# Weakly Nonlinear Stability Analysis of a Thin Magnetic Fluid during Spin Coating

Authors: Cha'o-Kuang Chen; Dong-Yu Lai
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

This paper investigates the stability of a thin electrically conductive fluid under an applied uniform magnetic filed during spin coating. A generalized nonlinear kinematic model is derived by the long-wave perturbation method to represent the physical system. After linearizing the nonlinear evolution equation, the method of normal mode is applied to study the linear stability. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The Ginzburg-Landau equation is determined to discuss the necessary conditions of the various critical flow states, namely, subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The study reveals that the rotation number and the radius of the rotating circular disk generate similar destabilizing effects but the Hartmann number gives a stabilizing effect. Moreover, the optimum conditions can be found to alter stability of the film flow by controlling the applied magnetic field.


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## Body

## 1. Introduction

The study of magnetohydrodynamic (MHD) effects is important for a wide range of situations, varying from plasma engineering, MEMS technology, and thin film materials technology. Due to various applications, attention has gradually been focused on this subject [1–12]. In fact, stabilization of the film flow by applying a magnetic field is advantageous as follows: (1) neither electrical nor mechanical contacts with the fluid are necessary; (2) active control of a technological process is simple.The spin coating process is one of the fundamental fabrication methods found in the chip manufacturing processes of integrated circuit (IC) such as wafer manufacturing, photolithography, and sputtering deposition. Therefore, it is highly desirable to develop suitable working techniques for homogeneous film growth, techniques that are able to adapt to various flow configurations and associated time-dependent properties. For these reasons, the ability to control and maintain uniform and stable thin films is an important research topic with regard to spin coating [13–17].Linear stability theories for various film flows have been presented by Chandrasekhar [18] and Lin [19]. Stuart [20] was the first to study the weakly nonlinear theory of hydrodynamic stability. This model has been widely employed in subsequent investigations. Long-wave perturbation method is an effective means of revealing the linear and nonlinear evolution equation of the film thickness. The wavelength of surface waves is assumed to be sufficiently large and the perturbation amplitude α≪1; so the long-wavelength modes that give the smallest wave number are most likely to induce flow instability. Yih [21] indicated that the long-wave perturbation method is appropriate for film flow with a small Reynolds number. Alekseenko et al. [22] investigated the wave formation of a vertical liquid film. The maximum growth rates, wave number, and wave velocities were found theoretically and compared with obtained experimentally at Re~30. Then, the hydrodynamic stability analysis of the thin liquid film by the long-wave perturbation method was performed [23–26]. Furthermore, numerous investigations of hydrodynamic stability have been studied by the experimental observations [27, 28].Several researchers presented hydromagnetic stability of film flows on a rotating disk [29–32] and the weakly nonlinear analysis of the MHD flow down a vertical plate was published [33]. In this study, the authors present a weakly nonlinear stability analysis of a thin electrically conductive fluid under an applied uniform magnetic field on a rotating disk, namely, spin coating. The induced magnetic field is neglected by assuming that the magnetic Reynolds number Rem≪1 [30] during spin coating. It is also assumed that the disk radius is much larger than the film thickness. Therefore, the peripheral effect is negligible in comparison to the total film area [15]. It is demonstrated that stability is revealed in the region near the rotating axis due to centrifugal forces and MHD effects. The influence of the rotational motion, disk size, and the Hartmann number on the equilibrium finite amplitude is studied and characterized mathematically. In an attempt to verify the computational results and to illustrate the effectiveness of the proposed modeling approach, several numerical examples are presented.


## 2. Mathematical Formulation

Under the assumption that no polarization voltage is applied (i.e.,E⃗=0⃗), the electromagnetic force Fm⃗ acting on the MHD flow is given: (2.1)Fm⃗=J⃗×B⃗=σ(V⃗×B⃗)×B⃗,
where E⃗ is the electric field, J⃗ the current density, B⃗ the magnetic flux, σ the electrical conductivity, and V⃗ the velocity vector.Consider axisymmetric flow of a thin electrically conductive fluid flowing on a rotating circular disk which rotates with constant angular velocityΩ* under an applied magnetic field B0*. The external uniform magnetic field is applied perpendicular to the plane of the disk (see Figure 1). A variable with a superscript “*” represents a dimensional quantity. Here the cylindrical polar coordinate axes r*,θ*,z* are chosen as the radial direction, the circumferential direction, and the direction perpendicular to the plate, respectively. All associated physical properties and the rate of film flow are assumed to be constant (i.e., time-invariant) because of the steady mass flow source at the center of rotation. Let u* and w* be the velocity components in the radial direction r* and the perpendicular direction z* of a circular disk, respectively. According to the experimental observation by Takama and Kobayashi [34], it is reasonable to assume negligible circumferential flow when the liquid film is very thin (h*≪r*). As a matter of simplification, we take the fluid velocity in the thinning film to be independent of θ*. For small magnetic Reynolds number (Rem≪1), the electromagnetic force Fm is σB0*2u* [30, 35], when imposed and induced electric fields are negligible and the only applied magnetic field is B0*. The MHD governing equations of motion can be expressed as(2.2)1r*∂(r*u*)∂r*+∂w*∂z*=0,ρ(∂u*∂t*+u*∂u*∂r*+w*∂u*∂z*-v*2r*)=-∂p*∂r*+μ(∂2u*∂r*2+1r*∂u*∂r*+∂2u*∂z*2-u*r*2)-σB0*2u*,∂w*∂t*+u*∂w*∂r*+w*∂w*∂z*=-1ρ∂p*∂z*-g+μρ(1r*∂∂r*(r*∂w*∂r*)+∂2w*∂z*2),
where v* is the azimuthal velocity, ρ the constant fluid density, p* the fluid pressure, g the acceleration due to gravity, μ the dynamic viscosity of the fluid, and B0* is the magnetic flux density.Figure 1
Schematic diagram of a thin MHD fluid flowing on a rotating circular disk.On the disk surfacez*=0, the boundary conditions are treated as no-slip as (2.3)u*=0,w*=0.
On the free surface z*=h*, the boundary condition approximated by the vanishing of the shear stress is expressed as(2.4)(∂u*∂z*+∂w*∂r*)(1-(∂h*∂r*)2)-2*(∂u*∂r*-∂w*∂z*)(∂h*∂r*)=0.
The normal stress condition obtained by solving the balance equation in the direction normal to the free surface is given as (2.5)p*+2μ(1+(∂h*∂r*)2)-1[∂h*∂r*(∂u*∂z*+∂w*∂r*)-∂w*∂z*-∂u*∂r*(∂h*∂r*)2]+S*∂2h*∂r*2[1+(∂h*∂r*)2]-3/2=pa*.
The kinematic condition for a material free surface can be given as(2.6)∂h*∂t*+∂h*∂r*u*-w*=0,
where h* is the local film thickness, S* is surface tension, and pa* is the atmospheric pressure. By introducing a stream function φ*, the dimensional velocity components can be expressed as (2.7)u*=1r*∂φ*∂z*,w*=-1r*∂φ*∂r*.
The following variables are used to form the dimensionless governing equations and boundary conditions(2.8)z=z*h0*,r=αr*h0*,t=αt*u0*h0*,h=h*h0*,φ=αφ*u0*h0*2,p=p*-pa*ρu0*2,Re=u0*h0*ν,Fr=gh0*u0*2,S=S*ρu0*2h0*,m=σB0*2h0*2μ,α=2πh0*λ,
where h0*  is the average film thickness, α the dimensionless wave number, u0* the scale of the velocity, ν the kinematic viscosity, Re the Reynolds number, Fr the Froude number, m the Hartmann number, and λ the wavelength.In order to investigate the effects of angular velocity,Ω* on the stability of the flow field, it is assumed that the azimuthal velocity is constant [30, 35, 36] throughout the radial direction in the thin film, that is, v*=r*Ω*. The dimensionless parameter Ro (rotation number) is defined as(2.9)Ro=Ω*h0*u0*.
In terms of these nondimensional variables, the equations of motion can be expressed as(2.10)r-1φzzz=-RerRo2+r-1mφz+αRe(pr+r-1φtz+r-2φzφrz-r-3φz2-r-2φrφzz)+O(α2),pz=-Fr+α(-Re-1r-1φrzz)+O(α2).
Using the nondimensional variables, the boundary conditions at the surface of the disk z=0 reduce to(2.11)φ=φr=φz=0,
and the boundary conditions at the free surface of the disk z=h become(2.12)r-1φzz=α2[r-1φrr-r-2φr+2hr(1-α2hr2)-1(2r-1φrz-r-2φz)],(2.13)p=-Sα2hrr(1+α2hr2)-3/2+α[-2Re-1(1+α2hr2)-1(r-1φrz+r-1φzzhr)]+O(α2),(2.14)ht+r-1hrφz+r-1φr=0.
In practice, the parameter S has a large value and α≪1; so the term α2S is taken to be of order one [33, 37]. Long-wave length modes are considered in the present analysis; this can be done by expanding the stream function and flow pressure in terms of some small wave number (α≪1) as(2.15)φ=φ0+αφ1+O(α2),p=p0+αp1+O(α2).The flow conditions of a film can be obtained by inserting the above expressions into (2.10)–(2.13) and then solving systematically the resulting equations. By collecting all terms of zeroth-order α0 and first-order α1 in the above governing equations and boundary conditions, the solutions of the zeroth-order and first-order equations were obtained and are given in Appendix A.The zeroth-order and first-order solutions are inserted into the dimensionless free surface kinematic equation to yield the following generalized nonlinear kinematic equation:(2.16)ht+A(h)hr+B(h)hrr+C(h)hrrr+D(h)hrrrr+E(h)hr2+F(h)hrhrrr=0,
where A(h),B(h),C(h),D(h),E(h), and F(h) are given in Appendix B. When the Hartmann number m approaches 0, the fluid flow is reduced to the typical Newtonian film flow with no applied magnetic field [38].


## 3. Stability Analysis

The dimensionless film thickness, when expressed in terms of the perturbed variables, can be given as(3.1)h(r,t)=1+η(r,t),
where η is a perturbed quantity to the stationary film thickness. Substituting the value of h(r,t) into the evolution equation (2.16) and all terms up to order η3 being collected, the evolution equation of η becomes (3.2)ηt+Aηr+Bηrr+Cηrrr+Dηrrrr+Eηr2+Fηrηrrr=(A′η+A′′2η2)ηr+(B′η+B′′2η2)ηrr+(C′η+C′′2η2)ηrrr+(D′η+D′′2η2)ηrrrr+(E+E′η)ηr2+(F+F′η)ηrηrrr+O(η4),
where the values of A,B,C,D,E,F and their derivatives are all evaluated at the dimensionless height of the film, h=1.


### 3.1. Linear Stability Analysis

When the nonlinear terms of (3.2) are neglected, the linearized equation is given as(3.3)ηt+Aηr+Bηrr+Cηrrr+Dηrrrr=0.
In order to use the normal mode analysis, we assume that(3.4)η=aexp[i(r-dt)]+c.c.,
where a is the perturbation amplitude, and c.c. is the complex conjugate counterpart. The complex wave celerity d is given as(3.5)d=dr+idi=(A-C)+i(B-D),
where dr and di are the linear wave speed and linear growth rate of the disturbance, respectively. The solution of the disturbance about h(r,t)=1 is asymptotically stable or unstable according as di<0 or di>0. This is equivalent to the inequality B<D or B>D.


### 3.2. Weakly Nonlinear Stability Analysis

Nonlinear effects, when they are sufficiently weak, do not fundamentally alter the nature of the motion. A weakly nonlinear solution can be usefully expressed as weak stability or instability, but the definition is restricted to some neighborhood of critical value [39]. In this paper the authors are interested in investigating the existence of the supercritical and subcritical regions. In order to characterize the weakly nonlinear behavior of the film flows, the method of multiple scales [40] is employed here; as a result,the Ginburg-Landau equation [41] can be derived following the same procedure as in [33] and Cheng and Lin [42]: (3.6)∂a∂t2+D1∂2a∂r12-ε-2dia+(E1+iF1)a2a¯=0,
where ε is a small perturbation parameter, t2=ε2t,r1=εr, and  (3.7)e=er+iei=(B′-D′+E-F)(16D-4B)+6C(A′-C′)(16D-4B)2+36C2+i6C(B′-D′+E-F)-(A′-C′)(16D-4B)(16D-4B)2+36C2,D1=[(B-6D)+i(3C)],E1=(-5B′+17D′+4E-10F)er-(A′-7C′)ei+(-32B′′+32D′′+E′-F′),F1=(-5B′+17D′+4E-10F)ei+(A′-7C′)er+12(A′′-C′′).
The overhead bar appearing in (3.6) stands for the complex conjugate of the same variable. Equation (3.6) can be used to investigate the weak nonlinear behavior of the fluid film flow. In order to solve (3.6), we assume a filtered wave with no spatial modulation; so the filtered wave can be expressed as(3.8)a=a0exp[-ib(t2)t2].
After substituting (3.8) into (3.6), one can obtain(3.9)∂a0∂t2=(ε-2di-E1a02)a0,(3.10)∂[b(t2)t2]∂t2=F1a02.
The associated wave amplitude εa0 in the supercritical stable region is derived and given as(3.11)εa0=diE1.
If E1=0, then (3.9) is reduced to a linear equation. The second term on the right-hand side of (3.9) is due to the nonlinearity and may moderate or accelerate the exponential growth of the linear disturbance according to the signs of di and E1. Equation (3.10) is used to modify the perturbed wave speed caused by infinitesimal disturbances appearing in the nonlinear system. The Ginzburg-Landau equation can be used to characterize various flow states, with the results summarized in Table 1.Table 1
Various states of Landau equation.
Linearly stable (subcritical region)di<0Subcritical instabilityE1<0εa0<(di/E1)1/2a0→0Conditional stabilityεa0>(di/E1)1/2a0↑Subcritical explosive stateSubcritical (absolute) stabilityE1>0a0→0Linearly unstable (supercritical region)di>0Supercritical explosive stateE1<0a0↑Supercritical stabilityE1>0εa0→(di/E1)1/2Ncr→dr+di(F1/E1)


## 3.1. Linear Stability Analysis

When the nonlinear terms of (3.2) are neglected, the linearized equation is given as(3.3)ηt+Aηr+Bηrr+Cηrrr+Dηrrrr=0.
In order to use the normal mode analysis, we assume that(3.4)η=aexp[i(r-dt)]+c.c.,
where a is the perturbation amplitude, and c.c. is the complex conjugate counterpart. The complex wave celerity d is given as(3.5)d=dr+idi=(A-C)+i(B-D),
where dr and di are the linear wave speed and linear growth rate of the disturbance, respectively. The solution of the disturbance about h(r,t)=1 is asymptotically stable or unstable according as di<0 or di>0. This is equivalent to the inequality B<D or B>D.


## 3.2. Weakly Nonlinear Stability Analysis

Nonlinear effects, when they are sufficiently weak, do not fundamentally alter the nature of the motion. A weakly nonlinear solution can be usefully expressed as weak stability or instability, but the definition is restricted to some neighborhood of critical value [39]. In this paper the authors are interested in investigating the existence of the supercritical and subcritical regions. In order to characterize the weakly nonlinear behavior of the film flows, the method of multiple scales [40] is employed here; as a result,the Ginburg-Landau equation [41] can be derived following the same procedure as in [33] and Cheng and Lin [42]: (3.6)∂a∂t2+D1∂2a∂r12-ε-2dia+(E1+iF1)a2a¯=0,
where ε is a small perturbation parameter, t2=ε2t,r1=εr, and  (3.7)e=er+iei=(B′-D′+E-F)(16D-4B)+6C(A′-C′)(16D-4B)2+36C2+i6C(B′-D′+E-F)-(A′-C′)(16D-4B)(16D-4B)2+36C2,D1=[(B-6D)+i(3C)],E1=(-5B′+17D′+4E-10F)er-(A′-7C′)ei+(-32B′′+32D′′+E′-F′),F1=(-5B′+17D′+4E-10F)ei+(A′-7C′)er+12(A′′-C′′).
The overhead bar appearing in (3.6) stands for the complex conjugate of the same variable. Equation (3.6) can be used to investigate the weak nonlinear behavior of the fluid film flow. In order to solve (3.6), we assume a filtered wave with no spatial modulation; so the filtered wave can be expressed as(3.8)a=a0exp[-ib(t2)t2].
After substituting (3.8) into (3.6), one can obtain(3.9)∂a0∂t2=(ε-2di-E1a02)a0,(3.10)∂[b(t2)t2]∂t2=F1a02.
The associated wave amplitude εa0 in the supercritical stable region is derived and given as(3.11)εa0=diE1.
If E1=0, then (3.9) is reduced to a linear equation. The second term on the right-hand side of (3.9) is due to the nonlinearity and may moderate or accelerate the exponential growth of the linear disturbance according to the signs of di and E1. Equation (3.10) is used to modify the perturbed wave speed caused by infinitesimal disturbances appearing in the nonlinear system. The Ginzburg-Landau equation can be used to characterize various flow states, with the results summarized in Table 1.Table 1
Various states of Landau equation.
Linearly stable (subcritical region)di<0Subcritical instabilityE1<0εa0<(di/E1)1/2a0→0Conditional stabilityεa0>(di/E1)1/2a0↑Subcritical explosive stateSubcritical (absolute) stabilityE1>0a0→0Linearly unstable (supercritical region)di>0Supercritical explosive stateE1<0a0↑Supercritical stabilityE1>0εa0→(di/E1)1/2Ncr→dr+di(F1/E1)


## 4. Numerical Examples

In order to study the effects of the dimensionless radius, rotation number, and Hartmann number on the stability of a thin flow, we select randomly but within specified ranges physical parameters for the numerical experiments. The ranges for these parameters are based on published reasonable ranges for these parameters [33, 36, 38, 43]. More specifically, these parameters and their values include (1) Reynolds number (range from 0 to 15), (2) dimensionless perturbation wave numbers (range from 0 to 0.12), (3) Rotation number (any one of the three values 0.15, 0.175, and 0.2), (4) dimensionless radius (any one of the three values 10, 15, and 20). (5) Hartman number (any one of the four values 0, 0.05, 0.1, and 0.15). The remaining parameters are treated as constants for all numerical computations since we are considering practical spin coating systems in which these variables are not expected to undergo significant variation. Further, for a simplified analysis, α2S, Re, and Fr are taken to be of the same order (O(1)) [34, 38, 43]; so the values of some dimensionless parameters are taken as constants, for example, the dimensionless surface tension S=6173.5 and the Froude number Fr=9.8.


### 4.1. Linear Stability Analysis

By settingdi=0 in the linear stability analysis, the neutral stability curve can be determined easily from (3.5). The α2S-Re plane is divided into two different characteristic regions by the neutral stability curve. One is the linearly stable region where small disturbances decay with time, and the other is the linearly unstable region where small perturbations grow as time increases. Figure 2(a) shows that the stable region decreases and unstable region increases with an increase of rotation number. Figure 2(b) shows that the stable region decreases and unstable region increases with increasing radius of the circular disk. The reason for this phenomenon is the existence of the centrifugal force term, which is a radius-related force in the governing equation. To increase the radius and the rotation number results in accelerated growth of the linear disturbance due to the centrifugal force. Figure 2(c) shows that the stable region increases and unstable region decreases with an increase of the Hartman number. The reason for this phenomenon is that the Lorentz forces can modify the velocity field and moderate the growth of linear disturbance. Hence one can say that in linear stability analysis the rotation number and the radius of the circular disk generate similar destabilizing effects but the Hartmann number gives a stabilizing effect.Linear neutral stability curves for three different Ro values atr=10 and m=0.1. Linear neutral stability curves for three different r values at m=0.1 and Ro=0.1. Linear neutral stability curves for three different m values at r=10 and Ro=0.1
(a)(b)(c)


### 4.2. Weakly Nonlinear Stability Analysis

Figures3(a) to 3(d) reveal various conditions for subcritical instability (di<0,E1<0), sub-critical stability (di<0,E1>0), supercritical stability (di>0,E1>0) and the supercritical explosion (di>0,E1<0). Figure 4(a) shows the threshold amplitude in the sub-critical instability region for various wave numbers with different Ro values at Re=3,r=10 and m=0.1. The results indicate that the threshold amplitude εa0 becomes smaller as the value of the rotation number (Ro) increases. Figure 4(b) shows the threshold amplitude in the sub-critical instability region for various wave numbers with different Hartmann number (m) values at Re=3,r=10, and Ro=0.15. The results indicate that the threshold amplitude εa0 becomes smaller as the value of m decreases. The film flow which holds the higher threshold amplitude value will become more stable than that holds the smaller one. If the initial finite-amplitude disturbance is less than the threshold amplitude, the system will become conditionally stable.Neutral stability curves of MHD film flows forr=10,Ro=0.15, and m=0.1. Neutral stability curves of MHD film flows for r=10,Ro=0.175, and m=0.1. Neutral stability curves of MHD film flows for r=15,Ro=0.15, and m=0.1. Neutral stability curves of MHD film flows for r=10,Ro=0.15, and m=0.05
(a)(b)(c)(d)Threshold amplitude in subcritical instability region for three different Ro values at Re = 3,r=10, and m=0.1. Threshold amplitude in subcritical instability region for three different m values at Re = 3, r=10, and Ro = 0.15
(a)(b)Figure5(a) shows the threshold amplitude in the supercritical stability region for various wave numbers with different Ro values at Re=6,r=10, and m=0.1. It is found that decreasing the rotation number will lower the threshold amplitude, whereupon the flow becomes relatively more stable. Figure 5(b) shows the threshold amplitude in the supercritical stability region for various wave numbers with different m values at Re=6,r=10 and Ro=0.15. It is found that increasing the Hartmann number will lower the threshold amplitude, and the flow will become relatively more stable.Threshold amplitude in supercritical stability region for two different Ro values at Re = 6,r=10 and m=0.1. Threshold amplitude in supercritical stability region for two different m values at Re = 6, r=10 and Ro = 0.15
(a)(b)


## 4.1. Linear Stability Analysis

By settingdi=0 in the linear stability analysis, the neutral stability curve can be determined easily from (3.5). The α2S-Re plane is divided into two different characteristic regions by the neutral stability curve. One is the linearly stable region where small disturbances decay with time, and the other is the linearly unstable region where small perturbations grow as time increases. Figure 2(a) shows that the stable region decreases and unstable region increases with an increase of rotation number. Figure 2(b) shows that the stable region decreases and unstable region increases with increasing radius of the circular disk. The reason for this phenomenon is the existence of the centrifugal force term, which is a radius-related force in the governing equation. To increase the radius and the rotation number results in accelerated growth of the linear disturbance due to the centrifugal force. Figure 2(c) shows that the stable region increases and unstable region decreases with an increase of the Hartman number. The reason for this phenomenon is that the Lorentz forces can modify the velocity field and moderate the growth of linear disturbance. Hence one can say that in linear stability analysis the rotation number and the radius of the circular disk generate similar destabilizing effects but the Hartmann number gives a stabilizing effect.Linear neutral stability curves for three different Ro values atr=10 and m=0.1. Linear neutral stability curves for three different r values at m=0.1 and Ro=0.1. Linear neutral stability curves for three different m values at r=10 and Ro=0.1
(a)(b)(c)


## 4.2. Weakly Nonlinear Stability Analysis

Figures3(a) to 3(d) reveal various conditions for subcritical instability (di<0,E1<0), sub-critical stability (di<0,E1>0), supercritical stability (di>0,E1>0) and the supercritical explosion (di>0,E1<0). Figure 4(a) shows the threshold amplitude in the sub-critical instability region for various wave numbers with different Ro values at Re=3,r=10 and m=0.1. The results indicate that the threshold amplitude εa0 becomes smaller as the value of the rotation number (Ro) increases. Figure 4(b) shows the threshold amplitude in the sub-critical instability region for various wave numbers with different Hartmann number (m) values at Re=3,r=10, and Ro=0.15. The results indicate that the threshold amplitude εa0 becomes smaller as the value of m decreases. The film flow which holds the higher threshold amplitude value will become more stable than that holds the smaller one. If the initial finite-amplitude disturbance is less than the threshold amplitude, the system will become conditionally stable.Neutral stability curves of MHD film flows forr=10,Ro=0.15, and m=0.1. Neutral stability curves of MHD film flows for r=10,Ro=0.175, and m=0.1. Neutral stability curves of MHD film flows for r=15,Ro=0.15, and m=0.1. Neutral stability curves of MHD film flows for r=10,Ro=0.15, and m=0.05
(a)(b)(c)(d)Threshold amplitude in subcritical instability region for three different Ro values at Re = 3,r=10, and m=0.1. Threshold amplitude in subcritical instability region for three different m values at Re = 3, r=10, and Ro = 0.15
(a)(b)Figure5(a) shows the threshold amplitude in the supercritical stability region for various wave numbers with different Ro values at Re=6,r=10, and m=0.1. It is found that decreasing the rotation number will lower the threshold amplitude, whereupon the flow becomes relatively more stable. Figure 5(b) shows the threshold amplitude in the supercritical stability region for various wave numbers with different m values at Re=6,r=10 and Ro=0.15. It is found that increasing the Hartmann number will lower the threshold amplitude, and the flow will become relatively more stable.Threshold amplitude in supercritical stability region for two different Ro values at Re = 6,r=10 and m=0.1. Threshold amplitude in supercritical stability region for two different m values at Re = 6, r=10 and Ro = 0.15
(a)(b)


## 5. Concluding Remarks

The stability of a thin electrically conductive fluid under the applied uniform magnetic field during spin coating is investigated using the method of long-wave perturbation. On the basis of the results of numerical modeling, several conclusions can be drawn.(1)
The modeling results indicate that the region of linear stability becomes smaller for increasing rotation number or increasing radius. Hence one can say that in the linear stability analysis the rotation number and the radius of the circular disk generate similar destabilizing effects.(2)
The modeling results also show that the stable region increases and unstable region decreases with an increase of the Hartmann number. In linear stability analysis Hartmann number gives a stabilizing effect.(3)
Weakly nonlinear stability analysis has successfully revealed sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion regions for the flow patterns of a thin film on a rotating disk under the applied uniform magnetic field. It is found that in sub-critical instability regions, the threshold amplitudeεa0 becomes smaller as the value of the rotation number becomes larger. When the initial finite-amplitude disturbance is less than the threshold amplitude, the flow will be conditionally stable.(4)
It is also shown that in sub-critical instability regions, the threshold amplitudeεa0 becomes larger as the value of the Hartman number becomes larger.  Optimum conditions can be obtained: the system is used to increase the stability of the film flow by controlling the applied magnetic field.



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