Inﬂuence of the aspect ratio of a drop in the spreading process over a horizontal surface

We study in this paper the inﬂuence of the aspect ratio of an axisymmetric drop on the spreading rate. For very small values of aspect ratio, the spreading rate is proportional to the cube of the aspect ratio as stated by Tanner’s law. However, as the value of the aspect ratio increases, the proportionality constant shows a weak dependence on the aspect ratio, ﬁrst decreasing and then increasing after reaching a minimum. Due to the fact that the aspect ratio of the drop decreases with time in the spreading drop, its inﬂuence decreases as time increases. (cid:64) S1063-651X (cid:126) 98 (cid:33) 09109-0 (cid:35)


I. INTRODUCTION
In Ref. ͓1͔ we deduce a simple first-order differential equation for the spreading evolution of a drop over or under a horizontal surface. Both gravity as well the disjoining pressure effects were considered. We assumed a fluid with a positive spreading parameter Sϭ SG Ϫ SL ϪϾ0, where SG , SL , and are the solid-gas, solid-liquid, and liquidgas free energy per unit area, respectively. is also called the surface tension. We show how the spreading velocity depends on the ratio of the van der Waals influence length to the actual drop size. Due to the changing drop size with time, the spreading rate also changes with time, modifying the drop size evolution. We also considered very thin drops ͑very small aspect ratio of drops ␦, ␦ϭh/R, where h is the thickness of the fluid drop and R is its radius͒, with a linearized form of the surface tension pressure gradients.
Using very simple arguments and assuming a small drop shape as a spherical cap, it is possible to derive a very simple theory for the spreading process ͓2͔. The pressure gradient generated by surface tension must be balanced by the viscous force, i.e., v/h 2 ϳh/R 3 , where is the viscosity of the fluid and v is the radial velocity of the fluid. It follows that v/ϭCaϳ␦ 3 . For ␦Ӷ1, ␦ϳ, where is the angle of the surface profile and Ca is the usual capillary number. This is the so-called Tanner's law ͓3͔ written as ϭC T Ca 1/3 , where C T is the Tanner's constant. Replacing v by dR/dt and h by V/C V R 2 where V is the volume of the drop and C V a constant of order unity, we obtain from Tanner's law a firstorder ordinary differential equation for R(t), which gives the well-known asymptotic behavior Rϳt 1/10 for t→ϱ. However, the constant C T is not universal and depends globally on the specific problem ͓4-8͔. In most of the analyses on the spreading process of small drops, the lubrication approximation has been employed, where the inertial terms are negli-gibly small compared with the surface tension and viscous terms. This is true if the Reynolds number associated with the spreading velocity and drop thickness is much lower than the inverse of the aspect ratio of the drop, Reϭvh/ Ӷ1/␦ where is the fluid density. From the order of magnitude written lines above, we know that the fluid velocity is vϳ␦ 3 /, therefore the associated Reynolds number will be Reϳ␦ 4 /Oh 2 , where Oh is the well-known Ohnesorge number defined by Ohϭ/ͱR. Thus, the lubrication approximation is still valid for values of ␦ such as ␦ӶOh 2/5 . Typical values for silicon oils are ͓9͔ Ϸ841 kg/m 3 , Ϸ0.0225 kg/ms, and Ϸ0.035 kg/s 2 . With these values, the Ohnesorge number ranges from 0.415 for Rϭ10 Ϫ4 m to 0.0415 for Rϭ10 Ϫ2 m. Therefore, the lubrication approximation is valid for ␦Ӷ0.7 for Rϭ10 Ϫ4 m and ␦Ӷ0.3 for The main purpose of this paper is to extend the analysis in ͓1͔, considering the full term arising from the Young-Laplace equation for the surface tension in order to evaluate the influence of a small but finite aspect ratio of the drop.

II. FORMULATION
Using the lubrication approximation, the nondimensional form of the equation for the evolution of a free surface of a fluid under gravity and capillary forces is given by ͓2͔ where D is a differential operator given by ␦ϭ␦ 0 G/F with ␦ 0 ϭH 0 /R 0 representing the aspect ratio of the drop at time tϭ0, is the ratio of the van der Waals length aϭͱ͉A͉/6 to the size of the drop, ϭ 0 F/G 2 , which is assumed to be very small compared with unity, with 0 ϭaR 0 /H 0 2 being its value at time tϭ0. BϭgR 2 / corresponds to the Bond number, which relates the gravity to the surface tension forces. Here, BϭB 0 F 2 , with B 0 being the corresponding value at time tϭ0. A is the Hamaker constant, which is negative for a wetting liquid. Here we used the following nondimensional variables: where HϭH 0 G() and RϭR 0 F() are the thickness at the center of the drop and the macroscopic radius of the drop, respectively. H 0 and R 0 are the corresponding values at time tϭ0. In a nondimensional form, the radial averaged velocity is then At the macroscopic edge of the drop, the nondimensional radial velocity KϭK f is therefore where K is related to the usual capillary number by K ϭ3Ca/␦ 0 3 , with Caϭv f /. The total volume of the drop that remains invariant during the spreading process is written by To solve Eq. ͑1͒ with the corresponding boundary and initial conditions, we divide the problem in a macroscopic ͑surface tension-viscous-gravity͒ region where /ӷ1, and a thin region of order close to the edge of the drop, (1Ϫ)ϳ, where the effect of the van der Waals forces must be considered in the analysis. Due to the disparity in the two spatial scales, →0, the solution in both regions is to be obtained and properly matched.

A. Macroscopic region †"1؊…ӷ ‡
Assuming a quasisteady self-similar solution to the macroscopic problem ͑where the effect of the van der Waals forces can be neglected͒, ϭ(), from the overall volume conservation ͑5͒, it follows that F 2 Gϭ1 and Vϭ2R 0 2 H 0 I. In this case, the macroscopic equation ͑3͒ reduces to where the averaged radial velocity related to its value at the edge drop is found to increase linearly with the radial coordinate, K r ϭK. The nondimensional boundary conditions needed to solve this equation are given by together with the result from matching with the precursor region.
B. Inner or precursor region †"1؊…ϳ ‡ At the edge of the drop, where is of order , there is a very thin region with (1Ϫ)ϳ, where the nonretarded van der Waals forces cannot be neglected. Introducing for this region the following inner variables of order unity, yϭ K 1/3 3 1/2 and xϭ the inner equation takes the nondimensional form Here yЈϭdy/dx. In the precursor film, y→0 as x→ϱ. To the left, the boundary conditions are to be properly matched with the macroscopic region.

III. RESULTS
We transform Eq. ͑6͒ to a nonlinear equation given by where ϭK 1/4 , ␦* is a reduced aspect ratio of the drop, ␦*ϭ␦K 1/4 , and B* is a reduced Bond number given by B* together with the matching condition to the precursor region. We integrate numerically Eqs. ͑11͒ and ͑12͒ using a fourthorder Runge-Kutta equation with an initial guess of d 2 /d 2 ͉ 0 until we reach the precursor film with y→0 as x→ϱ. The appropriate solution is obtained as we get the final condition in the precursor film, y→0 for x→ϱ. We used a step size of ⌬ϭ10 Ϫ10 for the macroscopic region and ⌬xϭ10 Ϫ4 for the inner region. For simplicity, we do not consider gravity effects here, included in Ref. ͓1͔. Figure 1 shows the reduced capillary number K, as a function of , for different values of the reduced aspect ratio ␦*. For very small values of ␦*, the reduced spreading rate K is lower than that corresponding to ␦*ϭ0. However, there is a value of ␦* around 0.25, where the above tendency inverts, generating a minimum on K. For relatively large values of ␦*, the solution shows an asymptotic behavior for Here, K*(*) is a reference value. In our case K*(10 Ϫ10 ) ϭ0.13963. In this figure it is clear that the universal behavior represented by the similar minimum value of K/K 0 , which suggests to introduce ␦/ 0.063 instead of ␦ as shown in Fig. 4.
A very good correlation for K is given by For large values of ␦*, the macroscopic region of the drop dictates the value of K without taking care of the precursor layer structure. To study this effect, we plotted in Fig   increases also very rapidly, thus indicating that the macroscopic shape dictates the form that can be managed in the precursor region to reach the appropriate condition y→0 as x→ϱ. Finally in Fig. 6 we show the asymptotic behavior of the nondimensional spreading rate as a function of ␦, for ϭ0, without considering the disjoining pressure effects. This in fact is very similar to that obtained with ϭ10 Ϫ10 in Fig.  3.
In summary, for values of ␦Ӷ1, the nondimensional spreading rate or capillary number can be well represented by

͑15͒
Replacing Eqs. ͑13͒ and ͑14͒ into the definition of K,K ϭF 9 dF/d, we obtain the evolution equation for the drop radius as  Figure 7 shows the values of ⍀ and ⌬ as a function of time. At the beginning, the actual aspect ratio of the drop is relatively large and the influence of ⌬ is strong compared with the influence of . As the drop radius increases the actual aspect ratio of the drop decreases, decreasing its influence in the spreading process. Therefore ⌬→1 for →ϱ. The influence of the aspect ratio is negligible for times larger than 10 3 s. The contrary occurs with the influence of , which increases always as the time increases. The asymptotic behavior of the solution for →ϱ is therefore almost the same as that without considering the aspect ratio effects. The solution to Eq. ͑16͒ is shown in Fig. 8. This result is compared with the solution obtained by neglecting any contribution of the aspect ratio of the drop and changing values of , that is, with ⍀ϭ⌬ϭ1, given by the classical form . ͑17͒ Using the log-log plot in Fig. 8 it is difficult to show any big difference between both curves. However, when plotting the difference between them we see that Eq. ͑17͒ underestimates the solution for the drop radius on 4% at times around the hour. The influence of the initial aspect ratio is negligible small at large times.

ACKNOWLEDGMENTS
C.T. acknowledges the Comisión de Investigaciones Científicas de la Provincia de Buenos Aires and the Facultad de Ciencias Exactas de la Universidad de Buenos Aires for supporting a short visit to Argentina.