wavenumber increases in the supercritical stable region. However, when the effect of the magnetic field is increased, the nonlinear wave speed decreases in magnitude and sketches an almost linear constant profile. The magnitude of in the supercritical stable region corresponding to remains inbetween the values corresponding to and .
4.2.2. Sideband Instability
Let us consider the quasimonochromatic wave of exhibiting no spatial modulation of the form where and are defined by and such that is the wave frequency and is the modulation wavenumber.
If one considers a band of frequencies centered around , the interaction of one sidemode with the second harmonic would be resonant with the other sidemode causing the frequency to amplify. This leads to an instability known as sideband instability [60, 61].
To investigate such an instability, is subjugated to sideband disturbances of bandwidth [12, 56, 57]. The explicit expression for the eigenvalues is found as (refer to Appendix C) where
It should be remarked that the above expression for is true only if . However, when , it is easily seen from (23) that and (when ), implying that the system is stable to the sideband disturbances as . For nonzero , the eigenvalues depend on the dimensionless flow parameters. If and is less than the absolute value of , the sideband modes stabilize the system as . However, if and is greater than the absolute value of , only one of the modes is sideband stable. On the other hand, if , again, one of the modes is sideband stable.
5. Equilibria of a Bimodal Dynamical System
Considering the initial thickness of the amplitude to be one, Gjevik [58] analyzed the amplitude equations by representing the amplitude using a truncated Fourier series and by imposing restrictions on its coefficients. The velocity along the mean flow direction and the corresponding surface deflection moving with this velocity were deduced by posing the problem into a dynamical system. Gottlieb and Oron [62] and Dandapat and Samanta [12] also expanded the evolution equation using a truncated Fourier series to derive a modal dynamical system. The results in Gottlieb and Oron [62] showed that a twomode model was found to coincide with the numerical solution along the Hopf bifurcation curve. Based on this confidence, the stability of the bimodal dynamical system was assessed in Dandapat and Samanta [12].
In this section, the stability of a truncated bimodal dynamical system is analyzed using the approach followed by the above authors, but using the assumptions considered in Gjevik [58]. The coupled dynamical system and its entries are listed in Appendix D.
It should be remarked that in the plane the equations and give the neutral stability curve and the curve corresponding to the maximum rate of amplification for linear disturbances. The expression measures the velocity along the direction of the mean flow of a steady finiteamplitude wave. The surface deflection, , moving with velocity along the mean flow direction in a coordinate system is calculated from and reads as
The steady solutions of the system and correspond to the fixed points of –. In addition to the trivial solution , , and (), system – offers nontrivial fixed points [62–64]. These fixed points can be classified as puremode fixed points (where one of the fixed points is zero and the others are nonzero), mixedmode fixed points (nonzero fixed points with a zero or nonzero phase difference), and traveling waves with nonzero constant phase difference.
5.1. PureMode
The fixed points in this case are obtained by setting , , in –. This gives and . The solution exists if . For convenience, is set. From , is obtained. From this condition, the fixed point for is derived as provided that the quantity within the square root is real and positive. The stability of the nonlinear dynamical system – can be locally evaluated using the eigenvalues of the matrix obtained after linearizing the system around the fixed points. The linear approximation of the dynamical system – can be represented in matrix notation as such that
The stability of the linear system (28) depends on the eigenvalues of the coefficient matrix with entries . The eigenvalues are found as
The Hopf bifurcation at the critical threshold is defined by , and this yields . This eigenvalue being independent of increases when the effects of surfacetension and Hartmann number increase. Therefore, at the critical threshold, the fixed points corresponding to the puremode are unstable. Beyond the neutral stability limit where the flow is linearly stable () and where the stable wavenumber region decreases when the direction of the shear offered by the wind changes from uphill to downhill direction, the eigenvalue may remain positive and still destabilize the system as shown in Figure 9. The stable wavenumbers corresponding to the linear stability threshold (refer to Figure 5 with and ) are considered to plot the eigenvalue . Although the magnitude of remains larger for than for , being positive, it plays a destabilizing role.
5.2. MixedMode
When , , the fixed points in this case correspond to mixedmode. Considering and , , and to be the fixed points of –, the nonlinear system can be linearized around the fixed points. Then, the stability (instability) of the fixed points demands all of the eigenvalues of the linearized Jacobian matrix to be negative (at least one of them to be positive). The nine entries of the 3 × 3 Jacobian matrix arising due to linearization are the following:
It should be remarked that Samanta [65] also discussed the mixedmode for the flow of a thin film on a nonuniformly heated vertical wall. However, it should be noted that the Jacobian was computed not by considering a zero phase difference with , but by evaluating the Jacobian first by imposing and then by substituting in the computed Jacobian. If one considers , an overdetermined system is obtained.
5.3. Traveling Waves
Fixed points of – with being a nonzero constant correspond to traveling waves. The modal amplitudes are considered small in the neighborhood of the neutral stability limit. After rescaling and , the following system is obtained:
To the , there are no nontrivial fixed points in the above system. The phase evolution is governed by equating the righthand side of (32c) to zero by considering terms up to : where , , and