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

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.

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 Re_𝑚≪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., 𝐸=0), the electromagnetic force 𝐹_𝑚 acting on the MHD flow is given: 𝐹_𝑚=𝐽×𝐵=𝜎(𝑉×𝐵)×𝐵, (2.1) where 𝐸 is the electric field, 𝐽 the current density, 𝐵 the magnetic flux, 𝜎 the electrical conductivity, and 𝑉 the velocity vector.

Consider axisymmetric flow of a thin electrically conductive fluid flowing on a rotating circular disk which rotates with constant angular velocity Ω^∗ under an applied magnetic field 𝐵^∗₀. 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 (Re_𝑚≪1), the electromagnetic force 𝐹_𝑚 is 𝜎𝐵₀^∗2𝑢^∗ [30, 35], when imposed and induced electric fields are negligible and the only applied magnetic field is 𝐵^∗₀. The MHD governing equations of motion can be expressed as1𝑟^∗𝜕(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^∗1=−𝜌𝜕𝑝^∗𝜕𝑧^∗𝜇−𝑔+𝜌(1𝑟^∗𝜕𝜕𝑟^∗(𝑟^∗𝜕𝑤^∗𝜕𝑟^∗𝜕)+²𝑤^∗𝜕𝑧^∗2), (2.2) 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 𝐵^∗₀ is the magnetic flux density.

Figure 1 

Schematic diagram of a thin MHD fluid flowing on a rotating circular disk.

On the disk surface 𝑧^∗=0, the boundary conditions are treated as no-slip as 𝑢^∗𝑤=0,_∗=0. (2.3) On the free surface 𝑧^∗=ℎ^∗, the boundary condition approximated by the vanishing of the shear stress is expressed as(𝜕𝑢^∗𝜕𝑧^∗+𝜕𝑤^∗𝜕𝑟^∗)(1−(𝜕ℎ^∗𝜕𝑟^∗)²)−2^∗(𝜕𝑢^∗𝜕𝑟^∗−𝜕𝑤^∗𝜕𝑧^∗)(𝜕ℎ^∗𝜕𝑟^∗)=0. (2.4) The normal stress condition obtained by solving the balance equation in the direction normal to the free surface is given as 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=𝑝^∗_𝑎. (2.5) The kinematic condition for a material free surface can be given as𝜕ℎ^∗𝜕𝑡^∗+𝜕ℎ^∗𝜕𝑟^∗𝑢^∗−𝑤^∗=0, (2.6) where ℎ^∗ is the local film thickness, 𝑆^∗ is surface tension, and 𝑝^∗_𝑎 is the atmospheric pressure. By introducing a stream function 𝜑^∗, the dimensional velocity components can be expressed as 𝑢^∗=1𝑟^∗𝜕𝜑^∗𝜕𝑧^∗,𝑤^∗1=−𝑟^∗𝜕𝜑^∗𝜕𝑟^∗. (2.7) The following variables are used to form the dimensionless governing equations and boundary conditions𝑧𝑧=^∗ℎ^∗₀,𝑟=𝛼𝑟^∗ℎ^∗₀,𝑡=𝛼𝑡^∗𝑢^∗₀ℎ^∗₀ℎ,ℎ=^∗ℎ^∗₀,𝜑=𝛼𝜑^∗𝑢^∗₀ℎ₀^∗2𝑝,𝑝=^∗−𝑝^∗_𝑎𝜌𝑢₀^∗2𝑢,Re=^∗0ℎ^∗₀𝜈,Fr=𝑔ℎ^∗₀𝑢₀^∗2𝑆,𝑆=^∗𝜌𝑢₀^∗2ℎ^∗₀,𝑚=𝜎𝐵₀^∗2ℎ₀^∗2𝜇,𝛼=2𝜋ℎ^∗₀𝜆, (2.8) where ℎ^∗₀ is the average film thickness, 𝛼 the dimensionless wave number, 𝑢^∗₀ 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 asΩRo=^∗ℎ^∗₀𝑢^∗₀. (2.9) In terms of these nondimensional variables, the equations of motion can be expressed as𝑟⁻¹𝜑_𝑧𝑧𝑧=−Re𝑟Ro²+𝑟⁻¹𝑚𝜑_𝑧+𝛼Re(𝑝_𝑟+𝑟⁻¹𝜑_𝑡𝑧+𝑟⁻²𝜑_𝑧𝜑_𝑟𝑧−𝑟⁻³𝜑²_𝑧−𝑟⁻²𝜑_𝑟𝜑_𝑧𝑧)+𝑂(𝛼²𝑝),_𝑧=−Fr+𝛼(−Re⁻¹𝑟⁻¹𝜑_𝑟𝑧𝑧)+𝑂(𝛼²). (2.10) Using the nondimensional variables, the boundary conditions at the surface of the disk 𝑧=0 reduce to𝜑=𝜑_𝑟=𝜑_𝑧=0, (2.11) and the boundary conditions at the free surface of the disk 𝑧=ℎ become𝑟⁻¹𝜑_𝑧𝑧=𝛼²[𝑟⁻¹𝜑_𝑟𝑟−𝑟⁻²𝜑_𝑟+2ℎ_𝑟(1−𝛼²ℎ²_𝑟)⁻¹(2𝑟⁻¹𝜑_𝑟𝑧−𝑟⁻²𝜑_𝑧)],𝑝=−𝑆𝛼²ℎ_𝑟𝑟(1+𝛼²ℎ²_𝑟)^−3/2+𝛼[−2Re⁻¹(1+𝛼²ℎ²_𝑟)⁻¹(𝑟⁻¹𝜑_𝑟𝑧+𝑟⁻¹𝜑_𝑧𝑧ℎ_𝑟)]+𝑂(𝛼²ℎ),_𝑡+𝑟⁻¹ℎ_𝑟𝜑_𝑧+𝑟⁻¹𝜑_𝑟=0. (2.12)(2.13)(2.14) 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) as𝜑=𝜑₀+𝛼𝜑₁+𝑂(𝛼²),𝑝=𝑝₀+𝛼𝑝₁+𝑂(𝛼²). (2.15)

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: ℎ_𝑡+𝐴(ℎ)ℎ_𝑟+𝐵(ℎ)ℎ_𝑟𝑟+𝐶(ℎ)ℎ_𝑟𝑟𝑟+𝐷(ℎ)ℎ_{𝑟𝑟𝑟𝑟}+𝐸(ℎ)ℎ²_𝑟+𝐹(ℎ)ℎ_𝑟ℎ_𝑟𝑟𝑟=0, (2.16) 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 asℎ(𝑟,𝑡)=1+𝜂(𝑟,𝑡), (3.1) 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 𝜂_𝑡+𝐴𝜂_𝑟+𝐵𝜂_𝑟𝑟+𝐶𝜂_𝑟𝑟𝑟+𝐷𝜂_{𝑟𝑟𝑟𝑟}+𝐸𝜂²_𝑟+𝐹𝜂_𝑟𝜂_𝑟𝑟𝑟=(𝐴^′𝐴𝜂+^′′2𝜂²)𝜂_𝑟+(𝐵^′𝐵𝜂+^′′2𝜂²)𝜂_𝑟𝑟+(𝐶^′𝐶𝜂+^′′2𝜂²)𝜂_𝑟𝑟𝑟+(𝐷^′𝐷𝜂+^′′2𝜂²)𝜂_{𝑟𝑟𝑟𝑟}+(𝐸+𝐸^′𝜂)𝜂²_𝑟+(𝐹+𝐹^′𝜂)𝜂_𝑟𝜂_𝑟𝑟𝑟+𝑂(𝜂⁴), (3.2) 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 as𝜂_𝑡+𝐴𝜂_𝑟+𝐵𝜂_𝑟𝑟+𝐶𝜂_𝑟𝑟𝑟+𝐷𝜂_{𝑟𝑟𝑟𝑟}=0. (3.3) In order to use the normal mode analysis, we assume that𝜂=𝑎exp[𝑖(𝑟−𝑑𝑡)]+c.c., (3.4) where 𝑎 is the perturbation amplitude, and c.c. is the complex conjugate counterpart. The complex wave celerity 𝑑 is given as𝑑=𝑑_𝑟+𝑖𝑑_𝑖=(𝐴−𝐶)+𝑖(𝐵−𝐷), (3.5) where 𝑑_𝑟 and 𝑑_𝑖 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 𝑑_𝑖<0 or 𝑑_𝑖>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]: 𝜕𝑎𝜕𝑡₂+𝐷₁𝜕²𝑎𝜕𝑟²₁−𝜀⁻²𝑑_𝑖𝑎+(𝐸₁+𝑖𝐹₁)𝑎²𝑎=0, (3.6) where 𝜀 is a small perturbation parameter, 𝑡₂=𝜀²𝑡,𝑟₁=𝜀𝑟, and 𝑒=𝑒_𝑟+𝑖𝑒_𝑖=(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,𝐷₁𝐸=[(𝐵−6𝐷)+𝑖(3𝐶)],₁=(−5𝐵^′+17𝐷^′+4𝐸−10𝐹)𝑒_𝑟−(𝐴^′−7𝐶^′)𝑒_𝑖3+(−2𝐵^′′+32𝐷^′′+𝐸^′−𝐹^′𝐹),₁=(−5𝐵^′+17𝐷^′+4𝐸−10𝐹)𝑒_𝑖+(𝐴^′−7𝐶^′)𝑒_𝑟+12(𝐴^′′−𝐶^′′). (3.7) 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 as𝑎=𝑎₀exp[−𝑖𝑏(𝑡₂)𝑡₂]. (3.8) After substituting (3.8) into (3.6), one can obtain𝜕𝑎₀𝜕𝑡₂=(𝜀⁻²𝑑_𝑖−𝐸₁𝑎²₀)𝑎₀,𝜕[𝑏(𝑡₂)𝑡₂]𝜕𝑡₂=𝐹₁𝑎²₀. (3.9)(3.10) The associated wave amplitude 𝜀𝑎₀ in the supercritical stable region is derived and given as𝜀𝑎₀=𝑑_𝑖𝐸₁. (3.11) 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 𝑑_𝑖 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.

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 setting 𝑑_𝑖=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.

(a)

(b)

(c)

Figure 2 

Linear neutral stability curves for three different Ro values at $r=10$ and $m=0.1.$ Linear neutral stability curves for three different $r$ values at $m=0.1$ and $\text{Ro}=0.1.$ Linear neutral stability curves for three different $m$ values at $r=10$ and $\text{Ro}=0.1$

4.2. Weakly Nonlinear Stability Analysis

Figures 3(a) to 3(d) reveal various conditions for subcritical instability (𝑑_𝑖<0,𝐸₁<0), sub-critical stability (𝑑_𝑖<0,𝐸₁>0), supercritical stability (𝑑_𝑖>0,𝐸₁>0) and the supercritical explosion (𝑑_𝑖>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.

(a)

(b)

(c)

(d)

Figure 3 

Neutral stability curves of MHD film flows for $r=10,\hspace{0.17em}\hspace{0.17em}\text{Ro}=0.15,$ and $m=0.1.$ Neutral stability curves of MHD film flows for $r=10,\hspace{0.17em}\hspace{0.17em}\text{Ro}=0.175,$ and $m=0.1.$ Neutral stability curves of MHD film flows for $r=15,\hspace{0.17em}\hspace{0.17em}\text{Ro}=0.15,$ and $m=0.1.$ Neutral stability curves of MHD film flows for $r=10,\hspace{0.17em}\hspace{0.17em}\text{Ro}=0.15,$ and $m=0.05$

(a)

(b)

Figure 4 

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

Figure 5(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.

(a)

(b)

Figure 5 

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

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 amplitude 𝜀𝑎₀ 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 amplitude 𝜀𝑎₀ 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.

Appendices

A. Zeroth-Order and First-Order Solutions

Zeroth-order solution:𝜑₀=Re⋅𝑟²⋅Ro²⋅Sech(ℎ√𝑚)[√𝑚⋅𝑧⋅Cosh(ℎ√𝑚)−Sinh(ℎ√𝑚)+Sinh((ℎ−𝑧)√𝑚)]𝑚^3/2, (A.1)

where the zero-order governing equation: 𝑟⁻¹𝜑_{0𝑧𝑧𝑧}=−Re𝑟Ro²+𝑟⁻¹𝑚𝜑_0𝑧,𝑝_0𝑧=−Fr, (A.2)

associated with boundary condition:𝑧=0,𝜑₀=𝜑_0𝑧𝑟=0,𝑧=ℎ,₋₁𝜑_0𝑧𝑧=0,𝑝₀=−𝛼²𝑆ℎ_𝑟𝑟. (A.3) First-order solution:𝜑₁=𝑘₁𝑧⁴+𝑘₂𝑧³+𝑘₃𝑧²+𝑘₄𝑧+𝑘₅, (A.4)

where𝑘₁𝑟=−²⋅Re⋅Ro⁴12𝑚²𝑘₂,=16𝑚^5/2×(𝑟(2𝑟⋅Re⋅Ro⁴(√𝑚(1+Sech(ℎ√𝑚)²)+Tanh(ℎ√𝑚))+(Fr⋅𝑚^5/2−𝑚⋅𝑟²⋅Re⋅Ro⁴⋅Sech(ℎ√𝑚)²⋅Tanh(ℎ√𝑚))ℎ_𝑟−𝑚^5/2𝑆⋅𝛼²ℎ_𝑟𝑟𝑟𝑘))₃=14𝑚³×(𝑟(−2𝑚²⋅𝑟⋅Re⋅Ro²⋅ℎ_0𝑡⋅Sech(ℎ√𝑚)+𝑟⋅Re⋅Ro⁴×(2(−2+ℎ)ℎ⋅𝑚+Sech(ℎ√√𝑚)×(−4ℎ⋅𝑚+(1−4ℎ⋅𝑚)Sech(ℎ𝑚)+4√𝑚(−ℎ+Sech(ℎ√𝑚))Sinh(ℎ√𝑚)))−2(Fr⋅ℎ⋅𝑚³+𝑚⋅𝑟²⋅Re⋅Ro⁴⋅Sech(ℎ√𝑚)²×(−ℎ√𝑚+Sinh(ℎ√𝑚))Tanh(ℎ√𝑚))ℎ_𝑟+2ℎ⋅𝑚³⋅𝑆⋅𝛼²ℎ_𝑟𝑟𝑟𝑘))₄=18𝑚^7/2×(𝑟²⋅Re⋅Ro⁴⋅Sech(ℎ√𝑚)³×(4√𝑚(13Cosh(ℎ√𝑚)+3Cosh(3ℎ√𝑚)+Cosh(√√𝑚(−3ℎ+𝑧))+2Cosh(𝑚(−ℎ+𝑧))+Cosh(√𝑚(ℎ+𝑧))−Sinh(ℎ√√𝑚)−Sinh(3ℎ𝑚)+4Sinh(√√𝑚(−3ℎ+𝑧))+2Sinh(𝑚(−ℎ+𝑧)+Sinh(√𝑚(ℎ+𝑧))−8𝑚𝑟Sinh(√𝑚ℎ_𝑟))×ℎ_𝑟+2ℎ⋅𝑚³⋅𝑆⋅𝛼²ℎ_𝑟𝑟𝑟𝑘)))₅=116𝑚⁴×(𝑟²⋅Re⋅Ro⁴⋅Sech(ℎ√𝑚)³×(16𝑚²Cosh(ℎ√𝑚)Cosh(𝑧√𝑚)ℎ_0𝑡−Ro²Cosh(√√𝑚(ℎ−2𝑧))+8Cosh(𝑚(−3ℎ+𝑧))+32Cosh(√√𝑚(−ℎ+𝑧))+24Cosh𝑚(ℎ+𝑧)+Cosh(√√𝑚(−3ℎ+2𝑧))+16𝑚(2Sinh(√𝑚(−3ℎ+𝑧))+5Sinh(√𝑚(−ℎ+𝑧))+3Sinh(√√𝑚(ℎ+𝑧))−16𝑚⋅𝑟⋅Sinh(ℎ𝑚)Sinh(𝑧√𝑚)ℎ_𝑟ℎ)))_0𝑡=−Re⋅Ro²(2(ℎ√𝑚−Tanh(ℎ√𝑚))+√𝑚⋅𝑟⋅Tanh(ℎ√𝑚)²ℎ_𝑟)𝑚^3/2 (A.5)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)where the first-order governing equation:𝑟⁻¹𝜑_{1𝑧𝑧𝑧}=Re(𝑝_0𝑟+𝑟⁻¹𝜑_0𝑡𝑧+𝑟⁻²𝜑_0𝑧𝜑_0𝑟𝑧−𝑟⁻³𝜑²_0𝑧−𝑟⁻²𝜑_0𝑟𝜑_0𝑧𝑧𝑝),_1𝑧=−Re⁻¹𝑟⁻¹𝜑_{0𝑟𝑧𝑧}. (A.6)

associated with boundary condition𝑧=0,𝜑₀=𝜑_0𝑧𝑟=0,𝑧=ℎ,₋₁𝜑_1𝑧𝑧=0,𝑝₁=−2Re⁻¹(𝑟⁻¹𝜑_0𝑟𝑧+𝑟⁻¹𝜑_0𝑧𝑧ℎ_𝑟). (A.7)

B. Generalized Nonlinear Kinematic Equation

One has1𝐴(ℎ)=12𝑚^7/2𝑟Sech(ℎ√𝑚)²×(2√𝑚(−3𝑚²𝑟²Ro²−Fr⋅ℎ³𝑚³𝛼+30𝑟²Re²Ro⁴𝛼+(ℎ−18)ℎ²𝑚⋅𝑟²Re²Ro⁴𝛼+(3𝑚²𝑟²Ro²−Fr⋅ℎ³𝑚³𝛼+ℎ²(5ℎ−6)𝑚⋅𝑟²Re²Ro⁴√𝛼)×Cosh(2ℎ𝑚))−𝑟²Re²Ro⁴√𝛼×(2ℎ⋅𝑚(42+ℎ(9−14ℎ⋅𝑚))−48Cosh(ℎ𝑚)+3(1+4ℎ²𝑚)Cosh(ℎ√√𝑚))×Tanh(ℎ1𝑚)),𝐵(ℎ)=3ℎ⋅𝛼⋅Re(−Fr⋅ℎ²+(3−ℎ²𝑚)𝑟²Re²⋅Ro⁴Sech(ℎ√𝑚)²Tanh(ℎ√𝑚)𝑚^5/2),𝐶(ℎ)=𝑆⋅Re⋅ℎ³𝛼³3𝑟,𝐷(ℎ)=𝑆⋅Re⋅ℎ³𝛼³31,𝐸(ℎ)=24𝑚^5/2×(𝛼⋅Sech(ℎ√𝑚)⁴×(ℎ√𝑚(−9Fr⋅ℎ⋅𝑚²+16(ℎ²𝑚−3)𝑟²Re³Ro⁴−4(3Fr⋅ℎ⋅𝑚²+2(ℎ⋅𝑚²−3)𝑟²ReRo⁴)Cosh(2ℎ√𝑚)−3Fr⋅ℎ⋅𝑚²Cosh(4ℎ√𝑚))+12(ℎ²𝑚−1)𝑟²Re³Ro⁴Sinh(2ℎ√𝑚))),𝐹(ℎ)=Re⋅𝑆⋅ℎ²𝛼³. (B.1)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)

Acknowledgment

The authors sincerely thank the reviewer for his useful comments and suggestions which have helped in improving the quality of this manuscript.