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Two-layer film flow on a rotating disk: A numerical study
International Journal of Non-Linear Mechanics 46 (2011) 272–277
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Two-layer film flow on a rotating disk: A numerical study B.S. Dandapat a,, S.K. Singh b a b
Department of Mathematics, Sikkim Manipal Institute of Technology, Majitar, East Sikkim, 737 132 Sikkim, India Physics and Applied Mathematics Unit, Indian Statistical Institute, 203-B. T. Road, Calcutta 700 108, India
a r t i c l e in f o
a b s t r a c t
Article history: Received 24 December 2008 Received in revised form 17 July 2010 Accepted 17 September 2010
Development of thin two-layer film over a uniformly rotating disk is studied numerically under the assumption of planar interface and free surface. Similarity transformation is applied to transform the Navier–Stokes equations into a set of coupled non-linear, unsteady partial differential equations. This set of equations are solved numerically by using the finite-difference technique. It is observed that the rate of film thickness varies at different time zone depending on the rate of rotational speed of the disk. A physical explanation is provided to justify this anomalous behaviour. It is observed that, smaller thickness on the top layer enhance the initial rate of film thinning. But the overall effect of density, viscosity and the initial film thickness ratio are found to be insensitive to the final film thickness at large time. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Double layered thin film flow Rotating disk Viscous flow Spin coating
1. Introduction Development of thin film on horizontal rotating disk is known as spin coating in the literature. This technique is widely used in microelectronics industries to coat the photoresist on silicon wafers for integrated circuits, for magnetic storage disks, for magnetic paint coating on the substrates, etc. Emslie et al. [1], first initiated the study on the development of thin film on a rotating disk. Assuming a balance between the centrifugal and viscous force during the process of rotation they were able to simplify the system of Navier–Stokes equations and finally, able to show that the uniformity of the film is maintained as it thins continuously more and more. Later on Meyerhofer [2], Lai [3], Chen [4], Ma and Hwang [5], and others have studied the different effects like solvent evaporation, surface tension, non-Newtonian effects, disk roughness etc. on film thinning. All these analyses were based on the typical hydrodynamical approximation as employed by Emslie et al. [1]. Full Navier–Stokes equations were considered first time by Higgins [6] to study the flow and film development through matched asymptotic expansion procedure and able to show that the uniformity of the film thickness remains as spinning process continues. Following Higgins’s analytical treatment Dandapat and Ray [7,8], Ray and Dandapat [9] have studied the effects of film cooling, thermocapillarity and magnetic field on film development. Dandapat [10] has observed that at the initial stage of rotation, the film thinning rate is more for slow, than the fast rotational speed, but at large time, film thinning rate increases with the faster rotational speed if
Corresponding author. Tel.: +91 9433391227.
E-mail address: [email protected] (B.S. Dandapat). 0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2010.09.007
the spinner starts from rest with impulsive rotation and maintains the speed for the rest of the period. However, faster rate of thinning can be obtained if the spinner starts impulsively and then increases its spinning rate continuously. It is to be noted here that in all the above studies the tacit assumption was that the disk is wet so that the classical no-slip boundary condition can be applied at every point on the disk surface and the film flows under a planar interface for entire period of spinning. It is to be remembered here that in general, an interface always separates two different physical media. So to understand the spincoating process in details we need to study the film thinning mechanism by considering a two-layer fluid model. This necessity we felt from our past experiences in connection with the other flow problems. There we have seen new phenomena appears when interface between two fluids is considered. This is because in the frame of one fluid model the movement of gas/liquid vapour is treated as passive, whereas in reality the motion of gas/liquid vapour can exert shear stress on the surface of the fluid. This neglected shear stress on the free surface may brings changes in the system. Multilayered approach has been applied for the investigation of several modern engineering processes. For example, the liquid encapsulation technique used in crystal growth [11]. Multilayered approach also used in multilayer coating for the production of photographic films and multilayer fibers for optoelectronics devices [12] and emulsified liquid membrane separation techniques widely used now in various extraction processes and industrial waste-water treatment [13]. Kitamura [14] has advocated the usefulness of multilayered coating in potentially hazardous environment. To the best of our knowledge, so far, no theoretical study has been reported in the literature of thin film development on a rotating disk considering two or multilayered initial deposition of the liquid.
B.S. Dandapat, S.K. Singh / International Journal of Non-Linear Mechanics 46 (2011) 272–277
In this article, we have studied the flow and film development over a rotating disk, when two immiscible Newtonian liquids having different fluid properties with different film thickness are deposited initially.
We consider a stable configuration of two horizontal layers of immiscible liquids with different physical properties having different uniform thickness over a disk of large diameter. The disk is made to rotate in its own plane with constant angular velocity O. Following Higgins [6] and Dandapat [10], we restrict our attention to the flows that maintains a planar film thickness throughout the entire process. In fact, earlier works of Emslie et al. [1], Kitamura [15] and Dandapat et al. [16] have shown that the initial non-uniformity of the film thickness dies out in course of time and a uniform planar final film thickness is the neat outcome of the process. Further, due to axisymmetric flow assumption, the dependent variables are independent of azimuthal co-ordinate y. It is to be pointed out here that Parter and Rajagopal [17] and Rajagopal [18] have shown that the axial symmetric solutions are not always possible when two disks at a finite distance are rotating about a common axis or about distinct axes at different speeds. The initial film thickness of the lower layer is assumed to be h10 and that of the upper layer is h20 h10. In other words, total film thickness is h20. A schematic diagram given in Fig. 1 describes the details of the system. All the variables corresponding to the bottom layer are marked by subscript 1 and that corresponding to the top layer are marked by 2. Vi, pi, ri , ni and mi denote, respectively, the velocity, pressure, density, kinematic and dynamic viscosity, of the ith (i¼1,2) fluid layer. Under this convention, we represent the velocity field as Vi ¼ ui ðr,z,tÞe^ r þvi ðr,z,tÞe^ y þwi ðr,z,tÞe^ z ,
The boundary conditions at the surface of the disk are u1 ðr,0,tÞ ¼ 0,
v1 ðr,0,tÞ ¼ Or,
w1 ðr,0,tÞ ¼ 0:
ð1Þ
where e^ r , e^ y , e^ z are the unit vectors along the co-ordinates axes. The governing equations of continuity and motion for both incompressible Newtonian fluid are ð2Þ
2 @ui @u v2 @u 1 @pi @ ui @ ui @2 ui þui i i þwi i ¼ þ ni þ þ , @r r @t @r r @z ri @r @r 2 @z2
ð3Þ
2 @vi @v uv @v @ vi @ vi @2 vi þ ui i þ i i þ wi i ¼ ni þ þ , @r r @t @r r @z @r 2 @z2
ð4Þ
2 @wi @wi @wi 1 @pi @ wi 1 @wi @2 wi þ ui þ wi ¼ þ ni þ þ : r @r @t @r @z ri @z @r 2 @z2
ð5Þ
ð7Þ
the balance of shear stress @w1 @u1 @w2 @u2 þ ¼ m2 þ , m1 @r @z @r @z
m1
@v1 @v2 ¼ m2 , @z @z
ð8Þ
ð9Þ
the balance of normal stress @w1 @w2 p1 p2 ¼ 2 m1 m2 , @z @z
ð10Þ
and the kinematic condition dh1 ¼ w1 ðr,h1 ,tÞ: dt
ð11Þ
The boundary conditions at the free surface z ¼h2(t) the normal and the shear stress must vanish and these are given by p2 þ 2m2
@w2 ¼ 0, @z
ð12Þ
@w2 @u2 þ ¼ 0, @r @z
ð13Þ
@v2 ¼ 0: @z
ð14Þ
In addition, the kinematic condition at the free surface z¼h2(t) reads as dh2 ¼ w2 ðr,h2 ,tÞ: dt
ð15Þ
The initial conditions for the velocity field and the film thickness satisfy Vi ¼ 0,
@ui ui @wi þ þ ¼ 0, @r r @z
ð6Þ
The boundary conditions at the liquid–liquid interface z ¼h1(t), are the following: the continuity of the velocity field V1 ¼ V2 ,
2. Mathematical formulation
273
hi ¼ hi0 ,
dhi ¼ 0: dt
ð16Þ
Following von Ka´rma´n [19], the solution of the above system may be assumed in the following well-known similarity form; ui ¼ rf i ðz,tÞ,
vi ¼ rg i ðz,tÞ, wi ¼ wi ðz,tÞ, 2 r pi ðr,z,tÞ ¼ ri Ai ðz,tÞ þ Bi ðz,tÞ: 2
ð17Þ
Substituting (17) into the system of Eqs. (2)–(5) and collecting the different orders of r, we get 2fi þ wi,z ¼ 0,
ð18Þ
fi,t þ fi2 gi2 þwi fi,z ni fi,zz ¼ ri Ai ðz,tÞ,
ð19Þ
gi,t þ 2fi gi þ wi gi,z ¼ ni gi,zz ,
ð20Þ
ri ðwi,t þwi wi,z ni wi,zz Þ ¼ Bi,z ,
ð21Þ
Ai,z ¼ 0:
ð22Þ
The suffixes except i indicate derivatives with respect to that variable. The viscous term in the normal stress boundary conditions (10) and (12) are independent of r, implying Ai ¼0. We therefore conclude that Ai ¼0 are valid for the entire depth of the liquid. Now, Bi(z,t) can be found after integrating (21) with respect to z, from z to z¼h(t), by using the conditions on z¼h2(t) Fig. 1. Systematic diagram.
B2 ¼ 2m2
@w2 , @z
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3. Method of solution
and on z¼h1(t) @w1 @w2 m2 , B1 B2 ¼ 2 m1 @z @z and thus we can evaluate the pressure from Eq. (17). Using (17) in the boundary conditions (6)–(15), we get On the surface of the disk: at z¼ 0 f1 ðt,0Þ ¼ 0,
g1 ðt,0Þ ¼ O,
w1 ðt,0Þ ¼ 0,
ð23Þ
On the interface between the two liquids: at z¼ h1(t) 9 w1 ðt,h1 Þ ¼ w2 ðt,h1 Þ, > = m1 f1,z ðt,h1 Þ ¼ m2 f2,z ðt,h1 Þ, m1 g1,z ðt,h1 Þ ¼ m2 g2,z ðt,h1 Þ, > ; m1 w1,z ðt,h1 Þ ¼ m2 w2,z ðt,h1 Þ,
f1 ðt,h1 Þ ¼ f2 ðt,h1 Þ,
g1 ðt,h1 Þ ¼ g2 ðt,h1 Þ,
ð24Þ On the free surface: z ¼h2(t) f2,z ðt,h2 Þ ¼ 0,
g2,z ðt,h2 Þ ¼ 0,
ð25Þ
The kinematic conditions for the evolution of the surfaces: hi,t ¼ wi ðt,hi Þ:
ð26Þ
The initial conditions at t ¼0 fi ð0,zÞ ¼ 0, hi ¼ hi0 ,
gi ð0,zÞ ¼ 0,
wi ð0,zÞ ¼ 0
) ð27Þ
hi,t ¼ 0:
9
r1 m h20 h10 > > , n¼ 1 , d¼ > r2 m2 h10 > > > > > > r1 h210 O > h1 h2 h2 > > H1 ¼ , H2 ¼ ¼ , Re1 ¼ > > h10 h20 h10 dh10 m1 > > > = 2 r2 ðh20 h10 Þ2 O nd Re1 , Re2 ¼ ¼ > > m2 m > > W1 ¼
Fi ¼
z , h10
w1 , h10 O
fi
O
,
m¼
W2 ¼
Gi ¼
gi
O
w2 w2 ¼ ðh20 h10 ÞO dh10 O
,
ð28Þ
> > > > > > > > > > > > > ;
into the system of Eqs. (18)–(20) and the corresponding boundary and initial conditions (23)–(27) we obtain the dimensionless governing equations as well as their boundary and initial conditions as 2Fi þ li Wi, Z ¼ 0,
ð29Þ
Re1 ðFi, t þFi2 þ li Wi Fi, Z G2i Þ ¼ ai Fi, ZZ ,
ð30Þ
Re1 ðGi, t þ 2Gi Fi þ li Wi Gi, Z Þ ¼ ai Gi, ZZ ,
ð31Þ
where li ¼ ð1, dÞ and ai ¼ ð1,m=nÞ. The boundary conditions are F1 ðt,0Þ ¼ 0,
G1 ðt,0Þ ¼ 1,
W1 ðt,0Þ ¼ 0,
ð32Þ
9 F1 ðt,H1 Þ ¼ F2 ðt,H1 Þ, G1 ðt,H1 Þ ¼ G2 ðt,H1 Þ, W1 ðt,H1 Þ ¼ dW2 ðt,H1 Þ, = 1 1 ; F1, Z ðt,H1 Þ ¼ F2, Z ðt,H1 Þ, G1, Z ðt,H1 Þ ¼ G2, Z ðt,H1 Þ, n n
ð33Þ F2, Z ðt, dH2 Þ ¼ 0,
G2, Z ðt, dH2 Þ ¼ 0:
ð34Þ
The initial conditions become Fi ð0, ZÞ ¼ 0,
Gi ð0, ZÞ ¼ 0,
Wi ð0, ZÞ ¼ 0,
Hi ð0Þ ¼ Hi0 ,
ð35Þ
and kinematic conditions read as H1t ¼ W1 ðt,H1 Þ,
H2t ¼ W2 ðt, dH2 Þ:
where ( uðZ, tÞ ¼
Using the following dimensionless variables
t ¼ t O, Z ¼
The above coupled non-linear system of Eqs. (29)–(36) with the corresponding boundary conditions can be solved by using the finite-difference technique. It is to be noted here that the conventional finite-difference method cannot be used in this problem as thickness of both the layers are continuously decreasing with time. Considering this fact, the time-dependent physical domain is transformed to a fixed computational domain ½0,1 þ d such that the film thickness will always remain in a fixed computational domain for all times. Further, care has been taken through a fine grid distribution for the large velocity gradients that may be present near the disk surface. It should be pointed out here that the same transformation will be useful for fine as well as uniform grid distribution. We choose the transformation as ! A2 H1ZðtÞ x ¼ 1A1 log þ ½1uðZ, tÞ Z B2 þ H1 ðtÞ 8 0 1 !) 1 ðtÞ < A2 dHZ2 H A2 H1ZðtÞ ðtÞH1 ðtÞ A @ , ð37Þ þA log dA1 dlog 1 Z 1 ðtÞ : B2 þ H1 ðtÞ B2 þ dHZ2 H ðtÞH1 ðtÞ
ð36Þ
1
for 0 r Z o H1 ðtÞ,
0
for H1 ðtÞ r Z r dH2 ðtÞ
ð38Þ
and A1 ¼ ½logðA2 =B2 Þ1 , with A2 ¼(c+ 1), B2 ¼(c 1). Here, c is the grid controlling parameter in the physical domain. Small values of c cluster grid points at the disk surface where as large values make grid spacing uniform throughout the two-layer film. The Crank–Nicholson scheme is used to solve the transformed nonlinear system of Eqs. (29)–(36) after approximating the non-linear terms according to the Newton’s linearization technique [20]. Computation is carried out in each time level on the following linear tridiagonal system of algebraic equations: þ1 PðGi Þkj1 þ Q ðGi Þkj þ 1 þ RðGi Þkj þþ11 ¼ ðS1 ÞkðjÞ
ð39Þ
þ1 þ Q ðFi Þkj þ 1 þ RðFi Þkj þþ11 ¼ ðS2 ÞkðjÞ : PðFi Þkj1
ð40Þ
Here i, j and k denote the position of the layer, spatial level of discretization and time level, respectively. For i¼1 or 2, j takes values from 1 to N 1+1 or N 1+1 to N 1+ N 2 + 1, respectively. Here, N 1 and N 2 denote the number of partitions in the lower and upper layer of the fluid. The expressions for P, Q, R, (S1)kj and (S2)kj are lengthy but their derivations are straightforward. (We are dropping these expressions to reduce the paper length and willing to supply under request.) At each and every time level (Gi)kj + 1 and (Fi)kj + 1 are computed (39) and (40) and then the axial velocity (Wi)kj + 1 is obtained from finite-difference representation of continuity equation by using the values of (Fi)kj + 1 at that time level. The iteration process continues until it attains the desired level of thickness. Numerical computation is carried out on 101 grids in the vertical direction with c¼104 this gives uniform grid distribution in physical and as well as in the computational domain. However, for higher Reynolds number one needs to increase the number of grid points in the vertical direction. Under this situation, the value of c must decrease depending on the rotational speed of the disk. The time step has been calculated by using
Dt r0:25 Dx2 :
ð41Þ
This relation comes from the CFL condition of numerical stability [20]. The domain of Dt has been chosen smaller than the stability domain for linear equation due to the coupled non-linear system.
B.S. Dandapat, S.K. Singh / International Journal of Non-Linear Mechanics 46 (2011) 272–277
4. Results and discussion Fig. 2 represents the effect of d, the ratio of the initial film thickness deposition between the two layers, on the gradual development of the final film thickness with respect to time t for fixed values of m, n and the Reynolds number Re1. It is clear from the figure that the change of d effects the film thickness at the initial stage but it is more or less insensitive to the final film thickness at large time. Fig. 3 shows the gradual variation of the film height in both the layers with time t when the fluid viscosity ratio n are changed. It is clear from the figure that variation of viscosity between the two layers has least effect on film thinning. Fig. 4 depicts the gradual development of film thickness when the fluids density ratio varies. It is evident from the figure that the
275
variation of density ratio effects just after the spin off stage but the final film thickness is independent on it. Fig. 5 shows the effects of the variation of the Reynolds number on film thinning for a particular set of values of d, n, and m. It is further clear that the thinning rate for both the layers increase with decrease in the Reynolds number during the time interval 0 o t o tc , where tc is a point on the time scale. On the other hand film thinning rate in both the layers increase with increase of the Reynolds number in the interval tc o t. This anomalous behaviour of the variation of the film height in different time interval can be explained by considering the two time regions t o tc and t 4 tc . The point tc in the time interval indicates that time, when total amount of fluid flowing out of a certain radial distance for two different Reynolds
2
2.5
1.8
H2 δ
1.4 Film height
2
Film height
1.6
H 2δ
1.5
1
1.2 1 0.8 0.6
H1
0.4
H1
0.5
0.2 0
0
0
1
2
3
4 τ
5
6
7
2
1.8
1.8
1.6
6 τ
1.6
H2δ
1.2 1 0.8
0.4
H1
2
12
1 H1
0.6 0.4
0.2 0
10
1.2
0.8
0.6
8
H2 δ
1.4 Film height
Film height
4
Fig. 4. Variation of top film’s height ðH2 dÞ and the bottom film (interface height H1) with t for two different values of m. Here, dotted and solid lines represent total film thickness for m¼ 1.4 and 2.0, respectively. Dashed and dotted–dashed lines represent lower layer thickness for m ¼1.4 and 2.0, respectively. While n¼ 2.5, Re1 ¼ 10.0 and d ¼ 1:0 are considered as fixed.
2
0
2
8
Fig. 2. Variation of top film’s height ðH2 dÞ and the bottom film (interface height H1) with t for three different values of d. Solid, dashed and dashed–dotted lines represent for d ¼ 1:2, 1.0 and 0.9, respectively. While m¼ 1.6, n¼ 1.4 and Re1 ¼6.0 are considered as fixed.
1.4
0
4
6
8
10
12
τ Fig. 3. Variation of top film’s height ðH2 dÞ and the bottom film (interface height H1) with t for two different values of n. For total layer, dotted and solid lines represent film thickness for n¼ 3.5 and 6.0, respectively. Dashed and dashed– dotted lines representing the lower layer thickness for n ¼3.5 and 6.0, respectively. While m¼2.0, Re1 ¼ 10.0 and d ¼ 1:0 are considered as fixed.
0.2
0
1
2
3
4
5
τ Fig. 5. Variation of top film’s height ðH2 dÞ and the bottom film (interface height H1) with t for two different Re1. Here, dotted and solid lines represent total film thickness for Re1 ¼ 0.5 and 0.9, respectively. Dashed and dotted–dashed lines represent lower layer thickness for Re1 ¼ 0.5 and 0.9, respectively. While m ¼2.0, n¼2.5 and d ¼ 1 are considered as fixed.
B.S. Dandapat, S.K. Singh / International Journal of Non-Linear Mechanics 46 (2011) 272–277
0.25 0.2 0.15 F2
numbers become equal. In other words we can say that the tc depends on the competitive Reynolds numbers. In interval t o tc , thinning rate for both layers increase with the decrease in the rotational speed. This seems to be inconsistent with common observation that film thins faster with faster rotation of the disk. A close observation of this physical process exposes that the azimuthal component of velocity for both layers develop along the entire depth of the liquid faster, with slower rotation of the disk. This occurred due to the fact that the Reynolds number is the ratio between the centrifugal and viscous forces. Increase in the Reynolds number reflects that centrifugal force dominates over viscous force. Finally, viscous action builds up azimuthal component of velocity field faster throughout the depth of the fluid for smaller Reynolds number. This result can be found in Fig. 6. It is expected that the faster development of azimuthal velocity field will introduce the corresponding radial velocity faster for smaller Reynolds numbers only. Fig. 7 depicts the radial component of velocity for both the layers that developed over the entire depth for smaller values of Reynolds number in the interval 0 o t o tc as result film thins quicker for the smaller value of the Reynolds number in this interval. In time interval t 4 tc , azimuthal velocities for both the layers have developed throughout the film depth for both large and small Reynolds numbers and the corresponding developed centrifugal force in both the layers drive away large amount of fluid when the value of the Reynolds number is large. In other words, fluid moves out of the disk quickly when rotation speed of the disk is high. This explains why films thin faster in the time interval tc o t for the large Reynolds number. Fig. 8 depicts the variation of the azimuthal components of velocity for both the layers when d, the ration of the initial film thickness deposition between the layers varies for fixed Reynolds number Re1, m and n. It is clear from the figure that thinner film deposition on the top layer in compare to the bottom layer, produces faster azimuthal in both the layers and enhance film
0.1 0.05 0
1
2
3
4
5
3
4
5
0.25 0.2 0.15 0.1 0.05 0
0
1
2
τ
Fig. 7. Variation of F2 at free surface ðH2 dÞ and F1 at interface (H1) with time t for two different Re1. Dashed and solid line represent Re1 ¼ 0.5 and Re1 ¼ 0.9, respectively. While m ¼2.0, n¼ 2.5 and d ¼ 1:0 are considered as fixed.
0.8
0.8
0.7
0.7
0.6
0.6
0.5 G2
0.5 G2
0
τ
F1
276
0.4
0.4 0.3
0.3
0.2
0.2
0.1
0.1
0
0 0
1
2
3
4
5
τ
0
1
2
3
4 τ
5
6
7
8
0
1
2
3
4 τ
5
6
7
8
1
1
0.8
0.8
0.6
G1
G1
0.6 0.4
0.4 0.2
0.2 0
0
0
1
2
3
4
5
τ Fig. 6. Variation of G2 at free surface ðH2 dÞ and G1 at interface (H1) with time t for two different Re1. Dashed and solid line represent Re1 ¼0.5 and Re1 ¼ 0.9, respectively. While m¼ 2.0, n¼2.5 and d ¼ 1:0 are considered as fixed.
Fig. 8. Variation of G2 at the free surface ðH2 dÞ and G1 at the interface (H1) with time t for three different d. Solid, dashed and dotted–dashed lines represent d ¼ 1:2, 1.0 and 0.9, respectively. While m ¼1.6, n ¼1.4 and Re1 ¼6.0 are considered as fixed.
B.S. Dandapat, S.K. Singh / International Journal of Non-Linear Mechanics 46 (2011) 272–277
0.25
277
the ratio of the deposited film thickness between the layers. Smaller the thickness on the top layer, faster the initial film thinning rate in both the layers. It is also found that change of viscosity or density has very minor effect on initial film thinning.
0.2
F2
0.15 0.1
Acknowledgement
0.05 0
We express our sincere thanks to the reviewer for his constructive and helpful suggestions in modifying the manuscript. 0
1
2
3
4
5
6
7
8
τ 0.35 0.3
F1
0.25 0.2 0.15 0.1 0.05 0
0
1
2
3
4
5
6
7
8
τ Fig. 9. Variation of F2 at free surface ðH2 dÞ and F1 at interface (H1) with time t for three different d. Solid, dashed and dotted–dashed line represent d ¼ 1:2, 1.0 and 0.9, respectively. While m¼ 1.6, n¼ 1.4 and Re1 ¼ 6.0 are considered as fixed.
thinning rate via faster radial velocity development as shown in Fig. 9.
5. Conclusion Gradual development of film thickness over a rotating disk is studied numerically when two uniform layers of fluid are deposited over the disk. Under the action of the centrifugal force the film started thinning in both the layers. It is further found that at the initial stage, the rate of film thinning in both the layers are more under smaller rotation of the disk. However, film thins faster with strong rotation as the spinning continues for a long time. This apparently anomalous behaviour of film thinning is due to the crucial role played by the viscosity. Further it is observed that the film thinning rate at the initial stage of rotation depends on
References [1] A.C. Emslie, F.D. Bonner, L.G. Peck, Flow of a viscous liquid on a rotating disk, J. Appl. Phys. 29 (1958) 858. [2] D. Meyerhofer, Characteristic of resist films produced by spinning, J. Appl. Phys. 49 (1978) 3993. [3] J.H. Lai, An investigation of spin coating of electron resists, Polym. Eng. Sci. 19 (1979) 1117. [4] B.T. Chen, Investigation of the solvent-evaporation effect on spin coating of thin films, Polym. Eng. Sci. 23 (1983) 399. [5] F. Ma, J.H. Hwang, The effect of air shear on the flow thin liquid film over a rough rotating disk, J. Appl. Phys. 68 (1990) 1265. [6] B.G. Higgins, Film flow on rotating disk, Phys. Fluids 29 (1986) 3522. [7] B.S. Dandapat, P.C. Ray, Film cooling on a rotating disk, Int. J. Non-Linear Mech. 29 (1990) 569. [8] B.S. Dandapat, P.C. Ray, The effect of thermocapillarity on the flow of thin liquid film on rotating disk, J. Phys D: Appl. Phys. 27 (1994) 2041. [9] P.C. Ray, B.S. Dandapat, Flow of thin liquid film on a rotating disk in the presence of a transverse magnetic field, Q. J. Mech. Appl. Math. 47 (1994) 297. [10] B.S. Dandapat, Unsteady flow of thin liquid film on a disk under ununiform rotation, Phys. Fluids 13 (2001) 1860. [11] P. Georis, J.-C. Legros, Pure thermocapillary convection in a multilayer system: first results from the IML-2mission, in: L. Ratke, H. Walter, B. Feuerbacher (Eds.), Materials and Fluids under Low Gravity, SpringerVerlag, Berlin1996, p. 299. [12] A.L. Yarin, Surface tension driven flows at low Reynolds number arising in opto-electronic technology, J. Fluid Mech. 286 (1995) 273. [13] R.D. Noble, J. Douglas Way, Liquid Membrane Technology: An Overview, in Liquid Membranes: Theory and Applications, ACS Symposium Series, American Chemical Society, Washington, DC, 1987. [14] A. Kitamura, Application of thin film in potentially hazardous environment, in: B.S. Dandapat, B.S. Mazumder (Eds.), ICA Fluid Mechanics in Industry and Environment, Research Publishing, Chennai and Singapore2006, p. 50. [15] A. Kitamura, Asymptotic solution for the film flowing on a rotating disk, Phys. Fluids 12 (2000) 2141. [16] B.S. Dandapat, B. Santra, A. Kitamura, Thermal effect on film development during spin coating, Phys. Fluids 17 (2005) 062102. [17] S.V. Parter, K.R. Rajagopal, Swirling flow between rotating plates, Arch. Ration. Mech. Anal. 86 (1984) 305. [18] K.R. Rajagopal, Flow of viscoelastic fluids between rotating disks, Theor. Comput. Fluid Dyn. 3 (1992) 185. [19] T. von Ka´rma´n, Uber laminare and turbulent reiburg, Z. Angew Math. Mech. 1 (1921) 233. [20] C.A.J. Fletcher, Computational Techniques for Fluid Dynamics, vol. II, Springer-Verlag, 1988.
Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Two-layer film flow on a rotating disk: A numerical study B.S. Dandapat a,, S.K. Singh b a b
Department of Mathematics, Sikkim Manipal Institute of Technology, Majitar, East Sikkim, 737 132 Sikkim, India Physics and Applied Mathematics Unit, Indian Statistical Institute, 203-B. T. Road, Calcutta 700 108, India
a r t i c l e in f o
a b s t r a c t
Article history: Received 24 December 2008 Received in revised form 17 July 2010 Accepted 17 September 2010
Development of thin two-layer film over a uniformly rotating disk is studied numerically under the assumption of planar interface and free surface. Similarity transformation is applied to transform the Navier–Stokes equations into a set of coupled non-linear, unsteady partial differential equations. This set of equations are solved numerically by using the finite-difference technique. It is observed that the rate of film thickness varies at different time zone depending on the rate of rotational speed of the disk. A physical explanation is provided to justify this anomalous behaviour. It is observed that, smaller thickness on the top layer enhance the initial rate of film thinning. But the overall effect of density, viscosity and the initial film thickness ratio are found to be insensitive to the final film thickness at large time. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Double layered thin film flow Rotating disk Viscous flow Spin coating
1. Introduction Development of thin film on horizontal rotating disk is known as spin coating in the literature. This technique is widely used in microelectronics industries to coat the photoresist on silicon wafers for integrated circuits, for magnetic storage disks, for magnetic paint coating on the substrates, etc. Emslie et al. [1], first initiated the study on the development of thin film on a rotating disk. Assuming a balance between the centrifugal and viscous force during the process of rotation they were able to simplify the system of Navier–Stokes equations and finally, able to show that the uniformity of the film is maintained as it thins continuously more and more. Later on Meyerhofer [2], Lai [3], Chen [4], Ma and Hwang [5], and others have studied the different effects like solvent evaporation, surface tension, non-Newtonian effects, disk roughness etc. on film thinning. All these analyses were based on the typical hydrodynamical approximation as employed by Emslie et al. [1]. Full Navier–Stokes equations were considered first time by Higgins [6] to study the flow and film development through matched asymptotic expansion procedure and able to show that the uniformity of the film thickness remains as spinning process continues. Following Higgins’s analytical treatment Dandapat and Ray [7,8], Ray and Dandapat [9] have studied the effects of film cooling, thermocapillarity and magnetic field on film development. Dandapat [10] has observed that at the initial stage of rotation, the film thinning rate is more for slow, than the fast rotational speed, but at large time, film thinning rate increases with the faster rotational speed if
Corresponding author. Tel.: +91 9433391227.
E-mail address: [email protected] (B.S. Dandapat). 0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2010.09.007
the spinner starts from rest with impulsive rotation and maintains the speed for the rest of the period. However, faster rate of thinning can be obtained if the spinner starts impulsively and then increases its spinning rate continuously. It is to be noted here that in all the above studies the tacit assumption was that the disk is wet so that the classical no-slip boundary condition can be applied at every point on the disk surface and the film flows under a planar interface for entire period of spinning. It is to be remembered here that in general, an interface always separates two different physical media. So to understand the spincoating process in details we need to study the film thinning mechanism by considering a two-layer fluid model. This necessity we felt from our past experiences in connection with the other flow problems. There we have seen new phenomena appears when interface between two fluids is considered. This is because in the frame of one fluid model the movement of gas/liquid vapour is treated as passive, whereas in reality the motion of gas/liquid vapour can exert shear stress on the surface of the fluid. This neglected shear stress on the free surface may brings changes in the system. Multilayered approach has been applied for the investigation of several modern engineering processes. For example, the liquid encapsulation technique used in crystal growth [11]. Multilayered approach also used in multilayer coating for the production of photographic films and multilayer fibers for optoelectronics devices [12] and emulsified liquid membrane separation techniques widely used now in various extraction processes and industrial waste-water treatment [13]. Kitamura [14] has advocated the usefulness of multilayered coating in potentially hazardous environment. To the best of our knowledge, so far, no theoretical study has been reported in the literature of thin film development on a rotating disk considering two or multilayered initial deposition of the liquid.
B.S. Dandapat, S.K. Singh / International Journal of Non-Linear Mechanics 46 (2011) 272–277
In this article, we have studied the flow and film development over a rotating disk, when two immiscible Newtonian liquids having different fluid properties with different film thickness are deposited initially.
We consider a stable configuration of two horizontal layers of immiscible liquids with different physical properties having different uniform thickness over a disk of large diameter. The disk is made to rotate in its own plane with constant angular velocity O. Following Higgins [6] and Dandapat [10], we restrict our attention to the flows that maintains a planar film thickness throughout the entire process. In fact, earlier works of Emslie et al. [1], Kitamura [15] and Dandapat et al. [16] have shown that the initial non-uniformity of the film thickness dies out in course of time and a uniform planar final film thickness is the neat outcome of the process. Further, due to axisymmetric flow assumption, the dependent variables are independent of azimuthal co-ordinate y. It is to be pointed out here that Parter and Rajagopal [17] and Rajagopal [18] have shown that the axial symmetric solutions are not always possible when two disks at a finite distance are rotating about a common axis or about distinct axes at different speeds. The initial film thickness of the lower layer is assumed to be h10 and that of the upper layer is h20 h10. In other words, total film thickness is h20. A schematic diagram given in Fig. 1 describes the details of the system. All the variables corresponding to the bottom layer are marked by subscript 1 and that corresponding to the top layer are marked by 2. Vi, pi, ri , ni and mi denote, respectively, the velocity, pressure, density, kinematic and dynamic viscosity, of the ith (i¼1,2) fluid layer. Under this convention, we represent the velocity field as Vi ¼ ui ðr,z,tÞe^ r þvi ðr,z,tÞe^ y þwi ðr,z,tÞe^ z ,
The boundary conditions at the surface of the disk are u1 ðr,0,tÞ ¼ 0,
v1 ðr,0,tÞ ¼ Or,
w1 ðr,0,tÞ ¼ 0:
ð1Þ
where e^ r , e^ y , e^ z are the unit vectors along the co-ordinates axes. The governing equations of continuity and motion for both incompressible Newtonian fluid are ð2Þ
2 @ui @u v2 @u 1 @pi @ ui @ ui @2 ui þui i i þwi i ¼ þ ni þ þ , @r r @t @r r @z ri @r @r 2 @z2
ð3Þ
2 @vi @v uv @v @ vi @ vi @2 vi þ ui i þ i i þ wi i ¼ ni þ þ , @r r @t @r r @z @r 2 @z2
ð4Þ
2 @wi @wi @wi 1 @pi @ wi 1 @wi @2 wi þ ui þ wi ¼ þ ni þ þ : r @r @t @r @z ri @z @r 2 @z2
ð5Þ
ð7Þ
the balance of shear stress @w1 @u1 @w2 @u2 þ ¼ m2 þ , m1 @r @z @r @z
m1
@v1 @v2 ¼ m2 , @z @z
ð8Þ
ð9Þ
the balance of normal stress @w1 @w2 p1 p2 ¼ 2 m1 m2 , @z @z
ð10Þ
and the kinematic condition dh1 ¼ w1 ðr,h1 ,tÞ: dt
ð11Þ
The boundary conditions at the free surface z ¼h2(t) the normal and the shear stress must vanish and these are given by p2 þ 2m2
@w2 ¼ 0, @z
ð12Þ
@w2 @u2 þ ¼ 0, @r @z
ð13Þ
@v2 ¼ 0: @z
ð14Þ
In addition, the kinematic condition at the free surface z¼h2(t) reads as dh2 ¼ w2 ðr,h2 ,tÞ: dt
ð15Þ
The initial conditions for the velocity field and the film thickness satisfy Vi ¼ 0,
@ui ui @wi þ þ ¼ 0, @r r @z
ð6Þ
The boundary conditions at the liquid–liquid interface z ¼h1(t), are the following: the continuity of the velocity field V1 ¼ V2 ,
2. Mathematical formulation
273
hi ¼ hi0 ,
dhi ¼ 0: dt
ð16Þ
Following von Ka´rma´n [19], the solution of the above system may be assumed in the following well-known similarity form; ui ¼ rf i ðz,tÞ,
vi ¼ rg i ðz,tÞ, wi ¼ wi ðz,tÞ, 2 r pi ðr,z,tÞ ¼ ri Ai ðz,tÞ þ Bi ðz,tÞ: 2
ð17Þ
Substituting (17) into the system of Eqs. (2)–(5) and collecting the different orders of r, we get 2fi þ wi,z ¼ 0,
ð18Þ
fi,t þ fi2 gi2 þwi fi,z ni fi,zz ¼ ri Ai ðz,tÞ,
ð19Þ
gi,t þ 2fi gi þ wi gi,z ¼ ni gi,zz ,
ð20Þ
ri ðwi,t þwi wi,z ni wi,zz Þ ¼ Bi,z ,
ð21Þ
Ai,z ¼ 0:
ð22Þ
The suffixes except i indicate derivatives with respect to that variable. The viscous term in the normal stress boundary conditions (10) and (12) are independent of r, implying Ai ¼0. We therefore conclude that Ai ¼0 are valid for the entire depth of the liquid. Now, Bi(z,t) can be found after integrating (21) with respect to z, from z to z¼h(t), by using the conditions on z¼h2(t) Fig. 1. Systematic diagram.
B2 ¼ 2m2
@w2 , @z
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3. Method of solution
and on z¼h1(t) @w1 @w2 m2 , B1 B2 ¼ 2 m1 @z @z and thus we can evaluate the pressure from Eq. (17). Using (17) in the boundary conditions (6)–(15), we get On the surface of the disk: at z¼ 0 f1 ðt,0Þ ¼ 0,
g1 ðt,0Þ ¼ O,
w1 ðt,0Þ ¼ 0,
ð23Þ
On the interface between the two liquids: at z¼ h1(t) 9 w1 ðt,h1 Þ ¼ w2 ðt,h1 Þ, > = m1 f1,z ðt,h1 Þ ¼ m2 f2,z ðt,h1 Þ, m1 g1,z ðt,h1 Þ ¼ m2 g2,z ðt,h1 Þ, > ; m1 w1,z ðt,h1 Þ ¼ m2 w2,z ðt,h1 Þ,
f1 ðt,h1 Þ ¼ f2 ðt,h1 Þ,
g1 ðt,h1 Þ ¼ g2 ðt,h1 Þ,
ð24Þ On the free surface: z ¼h2(t) f2,z ðt,h2 Þ ¼ 0,
g2,z ðt,h2 Þ ¼ 0,
ð25Þ
The kinematic conditions for the evolution of the surfaces: hi,t ¼ wi ðt,hi Þ:
ð26Þ
The initial conditions at t ¼0 fi ð0,zÞ ¼ 0, hi ¼ hi0 ,
gi ð0,zÞ ¼ 0,
wi ð0,zÞ ¼ 0
) ð27Þ
hi,t ¼ 0:
9
r1 m h20 h10 > > , n¼ 1 , d¼ > r2 m2 h10 > > > > > > r1 h210 O > h1 h2 h2 > > H1 ¼ , H2 ¼ ¼ , Re1 ¼ > > h10 h20 h10 dh10 m1 > > > = 2 r2 ðh20 h10 Þ2 O nd Re1 , Re2 ¼ ¼ > > m2 m > > W1 ¼
Fi ¼
z , h10
w1 , h10 O
fi
O
,
m¼
W2 ¼
Gi ¼
gi
O
w2 w2 ¼ ðh20 h10 ÞO dh10 O
,
ð28Þ
> > > > > > > > > > > > > ;
into the system of Eqs. (18)–(20) and the corresponding boundary and initial conditions (23)–(27) we obtain the dimensionless governing equations as well as their boundary and initial conditions as 2Fi þ li Wi, Z ¼ 0,
ð29Þ
Re1 ðFi, t þFi2 þ li Wi Fi, Z G2i Þ ¼ ai Fi, ZZ ,
ð30Þ
Re1 ðGi, t þ 2Gi Fi þ li Wi Gi, Z Þ ¼ ai Gi, ZZ ,
ð31Þ
where li ¼ ð1, dÞ and ai ¼ ð1,m=nÞ. The boundary conditions are F1 ðt,0Þ ¼ 0,
G1 ðt,0Þ ¼ 1,
W1 ðt,0Þ ¼ 0,
ð32Þ
9 F1 ðt,H1 Þ ¼ F2 ðt,H1 Þ, G1 ðt,H1 Þ ¼ G2 ðt,H1 Þ, W1 ðt,H1 Þ ¼ dW2 ðt,H1 Þ, = 1 1 ; F1, Z ðt,H1 Þ ¼ F2, Z ðt,H1 Þ, G1, Z ðt,H1 Þ ¼ G2, Z ðt,H1 Þ, n n
ð33Þ F2, Z ðt, dH2 Þ ¼ 0,
G2, Z ðt, dH2 Þ ¼ 0:
ð34Þ
The initial conditions become Fi ð0, ZÞ ¼ 0,
Gi ð0, ZÞ ¼ 0,
Wi ð0, ZÞ ¼ 0,
Hi ð0Þ ¼ Hi0 ,
ð35Þ
and kinematic conditions read as H1t ¼ W1 ðt,H1 Þ,
H2t ¼ W2 ðt, dH2 Þ:
where ( uðZ, tÞ ¼
Using the following dimensionless variables
t ¼ t O, Z ¼
The above coupled non-linear system of Eqs. (29)–(36) with the corresponding boundary conditions can be solved by using the finite-difference technique. It is to be noted here that the conventional finite-difference method cannot be used in this problem as thickness of both the layers are continuously decreasing with time. Considering this fact, the time-dependent physical domain is transformed to a fixed computational domain ½0,1 þ d such that the film thickness will always remain in a fixed computational domain for all times. Further, care has been taken through a fine grid distribution for the large velocity gradients that may be present near the disk surface. It should be pointed out here that the same transformation will be useful for fine as well as uniform grid distribution. We choose the transformation as ! A2 H1ZðtÞ x ¼ 1A1 log þ ½1uðZ, tÞ Z B2 þ H1 ðtÞ 8 0 1 !) 1 ðtÞ < A2 dHZ2 H A2 H1ZðtÞ ðtÞH1 ðtÞ A @ , ð37Þ þA log dA1 dlog 1 Z 1 ðtÞ : B2 þ H1 ðtÞ B2 þ dHZ2 H ðtÞH1 ðtÞ
ð36Þ
1
for 0 r Z o H1 ðtÞ,
0
for H1 ðtÞ r Z r dH2 ðtÞ
ð38Þ
and A1 ¼ ½logðA2 =B2 Þ1 , with A2 ¼(c+ 1), B2 ¼(c 1). Here, c is the grid controlling parameter in the physical domain. Small values of c cluster grid points at the disk surface where as large values make grid spacing uniform throughout the two-layer film. The Crank–Nicholson scheme is used to solve the transformed nonlinear system of Eqs. (29)–(36) after approximating the non-linear terms according to the Newton’s linearization technique [20]. Computation is carried out in each time level on the following linear tridiagonal system of algebraic equations: þ1 PðGi Þkj1 þ Q ðGi Þkj þ 1 þ RðGi Þkj þþ11 ¼ ðS1 ÞkðjÞ
ð39Þ
þ1 þ Q ðFi Þkj þ 1 þ RðFi Þkj þþ11 ¼ ðS2 ÞkðjÞ : PðFi Þkj1
ð40Þ
Here i, j and k denote the position of the layer, spatial level of discretization and time level, respectively. For i¼1 or 2, j takes values from 1 to N 1+1 or N 1+1 to N 1+ N 2 + 1, respectively. Here, N 1 and N 2 denote the number of partitions in the lower and upper layer of the fluid. The expressions for P, Q, R, (S1)kj and (S2)kj are lengthy but their derivations are straightforward. (We are dropping these expressions to reduce the paper length and willing to supply under request.) At each and every time level (Gi)kj + 1 and (Fi)kj + 1 are computed (39) and (40) and then the axial velocity (Wi)kj + 1 is obtained from finite-difference representation of continuity equation by using the values of (Fi)kj + 1 at that time level. The iteration process continues until it attains the desired level of thickness. Numerical computation is carried out on 101 grids in the vertical direction with c¼104 this gives uniform grid distribution in physical and as well as in the computational domain. However, for higher Reynolds number one needs to increase the number of grid points in the vertical direction. Under this situation, the value of c must decrease depending on the rotational speed of the disk. The time step has been calculated by using
Dt r0:25 Dx2 :
ð41Þ
This relation comes from the CFL condition of numerical stability [20]. The domain of Dt has been chosen smaller than the stability domain for linear equation due to the coupled non-linear system.
B.S. Dandapat, S.K. Singh / International Journal of Non-Linear Mechanics 46 (2011) 272–277
4. Results and discussion Fig. 2 represents the effect of d, the ratio of the initial film thickness deposition between the two layers, on the gradual development of the final film thickness with respect to time t for fixed values of m, n and the Reynolds number Re1. It is clear from the figure that the change of d effects the film thickness at the initial stage but it is more or less insensitive to the final film thickness at large time. Fig. 3 shows the gradual variation of the film height in both the layers with time t when the fluid viscosity ratio n are changed. It is clear from the figure that variation of viscosity between the two layers has least effect on film thinning. Fig. 4 depicts the gradual development of film thickness when the fluids density ratio varies. It is evident from the figure that the
275
variation of density ratio effects just after the spin off stage but the final film thickness is independent on it. Fig. 5 shows the effects of the variation of the Reynolds number on film thinning for a particular set of values of d, n, and m. It is further clear that the thinning rate for both the layers increase with decrease in the Reynolds number during the time interval 0 o t o tc , where tc is a point on the time scale. On the other hand film thinning rate in both the layers increase with increase of the Reynolds number in the interval tc o t. This anomalous behaviour of the variation of the film height in different time interval can be explained by considering the two time regions t o tc and t 4 tc . The point tc in the time interval indicates that time, when total amount of fluid flowing out of a certain radial distance for two different Reynolds
2
2.5
1.8
H2 δ
1.4 Film height
2
Film height
1.6
H 2δ
1.5
1
1.2 1 0.8 0.6
H1
0.4
H1
0.5
0.2 0
0
0
1
2
3
4 τ
5
6
7
2
1.8
1.8
1.6
6 τ
1.6
H2δ
1.2 1 0.8
0.4
H1
2
12
1 H1
0.6 0.4
0.2 0
10
1.2
0.8
0.6
8
H2 δ
1.4 Film height
Film height
4
Fig. 4. Variation of top film’s height ðH2 dÞ and the bottom film (interface height H1) with t for two different values of m. Here, dotted and solid lines represent total film thickness for m¼ 1.4 and 2.0, respectively. Dashed and dotted–dashed lines represent lower layer thickness for m ¼1.4 and 2.0, respectively. While n¼ 2.5, Re1 ¼ 10.0 and d ¼ 1:0 are considered as fixed.
2
0
2
8
Fig. 2. Variation of top film’s height ðH2 dÞ and the bottom film (interface height H1) with t for three different values of d. Solid, dashed and dashed–dotted lines represent for d ¼ 1:2, 1.0 and 0.9, respectively. While m¼ 1.6, n¼ 1.4 and Re1 ¼6.0 are considered as fixed.
1.4
0
4
6
8
10
12
τ Fig. 3. Variation of top film’s height ðH2 dÞ and the bottom film (interface height H1) with t for two different values of n. For total layer, dotted and solid lines represent film thickness for n¼ 3.5 and 6.0, respectively. Dashed and dashed– dotted lines representing the lower layer thickness for n ¼3.5 and 6.0, respectively. While m¼2.0, Re1 ¼ 10.0 and d ¼ 1:0 are considered as fixed.
0.2
0
1
2
3
4
5
τ Fig. 5. Variation of top film’s height ðH2 dÞ and the bottom film (interface height H1) with t for two different Re1. Here, dotted and solid lines represent total film thickness for Re1 ¼ 0.5 and 0.9, respectively. Dashed and dotted–dashed lines represent lower layer thickness for Re1 ¼ 0.5 and 0.9, respectively. While m ¼2.0, n¼2.5 and d ¼ 1 are considered as fixed.
B.S. Dandapat, S.K. Singh / International Journal of Non-Linear Mechanics 46 (2011) 272–277
0.25 0.2 0.15 F2
numbers become equal. In other words we can say that the tc depends on the competitive Reynolds numbers. In interval t o tc , thinning rate for both layers increase with the decrease in the rotational speed. This seems to be inconsistent with common observation that film thins faster with faster rotation of the disk. A close observation of this physical process exposes that the azimuthal component of velocity for both layers develop along the entire depth of the liquid faster, with slower rotation of the disk. This occurred due to the fact that the Reynolds number is the ratio between the centrifugal and viscous forces. Increase in the Reynolds number reflects that centrifugal force dominates over viscous force. Finally, viscous action builds up azimuthal component of velocity field faster throughout the depth of the fluid for smaller Reynolds number. This result can be found in Fig. 6. It is expected that the faster development of azimuthal velocity field will introduce the corresponding radial velocity faster for smaller Reynolds numbers only. Fig. 7 depicts the radial component of velocity for both the layers that developed over the entire depth for smaller values of Reynolds number in the interval 0 o t o tc as result film thins quicker for the smaller value of the Reynolds number in this interval. In time interval t 4 tc , azimuthal velocities for both the layers have developed throughout the film depth for both large and small Reynolds numbers and the corresponding developed centrifugal force in both the layers drive away large amount of fluid when the value of the Reynolds number is large. In other words, fluid moves out of the disk quickly when rotation speed of the disk is high. This explains why films thin faster in the time interval tc o t for the large Reynolds number. Fig. 8 depicts the variation of the azimuthal components of velocity for both the layers when d, the ration of the initial film thickness deposition between the layers varies for fixed Reynolds number Re1, m and n. It is clear from the figure that thinner film deposition on the top layer in compare to the bottom layer, produces faster azimuthal in both the layers and enhance film
0.1 0.05 0
1
2
3
4
5
3
4
5
0.25 0.2 0.15 0.1 0.05 0
0
1
2
τ
Fig. 7. Variation of F2 at free surface ðH2 dÞ and F1 at interface (H1) with time t for two different Re1. Dashed and solid line represent Re1 ¼ 0.5 and Re1 ¼ 0.9, respectively. While m ¼2.0, n¼ 2.5 and d ¼ 1:0 are considered as fixed.
0.8
0.8
0.7
0.7
0.6
0.6
0.5 G2
0.5 G2
0
τ
F1
276
0.4
0.4 0.3
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0.2
0.2
0.1
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0
0 0
1
2
3
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5
τ
0
1
2
3
4 τ
5
6
7
8
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3
4 τ
5
6
7
8
1
1
0.8
0.8
0.6
G1
G1
0.6 0.4
0.4 0.2
0.2 0
0
0
1
2
3
4
5
τ Fig. 6. Variation of G2 at free surface ðH2 dÞ and G1 at interface (H1) with time t for two different Re1. Dashed and solid line represent Re1 ¼0.5 and Re1 ¼ 0.9, respectively. While m¼ 2.0, n¼2.5 and d ¼ 1:0 are considered as fixed.
Fig. 8. Variation of G2 at the free surface ðH2 dÞ and G1 at the interface (H1) with time t for three different d. Solid, dashed and dotted–dashed lines represent d ¼ 1:2, 1.0 and 0.9, respectively. While m ¼1.6, n ¼1.4 and Re1 ¼6.0 are considered as fixed.
B.S. Dandapat, S.K. Singh / International Journal of Non-Linear Mechanics 46 (2011) 272–277
0.25
277
the ratio of the deposited film thickness between the layers. Smaller the thickness on the top layer, faster the initial film thinning rate in both the layers. It is also found that change of viscosity or density has very minor effect on initial film thinning.
0.2
F2
0.15 0.1
Acknowledgement
0.05 0
We express our sincere thanks to the reviewer for his constructive and helpful suggestions in modifying the manuscript. 0
1
2
3
4
5
6
7
8
τ 0.35 0.3
F1
0.25 0.2 0.15 0.1 0.05 0
0
1
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3
4
5
6
7
8
τ Fig. 9. Variation of F2 at free surface ðH2 dÞ and F1 at interface (H1) with time t for three different d. Solid, dashed and dotted–dashed line represent d ¼ 1:2, 1.0 and 0.9, respectively. While m¼ 1.6, n¼ 1.4 and Re1 ¼ 6.0 are considered as fixed.
thinning rate via faster radial velocity development as shown in Fig. 9.
5. Conclusion Gradual development of film thickness over a rotating disk is studied numerically when two uniform layers of fluid are deposited over the disk. Under the action of the centrifugal force the film started thinning in both the layers. It is further found that at the initial stage, the rate of film thinning in both the layers are more under smaller rotation of the disk. However, film thins faster with strong rotation as the spinning continues for a long time. This apparently anomalous behaviour of film thinning is due to the crucial role played by the viscosity. Further it is observed that the film thinning rate at the initial stage of rotation depends on
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