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Viscous-gravity spreading of time-varying liquid drop volumes on solid surfaces
Journal of Colloid and Interface Science 300 (2006) 688–696 www.elsevier.com/locate/jcis
Viscous-gravity spreading of time-varying liquid drop volumes on solid surfaces Rachid Chebbi Department of Chemical and Petroleum Engineering, United Arab Emirates University, P.O. Box 17555, Al Ain, United Arab Emirates Received 28 November 2005; accepted 1 April 2006 Available online 7 April 2006
Abstract Viscous-gravity spreading of liquid drops of time-dependent volume over a solid surface is considered. A self-similar solution for the drop configuration is obtained, in the case the liquid drop volume varies as a power-law function of time, along with the spreading laws in both cases of cylindrical and axisymmetric geometries. Results compare favorably with published experimental results and previous theoretical work. The limitations of the model are discussed, along with a comparison with viscous gravity spreading of oil on water. The validity of using approximate spreading laws is considered, and an approximate method is suggested to provide the dynamics of spreading in the general case where the drop volume does not necessarily vary as a power-law function of time. © 2006 Elsevier Inc. All rights reserved. Keywords: Drop spreading; Liquid spreading; Viscous-gravity spreading; Spontaneous spreading; Spreading laws; Variable volume
1. Introduction Spreading of liquid drops over solid substrates has been studied extensively in the literature. Reviews can be found in [1–3] and in the literature cited therein. When viscous-gravity spreading occurs, the driving force for spreading is gravity, resisted by viscous forces. This problem was addressed in [4] in the case of the constant liquid volume case. The spreading law was found as [4] x0 =
3ζ + 5 V 3 gt 36wζ3+1 ν
1/(3ζ +5) ,
(1)
in which x0 is the drop edge coordinate, t is time, V is the constant volume of the drop (per unit width in the case of cylindrical drops), ν is the liquid kinematic viscosity, g represents the acceleration of gravity, ζ is zero for cylindrical drops and equal to one in the case of axisymmetric geometry, and w1 =
π 1/2 Γ (1/3) , 5 Γ (5/6)
w2 =
3π . 8
E-mail address: [email protected]. 0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.04.018
(2)
The self-similar profile is found as 2 1/3 V x h= , 1− ζ +1 x 0 2wζ +1 x
(3)
0
in which h is the drop thickness, and x is the coordinate in the direction of spreading, which is Cartesian in the case of unidirectional spreading and radial in the axisymmetric case. A numerical solution in Chebbi and Selim [5] showed spreading to occur after a drop shape rearrangement takes place without a significant change in the radius of the drop. The study [5] showed the similarity solution [4] to be valid even before a substantial increase in the size of the drop occurs [5]. In addition, losses of potential energy were shown to be converted into viscous dissipation [5]. Brochard-Wyart et al. [6] showed that viscous-gravity spreading of a wetting liquid on a flat solid surface occurs when the drop radius R satisfies R Rc , in which Rc = κ −1 ln(1/κa), where a is the molecular length and κ −1 = (γ /ρg)1/2 is the capillary length, with most of the profile described by the selfsimilar solution in [4], except for a foot of size κ −1 at the edge, and with the viscous dissipation dominant in the bulk [6]. The problem of two-dimensional and axisymmetric viscousgravity currents over a rigid horizontal surface was studied the-
R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
oretically in both cases of constant volume and variable volume with the fluid being continually introduced at the origin [7]. Huppert [7] also studied experimentally both the constant volume and the constant rate of release cases using silicone oils spreading axisymmetrically on a horizontal sheet of Perspex. The viscous-gravity current problem studied in [7] is found similar to the problem addressed here of viscous-gravity spreading of liquid drops. The liquid volume was assumed in [7] to vary as a power-law function of time (V = qt n with our notations), and the liquid thickness profile was assumed of the form [7] 2 1/5 2/3 3(q/2) ν t (2n−1)/5 φ(x/x0 ) (ζ = 0), h = ηN (4) g 1/4 2/3 3qν t (n−1)/4 ψ(x/x0 ) (ζ = 1). h = ξN (5) g In Eq. (4), q/2 is shown instead of q since in our notation q represents double the liquid volume in [7] given that Huppert [7] considered the case in which spreading occurs in only one side (half of the domain). Both problems are obviously similar because of symmetry. Solving the equations of motion along with the continuity equation, Huppert [7] ended up with two second-order differential equations, one for φ and a second one for ψ , which he solved starting nearly from the leading edge, by using an asymptotic solution consisting of two terms. In the particular case of constant volume (n = 0), Huppert [7] mentioned that there are analytical solutions in [8] for φ and ψ satisfying the differential equations he obtained. By comparing the constant-volume analytical solutions for the spreading laws and thickness profiles obtained in [7] by using the analytical solutions for φ and ψ in [8], we found them consistent with those in Lopez et al. [4]. Based on order of magnitude analysis arguments, in which viscous and gravity forces are balanced, Huppert [7] used the following forms for the spreading size: (q/2)3 g 1/5 (3n+1)/5 x0 = ηN (6) t (ζ = 0), 3ν 3 1/8 q g t (3n+1)/8 (ζ = 1). x0 = ξN (7) 3ν Here ηN and ξN need to be determined using the overall mass balance and results for φ and ψ , respectively. Order of magnitude arguments were also used in Fay [9], Hoult [10], Chebbi [11], and Chebbi et al. [12,13] (with the treatment in [11] extended in [14]) in the case of spreading of constant oil volume spill on calm water, and in Didden and Maxworthy [15] for viscous spreading of gravity currents in the case of fluids of close dynamic viscosities. The present work extends the approach in [4] to the case where the drop volume varies as a power-law function of time, and results are compared with those in [7]. We seek a self-similar solution for the bulk of the drop in which the predominant driving force for spreading is gravity, and the main resisting force is the viscous force. A detailed similarity solution is presented. Both the order of magnitude analysis and the numerical solution are obtained
689
differently compared to [7]. The form of the solution we obtained is compared with the results from the order of magnitude analysis. In addition, the validity of the solution is discussed extensively, and a method is proposed to treat the general case in which the liquid volume does not necessarily increase as a power-law function of time. The governing equations are presented in Section 2, followed by the self-similar solution presented in Section 3. Asymptotic solutions at the origin and near the drop edge are given in Section 4, with final results presented, discussed and compared with published experimental and theoretical results in Section 5. Using the results for the special case in which the liquid volume varies as a power-law function of time, an approximate method is proposed in Section 5 to calculate the dynamics of spreading in the general case. The limitations of the model are discussed in Section 6 in the light of the results obtained in this work. The order of magnitude analysis, given in Appendix A, provides more insight into the physical justification of the spreading law and form of the mathematical solution. In addition it provides the order of magnitude for the transition time from inertial to viscous spreading or from viscous to inertial spreading depending on the value of n. 2. Governing equations We address the problem of viscous-gravity spreading of cylindrical and axisymmetric drops of a Newtonian liquid on solid surfaces. As in [7], we neglect capillary and inertial effects, i.e., we assume large Bond number and small Reynolds number. In addition, the thickness and slope are assumed sufficiently small so that the lubrication theory becomes applicable. A schematic representation is given in Fig. 1, showing continuous inflow at the origin and the average velocity u¯ 0 at that location. Under the lubrication theory approximation, and with the assumption of small capillary and inertial effects, the momentum equations reduce to [4] ∂p ∂ 2u + μ 2 = 0, ∂x ∂z ∂p − − ρg = 0, ∂z −
(8) (9)
Fig. 1. Schematic figure of a spreading drop with continuous inflow at the origin.
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in which u is the velocity, p denotes the pressure, ρ is the liquid density, and z is the upward vertical coordinate. The velocity is subject to the boundary conditions u = 0 at z = 0, (10) ∂u (11) = 0 at z = h, ∂z in which h is the drop thickness function of x and t . Equation (9) shows that the variation of pressure in the vertical direction is hydrostatic given by [7] p = pG + ρg(h − z),
(12)
in which pG is the gas pressure. The continuity equation yields
(14)
which after substituting into Eq. (13) gives [4,5,7] ∂h ρg ∂ ζ 3 ∂h h = x . ∂t 3μx ζ ∂x ∂x
(15)
We seek a similarity solution generalizing the solution in [4] valid in the case of a constant drop volume. The solution sought is of the form H = H˜ (T )ϕ(η),
ζ
x h dx,
(16)
0
in which the drop volume V , defined per unit width in the case of cylindrical drops, is assumed to change as a power-law function of time as qt n . Using an overall material balance shows that the rate of input at the origin is dV /dt = nqt n−1 . The rate of input at the origin is also equal to the product uhA ¯ at the origin, with A representing the area for flow, taken per unit width in the case of cylindrical geometry; therefore we have nqt n−1 = −2(πx)ζ
ρg 3 ∂h h 3μ ∂x
at x = 0.
(17)
η = x/x0 = X/R,
h = 0 at x = x0 .
R = x0 /x. ¯
(25)
R = αT p ,
(26)
in which both α, the spreading law prefactor, and p, the exponent, need to be determined. To solve the problem, it is convenient to define ξ = αη.
(27)
Substituting for H into Eqs. (21) and (22) yields
d2 ϕ 3 dϕ = n − (ζ + 1)p ϕ − pξ dξ dξ 2 ϕ3 2 3 dϕ ζ dϕ , − − ξ dξ ϕ dξ α 1 ξ ζ ϕ dξ = , 2π ζ
(28) (29)
with (18)
The use of Eq. (18), along with the no-slip condition, Eq. (10), is expected to yield an infinite shear stress at the leading edge. This aspect was discussed in [1,3] and in the references therein. The validity of the solution at the contact line is discussed in Section 5 and a summary is given in Section 6.
H˜ = T n−(ζ +1)p
3n + 1 (31) . 5 + 3ζ To get ϕ and α, we solve Eqs. (28) and (29) using the two boundary conditions
p=
n=− New dimensionless variables are defined using the following transformations, ¯ H = h/h,
(19)
(30)
and
3. Similarity solution
X = x/x, ¯
(24)
0
In addition, h satisfies at the drop edge
T = t/t¯,
(23)
We seek a solution in which R is a power law of time,
x0 V = qt = 2π
(22)
R being the dimensionless edge position defined as
Neglecting evaporation, the conservation-of-mass condition gives [7] ζ
(21)
in which the combined variable η is defined as
ρg 2 ∂h h , 3μ ∂x
n
(20)
0
∂h 1 ∂ ζ (13) ¯ =− ζ (x uh). ∂t x ∂x Integrating the x-momentum equation, while substituting for p, yields [4] u¯ = −
in which t¯, x, ¯ and h¯ are (7−2ζ )/(2−ζ ) (2−ζ )/(6n−3ζ n−3) ν , x¯ = (ν t¯ )1/2 , t¯ = q 3g q t¯n h¯ = ζ +1 . x¯ In dimensionless form, Eqs. (15) and (16) become 1 ∂ ∂H ζ 3 ∂H H = X , ∂T 3X ζ ∂X ∂X R ζ X ζ H dX = T n . 2π
2(πξ )ζ 3 dϕ ϕ 3 dξ
at ξ = 0
(32)
and ϕ = 0 at ξ = α.
(33)
R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
4. Asymptotic solutions x0 = α 4.1. Cylindrical drops and The asymptotic solutions are given by h=
• Near the origin 1/4 4 ϕ = ϕ(0) − 6nξ • Near the drop edge 1/3 ϕ = 9pα(α − ξ )
as ξ → 0.
as ξ → α.
(34)
(35)
4.2. Axisymmetric drops The asymptotic solutions are given by • Near the origin 1/4 ϕ = ln(kξ −6n/π )
as ξ → 0,
(36)
in which k is integration constant. • Near the drop edge Equation (35) is also valid in the case of axisymmetric geometry. Equation (35) is found to be consistent with the first of the two terms in the asymptotic solution near the edge given in [7]. 5. Results 5.1. Cylindrical drops Using the definitions in Eq. (20), we get 3n+2 1/(6n−3) 7/2 2/(6n−3) ν ν ¯t = , x¯ = , q 3g q 3g 2 1/3 ν . h¯ = g
(37)
Substituting, while using Eqs. (19), (23)–(27), (30), and (31), yields 3 1/5 q g t (3n+1)/5 x0 = α (38) ν and h=
q 2ν g
1/5 t (2n−1)/5 ϕ(ξ ).
(39)
5.2. Axisymmetric drops Following the same steps leads to 3n+2 1/(6n−6) 5 1/(3n−3) ν ν , x ¯ = , t¯ = q 3g q 3g 2 1/3 ν , h¯ = g
(40)
q 3g ν
qν g
691
1/8 t (3n+1)/8
(41)
1/4 t (n−1)/4 ϕ(ξ ).
(42)
Equations (38), (39), (41), and (42) are found to be consistent with the order of magnitude analysis (Eqs. (A.5) and (A.6)) in Appendix A, and also with the form of the solution assumed in Huppert [7]: Eqs. (4)–(7) listed above in Section 1. In particular, comparing Eqs. (39) and (42) with the order of magnitude analysis, Eq. (A.6), it is clear that hr , the representative thickness, can be considered as the value of h at some representative value of the similarity variable ξr . The order of magnitude analysis shows that inertia terms can be neglected compared to viscous terms for large times compared with the transition time tc (t tc ) for n < nc , with nc = 7/4 in the case of cylindrical geometry, and nc = 3 for axisymmetric geometry as in [7]. For n > nc , inertia terms can be neglected for t tc , also as in [7]. Finally for n = nc , viscous terms are predominant compared to inertia terms if q (ν 3 g 2 )1/4 for the cylindrical geometry case, and q νg in the case of axisymmetric configuration. A numerical solution is sought to integrate Eq. (28) subject to the boundary conditions, Eqs. (32) and (33), along with the conservation of mass condition, Eq. (29). The unknowns are the thickness profile ϕ and the spreading law prefactor α. For integration, the asymptotic solutions near the origin given by Eqs. (34) and (36) are used to start. The values ϕ(0) and k are found by trial and error so that Eq. (29) is satisfied. For the cylindrical case, integration is started at ξ = 0, however for the axisymmetric case, integration is started at a small but nonzero value of ξ . A value of 10−5 is found suitable, with results insensitive to the small value used. Integration of Eq. (28) is continued till Eq. (33) is satisfied with ϕ becoming very small at the end of integration. Equation (29) is not necessarily satisfied, and a new integration is started using a new value for ϕ(0) (cylindrical case) or k (axisymmetric case). A number of iterations is needed till the value of ϕ(0) or k is found to yield a solution satisfying Eq. (29). After this solution is reached, the end value of ξ found is the spreading law prefactor α. In contrast to the present work, the asymptotic solution at the leading edge was used to start the numerical integration in [7] instead of the asymptotic solution at the origin. The prefactor results for both cylindrical and axisymmetric geometries are shown in Fig. 2 for different values of n. The values found for ϕ(0) and k are plotted in Fig. 3. The results for the prefactor α for n equal to zero are found to match exactly those in [4,7], i.e., 0.7474 in the case of cylindrical drops and 0.7792 for axisymmetric drops. Also, for all values of n, the prefactor values are found to match exactly those in [7], as seen from Fig. 2. The drop profiles are shown for three values of n (0, 1 and 2) as examples in Figs. 4 and 5 for cylindrical and axisymmetric drop geometries, respectively. The asymptotic solutions given by Eqs. (34)–(36), plotted in Figs. 4 and 5, are also seen to
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R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
(a)
Fig. 2. Results for the spreading law prefactor α as a function of n.
(b) Fig. 3. Results for ϕ(0) (cylindrical geometry) and k (axisymmetric geometry).
match asymptotically the solutions at both ends. For n = 0, the solutions are found to match the analytical solution in [4]. For all the cases (n = 1, 2) shown in Figs. 4 and 5, the drop profiles exactly match those in [7]. For the unidirectional case, and n = 1 and 2, the asymptotic solution at the edge with only one term is seen to approximate very closely the solution in the whole domain. A similar result was obtained in [7] with two terms in the asymptotic solution. At the edge, all the drop profiles are seen to exhibit infinite slope. This can also be seen from the asymptotic solution at the leading edge, by differentiating once Eq. (35), and also from Eqs. (2.15) and (2.27) in [7]. Differentiating twice shows that the curvature also becomes infinity at the leading edge. The use of the no-slip condition, Eq. (10) along with Eq. (18), is expected to yield infinite shear stress at the contact line. Details can be found in [1,3] and in the references therein. Following Huh and Scriven [16], and assuming the particular case of flow pattern in a simple wedge, advancing with a constant velocity and small angle, de Gennes [3] showed that the dissipation would diverge. Extending this approach of de Gennes [3], and a previous work of Chebbi [17] on capillary spreading over smooth solid surfaces, we study viscous dissipation in this case.
(c) Fig. 4. Drop shapes for the cylindrical geometry case: (a) n = 0, (b) n = 1, and (c) n = 2.
The velocity profile is known to be parabolic (see [5], for instance): u=
ρg ∂h 2 (y − 2hy). 2μ ∂x
(43)
R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
in which
Ω0 (t) = (9pα)1/3 Ω1 (t) = (9pα)
1/3
q 3 ν 4 3n−4 t g4 q 3 ν 7 3n−7 t g7
693
1/15 , 1/24 .
Using Eqs. (43) and (44) gives
2 ∂u
1 = ρg Ωζ (t) (x0 − x)−1/3 , τw = μ ∂y y=0 3
(45)
(46)
which shows an infinite shear stress at the contact line as expected. Substituting for u from Eq. (43), the rate of viscous dissipation per unit contact area between the liquid and the solid is given by
(a)
h 2 ∂u 1 (ρg)2 3 ∂h 2 μ dy = . h ∂y 3 μ ∂x
(47)
0
Substituting for h from Eq. (44) yields h 2 5 ∂u 1 (ρg)2 μ dy = Ω0 (t) (x0 − x)−1/3 , ∂y 27 μ
(48)
0
(b)
(c) Fig. 5. Drop shapes for the axisymmetric geometry case: (a) n = 0, (b) n = 1, and (c) n = 2.
In dimensional variables, the asymptotic solution near the edge, Eq. (35), becomes h = Ωζ (t)(x0 − x)1/3 ,
(44)
which is infinite at the contact line. Although both the shear stress force and the rate of dissipation, obtained by integration (with respect to contact area) of the shear stress and rate of viscous dissipation per unit contact area, do not diverge (power exponents of x0 − x both equal to −1/3 (greater than −1), which means the improper integrals converge), the solution is unacceptable very near the edge, because of the large curvatures, with surface tension forces expected to come into play, and also because of the infinite slope, which is inconsistent with the lubrication approximation used in the model. In contrast, in the derivation of de Gennes [3], the slope at the common line is assumed small, and the power exponent in the rate of viscous dissipation per unit contact area is found to vary as (x0 − x)−1 (instead of (x0 − x)−1/3 ) with the improper integral obviously diverging. In the work on capillary spreading over a smooth solid surface of Chebbi [17], the slope was found not to violate the lubrication theory for thicknesses as small as 1 nm. Even if we argue in the case studied here of viscous gravity spread that the thickness is not expected to reach zero, it is clear that surface tension effects cannot be neglected as mentioned above. It was already mentioned by BrochardWyart et al. [6] that for a wetting liquid, Lopez et al. solution [4] for a constant liquid volume does not apply in a foot size region at the droplet edge. It is clear that to study the foot region, disjoining effects should be included as in [6]. At the origin and in the axisymmetric case, there is singularity for n different from zero as noted previously in [7]. Both the slope and curvature become infinite as can be seen by differentiating once and twice Eq. (36). For the axisymmetric case and n = 0 (constant volume), experimental work is available in [7] and the best fit power law for the spreading radius was found in good agreement with the
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R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
analytical result in [4,7]. The results for the apex height were found to be in reasonable agreement with the theoretical results [7]. Scattering of the apex thickness data was attributed to measurement difficulties [7]. For the spreading of axisymmetric drops and constant inflow rate (n = 1), the experimental result is [7] 3 1/8 q g x0 = (0.694 ± 0.004) t 0.499±0.001 3ν (ζ = 1, n = 1). (49) Our results (with a prefactor of 0.623) match exactly those in [7], and are in good agreement with the above experimental result. As in [7], we have 3 1/8 q g x0 = 0.715 (50) t 0.5 (ζ = 1, n = 1). 3ν The solution presented above was developed for power-law time-dependent liquid volumes. In the general case, the volume does not necessarily increase as a power-law function of time. One way to approximate the spreading dynamics in the general case of variable oil spill volume spreading on water [11–13,18] consists in obtaining the rate of spreading using the spreading law for a constant volume, while substituting for the actual volume, function of time, in the spreading rate formula obtained. Applying this approximation was found to yield acceptable values compared to experimental values [12,13] in the case of constant rate of release of oil on water (n = 1) for both cylindrical and axisymmetric geometries. Extending this approach to the case of viscous-gravity spreading over a solid surface gives for the axisymmetric case and for n equal to zero after using Eq. (41) dx00 α(0) gV 3 1/8 −7/8 (51) t (ζ = 1), = dt 8 ν in which α(0) is the prefactor for n = 0 and superscript 0 in x00 refers to the spreading law for a constant liquid volume (i.e., n equal to zero). After substituting for V from Eq. (16) and integrating, we get the approximate spreading law for the power-law case (n different from zero): α(0) gq 3 1/8 (3n+1)/8 n x0,app = (52) t (ζ = 1), 3n + 1 ν and therefore α(0) (ζ = 1). (53) 3n + 1 Using the theoretical value for axisymmetric spreading of constant liquid volume αth (0) = 0.7792 (in good agreement with 0 (1) = 0.1948, clearly showthe experimental value) yields αapp ing that the approximation, Eq. (53), is absolutely inadequate since the theoretical value αth (1) (matching the theoretical result in [7], and in good agreement with the experimental result in [7]) is 0.6233. As n tends to zero, both the theoretical value 0 (n) tend to α (0), and thereαth (n) and approximate value αapp th fore become close. In the same way, it is easy to see that for 0 (n) = αapp
n
0 (n) based on rea given value n0 , the approximate value αapp sults for spreading for the power-law case (n0 case) is a good approximation for αth (n) when n is close to n0 . This leads us to use not only a single value like α(0) to get the approximate rate of spreading in the general case of a variable liquid volume, but instead we suggest to use the proper value α(n) at any given time t . To determine the appropriate value for n, we select the values of n and q for a power-law function of time that would provide the same values for both the actual volume V and actual rate of volume increase V˙ (this method is different from the approximate technique used for oil spreading on water indicated above in which the spreading rate is obtained from the spreading law of a constant volume that would be equal to the actual volume at the given time t ). Using the proposed approximate method yields
n=
V˙ t , V
q=
V t V˙ t/V
(54)
.
The theoretical spreading law for the corresponding power-law case for that particular value of n is given by dx0n 3n + 1 gq 3 1/5 (3n−4)/5 t (ζ = 0), = α(n) (55) dt 5 ν dx0n 3n + 1 gq 3 1/8 (3n−7)/8 (56) t (ζ = 1). = α(n) dt 8 ν With time, n and q are expected to change in the general case. Depending on the geometry considered, Eq. (55) or Eq. (56) is used with the calculated values of n and q in Eq. (54), assuming they give the same spreading rate, and therefore we propose in the general case t x0,app = 0
3 1/5 (3V˙ t/V ) + 1 g V ˙ α(V t/V ) 5 ν t V˙ t/V ˙
× t [(3V t/V )−4]/5 dt (ζ = 0), 1/8 t ˙ t/V ) + 1 g V 3 (3 V x0,app = α(V˙ t/V ) 8 ν t V˙ t/V
(57)
0
˙
× t [(3V t/V )−7]/8 dt
(ζ = 1),
(58)
with the notation α(V˙ t/V ) representing the value of α for a value of n equal to V˙ t/V . It is clear that Eqs. (57) and (58) reproduce the spreading laws when V is either constant (n = 0) or varies as a power-law function of time. 6. Conclusion The presented work provides a detailed similarity solution which matches the expected form obtained by order of magnitude arguments and the theoretical results in [7], and agrees with the experimental results in [7]. An approximate method is proposed to treat the general case of time increasing liquid volume, using results for the case where the liquid volume changes as a power-law function of time.
R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
Near the edge of the drop, the self-similar solution varies as the cubic root of the distance from the leading edge, with the proportionality factors being function of the geometry. The infinite slope at the leading edge is inconsistent with the lubrication approximation, which requires a small vertical component of the velocity, and therefore small slopes for the drop configuration. There is a similar inconsistency with the lubrication approximation at the origin in the case of axisymmetric geometry for nonzero values of n. Large curvatures suggest that capillary effects cannot be neglected near the edge of the drop in all cases and at the origin for the axisymmetric geometry in the case n = 0. For these reasons, we do not expect the solution to be valid in the vicinity of the edge in all cases, and at the origin in the case previously mentioned: axisymmetric geometry and n different from zero. However, the good agreement with the available experimental work in the axisymmetric geometry case for n = 0 and 1 suggests that the solution can be considered as basically valid to describe the dynamics of spreading, at the least for these two cases for which experimental data are available. The solution assumes large Bond number and negligible inertia terms compared to viscous terms. The present analysis is consistent with the one in [7], and provides the required condition for neglecting inertia terms depending on the value of n. One more condition of validity of the solution: R Rc is provided by Brochard-Wyart et al. [6], at least for the case of spreading of a constant volume of a wetting liquid. The similarity solution was already mentioned to be invalid at t equal to zero [4]; however, the experimental results of Huppert [7] suggest that this is not a serious limitation since the mode of initiation was found to make no difference after a few seconds from release of liquid, at least for the constant liquid volume case [7]. Also, the numerical simulations of Chebbi and Selim [5] showed the similarity solution in [4] to be valid even before a substantial increase in the size of the drop occurs [5] in the constant volume case. Additional experimental work is needed to confirm the limitations of the present model and to estimate the accuracy of the proposed approximate method to calculate the dynamics of spread. The spreading prefactors are found to decrease with n, with rates of changes also becoming smaller as n increases for both geometries. The theoretical value is found to overestimate the experimental value, and this is also the case for viscous-gravity spreading of oil on water in which the theoretical values in Buckmaster [19] and Chebbi [20] were also found to overestimate the constant oil volume experimental value in Hoult [10]. In comparing the two different spreading cases, we note that viscous dissipation occurs within the liquid in the case of spreading over a solid, and in the case of oil spreading on water, the oil profile is assumed flat, and only dependent on the position in the direction of spread [10,19,20], with viscous dissipation assumed to occur only within the water phase (in a boundary layer starting from the oil–water interface) and not in the spreading phase (oil). Even in the case n different from zero (axisymmetric geometry and n = 1), the theoretical prefactor was found to exceed
695
the experimental value in [7] for spreading over a solid, and somewhat similarly, approximate prefactors obtained from theoretical values for the constant volume case were also found to overestimate results obtained experimentally in the case of constant rate of release of oil on water (n = 1) in both cylindrical and axisymmetric geometry cases [12,13]. Appendix A The order of magnitude analysis is a tool that allows us to find the form of the spreading laws. However, it leaves undetermined the prefactors. Such an analysis was performed by Huppert [7]. Analogous ideas were used in Fay [9], Hoult [10], Chebbi [11], and Chebbi et al. [12,13] in the case of spreading of oil on water, and in Didden and Maxworthy [15] for liquids of close viscosities. These ideas are based on equating the overall promoting and resisting forces balancing each other, while satisfying the conservation of volume condition. Following similar steps to those in [7,9–13,15] we have u∼
t
(A.1)
and hr ζ +1 ∼ qt n .
(A.2)
Instead of equating the order of magnitude for the overall promoting and resisting forces as in [7,9–13,15], we equate the order of magnitudes of the relevant terms in the momentum equations. Using Eq. (8) along with Eq. (A.1) yields g
pr ∼μ 2 , hr t
(A.3) g
in which hr is a representative thickness [7,9,10] and pr is the driving force for flow represented by the difference between the pressure at the solid–liquid interface for that representative thickness and the pressure at the edge pr − pG . Integration of Eq. (9) gives g
pr = ρghr .
(A.4)
Eliminating (A.4) yields ∼
ρgq 3 μ
g pr
and hr from Eq. (A.3) by using Eqs. (A.2) and
1/(3ζ +5)
t (3n+1)/(3ζ +5) .
(A.5)
Substituting into Eq. (A.2) gives hr ∼ q
ρgq 3 μ
−(ζ +1)/(3ζ +5)
t n−((ζ +1)(3n+1)/(3ζ +5)) .
(A.6)
Since the present analysis considers the case in which viscous effects are predominant as a resisting force, it is important to evaluate the order of magnitude of the ratio of the two resisting forces: inertia and viscous forces. The order of magnitude of the ratio is also obtained from the terms in the x-momentum
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R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
balance as viscous term ∼ inertia term
μ(/t)/ h2r ρ(/t 2 )
=
νt , h2r
(A.7)
in which ν is the kinematic viscosity μ/ρ. Substituting for hr from Eq. (A.6) yields the following results: • Cylindrical drops 3 2 1/5 viscous term ν g t (7−4n)/5 . ∼ inertia term q4
Results in Eqs. (A.5) and (A.8)–(A.11) are found in agreement with those in [7], in which there is an evident misprint in Eq. (A7b). References [1] [2] [3] [4]
(A.8)
[5] [6]
Therefore viscous and inertia terms become of the same order of magnitude for 4 1/(7−4n) q t ∼ tc = (A.9) . ν 3g2
[7] [8] [9]
• Axisymmetric drops 1/2 viscous term νg t (3−n)/2 . ∼ inertia term q
(A.10)
Similarly in this case, viscous and inertia terms become of the same order of magnitude for 1/(3−n) q t ∼ tc = (A.11) . νg
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
E.B. Dussan V, Ann. Rev. Fluid Mech. 11 (1979) 371. A. Marmur, Adv. Colloid Interface Sci. 19 (1983) 75. P.G. de Gennes, Rev. Mod. Phys. 57 (1985) 827. J. Lopez, C.A. Miller, E. Ruckenstein, J. Colloid Interface Sci. 56 (1976) 460. R. Chebbi, M.S. Selim, J. Colloid Interface Sci. 172 (1995) 14. F. Brochard-Wyart, H. Hervet, C. Redon, F. Rondelez, J. Colloid Interface Sci. 142 (1991) 518. H.E. Huppert, J. Fluid Mech. 121 (1982) 43. R.E. Pattle, Q. J. Mech. Appl. Math. 12 (1959) 407. J.A. Fay, in: D.P. Hoult (Ed.), Oil on the Sea, Plenum Press, New York, 1969, pp. 53–64. D.P. Hoult, Ann. Rev. Fluid Mech. 4 (1972) 341. R. Chebbi, Chem. Eng. Sci. 55 (2000) 4953. R. Chebbi, A.M. Abubakr, A.Y. Al-Abdul Jabbar, A.M. Al-Qatabri, J. Chem. Eng. Jpn. 35 (2002) 304. R. Chebbi, S.A. Abd Elrahman, H.K. Ahmed, J. Chem. Eng. Jpn 35 (2002) 1330. R. Chebbi, Chem. Eng. Sci. 60 (2005) 6806. N. Didden, T. Maxworthy, J. Fluid Mech. 121 (1982) 27. C. Huh, L.E. Scriven, J. Colloid Interface Sci. 35 (1971) 85. R. Chebbi, J. Colloid Interface Sci. 229 (2000) 155. H.T. Shen, P.D. Yapa, M.E. Petroski, Water Resour. Res. 23 (1987) 1949. J. Buckmaster, J. Fluid Mech. 59 (1973) 481. R. Chebbi, AIChE J. 47 (2001) 288.
Viscous-gravity spreading of time-varying liquid drop volumes on solid surfaces Rachid Chebbi Department of Chemical and Petroleum Engineering, United Arab Emirates University, P.O. Box 17555, Al Ain, United Arab Emirates Received 28 November 2005; accepted 1 April 2006 Available online 7 April 2006
Abstract Viscous-gravity spreading of liquid drops of time-dependent volume over a solid surface is considered. A self-similar solution for the drop configuration is obtained, in the case the liquid drop volume varies as a power-law function of time, along with the spreading laws in both cases of cylindrical and axisymmetric geometries. Results compare favorably with published experimental results and previous theoretical work. The limitations of the model are discussed, along with a comparison with viscous gravity spreading of oil on water. The validity of using approximate spreading laws is considered, and an approximate method is suggested to provide the dynamics of spreading in the general case where the drop volume does not necessarily vary as a power-law function of time. © 2006 Elsevier Inc. All rights reserved. Keywords: Drop spreading; Liquid spreading; Viscous-gravity spreading; Spontaneous spreading; Spreading laws; Variable volume
1. Introduction Spreading of liquid drops over solid substrates has been studied extensively in the literature. Reviews can be found in [1–3] and in the literature cited therein. When viscous-gravity spreading occurs, the driving force for spreading is gravity, resisted by viscous forces. This problem was addressed in [4] in the case of the constant liquid volume case. The spreading law was found as [4] x0 =
3ζ + 5 V 3 gt 36wζ3+1 ν
1/(3ζ +5) ,
(1)
in which x0 is the drop edge coordinate, t is time, V is the constant volume of the drop (per unit width in the case of cylindrical drops), ν is the liquid kinematic viscosity, g represents the acceleration of gravity, ζ is zero for cylindrical drops and equal to one in the case of axisymmetric geometry, and w1 =
π 1/2 Γ (1/3) , 5 Γ (5/6)
w2 =
3π . 8
E-mail address: [email protected]. 0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2006.04.018
(2)
The self-similar profile is found as 2 1/3 V x h= , 1− ζ +1 x 0 2wζ +1 x
(3)
0
in which h is the drop thickness, and x is the coordinate in the direction of spreading, which is Cartesian in the case of unidirectional spreading and radial in the axisymmetric case. A numerical solution in Chebbi and Selim [5] showed spreading to occur after a drop shape rearrangement takes place without a significant change in the radius of the drop. The study [5] showed the similarity solution [4] to be valid even before a substantial increase in the size of the drop occurs [5]. In addition, losses of potential energy were shown to be converted into viscous dissipation [5]. Brochard-Wyart et al. [6] showed that viscous-gravity spreading of a wetting liquid on a flat solid surface occurs when the drop radius R satisfies R Rc , in which Rc = κ −1 ln(1/κa), where a is the molecular length and κ −1 = (γ /ρg)1/2 is the capillary length, with most of the profile described by the selfsimilar solution in [4], except for a foot of size κ −1 at the edge, and with the viscous dissipation dominant in the bulk [6]. The problem of two-dimensional and axisymmetric viscousgravity currents over a rigid horizontal surface was studied the-
R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
oretically in both cases of constant volume and variable volume with the fluid being continually introduced at the origin [7]. Huppert [7] also studied experimentally both the constant volume and the constant rate of release cases using silicone oils spreading axisymmetrically on a horizontal sheet of Perspex. The viscous-gravity current problem studied in [7] is found similar to the problem addressed here of viscous-gravity spreading of liquid drops. The liquid volume was assumed in [7] to vary as a power-law function of time (V = qt n with our notations), and the liquid thickness profile was assumed of the form [7] 2 1/5 2/3 3(q/2) ν t (2n−1)/5 φ(x/x0 ) (ζ = 0), h = ηN (4) g 1/4 2/3 3qν t (n−1)/4 ψ(x/x0 ) (ζ = 1). h = ξN (5) g In Eq. (4), q/2 is shown instead of q since in our notation q represents double the liquid volume in [7] given that Huppert [7] considered the case in which spreading occurs in only one side (half of the domain). Both problems are obviously similar because of symmetry. Solving the equations of motion along with the continuity equation, Huppert [7] ended up with two second-order differential equations, one for φ and a second one for ψ , which he solved starting nearly from the leading edge, by using an asymptotic solution consisting of two terms. In the particular case of constant volume (n = 0), Huppert [7] mentioned that there are analytical solutions in [8] for φ and ψ satisfying the differential equations he obtained. By comparing the constant-volume analytical solutions for the spreading laws and thickness profiles obtained in [7] by using the analytical solutions for φ and ψ in [8], we found them consistent with those in Lopez et al. [4]. Based on order of magnitude analysis arguments, in which viscous and gravity forces are balanced, Huppert [7] used the following forms for the spreading size: (q/2)3 g 1/5 (3n+1)/5 x0 = ηN (6) t (ζ = 0), 3ν 3 1/8 q g t (3n+1)/8 (ζ = 1). x0 = ξN (7) 3ν Here ηN and ξN need to be determined using the overall mass balance and results for φ and ψ , respectively. Order of magnitude arguments were also used in Fay [9], Hoult [10], Chebbi [11], and Chebbi et al. [12,13] (with the treatment in [11] extended in [14]) in the case of spreading of constant oil volume spill on calm water, and in Didden and Maxworthy [15] for viscous spreading of gravity currents in the case of fluids of close dynamic viscosities. The present work extends the approach in [4] to the case where the drop volume varies as a power-law function of time, and results are compared with those in [7]. We seek a self-similar solution for the bulk of the drop in which the predominant driving force for spreading is gravity, and the main resisting force is the viscous force. A detailed similarity solution is presented. Both the order of magnitude analysis and the numerical solution are obtained
689
differently compared to [7]. The form of the solution we obtained is compared with the results from the order of magnitude analysis. In addition, the validity of the solution is discussed extensively, and a method is proposed to treat the general case in which the liquid volume does not necessarily increase as a power-law function of time. The governing equations are presented in Section 2, followed by the self-similar solution presented in Section 3. Asymptotic solutions at the origin and near the drop edge are given in Section 4, with final results presented, discussed and compared with published experimental and theoretical results in Section 5. Using the results for the special case in which the liquid volume varies as a power-law function of time, an approximate method is proposed in Section 5 to calculate the dynamics of spreading in the general case. The limitations of the model are discussed in Section 6 in the light of the results obtained in this work. The order of magnitude analysis, given in Appendix A, provides more insight into the physical justification of the spreading law and form of the mathematical solution. In addition it provides the order of magnitude for the transition time from inertial to viscous spreading or from viscous to inertial spreading depending on the value of n. 2. Governing equations We address the problem of viscous-gravity spreading of cylindrical and axisymmetric drops of a Newtonian liquid on solid surfaces. As in [7], we neglect capillary and inertial effects, i.e., we assume large Bond number and small Reynolds number. In addition, the thickness and slope are assumed sufficiently small so that the lubrication theory becomes applicable. A schematic representation is given in Fig. 1, showing continuous inflow at the origin and the average velocity u¯ 0 at that location. Under the lubrication theory approximation, and with the assumption of small capillary and inertial effects, the momentum equations reduce to [4] ∂p ∂ 2u + μ 2 = 0, ∂x ∂z ∂p − − ρg = 0, ∂z −
(8) (9)
Fig. 1. Schematic figure of a spreading drop with continuous inflow at the origin.
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in which u is the velocity, p denotes the pressure, ρ is the liquid density, and z is the upward vertical coordinate. The velocity is subject to the boundary conditions u = 0 at z = 0, (10) ∂u (11) = 0 at z = h, ∂z in which h is the drop thickness function of x and t . Equation (9) shows that the variation of pressure in the vertical direction is hydrostatic given by [7] p = pG + ρg(h − z),
(12)
in which pG is the gas pressure. The continuity equation yields
(14)
which after substituting into Eq. (13) gives [4,5,7] ∂h ρg ∂ ζ 3 ∂h h = x . ∂t 3μx ζ ∂x ∂x
(15)
We seek a similarity solution generalizing the solution in [4] valid in the case of a constant drop volume. The solution sought is of the form H = H˜ (T )ϕ(η),
ζ
x h dx,
(16)
0
in which the drop volume V , defined per unit width in the case of cylindrical drops, is assumed to change as a power-law function of time as qt n . Using an overall material balance shows that the rate of input at the origin is dV /dt = nqt n−1 . The rate of input at the origin is also equal to the product uhA ¯ at the origin, with A representing the area for flow, taken per unit width in the case of cylindrical geometry; therefore we have nqt n−1 = −2(πx)ζ
ρg 3 ∂h h 3μ ∂x
at x = 0.
(17)
η = x/x0 = X/R,
h = 0 at x = x0 .
R = x0 /x. ¯
(25)
R = αT p ,
(26)
in which both α, the spreading law prefactor, and p, the exponent, need to be determined. To solve the problem, it is convenient to define ξ = αη.
(27)
Substituting for H into Eqs. (21) and (22) yields
d2 ϕ 3 dϕ = n − (ζ + 1)p ϕ − pξ dξ dξ 2 ϕ3 2 3 dϕ ζ dϕ , − − ξ dξ ϕ dξ α 1 ξ ζ ϕ dξ = , 2π ζ
(28) (29)
with (18)
The use of Eq. (18), along with the no-slip condition, Eq. (10), is expected to yield an infinite shear stress at the leading edge. This aspect was discussed in [1,3] and in the references therein. The validity of the solution at the contact line is discussed in Section 5 and a summary is given in Section 6.
H˜ = T n−(ζ +1)p
3n + 1 (31) . 5 + 3ζ To get ϕ and α, we solve Eqs. (28) and (29) using the two boundary conditions
p=
n=− New dimensionless variables are defined using the following transformations, ¯ H = h/h,
(19)
(30)
and
3. Similarity solution
X = x/x, ¯
(24)
0
In addition, h satisfies at the drop edge
T = t/t¯,
(23)
We seek a solution in which R is a power law of time,
x0 V = qt = 2π
(22)
R being the dimensionless edge position defined as
Neglecting evaporation, the conservation-of-mass condition gives [7] ζ
(21)
in which the combined variable η is defined as
ρg 2 ∂h h , 3μ ∂x
n
(20)
0
∂h 1 ∂ ζ (13) ¯ =− ζ (x uh). ∂t x ∂x Integrating the x-momentum equation, while substituting for p, yields [4] u¯ = −
in which t¯, x, ¯ and h¯ are (7−2ζ )/(2−ζ ) (2−ζ )/(6n−3ζ n−3) ν , x¯ = (ν t¯ )1/2 , t¯ = q 3g q t¯n h¯ = ζ +1 . x¯ In dimensionless form, Eqs. (15) and (16) become 1 ∂ ∂H ζ 3 ∂H H = X , ∂T 3X ζ ∂X ∂X R ζ X ζ H dX = T n . 2π
2(πξ )ζ 3 dϕ ϕ 3 dξ
at ξ = 0
(32)
and ϕ = 0 at ξ = α.
(33)
R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
4. Asymptotic solutions x0 = α 4.1. Cylindrical drops and The asymptotic solutions are given by h=
• Near the origin 1/4 4 ϕ = ϕ(0) − 6nξ • Near the drop edge 1/3 ϕ = 9pα(α − ξ )
as ξ → 0.
as ξ → α.
(34)
(35)
4.2. Axisymmetric drops The asymptotic solutions are given by • Near the origin 1/4 ϕ = ln(kξ −6n/π )
as ξ → 0,
(36)
in which k is integration constant. • Near the drop edge Equation (35) is also valid in the case of axisymmetric geometry. Equation (35) is found to be consistent with the first of the two terms in the asymptotic solution near the edge given in [7]. 5. Results 5.1. Cylindrical drops Using the definitions in Eq. (20), we get 3n+2 1/(6n−3) 7/2 2/(6n−3) ν ν ¯t = , x¯ = , q 3g q 3g 2 1/3 ν . h¯ = g
(37)
Substituting, while using Eqs. (19), (23)–(27), (30), and (31), yields 3 1/5 q g t (3n+1)/5 x0 = α (38) ν and h=
q 2ν g
1/5 t (2n−1)/5 ϕ(ξ ).
(39)
5.2. Axisymmetric drops Following the same steps leads to 3n+2 1/(6n−6) 5 1/(3n−3) ν ν , x ¯ = , t¯ = q 3g q 3g 2 1/3 ν , h¯ = g
(40)
q 3g ν
qν g
691
1/8 t (3n+1)/8
(41)
1/4 t (n−1)/4 ϕ(ξ ).
(42)
Equations (38), (39), (41), and (42) are found to be consistent with the order of magnitude analysis (Eqs. (A.5) and (A.6)) in Appendix A, and also with the form of the solution assumed in Huppert [7]: Eqs. (4)–(7) listed above in Section 1. In particular, comparing Eqs. (39) and (42) with the order of magnitude analysis, Eq. (A.6), it is clear that hr , the representative thickness, can be considered as the value of h at some representative value of the similarity variable ξr . The order of magnitude analysis shows that inertia terms can be neglected compared to viscous terms for large times compared with the transition time tc (t tc ) for n < nc , with nc = 7/4 in the case of cylindrical geometry, and nc = 3 for axisymmetric geometry as in [7]. For n > nc , inertia terms can be neglected for t tc , also as in [7]. Finally for n = nc , viscous terms are predominant compared to inertia terms if q (ν 3 g 2 )1/4 for the cylindrical geometry case, and q νg in the case of axisymmetric configuration. A numerical solution is sought to integrate Eq. (28) subject to the boundary conditions, Eqs. (32) and (33), along with the conservation of mass condition, Eq. (29). The unknowns are the thickness profile ϕ and the spreading law prefactor α. For integration, the asymptotic solutions near the origin given by Eqs. (34) and (36) are used to start. The values ϕ(0) and k are found by trial and error so that Eq. (29) is satisfied. For the cylindrical case, integration is started at ξ = 0, however for the axisymmetric case, integration is started at a small but nonzero value of ξ . A value of 10−5 is found suitable, with results insensitive to the small value used. Integration of Eq. (28) is continued till Eq. (33) is satisfied with ϕ becoming very small at the end of integration. Equation (29) is not necessarily satisfied, and a new integration is started using a new value for ϕ(0) (cylindrical case) or k (axisymmetric case). A number of iterations is needed till the value of ϕ(0) or k is found to yield a solution satisfying Eq. (29). After this solution is reached, the end value of ξ found is the spreading law prefactor α. In contrast to the present work, the asymptotic solution at the leading edge was used to start the numerical integration in [7] instead of the asymptotic solution at the origin. The prefactor results for both cylindrical and axisymmetric geometries are shown in Fig. 2 for different values of n. The values found for ϕ(0) and k are plotted in Fig. 3. The results for the prefactor α for n equal to zero are found to match exactly those in [4,7], i.e., 0.7474 in the case of cylindrical drops and 0.7792 for axisymmetric drops. Also, for all values of n, the prefactor values are found to match exactly those in [7], as seen from Fig. 2. The drop profiles are shown for three values of n (0, 1 and 2) as examples in Figs. 4 and 5 for cylindrical and axisymmetric drop geometries, respectively. The asymptotic solutions given by Eqs. (34)–(36), plotted in Figs. 4 and 5, are also seen to
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(a)
Fig. 2. Results for the spreading law prefactor α as a function of n.
(b) Fig. 3. Results for ϕ(0) (cylindrical geometry) and k (axisymmetric geometry).
match asymptotically the solutions at both ends. For n = 0, the solutions are found to match the analytical solution in [4]. For all the cases (n = 1, 2) shown in Figs. 4 and 5, the drop profiles exactly match those in [7]. For the unidirectional case, and n = 1 and 2, the asymptotic solution at the edge with only one term is seen to approximate very closely the solution in the whole domain. A similar result was obtained in [7] with two terms in the asymptotic solution. At the edge, all the drop profiles are seen to exhibit infinite slope. This can also be seen from the asymptotic solution at the leading edge, by differentiating once Eq. (35), and also from Eqs. (2.15) and (2.27) in [7]. Differentiating twice shows that the curvature also becomes infinity at the leading edge. The use of the no-slip condition, Eq. (10) along with Eq. (18), is expected to yield infinite shear stress at the contact line. Details can be found in [1,3] and in the references therein. Following Huh and Scriven [16], and assuming the particular case of flow pattern in a simple wedge, advancing with a constant velocity and small angle, de Gennes [3] showed that the dissipation would diverge. Extending this approach of de Gennes [3], and a previous work of Chebbi [17] on capillary spreading over smooth solid surfaces, we study viscous dissipation in this case.
(c) Fig. 4. Drop shapes for the cylindrical geometry case: (a) n = 0, (b) n = 1, and (c) n = 2.
The velocity profile is known to be parabolic (see [5], for instance): u=
ρg ∂h 2 (y − 2hy). 2μ ∂x
(43)
R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
in which
Ω0 (t) = (9pα)1/3 Ω1 (t) = (9pα)
1/3
q 3 ν 4 3n−4 t g4 q 3 ν 7 3n−7 t g7
693
1/15 , 1/24 .
Using Eqs. (43) and (44) gives
2 ∂u
1 = ρg Ωζ (t) (x0 − x)−1/3 , τw = μ ∂y y=0 3
(45)
(46)
which shows an infinite shear stress at the contact line as expected. Substituting for u from Eq. (43), the rate of viscous dissipation per unit contact area between the liquid and the solid is given by
(a)
h 2 ∂u 1 (ρg)2 3 ∂h 2 μ dy = . h ∂y 3 μ ∂x
(47)
0
Substituting for h from Eq. (44) yields h 2 5 ∂u 1 (ρg)2 μ dy = Ω0 (t) (x0 − x)−1/3 , ∂y 27 μ
(48)
0
(b)
(c) Fig. 5. Drop shapes for the axisymmetric geometry case: (a) n = 0, (b) n = 1, and (c) n = 2.
In dimensional variables, the asymptotic solution near the edge, Eq. (35), becomes h = Ωζ (t)(x0 − x)1/3 ,
(44)
which is infinite at the contact line. Although both the shear stress force and the rate of dissipation, obtained by integration (with respect to contact area) of the shear stress and rate of viscous dissipation per unit contact area, do not diverge (power exponents of x0 − x both equal to −1/3 (greater than −1), which means the improper integrals converge), the solution is unacceptable very near the edge, because of the large curvatures, with surface tension forces expected to come into play, and also because of the infinite slope, which is inconsistent with the lubrication approximation used in the model. In contrast, in the derivation of de Gennes [3], the slope at the common line is assumed small, and the power exponent in the rate of viscous dissipation per unit contact area is found to vary as (x0 − x)−1 (instead of (x0 − x)−1/3 ) with the improper integral obviously diverging. In the work on capillary spreading over a smooth solid surface of Chebbi [17], the slope was found not to violate the lubrication theory for thicknesses as small as 1 nm. Even if we argue in the case studied here of viscous gravity spread that the thickness is not expected to reach zero, it is clear that surface tension effects cannot be neglected as mentioned above. It was already mentioned by BrochardWyart et al. [6] that for a wetting liquid, Lopez et al. solution [4] for a constant liquid volume does not apply in a foot size region at the droplet edge. It is clear that to study the foot region, disjoining effects should be included as in [6]. At the origin and in the axisymmetric case, there is singularity for n different from zero as noted previously in [7]. Both the slope and curvature become infinite as can be seen by differentiating once and twice Eq. (36). For the axisymmetric case and n = 0 (constant volume), experimental work is available in [7] and the best fit power law for the spreading radius was found in good agreement with the
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R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
analytical result in [4,7]. The results for the apex height were found to be in reasonable agreement with the theoretical results [7]. Scattering of the apex thickness data was attributed to measurement difficulties [7]. For the spreading of axisymmetric drops and constant inflow rate (n = 1), the experimental result is [7] 3 1/8 q g x0 = (0.694 ± 0.004) t 0.499±0.001 3ν (ζ = 1, n = 1). (49) Our results (with a prefactor of 0.623) match exactly those in [7], and are in good agreement with the above experimental result. As in [7], we have 3 1/8 q g x0 = 0.715 (50) t 0.5 (ζ = 1, n = 1). 3ν The solution presented above was developed for power-law time-dependent liquid volumes. In the general case, the volume does not necessarily increase as a power-law function of time. One way to approximate the spreading dynamics in the general case of variable oil spill volume spreading on water [11–13,18] consists in obtaining the rate of spreading using the spreading law for a constant volume, while substituting for the actual volume, function of time, in the spreading rate formula obtained. Applying this approximation was found to yield acceptable values compared to experimental values [12,13] in the case of constant rate of release of oil on water (n = 1) for both cylindrical and axisymmetric geometries. Extending this approach to the case of viscous-gravity spreading over a solid surface gives for the axisymmetric case and for n equal to zero after using Eq. (41) dx00 α(0) gV 3 1/8 −7/8 (51) t (ζ = 1), = dt 8 ν in which α(0) is the prefactor for n = 0 and superscript 0 in x00 refers to the spreading law for a constant liquid volume (i.e., n equal to zero). After substituting for V from Eq. (16) and integrating, we get the approximate spreading law for the power-law case (n different from zero): α(0) gq 3 1/8 (3n+1)/8 n x0,app = (52) t (ζ = 1), 3n + 1 ν and therefore α(0) (ζ = 1). (53) 3n + 1 Using the theoretical value for axisymmetric spreading of constant liquid volume αth (0) = 0.7792 (in good agreement with 0 (1) = 0.1948, clearly showthe experimental value) yields αapp ing that the approximation, Eq. (53), is absolutely inadequate since the theoretical value αth (1) (matching the theoretical result in [7], and in good agreement with the experimental result in [7]) is 0.6233. As n tends to zero, both the theoretical value 0 (n) tend to α (0), and thereαth (n) and approximate value αapp th fore become close. In the same way, it is easy to see that for 0 (n) = αapp
n
0 (n) based on rea given value n0 , the approximate value αapp sults for spreading for the power-law case (n0 case) is a good approximation for αth (n) when n is close to n0 . This leads us to use not only a single value like α(0) to get the approximate rate of spreading in the general case of a variable liquid volume, but instead we suggest to use the proper value α(n) at any given time t . To determine the appropriate value for n, we select the values of n and q for a power-law function of time that would provide the same values for both the actual volume V and actual rate of volume increase V˙ (this method is different from the approximate technique used for oil spreading on water indicated above in which the spreading rate is obtained from the spreading law of a constant volume that would be equal to the actual volume at the given time t ). Using the proposed approximate method yields
n=
V˙ t , V
q=
V t V˙ t/V
(54)
.
The theoretical spreading law for the corresponding power-law case for that particular value of n is given by dx0n 3n + 1 gq 3 1/5 (3n−4)/5 t (ζ = 0), = α(n) (55) dt 5 ν dx0n 3n + 1 gq 3 1/8 (3n−7)/8 (56) t (ζ = 1). = α(n) dt 8 ν With time, n and q are expected to change in the general case. Depending on the geometry considered, Eq. (55) or Eq. (56) is used with the calculated values of n and q in Eq. (54), assuming they give the same spreading rate, and therefore we propose in the general case t x0,app = 0
3 1/5 (3V˙ t/V ) + 1 g V ˙ α(V t/V ) 5 ν t V˙ t/V ˙
× t [(3V t/V )−4]/5 dt (ζ = 0), 1/8 t ˙ t/V ) + 1 g V 3 (3 V x0,app = α(V˙ t/V ) 8 ν t V˙ t/V
(57)
0
˙
× t [(3V t/V )−7]/8 dt
(ζ = 1),
(58)
with the notation α(V˙ t/V ) representing the value of α for a value of n equal to V˙ t/V . It is clear that Eqs. (57) and (58) reproduce the spreading laws when V is either constant (n = 0) or varies as a power-law function of time. 6. Conclusion The presented work provides a detailed similarity solution which matches the expected form obtained by order of magnitude arguments and the theoretical results in [7], and agrees with the experimental results in [7]. An approximate method is proposed to treat the general case of time increasing liquid volume, using results for the case where the liquid volume changes as a power-law function of time.
R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
Near the edge of the drop, the self-similar solution varies as the cubic root of the distance from the leading edge, with the proportionality factors being function of the geometry. The infinite slope at the leading edge is inconsistent with the lubrication approximation, which requires a small vertical component of the velocity, and therefore small slopes for the drop configuration. There is a similar inconsistency with the lubrication approximation at the origin in the case of axisymmetric geometry for nonzero values of n. Large curvatures suggest that capillary effects cannot be neglected near the edge of the drop in all cases and at the origin for the axisymmetric geometry in the case n = 0. For these reasons, we do not expect the solution to be valid in the vicinity of the edge in all cases, and at the origin in the case previously mentioned: axisymmetric geometry and n different from zero. However, the good agreement with the available experimental work in the axisymmetric geometry case for n = 0 and 1 suggests that the solution can be considered as basically valid to describe the dynamics of spreading, at the least for these two cases for which experimental data are available. The solution assumes large Bond number and negligible inertia terms compared to viscous terms. The present analysis is consistent with the one in [7], and provides the required condition for neglecting inertia terms depending on the value of n. One more condition of validity of the solution: R Rc is provided by Brochard-Wyart et al. [6], at least for the case of spreading of a constant volume of a wetting liquid. The similarity solution was already mentioned to be invalid at t equal to zero [4]; however, the experimental results of Huppert [7] suggest that this is not a serious limitation since the mode of initiation was found to make no difference after a few seconds from release of liquid, at least for the constant liquid volume case [7]. Also, the numerical simulations of Chebbi and Selim [5] showed the similarity solution in [4] to be valid even before a substantial increase in the size of the drop occurs [5] in the constant volume case. Additional experimental work is needed to confirm the limitations of the present model and to estimate the accuracy of the proposed approximate method to calculate the dynamics of spread. The spreading prefactors are found to decrease with n, with rates of changes also becoming smaller as n increases for both geometries. The theoretical value is found to overestimate the experimental value, and this is also the case for viscous-gravity spreading of oil on water in which the theoretical values in Buckmaster [19] and Chebbi [20] were also found to overestimate the constant oil volume experimental value in Hoult [10]. In comparing the two different spreading cases, we note that viscous dissipation occurs within the liquid in the case of spreading over a solid, and in the case of oil spreading on water, the oil profile is assumed flat, and only dependent on the position in the direction of spread [10,19,20], with viscous dissipation assumed to occur only within the water phase (in a boundary layer starting from the oil–water interface) and not in the spreading phase (oil). Even in the case n different from zero (axisymmetric geometry and n = 1), the theoretical prefactor was found to exceed
695
the experimental value in [7] for spreading over a solid, and somewhat similarly, approximate prefactors obtained from theoretical values for the constant volume case were also found to overestimate results obtained experimentally in the case of constant rate of release of oil on water (n = 1) in both cylindrical and axisymmetric geometry cases [12,13]. Appendix A The order of magnitude analysis is a tool that allows us to find the form of the spreading laws. However, it leaves undetermined the prefactors. Such an analysis was performed by Huppert [7]. Analogous ideas were used in Fay [9], Hoult [10], Chebbi [11], and Chebbi et al. [12,13] in the case of spreading of oil on water, and in Didden and Maxworthy [15] for liquids of close viscosities. These ideas are based on equating the overall promoting and resisting forces balancing each other, while satisfying the conservation of volume condition. Following similar steps to those in [7,9–13,15] we have u∼
t
(A.1)
and hr ζ +1 ∼ qt n .
(A.2)
Instead of equating the order of magnitude for the overall promoting and resisting forces as in [7,9–13,15], we equate the order of magnitudes of the relevant terms in the momentum equations. Using Eq. (8) along with Eq. (A.1) yields g
pr ∼μ 2 , hr t
(A.3) g
in which hr is a representative thickness [7,9,10] and pr is the driving force for flow represented by the difference between the pressure at the solid–liquid interface for that representative thickness and the pressure at the edge pr − pG . Integration of Eq. (9) gives g
pr = ρghr .
(A.4)
Eliminating (A.4) yields ∼
ρgq 3 μ
g pr
and hr from Eq. (A.3) by using Eqs. (A.2) and
1/(3ζ +5)
t (3n+1)/(3ζ +5) .
(A.5)
Substituting into Eq. (A.2) gives hr ∼ q
ρgq 3 μ
−(ζ +1)/(3ζ +5)
t n−((ζ +1)(3n+1)/(3ζ +5)) .
(A.6)
Since the present analysis considers the case in which viscous effects are predominant as a resisting force, it is important to evaluate the order of magnitude of the ratio of the two resisting forces: inertia and viscous forces. The order of magnitude of the ratio is also obtained from the terms in the x-momentum
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R. Chebbi / Journal of Colloid and Interface Science 300 (2006) 688–696
balance as viscous term ∼ inertia term
μ(/t)/ h2r ρ(/t 2 )
=
νt , h2r
(A.7)
in which ν is the kinematic viscosity μ/ρ. Substituting for hr from Eq. (A.6) yields the following results: • Cylindrical drops 3 2 1/5 viscous term ν g t (7−4n)/5 . ∼ inertia term q4
Results in Eqs. (A.5) and (A.8)–(A.11) are found in agreement with those in [7], in which there is an evident misprint in Eq. (A7b). References [1] [2] [3] [4]
(A.8)
[5] [6]
Therefore viscous and inertia terms become of the same order of magnitude for 4 1/(7−4n) q t ∼ tc = (A.9) . ν 3g2
[7] [8] [9]
• Axisymmetric drops 1/2 viscous term νg t (3−n)/2 . ∼ inertia term q
(A.10)
Similarly in this case, viscous and inertia terms become of the same order of magnitude for 1/(3−n) q t ∼ tc = (A.11) . νg
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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