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An investigation into the spreading of a thin liquid drop under gravity on a slowly rotating disk
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International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
An investigation into the spreading of a thin liquid drop under gravity on a slowly rotating disk E. Momoniata;∗ , S. Abelmanb a Centre
for Di erential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa b Centre for Numerical Analysis and Computational Mathematics, School of Computational and Applied Mathematics University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Received 2 January 2002; received in revised form 18 September 2002; accepted 24 September 2002
Abstract The axisymmetric spreading of a thin liquid drop under the in4uence of gravity and rotation is investigated. The e5ects of the Coriolis force and surface tension are ignored. The Lie group method is used to analyse the non-linear di5usion-convection equation modelling the spreading of the liquid drop under gravity and rotation. A stationary group invariant solution is obtained. The case when rotation is small is considered next. A straightforward perturbation approach is used to determine the e5ects of the small rotation on the solution given for spreading under gravity only. Over a short period of time no real di5erence is observed between the approximate solution and the solution for spreading under gravity only. After a long period of time, the approximate solution tends toward a dewetting solution. We 9nd that the approximate solution is valid only in the interval t ∈ [0; t ∗ ), where t ∗ is the time when dewetting takes place. An approximation to t ∗ is obtained. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thin 9lm; Rotating disk; Dewetting; Lie group method
1. Introduction In this paper, we use the classical Lie group method [1–3] to determine group invariant solutions which model the spreading of a thin viscous liquid drop under the in4uence of gravity and rotation. The non-linear partial di5erential equation modelling the spreading of ∗
Corresponding author. Centre for Di5erential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa. Tel.: +2711-717-6137; fax: +27-11-403-9317. E-mail address: [email protected] (E. Momoniat).
a thin liquid has been derived by Myers and Charpin [4]. In this paper, we change the scaling of the partial di5erential equation by dividing through by the dimensionless number B as given by equation (16) in Myers and Charpin [4]. Ignoring surface tension, the free surface equation is given by @h 1 @ 3 @h 2 3 rh − r h ; (1.1) = @t 3r @r @r where is a dimensionless number measuring the ratio of the rotational speed of the disk and gravitational acceleration. This ratio has been given as
0020-7462/04/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 2 ) 0 0 1 7 3 - 7
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E. Momoniat, S. Abelman / International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
(see [5]) 2 Fr ; = Ro
(1.2)
where Fr is the Froude number and Ro the Rossby number. The scaling presented above is more appropriate for the analysis pursued in this paper, since we want to know what e5ect the rotation has on the solution for spreading under gravity only. A group invariant solution for the case of spreading under gravity only has been obtained by Momoniat et al. [6]. The e5ects of the Coriolis force can be ignored (see [4,7]). Huppert [8] has derived (1.1) when → 0 to model the axisymmetric propagation of two-dimensional viscous gravity currents over a rigid horizontal surface. Sherman [9] and Middleman [10] have considered similarity solutions for the case of spreading under gravity alone ( → 0). We 9rstly consider a group invariant solution admitted by the exact equation (1.1). Using the classical Lie group method, it is shown that (1.1) admits a stationary solution. We next consider the case when the disk is rotating slowly, i.e. = 1. We then look for approximate solutions of the form h(t; r; ) = h0 (t; r) + h1 (t; r) + · · ·
(1.3)
admitted by (1.1). The group invariant solution determined by Momoniat et al. [6] is used as the zeroth-order solution, h0 (t; r). The function h1 (t; r) is calculated using the classical Lie group method. As a result of approximation (1.3) we 9nd that the precursor 9lm height is O( 2 ). This approach has been presented by Fushchich et al. [11] as a way of using the Lie group method to solve di5erential equations with small parameters. It has successfully been used by Euler et al. [12] and Euler et al. [13,14]. We 9nd that after a short period of time, the solution (1.3) does not di5er signi9cantly from the zeroth-order solution, h0 (t; r). After a long period of time, however, solution (1.3) tends to a dewetting solution. The approximate solution is valid only in the interval t ∈ [0; t ∗ ), where t ∗ is the time when dewetting takes place. An approximation to t ∗ is obtained. The understanding of thin 9lm 4ows is important in the industrial application of spin coating (see e.g. [14–17]). Important references are also contained in Momoniat and Mason [7] and Myers and Charpin [4]. It is also important to obtain analytical results which
can be used to verify numerical computations and experimental results (see. e.g. [4,18,19]). The Lie group method provides a mechanism for obtaining these analytical results. The paper is divided up as follows: In Section 2 group invariant solutions for the exact equation (1.1) are calculated. In Section 3 approximate solutions of form (1.3) admitted by (1.1) when = 1 are calculated. Concluding remarks are made in Section 4. 2. Lie group analysis of exact equation To obtain the Lie point symmetry generators, X = 1 (t; r; h)@t + 2 (t; r; h)@r + (t; r; h)@h ;
(2.1)
admitted by (1.1), we solve the determining equation 1 @ @h @h − rh3 − r 2 h3 = 0: X [2] @t 3r @r @r (1:1) (2.2) The second prolongation, X [2] , of X is de9ned by X [2] = X + 1 @ht + 2 @hr + 11 @htt +12 @htr + 22 @hrr ;
(2.3)
where i = Di () − hk Di (k ); ij = Dj (i ) − hik Dj (k );
i = 1; 2; i; j = 1; 2;
(2.4) (2.5)
with summation over the repeated index k from k = 1 to k = 2 and D1 = Dt = @t + ht @h + htt @ht + hrt @hr + · · · ;
(2.6)
D2 = Dr = @r + hr @h + htr @ht + hrr @hr + · · · :
(2.7)
The coeIcients i and ij depend on ht which is eliminated from (2.2) using (1.1). It is easy to show that the Lie point symmetry generators admitted by (1.1) are given by 1 X1 = @t ; X2 = −2t@t + r@r + h@h : (2.8) 2 We consider a linear combination of the Lie point symmetries of the form X = a1 X1 + a2 X2 :
(2.9)
To determine a group invariant solution corresponding to the Lie point symmetry generator (2.9) we solve
E. Momoniat, S. Abelman / International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
the 9rst-order quasi-linear partial di5erential equation given by X (h − (t; r))|h=(t; r) = 0:
(2.10)
Solving (2.10) we obtain (2.11)
Substituting (2.11) into (1.1) we obtain the non-linear ordinary di5erential equation 1 a2 F() − b2 F () 2 1 d = (F 3 ()F () − 2 F 3 ()): 3 d
(2.12)
We can solve (2.12) using Laplace transforms. An important boundary condition comes from the fact that h(t; R(t)) = 0:
(2.14)
provided R(t) = (a1 − 2a2 t)−1=4 :
h(t; r) = −
C 2 r +D 6
(2.21)
since r ∈ [0; R(t)]. The constants C and D are determined by substituting (2.20) into (1.1). We 9nd that C = −3 and therefore h(t; r) =
1 2 r + D: 2
(2.22)
Solution (2.22) is a stationary solution admitted by (1.1). A non-stationary solution is determined in the next section.
(2.13)
This boundary condition emerges as a property of the group invariant solution determined by Momoniat et al. [6]. This boundary condition does not take account of the pre-cursor 9lm ejected from the base of the liquid drop (see e.g. [4,10]). This aspect of the problem will be dealt with in the next section. Imposing (2.13) on (2.11) implies F(1) = 0;
Since (2.15) holds, and r ∈ [0; R(t)] we let k =1. From (2.11) we 9nd that C (2.20) h(t; r) = − r 2 + D 1 r(a1 − 2a2 t)1=4 6 which simpli9es to
h(t; r) = (a1 − 2a2 t)−1=2 F(); = r(a1 − 2a2 t)1=4 :
267
(2.15)
Taking Laplace transforms of (2.12) and imposing (2.14) we obtain 4 (2.16) f (s) + f (s) = 0 s which can be easily solved to give C (2.17) f(s) = − 3 + D; 3s where C and D are constants. Taking inverse Laplace transforms of (2.17) we obtain C F() = − 2 + D k (); (2.18) 6 where k () is the Dirac-delta function de9ned as 1 0 6 6 k; k ¿ 0; (2.19) k () = k 0 ¿ k; k ¿ 0:
3. Approximate solution In this section we consider the case when the disk is rotating slowly, i.e. = 1. Substituting (1.3) into (1.1) and separating by coeIcients of we obtain 1 @ @h0 3 @h0 = rh0 ; (3.1) @t 3r @r @r @h1 1 @ @ 3 2 3 = r (h0 h1 ) − r h0 : (3.2) @t 3r @r @r The group invariant solution (see [6]) admitted by (3.1) is given by 1=3 r2 1 1− 2 h0 (t; r) = 2 ; (3.3) R (t) R (t) where R(t) is the radius of the liquid drop as a function of time de9ned as 1=8 16 R(t) = 1 + : (3.4) t 9 We substitute (3.3) with (3.4) into (3.2) and calculate the Lie point symmetry generators admitted by the resulting equation. Only one Lie point symmetry generator is admitted and is given by 2 4 16 (3.5) Y = 1+ t @t + r@r + h1 @h1 : 9 9 9
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E. Momoniat, S. Abelman / International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
The group invariant solution corresponding to the symmetry generator Y is given by r h1 (t; r) = R2 (t)P(!); ! = : (3.6) R(t)
where we have assumed the precursor 9lm has a height O( 2 ) (see e.g. [4,10] and the references therein). We make a further simplifying assumption that
Substituting (3.6) into (3.2) when (3.3) and (3.4) hold we obtain the second-order ordinary di5erential equation
We 9nd that
2
2
3!(2 − 4! + (−1 + ! )P (!)) +(−6 + 16!2 )P (!) + 22!P(!) = 0:
(3.7)
Eq. (3.7) admits the linear solution P(!) =
1 2 3 ! − : 4 16
(3.8)
Therefore h1 (t; r) =
1 2 3 2 r − R (t): 4 16
(3.9)
Therefore, for a slowly rotating disk, (1.1) admits the approximate solution 1=3 r2 1 1− 2 h(t; r) = 2 R (t) R (t) 1 2 3 2 r − R (t) : + (3.10) 4 16 As a result of the disk rotating, the radius of the liquid drop will no longer be given by R(t). Instead, the drop will have a new radius given by R∗ (t). To prevent (3.10) from becoming imaginary we must have that 1=3 1 r2 R2 (t) 1 − R2 (t) + 1 r 2 − 3 R2 (t) ; r 6 R(t); 4 16 h(t; r) = (3.11) 2 1=3 r 1 − 2 −1 R (t) R2 (t) 1 2 3 2 r R + − (t) ; 4 16 R(t) ¡ r 6 R∗ (t): We impose the condition h(t; R∗ (t)) = O( 2 );
(3.12)
R∗ (t) = R(t)(1 + n K(t)):
R12 (t) R (t) = R(t) 1 + 213 ∗
3
(3.13) :
(3.14)
It is important to note that there is no freedom of choice for the initial drop pro9le when using the Lie group method. This is because the initial pro9le must admit the same invariance as the partial di5erential equation. The approximate solution (3.11) admits the initial drop pro9le
1 2 3 2 1=3 1 − r r ; + − 4 16 r 6 R(t); h(0; r) = (3.15) 1 2 3 2 1=3 −(r − 1) + 4 r − 16 ; R(t) ¡ r 6 R∗ (t): An important aspect of the group invariant solution for spreading under gravity only presented in Momoniat et al. [6] is that the total volume, V , where R(t) V (t)s = 2& rh(t; r) dr (3.16) 0
is conserved, i.e. dV (t) = 0: (3.17) dt The equation for the total volume (3.16) can is written as
R∗ (t) R(t) V = 2& rh(t; r) dr + rh(t; r) dr : 0
R(t)
(3.18) To O( ), (3.18) reduces to 1 3 4 − R (t) : V = 2& 8 32 Substituting (3.19) into (3.12) we obtain dV (t) & = − R−4 (t): dt 18
(3.19)
(3.20)
E. Momoniat, S. Abelman / International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
1
269
0.03
0.8
0.025
t=0 h
0.02 0.6
t = 120000
h
0.015 0.4
0.01 t = 1000
0.2
0.5
1.5
1
2
t = 150000
0.005 1
2.5
Taking limits, we 9nd that lim
t→∞
dV (t) = 0: dt
(3.21)
Therefore, we can conclude that to O( ), (3.11) with (3.14) conserves the total volume of the liquid drop after a long period of time. After a long period of time we note that the approximate solution (3.10) models the behaviour of a liquid drop tending toward dewetting (see [10]). We can get an estimated time for dewetting by solving h(t ∗ ; 0) = 0 which gives 1 256 ∗ t = −9 : 16 2
(3.22)
(3.23)
Hence solution (3.10) is only valid in the interval t ∈ [0; t ∗ ). At t = t ∗ dewetting takes place and the approximate solution given by (3.10) is no longer physical. The approximate solution (3.10) is plotted for large time in Fig. 2. Eq. (3.21) can be rewritten to give lim∗
t→t
dV (t) 2 & =− ≈ 0: dt 96
(3.24)
We observe that after a short period of time (time is non-dimensional [4–7]) the approximate solution (3.10) is not very di5erent from the zeroth-order solution for spreading under gravity only. This is plotted in Fig. 1 (Graphs are plotted using MATHEMATICA [20]). As t → t ∗ we observe that the solution tends
3
4
r
r Fig. 1. Plot of the h0 (t; r)(: : :) and h(t; r) for t = 0 and t = 1000 where = 0:01.
2
Fig. 2. Plot of the h(t; r) for t = 120000 and t = 150000 where = 0:01.
to dewetting. This is plotted in Fig. 2. Note that for = 0:01, t ∗ = 159999:4375. This dewetting behaviour is counterbalanced by the surface tension in the results presented by Myers and Charpin [4]. Also, the initial pro9le chosen by Myers and Charpin [4] is given by h(0; r) = 1 − r 2 . The results presented in this paper are very dependent on the initial pro9le (3.15) of the liquid drop. The dewetting behaviour may not occur for any initial pro9le. Further work needs to be done to investigate the properties of the initial surface pro9les which lead to dewetting. 4. Concluding remarks We have derived an approximate group invariant solution which models the spreading of a thin liquid drop under gravity and rotation. The solution is valid in the interval t ∈ [0; t ∗ ) where t ∗ is the time when dewetting takes place. The solution also preserves the volume to O( ). We note that the solutions obtained using the Lie group approach are symmetric. This is not always the case for spreading on a rotating disk. Lai et al. [21,22] have determined asymmetric solutions for rotating 4ows. Future work involves using the Lie group approach to determine possible asymmetric solutions. References [1] L.V. Ovsiannikov, Group Properties of Di5erential Equations, Izdat. Sibirsk. Otdel. AN SSSR, Novosibirsk, 1962 (English translation by G. Bluman, 1967).
270
E. Momoniat, S. Abelman / International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
[2] L.V. Ovsiannikov, Group Analysis of Di5erential Equations, Academic Press, New York, 1982. [3] P.J. Olver, Applications of Lie Groups to Di5erential Equations, Springer, New York, 1986. [4] T.G. Myers, J.P.F. Charpin, The e5ect of the Coriolis force on axisymmetric rotating thin 9lm 4ows, Int. J. Non-Linear Mech. 36 (2001) 629. [5] E. Momoniat, An application of Lie group theory and computational methods to axisymmetric thin 9lm 4ow on a rotating disk, Ph.D. Thesis, University of the Witwatersrand, Johannesburg, 1999. [6] E. Momoniat, D.P. Mason, F.M. Mahomed, Non-linear di5usion of an axisymmetric thin liquid drop: group invariant solution and conservation law, Int. J. Non-Linear Mech. 36 (2001) 879. [7] E. Momoniat, D.P. Mason, Investigation of the e5ect of the Coriolis force on a thin 4uid 9lm on a rotating disk, Int. J. Non-Linear Mech. 33 (1998) 1069. [8] H.E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface, J. Fluid Mech. 121 (1982) 43. [9] F.S. Sherman, Viscous Flow, McGraw-Hill, New York, 1990. [10] S. Middleman, Modeling Axisymmetric Flows, Academic Press, New York, 1995. [11] W.I. Fushchich, W.M. Shtelen, On approximate symmetry and approximate solutions of the non-linear wave equation with a small parameter, J. Phys. A: Math. Gen. 22 (1989) L887.
[12] M. Euler, N. Euler, A. KOohler, On the construction of approximate solutions for a multidimensional non-linear heat equation, J. Phys. A: Math. Gen. 27 (1994) 2083. [13] N. Euler, M.W. Shul’ga, W.H. Steeb, Approximate symmetries and approximate solutions for a multidimensional Landau–Ginzburg equation, J. Phys. A: Math. Gen. 25 (1992) L1095. [14] N. Euler, M. Euler, Symmetry properties of the approximations of multidimensional generalized van der Pol equations, Nonlinear Math. Phys. 1 (1994) 41. [15] W.W. Flack, D.S. Soong, A.T. Bell, D.W. Hess, A mathematical model for spin coating of polymer resists, J. Appl. Phys. 56 (1984) 1199. [16] K.J. Ruschak, Coating Flows, Annu. Rev. Fluid Mech. 17 (1985) 65. [17] P.C. Sukanek, Spin coating, J. Imaging Technol. 11 (1985) 184. [18] R. Usha, R. Ravindran, Numerical study of 9lm cooling on a rotating disk, Int. J. Non-Linear Mech. 36 (2001) 147. [19] S.K. Wilson, R. Hunt, B.R. Du5y, The rate of spreading in spin coating, J. Fluid Mech. 413 (2000) 65. [20] S. Wolfram, The Mathematica Book, 3rd Edition, Wolfram Media/Cambridge University Press, Cambridge, 1996. [21] C.Y. Lai, K.R. Rajagopal, A.Z. Szeri, Asymmetric 4ow between parallel rotating disks, J. Fluid Mech. 146 (1984) 203. [22] C.Y. Lai, K.R. Rajagopal, A.Z. Szeri, Asymmetric 4ow above a rotating disk, J. Fluid Mech. 157 (1985) 471.
International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
An investigation into the spreading of a thin liquid drop under gravity on a slowly rotating disk E. Momoniata;∗ , S. Abelmanb a Centre
for Di erential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa b Centre for Numerical Analysis and Computational Mathematics, School of Computational and Applied Mathematics University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Received 2 January 2002; received in revised form 18 September 2002; accepted 24 September 2002
Abstract The axisymmetric spreading of a thin liquid drop under the in4uence of gravity and rotation is investigated. The e5ects of the Coriolis force and surface tension are ignored. The Lie group method is used to analyse the non-linear di5usion-convection equation modelling the spreading of the liquid drop under gravity and rotation. A stationary group invariant solution is obtained. The case when rotation is small is considered next. A straightforward perturbation approach is used to determine the e5ects of the small rotation on the solution given for spreading under gravity only. Over a short period of time no real di5erence is observed between the approximate solution and the solution for spreading under gravity only. After a long period of time, the approximate solution tends toward a dewetting solution. We 9nd that the approximate solution is valid only in the interval t ∈ [0; t ∗ ), where t ∗ is the time when dewetting takes place. An approximation to t ∗ is obtained. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thin 9lm; Rotating disk; Dewetting; Lie group method
1. Introduction In this paper, we use the classical Lie group method [1–3] to determine group invariant solutions which model the spreading of a thin viscous liquid drop under the in4uence of gravity and rotation. The non-linear partial di5erential equation modelling the spreading of ∗
Corresponding author. Centre for Di5erential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa. Tel.: +2711-717-6137; fax: +27-11-403-9317. E-mail address: [email protected] (E. Momoniat).
a thin liquid has been derived by Myers and Charpin [4]. In this paper, we change the scaling of the partial di5erential equation by dividing through by the dimensionless number B as given by equation (16) in Myers and Charpin [4]. Ignoring surface tension, the free surface equation is given by @h 1 @ 3 @h 2 3 rh − r h ; (1.1) = @t 3r @r @r where is a dimensionless number measuring the ratio of the rotational speed of the disk and gravitational acceleration. This ratio has been given as
0020-7462/04/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 2 ) 0 0 1 7 3 - 7
266
E. Momoniat, S. Abelman / International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
(see [5]) 2 Fr ; = Ro
(1.2)
where Fr is the Froude number and Ro the Rossby number. The scaling presented above is more appropriate for the analysis pursued in this paper, since we want to know what e5ect the rotation has on the solution for spreading under gravity only. A group invariant solution for the case of spreading under gravity only has been obtained by Momoniat et al. [6]. The e5ects of the Coriolis force can be ignored (see [4,7]). Huppert [8] has derived (1.1) when → 0 to model the axisymmetric propagation of two-dimensional viscous gravity currents over a rigid horizontal surface. Sherman [9] and Middleman [10] have considered similarity solutions for the case of spreading under gravity alone ( → 0). We 9rstly consider a group invariant solution admitted by the exact equation (1.1). Using the classical Lie group method, it is shown that (1.1) admits a stationary solution. We next consider the case when the disk is rotating slowly, i.e. = 1. We then look for approximate solutions of the form h(t; r; ) = h0 (t; r) + h1 (t; r) + · · ·
(1.3)
admitted by (1.1). The group invariant solution determined by Momoniat et al. [6] is used as the zeroth-order solution, h0 (t; r). The function h1 (t; r) is calculated using the classical Lie group method. As a result of approximation (1.3) we 9nd that the precursor 9lm height is O( 2 ). This approach has been presented by Fushchich et al. [11] as a way of using the Lie group method to solve di5erential equations with small parameters. It has successfully been used by Euler et al. [12] and Euler et al. [13,14]. We 9nd that after a short period of time, the solution (1.3) does not di5er signi9cantly from the zeroth-order solution, h0 (t; r). After a long period of time, however, solution (1.3) tends to a dewetting solution. The approximate solution is valid only in the interval t ∈ [0; t ∗ ), where t ∗ is the time when dewetting takes place. An approximation to t ∗ is obtained. The understanding of thin 9lm 4ows is important in the industrial application of spin coating (see e.g. [14–17]). Important references are also contained in Momoniat and Mason [7] and Myers and Charpin [4]. It is also important to obtain analytical results which
can be used to verify numerical computations and experimental results (see. e.g. [4,18,19]). The Lie group method provides a mechanism for obtaining these analytical results. The paper is divided up as follows: In Section 2 group invariant solutions for the exact equation (1.1) are calculated. In Section 3 approximate solutions of form (1.3) admitted by (1.1) when = 1 are calculated. Concluding remarks are made in Section 4. 2. Lie group analysis of exact equation To obtain the Lie point symmetry generators, X = 1 (t; r; h)@t + 2 (t; r; h)@r + (t; r; h)@h ;
(2.1)
admitted by (1.1), we solve the determining equation 1 @ @h @h − rh3 − r 2 h3 = 0: X [2] @t 3r @r @r (1:1) (2.2) The second prolongation, X [2] , of X is de9ned by X [2] = X + 1 @ht + 2 @hr + 11 @htt +12 @htr + 22 @hrr ;
(2.3)
where i = Di () − hk Di (k ); ij = Dj (i ) − hik Dj (k );
i = 1; 2; i; j = 1; 2;
(2.4) (2.5)
with summation over the repeated index k from k = 1 to k = 2 and D1 = Dt = @t + ht @h + htt @ht + hrt @hr + · · · ;
(2.6)
D2 = Dr = @r + hr @h + htr @ht + hrr @hr + · · · :
(2.7)
The coeIcients i and ij depend on ht which is eliminated from (2.2) using (1.1). It is easy to show that the Lie point symmetry generators admitted by (1.1) are given by 1 X1 = @t ; X2 = −2t@t + r@r + h@h : (2.8) 2 We consider a linear combination of the Lie point symmetries of the form X = a1 X1 + a2 X2 :
(2.9)
To determine a group invariant solution corresponding to the Lie point symmetry generator (2.9) we solve
E. Momoniat, S. Abelman / International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
the 9rst-order quasi-linear partial di5erential equation given by X (h − (t; r))|h=(t; r) = 0:
(2.10)
Solving (2.10) we obtain (2.11)
Substituting (2.11) into (1.1) we obtain the non-linear ordinary di5erential equation 1 a2 F() − b2 F () 2 1 d = (F 3 ()F () − 2 F 3 ()): 3 d
(2.12)
We can solve (2.12) using Laplace transforms. An important boundary condition comes from the fact that h(t; R(t)) = 0:
(2.14)
provided R(t) = (a1 − 2a2 t)−1=4 :
h(t; r) = −
C 2 r +D 6
(2.21)
since r ∈ [0; R(t)]. The constants C and D are determined by substituting (2.20) into (1.1). We 9nd that C = −3 and therefore h(t; r) =
1 2 r + D: 2
(2.22)
Solution (2.22) is a stationary solution admitted by (1.1). A non-stationary solution is determined in the next section.
(2.13)
This boundary condition emerges as a property of the group invariant solution determined by Momoniat et al. [6]. This boundary condition does not take account of the pre-cursor 9lm ejected from the base of the liquid drop (see e.g. [4,10]). This aspect of the problem will be dealt with in the next section. Imposing (2.13) on (2.11) implies F(1) = 0;
Since (2.15) holds, and r ∈ [0; R(t)] we let k =1. From (2.11) we 9nd that C (2.20) h(t; r) = − r 2 + D 1 r(a1 − 2a2 t)1=4 6 which simpli9es to
h(t; r) = (a1 − 2a2 t)−1=2 F(); = r(a1 − 2a2 t)1=4 :
267
(2.15)
Taking Laplace transforms of (2.12) and imposing (2.14) we obtain 4 (2.16) f (s) + f (s) = 0 s which can be easily solved to give C (2.17) f(s) = − 3 + D; 3s where C and D are constants. Taking inverse Laplace transforms of (2.17) we obtain C F() = − 2 + D k (); (2.18) 6 where k () is the Dirac-delta function de9ned as 1 0 6 6 k; k ¿ 0; (2.19) k () = k 0 ¿ k; k ¿ 0:
3. Approximate solution In this section we consider the case when the disk is rotating slowly, i.e. = 1. Substituting (1.3) into (1.1) and separating by coeIcients of we obtain 1 @ @h0 3 @h0 = rh0 ; (3.1) @t 3r @r @r @h1 1 @ @ 3 2 3 = r (h0 h1 ) − r h0 : (3.2) @t 3r @r @r The group invariant solution (see [6]) admitted by (3.1) is given by 1=3 r2 1 1− 2 h0 (t; r) = 2 ; (3.3) R (t) R (t) where R(t) is the radius of the liquid drop as a function of time de9ned as 1=8 16 R(t) = 1 + : (3.4) t 9 We substitute (3.3) with (3.4) into (3.2) and calculate the Lie point symmetry generators admitted by the resulting equation. Only one Lie point symmetry generator is admitted and is given by 2 4 16 (3.5) Y = 1+ t @t + r@r + h1 @h1 : 9 9 9
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The group invariant solution corresponding to the symmetry generator Y is given by r h1 (t; r) = R2 (t)P(!); ! = : (3.6) R(t)
where we have assumed the precursor 9lm has a height O( 2 ) (see e.g. [4,10] and the references therein). We make a further simplifying assumption that
Substituting (3.6) into (3.2) when (3.3) and (3.4) hold we obtain the second-order ordinary di5erential equation
We 9nd that
2
2
3!(2 − 4! + (−1 + ! )P (!)) +(−6 + 16!2 )P (!) + 22!P(!) = 0:
(3.7)
Eq. (3.7) admits the linear solution P(!) =
1 2 3 ! − : 4 16
(3.8)
Therefore h1 (t; r) =
1 2 3 2 r − R (t): 4 16
(3.9)
Therefore, for a slowly rotating disk, (1.1) admits the approximate solution 1=3 r2 1 1− 2 h(t; r) = 2 R (t) R (t) 1 2 3 2 r − R (t) : + (3.10) 4 16 As a result of the disk rotating, the radius of the liquid drop will no longer be given by R(t). Instead, the drop will have a new radius given by R∗ (t). To prevent (3.10) from becoming imaginary we must have that 1=3 1 r2 R2 (t) 1 − R2 (t) + 1 r 2 − 3 R2 (t) ; r 6 R(t); 4 16 h(t; r) = (3.11) 2 1=3 r 1 − 2 −1 R (t) R2 (t) 1 2 3 2 r R + − (t) ; 4 16 R(t) ¡ r 6 R∗ (t): We impose the condition h(t; R∗ (t)) = O( 2 );
(3.12)
R∗ (t) = R(t)(1 + n K(t)):
R12 (t) R (t) = R(t) 1 + 213 ∗
3
(3.13) :
(3.14)
It is important to note that there is no freedom of choice for the initial drop pro9le when using the Lie group method. This is because the initial pro9le must admit the same invariance as the partial di5erential equation. The approximate solution (3.11) admits the initial drop pro9le
1 2 3 2 1=3 1 − r r ; + − 4 16 r 6 R(t); h(0; r) = (3.15) 1 2 3 2 1=3 −(r − 1) + 4 r − 16 ; R(t) ¡ r 6 R∗ (t): An important aspect of the group invariant solution for spreading under gravity only presented in Momoniat et al. [6] is that the total volume, V , where R(t) V (t)s = 2& rh(t; r) dr (3.16) 0
is conserved, i.e. dV (t) = 0: (3.17) dt The equation for the total volume (3.16) can is written as
R∗ (t) R(t) V = 2& rh(t; r) dr + rh(t; r) dr : 0
R(t)
(3.18) To O( ), (3.18) reduces to 1 3 4 − R (t) : V = 2& 8 32 Substituting (3.19) into (3.12) we obtain dV (t) & = − R−4 (t): dt 18
(3.19)
(3.20)
E. Momoniat, S. Abelman / International Journal of Non-Linear Mechanics 39 (2004) 265 – 270
1
269
0.03
0.8
0.025
t=0 h
0.02 0.6
t = 120000
h
0.015 0.4
0.01 t = 1000
0.2
0.5
1.5
1
2
t = 150000
0.005 1
2.5
Taking limits, we 9nd that lim
t→∞
dV (t) = 0: dt
(3.21)
Therefore, we can conclude that to O( ), (3.11) with (3.14) conserves the total volume of the liquid drop after a long period of time. After a long period of time we note that the approximate solution (3.10) models the behaviour of a liquid drop tending toward dewetting (see [10]). We can get an estimated time for dewetting by solving h(t ∗ ; 0) = 0 which gives 1 256 ∗ t = −9 : 16 2
(3.22)
(3.23)
Hence solution (3.10) is only valid in the interval t ∈ [0; t ∗ ). At t = t ∗ dewetting takes place and the approximate solution given by (3.10) is no longer physical. The approximate solution (3.10) is plotted for large time in Fig. 2. Eq. (3.21) can be rewritten to give lim∗
t→t
dV (t) 2 & =− ≈ 0: dt 96
(3.24)
We observe that after a short period of time (time is non-dimensional [4–7]) the approximate solution (3.10) is not very di5erent from the zeroth-order solution for spreading under gravity only. This is plotted in Fig. 1 (Graphs are plotted using MATHEMATICA [20]). As t → t ∗ we observe that the solution tends
3
4
r
r Fig. 1. Plot of the h0 (t; r)(: : :) and h(t; r) for t = 0 and t = 1000 where = 0:01.
2
Fig. 2. Plot of the h(t; r) for t = 120000 and t = 150000 where = 0:01.
to dewetting. This is plotted in Fig. 2. Note that for = 0:01, t ∗ = 159999:4375. This dewetting behaviour is counterbalanced by the surface tension in the results presented by Myers and Charpin [4]. Also, the initial pro9le chosen by Myers and Charpin [4] is given by h(0; r) = 1 − r 2 . The results presented in this paper are very dependent on the initial pro9le (3.15) of the liquid drop. The dewetting behaviour may not occur for any initial pro9le. Further work needs to be done to investigate the properties of the initial surface pro9les which lead to dewetting. 4. Concluding remarks We have derived an approximate group invariant solution which models the spreading of a thin liquid drop under gravity and rotation. The solution is valid in the interval t ∈ [0; t ∗ ) where t ∗ is the time when dewetting takes place. The solution also preserves the volume to O( ). We note that the solutions obtained using the Lie group approach are symmetric. This is not always the case for spreading on a rotating disk. Lai et al. [21,22] have determined asymmetric solutions for rotating 4ows. Future work involves using the Lie group approach to determine possible asymmetric solutions. References [1] L.V. Ovsiannikov, Group Properties of Di5erential Equations, Izdat. Sibirsk. Otdel. AN SSSR, Novosibirsk, 1962 (English translation by G. Bluman, 1967).
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[2] L.V. Ovsiannikov, Group Analysis of Di5erential Equations, Academic Press, New York, 1982. [3] P.J. Olver, Applications of Lie Groups to Di5erential Equations, Springer, New York, 1986. [4] T.G. Myers, J.P.F. Charpin, The e5ect of the Coriolis force on axisymmetric rotating thin 9lm 4ows, Int. J. Non-Linear Mech. 36 (2001) 629. [5] E. Momoniat, An application of Lie group theory and computational methods to axisymmetric thin 9lm 4ow on a rotating disk, Ph.D. Thesis, University of the Witwatersrand, Johannesburg, 1999. [6] E. Momoniat, D.P. Mason, F.M. Mahomed, Non-linear di5usion of an axisymmetric thin liquid drop: group invariant solution and conservation law, Int. J. Non-Linear Mech. 36 (2001) 879. [7] E. Momoniat, D.P. Mason, Investigation of the e5ect of the Coriolis force on a thin 4uid 9lm on a rotating disk, Int. J. Non-Linear Mech. 33 (1998) 1069. [8] H.E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface, J. Fluid Mech. 121 (1982) 43. [9] F.S. Sherman, Viscous Flow, McGraw-Hill, New York, 1990. [10] S. Middleman, Modeling Axisymmetric Flows, Academic Press, New York, 1995. [11] W.I. Fushchich, W.M. Shtelen, On approximate symmetry and approximate solutions of the non-linear wave equation with a small parameter, J. Phys. A: Math. Gen. 22 (1989) L887.
[12] M. Euler, N. Euler, A. KOohler, On the construction of approximate solutions for a multidimensional non-linear heat equation, J. Phys. A: Math. Gen. 27 (1994) 2083. [13] N. Euler, M.W. Shul’ga, W.H. Steeb, Approximate symmetries and approximate solutions for a multidimensional Landau–Ginzburg equation, J. Phys. A: Math. Gen. 25 (1992) L1095. [14] N. Euler, M. Euler, Symmetry properties of the approximations of multidimensional generalized van der Pol equations, Nonlinear Math. Phys. 1 (1994) 41. [15] W.W. Flack, D.S. Soong, A.T. Bell, D.W. Hess, A mathematical model for spin coating of polymer resists, J. Appl. Phys. 56 (1984) 1199. [16] K.J. Ruschak, Coating Flows, Annu. Rev. Fluid Mech. 17 (1985) 65. [17] P.C. Sukanek, Spin coating, J. Imaging Technol. 11 (1985) 184. [18] R. Usha, R. Ravindran, Numerical study of 9lm cooling on a rotating disk, Int. J. Non-Linear Mech. 36 (2001) 147. [19] S.K. Wilson, R. Hunt, B.R. Du5y, The rate of spreading in spin coating, J. Fluid Mech. 413 (2000) 65. [20] S. Wolfram, The Mathematica Book, 3rd Edition, Wolfram Media/Cambridge University Press, Cambridge, 1996. [21] C.Y. Lai, K.R. Rajagopal, A.Z. Szeri, Asymmetric 4ow between parallel rotating disks, J. Fluid Mech. 146 (1984) 203. [22] C.Y. Lai, K.R. Rajagopal, A.Z. Szeri, Asymmetric 4ow above a rotating disk, J. Fluid Mech. 157 (1985) 471.