The main object of this paper is to study the weakly nonlinear hydrodynamic stability of the thin Newtonian fluid flowing on a rotating circular disk. A long-wave perturbation method is used to derive the nonlinear evolution equation for the film flow. The linear behaviors of the spreading wave are investigated by normal mode approach, and its weakly nonlinear behaviors are explored by the method of multiple scales. The Ginzburg-Landau equation is determined to discuss the necessary condition for the existence of such flow pattern. The results indicate that the superctitical instability region increases, and the subcritical stability region decreases with the increase of the rotation number or the radius of circular disk. It is found that the rotation number and the radius of circular disk not only play the significant roles in destabilizing the flow in the linear stability analysis but also shrink the area of supercritical stability region at high Reynolds number in the weakly nonlinear stability analysis.
1. Introduction
The study of the hydrodynamic stability of a thin liquid film is important for a wide range of situations, varying from engineering science and chemical science. Due to various applications, attention is gradually focused on this subject. For instance, a process for coating a surface on a spinning substrate, wherein a liquid coating material is dispensed radially from the center to the edge or from the edge to the center above the surface, and after application of the coating material the coating is cured. This process usually referred to as spin coating. To control the uniform and stable thin film is an interesting subject in the technological development for photolithography in wafer manufacturing [1, 2].
The linear stability theories for various film flows have
been clearly presented by Lin [3] and Chandrasekhar [4]. Kapitza [5] was the
first one to study the stability of
a film flow over inclined planes. Various stability behaviors of the layer
flows were analyzed. Benney [6] studied
the nonlinear evolution equation for free surfaces of the film flows by using the
method of small parameters. The
theory of Kapitza was later found not so compatible with the nonlinear theory of
Landau [7]. The same film flow stability problem was studied by Yih [8] using a
numerical approach. The transition mechanism from laminar flow to turbulent
flow was elegantly explained by the Landau equation. This study sheds a light
later for further development on the theory of nonlinear film stability. The
Landau equation was rederived by Stuart [9] using the disturbed energy balance
equation and Reynolds stresses. Pumir et al. [10] further included the effect
of surface tension on the film flow model and solved for the solitary wave
solutions. Hwang and Weng [11] showed that the conditions of both supercritical
stability and subcritical instability are possible to occur for a film flow
system. Atherton and Homsy [12] discussed the derivation of complicated
nonlinear partial equations, evolution equations that describe the movement of
a fluid-fluid interface. Ruyer-Quil and Manneville [13–15] derived several models
of the film flows down an inclined plane by the numerical simulation. The
results indicate that it allows to better capture the viscous effects which are
dominant at small Reynolds numbers. Amaouche et al. [16] showed an accurate
modeling of a wavy film flow down an inclined plane in the inertia-dominated
regimes by using the weighted residual technique. It is shown that the model follows
quite closely, for a suitable choice of α,
the Orr-Sommerfeld equation for all Weber and
Kapitza numbers, in linear stability analysis. Samanta [17] derived the wave
solution of a viscous film flowing down on a vertical nonuniformly heated
wall. The results indicate that supercritical unstable region increases, and the
subcritical stable region decreases with the increase of Peclet number.
Emslie et al. [18] were the pioneers who analyzed a Newtonian liquid flowing on rotating disk. The flow is governed by a balance between the centrifugal force and the viscous resisting force. It was shown that the nonuniform distribution in the initial film profile tends to become uniform during spinning. This model has been widely employed in the subsequent investigations. Higgins [19] analyzed the flow of a Newtonian liquid
placed on an impulsively started rotating disk. In this work, a uniform film thickness is assumed, and the method of matched asymptotic expansions is adopted. It is showed that for films that maintains a planar interface there exists a self-similar form for the velocity field that allows the radial dependence to be factored out of the Navier-Stokes equations and boundary conditions. Kitamura et al. [20] solved the unsteady thin liquid film flow of nonuniform thickness on a rotating disk by asymptotic methods.
In this paper the authors are interested in
investigating the weakly nonlinear hydrodynamic stability of the thin Newtonian
liquid flowing on a rotating circular disk. It is assumed that the disk radius is much larger than
film thickness. Therefore, the peripheral effects are neglected by comparing
with total film area [20]. It is focused that the effects
of instability due to inertia and centrifugal forces were revealed in the region near the rotating axis. The influence of the rotating motion and the disk size effect on the equilibrium finite amplitude is studied and characterized mathematically. In an attempt to verify computational results and to illustrate the effectiveness of the proposed modeling approach, several numerical examples are also presented.
2. Mathematical Formulation
Consider a two-dimensional incompressible, viscous liquid film flowing on a rotating circular disk which rotates with constant velocity Ω* (see Figure 1). The variable with a superscript ^{“*”} stands
for a dimensional quantity. Here the cylindrical polar coordinate axes r*,θ*,andz* are chosen as the radial
direction, the circumferential direction, and the axial
direction, respectively. All associated physical
properties and the rate of film flow are assumed to be constant (i.e.,
time-invariant). Let u* and w* be the velocity components in the radial
direction r* and the axial direction z*.
The governing equations of motion are
1r*∂(r*u*)∂r*+∂w*∂z*=0,∂u*∂t*+u*∂u*∂r*+w*∂u*∂z*−v*2r*=−1ρ∂p*∂r*+μρ(1r*∂∂r*(r*∂u*∂r*)−u*r*2+∂2u*∂z*2),∂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 tangential
velocity, ρ is the constant
fluid density, p* is the fluid pressure, g is the acceleration due to gravity, and μ is the fluid dynamic viscosity. On the disk surface z*=0,
the no-slip boundary conditions for the velocity fields areu*=0,w*=0. On the free
surface z*=h*,
the boundary condition approximated by the vanishing of shear stress is
expressed as
(∂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
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*,
where h* is the local film thickness, S* is surface tension, and pa* is the atmosphere pressure. The kinematic condition that the flow
does not travel across a free surface can be given as
∂h*∂t*+∂h*∂r*u*−w*=0. By introducing
a stream function φ*,
the dimensional velocity components can be expressed as
u*=1r*∂φ*∂z*,w*=−1r*∂φ*∂r*. Now the
following variables are used to form the dimensionless governing equations and
boundary conditions:
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*,α=2πh0*λ, where h0*, α, u0*, Re, ν, Fr, and λ are the average film thickness, perturbed
wavelength, scale of velocity, Reynolds number, the kinematic viscosity, Froude
number, and dimensionless wavenumber, respectively. For simplification, it is assumed
that Newtonian liquid film is very thin (h*≪r*).
In consequence, it is reasonable to assume that the tangential velocity is
constant throughout the radial direction in the thin film, that is, v*=r*Ω*.
In order to investigate the effect of angular
velocity, Ω*, on the stability
of the flow field, the dimensionless rotation number is introduced as follows:Ro=Ω*h0*u0*. In terms of
these nondimensional variables, the equations of motion become
r−1φzzz=−RerRo2+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 disk z=0 are reduced to
φ=φr=φz=0. And the
boundary conditions at the free surface of disk z=h become
r−1φzz=α2[r−1φrr−r−2φr+2hr(1−α2hr2)−1(2r−1φrz−r−2φz)],p=−Sα2hrr(1+α2hr2)−3/2+α[−2Re−1(1+α2hr2)−1(r−1φrz+r−1φzzhr)]+O(α2),ht+r−1hrφz+r−1φr=0. Hence the term α2S can be treated as a quantity of zeroth order [21]. Since
the modes of long-wavelength that give the smallest wave number are most
likely to induce flow instability for the film flow, this can be done by
expanding the stream function and flow pressure in terms of some small
wave number (α≪1) as
φ=φ0+αφ1+O(α2),p=p0+αp1+O(α2). Following procedure described in [6], the
complicated nonlinear system of (2.9) and boundary
conditions (2.10)–(2.13) is reduced to a single nonlinear evolution
equation for the film thickness h(r,t). After inserting the above two equations into the system
of equations and boundary conditions, the zeroth order (α0) terms in
the governing equations can be expressed as
r−1φ0zzz=−Re·r·Ro2,p0z=−Fr. The boundary
conditions associated with the equations of zeroth order are given as
z=0,φ0=φ0z=0,z=h,r−1φ0zz=0,p0=−α2Shrr. The solutions of
the zeroth-order equations are
φ0=16r2·Re·Ro2(3h−z)z2,p0=Fr(h−z)−Sα2hrr. After
considering all terms of first order (α1) from the
system of equations and boundary conditions, the first-order
equations can be achieved as
r−1φ1zzz=Re(p0r+r−1φ0tz+r−2φ0zφ0rz−r−3φ0z2−r−2φ0rφ0zz),p1z=−Re−1r−1φ0rzz, and boundary
conditions
can be achieved asz=0,φ1=φ1z=0,z=h,r−1φ1zz=0,p1=−2Re−1(r−1φ0rz+r−1φ0zzhr).
Schematic diagram of thin Newtonian liquid flow upon a rotating circular disk.
The
solutions of the first-order equations are given as
φ1=−16r·Re·z2(z−3h)(rRo2−Fr·hr+α2Shrrr),p1=Ro2(3z2−6z·h+h2−r(z+h))hr. By substituting the solutions of the zeroth-order and
first-order equations into the dimensionless-free surface kinematic equation of (2.13), the nonlinear evolution equation is derived and expressed as
ht+A(h)hr+B(h)hrr+C(h)hrrr+D(h)hrrrr+E(h)hr2+F(h)hrhrrr=0, where
A(h)=h2·Re(−60Fr·h·α+r2·Ro2(180+136h4·Re2·Ro2·α+73h5·Re2·Ro2·α))180r,B(h)=115(−5Fr·h3·Re·α+2h6·r2·Re3·Ro4·α),C(h)=13·Re·S·αr3·h3,D(h)=13·Re·S·α3·h3,E(h)=−Re·Fr·α·h2+45h5·r2·Re3·Ro4·α,F(h)=Re·S·α3·h2.
3. Stability Analysis
The dimensionless film thickness when expressed in perturbed state can be given as 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 (2.21) and all terms
up to the order of η3 are collected, the evolution equation of η becomes
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 ofA,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
As the nonlinear terms of (3.2) are neglected, the
linearized equation is given as
ηt+Aηr+Bηrr+Cηrrr+Dηrrrr=0. In order to use the normal mode analysis, we assume that
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
d=dr+idi=(A−C)+i(B−D), where dr and di are regarded as 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 to di<0 or di>0.
This is equivalent to the inequality of B<D or B>D.
3.2. Weakly Nonlinear Stability Analysis
Nonlinear effects,
when they are weak enough, do not fundamentally alter the nature of the motion.
A weakly nonlinear solution can still be usefully expressed as a superposition
of plane waves, but the amplitudes of these waves do not remain constant; they
are modulated by nonlinear interactions. In
order to characterize the weakly nonlinear behaviors of thin film flows, the method
of multiple scales [22] is employed here, and the resulting Ginburg-Landau equation [23] can
be derived as
∂a∂t2+D1∂2a∂r12−ε−2dia+(E1+iF1)a2a¯=0, where
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′′),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. Equation (3.6) can be used to
investigate the weak nonlinear behavior of the fluid film flow. In order to
solve for (3.6), solution is taken for a
filtered wave in which spatial modulation does not exist. So for a filtered
wave a can be given as a=a0exp[−ib(t2)t2]. After substituting
(3.8) into (3.6), one can obtain
∂a0∂t2=(ε−2di−E1a02)a0,∂[b(t2)t2]∂t2=F1a02. The threshold
amplitude εa0 in the supercritical stable region is given as
εa0=diE1, and the
nonlinear wave speed Ncr is given as
Ncr=dr+di(F1E1). The
detail derivation of the earlier mentioned equations can refer to Cheng and Lai
[24]. If E1=0,
(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 condition for
the film flow to present the behavior of subcritical instability in the
linearly stable region (di<0) is given as E1<0, and the
threshold amplitude of the wave is given as εa0. The
subcritical stable region can only be found as E1>0. The neutral
stability curve can only be derived and plotted for the condition of E1=0. On the
basis of earlier mentioned discussion, it is obvious that the Ginzburg-Landau equation
can be used to characterize various flow states. The Landau equation can be summarized
and presented in Table 1.
Various states of Landau equation.
Linearly
stable (subcritical region)
di<0
Subcritical instability
E1<0
εa0<(diE1)1/2
a0→0
Conditional stability
εa0>(diE1)1/2
a0↑
Subcritical
explosive state
Subcritical (absolute)
stability E1>0
a0→0
Linearly unstable
(supercritical region)
di>0
Supercritical
explosive
state E1<0
a0↑
Supercritical stability
E1>0
εa0→(diE1)1/2
Ncr→dr+di(F1E1)
4. Numerical Examples
Based on modeling results, the condition for the
stability behaviors of a thin film flow can be expressed as a function of
Reynolds number, Re, rotation number, Ro, dimensionless perturbation wave number, α, and
dimensionless radius of disk, r, respectively.
In order to study the effects of dimensionless
radius and rotation number on the stability of a thin flow, we select randomly
but within specified ranges physical parameters for numerical experiment. Physical
parameters that are selected for study include (1) Reynolds numbers ranging
from 0 to 15, (2) the dimensionless perturbation wave numbers ranging from 0 to
0.12, (3) rotation number including 0.1, 0.12, and 0.15, and (4) the values of
dimensionless radius including 50, 75, and 100. The ranges for these above
parameters are based on published reasonable ranges for these parameters [25].
Other of our 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. In
practice, the parameter S is a large value. Further, for
simplification analysis, Re and Fr are taken to be of the same order (O(1)) [21, 25, 26], so the values of some dimensionless parameters are taken as, a constant dimensionless
surface tension S=6173.5 and Fr=9.8.
4.1. Linear Stability Analysis
The neutral
stability curve is obtained by substituting di=0 from
(3.5). The α-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 the rotation number. Figure 2(b) shows
that the stable region decreases and unstable region increases with an increase
of the radius of circular disk. Hence one can say
that in linear stability analysis rotation number and the radius of circular disk give the same destabilizing effects.
(a) Linear
neutral stability curves for three different Ro values at r=50. (b) Linear
neutral stability curves for three different r values at Ro=0.1.
4.2. Weakly Nonlinear Stability Analysis
The main purpose of the nonlinear stability analysis is to study the
weakly nonlinear analysis of the evolution equation. Figures 3(a) to 3(c) indicates
that the area of shaded subcritical instability region and subcritical stability
region decreases, and the area of shaded supercritical instability region increases
with an increase of rotation number or the radius of circular disk. Also, from these
figures, it is interesting to find, in high Reynolds regime, that the area of
supercritical stability region decreases with an increase of rotation number or an increase of circular disk radius. It is found that the rotation number and the radius of circular disk not only play the significant roles in
destabilizing the flow in the linear stability analysis but also decrease the
ranges of supercritical stability region at large Reynolds number in the weakly nonlinear stability analysis. The reason for this phenomenon
is the existence of the centrifugal force term, which is a radius-related force
in the governing equation. As the size of radius and Reynolds number gradually increased, the centrifugal force is
enhanced significantly. Due to inertia forces, that may accelerate the growth of
the linear disturbance, the trend of instability for the flow with larger radius
and larger Reynolds number is higher than
those with smaller ones.
(a) Neutral
stability curves of Newtonian film flows for Ro=0.1 and r=50. (b) Neutral
stability curves of Newtonian film flows for Ro=0.15 and r=50. (c) Neutral
stability curves of Newtonian film flows for Ro=0.1 and r=100.
Figure 4(a) shows the
threshold amplitude in subcritical instability region for various wave numbers with
different Ro values at Re=6 and r=50. The results indicate
that the threshold amplitude εa0 becomes smaller as the value of rotation number increases. Figure 4(b)
shows the threshold amplitude in subcritical instability region for various
wave numbers with different r values at Re=6 and Ro=0.1. The results indicate that the threshold amplitude εa0 becomes smaller as the value of the radius increases. In such
situations, the film flow which holds the higher threshold amplitude value will
become more stable than that which holds smaller one. If the initial finite amplitude
disturbance is less than the threshold amplitude, the system will become
conditionally stable.
(a) Threshold amplitude in subcritical instability region for
three different Ro values at Re=6 and r=50. (b) Threshold amplitude in subcritical instability region for
three different r values at Re=6 and Ro=0.1.
Figure 5(a) shows the
threshold amplitude in the supercritical stability region for various wave
numbers with different Ro values at Re=6 and r=50. Figure 5(b) shows the threshold amplitude in the
supercritical stability region for various wave numbers with different r values
at Re=6 and Ro=0.1. It is found that the
decrease of rotation number or the radius of circular disk will lower the threshold
amplitude, and the flow will become relatively more stable.
(a) Threshold amplitude in supercritical stability region for
two different Ro values at Re=6 and r=50. (b) Threshold
amplitude in supercritical stability region for two different r values at Re=6 and Ro=0.1.
The wave speed of (3.5) predicted by using the linear theory is a
constant value for all wave number and rotation number. However, the wave speed of (3.12) predicted by using nonlinear theory is no longer a constant. It
is actually a function of wave number, Reynolds number, rotation number, and the
radius of disk. Figure 6(a) shows the nonlinear wave speed in the supercritical
region for various perturbed wave numbers and different Ro values at Re=6 and r=50. Figure 6(b)
shows the nonlinear wave speed in the supercritical stable region for various perturbed
wave numbers and different r values at Re=6 and Ro=0.1. It is found that the nonlinear wave speed increases
as the value of rotation number or the radius of circular disk increases.
(a) Nonlinear wave speed in supercritical stability region for two different Ro values
at Re=6 and r=50. (b) Nonlinear wave speed in supercritical stability region for two different r values
at Re=6 and Ro=0.1.
5. Concluding Remarks
The stability of thin Newtonian
fluid flowing on a rotating circular disk is thoroughly investigated by using the method of long-wave
perturbation. Based on the results of numerical modeling, several conclusions
are given as follows.
In the linear stability
analysis, the rotation number and the radius of circular disk give the same destabilizing effects in the thin film flow. Because of the
different order effects, an increase of rotation number is more rapidly
unstable than an increase the radius of circular disk.
In the weakly linear
stability analysis, it is noted that the area of shaded subcritical instability
region and subcritical stability region decreases, and the area of shaded supercritical
instability region increases with an increase of rotation number or the radius of circular disk. It is also noted that the threshold amplitude in
the subcritical instability region decreases as the value of rotation number or the radius of circular disk increases. If the initial finite amplitude
disturbance is less than the threshold amplitude, the system will become
conditionally stable.
Due to inertia forces, that
may accelerate the growth of the linear disturbance, the trend of instability
for the flow with larger radius and larger Reynolds number is higher than
that with smaller ones. In weakly nonlinear stability analysis, we also find that
high Reynolds number will shrink the area of supercritical stability
region. In other words, high Reynolds number plays a
detrimental role to the rotating coating process described in this paper.
YangY.-K.ChangT.-C.Experimental analysis and optimization of a photo resist coating process for photolithography in wafer fabricationKuoY.-K.ChaoC.-G.Control ability of spin coating planarization of resist filmand optimal control of developersLinC. C.ChandrasekharS.KapitzaP. L.Wave flow of thin viscous liquid filmsBenneyD. J.Long waves on liquid filmLandauL. D.On the problem of turbulenceYihC.-S.Stability of liquid flow down an inclined planeStuartJ. T.On the role of Reynolds stresses in stability theoryPumirA.MannevilleP.PomeauY.On solitary waves running down an inclined planeHwangC.-C.WengC.-I.Finite-amplitude stability analysis of liquid films down a vertical wall with and without interfacial phase changeAthertonR. W.HomsyG. M.On the derivation of evolution equations for interfacial wavesRuyer-QuilC.MannevilleP.Modeling film flows down inclined planesRuyer-QuilC.MannevilleP.Improved modeling of flows down inclined planesRuyer-QuilC.MannevilleP.Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximationsAmaoucheM.m_amaouche@yahoo.frMehidiN.nadbouam@yahoo.frAmatousseN.amatousse@yahoo.frAn accurate modeling of thin film flows down an incline for inertia dominated regimesSamantaA.Stability of liquid film falling down a vertical non-uniformly heated wallEmslieA. G.BonnerF. T.PeckL. G.Flow of a viscous liquid on a rotating diskHigginsB. G.Film flow on a rotating diskKitamuraA.HasegawaE.YoshizawaM.Asymptotic analysis of the formation of thin liquid film in spin coatingLinJ. S.WengC. I.Linear stability analysis of condensate film flow down a vertical cylinderKrishnaM. V. G.LinS. P.Nonlinear stability of a viscous film with respect to three-dimensional side-band disturbancesGinzburgV. L.LandauL. D.Theory of superconductivityChengP.-J.LaiH.-Y.Nonlinear stability analysis of thin film flow from a liquid jet impinging on a circular concentric diskChenC.-I.eddychen@isu.edu.twNon-linear stability characterization of the thin micropolar liquid film flowing down the inner surface of a rotating vertical cylinderHungC.-I.ChenC.-K.TsaiJ.-S.Weakly nonlinear stability analysis of condensate film flow down a vertical cylinder