Results and Discussion reSultS aNd diScuSSioN droplet SpreadiNg witH gravitatioNal effect

ISSN: 2180-1053 Vol. 2 No. 2 July-December 2010 Numerical Study of Droplet Dynamics on Solid Surface 95 be described by a Landau free energy functional as follow [20] − = s x dS T dx 2 2 , − = 8 The right hand side terms of Equation 8 represent free energy density of the bulk phase, free energy from density gradient and contribution from interaction between luid and solid [21] respectively. k is a constant related to the surface tension. Following Gennes [22] and Seppecher P. Seppecher, 1996, we expand ted to the surface we expand s as be sufficient for th e energy functional can − = = − − = = − − − = = − = − as a power series in Following Ge series in s and id-solid interactio rewritten as − = = − − = = − − − = = − = − and keep only the irst order term since this turn out to be suicient for the liquid-solid interaction scenarios that we want to consider. Therefore, free energy functional can be rewriten as s x T dx 1 2 2 , − = It then follows that J. S. Rowlinson = − − = = − − − = = − = − − = 9 = − − = = − − − = = − = − It then follows that J. S. Rowlinson and B. Widom, 1982 − = It then follows that J. S. Rowlinson = x x p P with − − = = − − − = = − = − − = = 10 − − = = − − − = = − = − with − = = 2 2 − − =p p x = − − − = = − = − − = = − − = 11 = − − − = = − = − where − = = − − = where p = T , − T, is t The Cahn model is used to re tangent plane to the droplet and th − − = = − = − is the equation of state of the luid. The Cahn model is used to relate − = = − − = = − is the equation of to relate 1 to , th and the substrate. Th − − = = − = − , the contact angle deined as the angle between the tangent plane to the droplet and the substrate. This gives − = = − − = = − − − = 3 cos 1 3 cos 2 sign 2 2 1 c p p where = − , = − and i − = = − − = = − − − = 12 = − = − − = = − − = = − − − = where p = T c −T T c , = cos −1 sin 2 and is a constant typically equal to 0.1.

3.0 Results and Discussion

ISSN: 2180-1053 Vol. 2 No. 2 July-December 2010 Journal of Mechanical Engineering and Technology 96

3.0 reSultS aNd diScuSSioN

Our irst numerical test is the simulation of droplet spreading phenomenon on a horizontal lat plate and compared the ‘benchmark’ result F. A. L. Dullien, 1979. Initially, the droplet was set at 180 contact angle or in non-weting conditions. The droplet was then let to spread until it reached the equilibrium contact angle = − = − on-wetting cond contact angle w . FI In order to verif droplet height b . FIGURE 1 shows the droplet on lat surface at contact angles of 70 and 104 . In order to verify the simulated results, the graph of the ratio of the droplet wet length a to the droplet height b was ploted and compared with the analytical results and shown in Fig. 2. = − = − w = 70 w = 104 FIGURE 1: Droplet at equilibrium contact angle FIGURE 1: Droplet at equilibrium contact angle Results of the comparison in Fig. 2 clearly show that the droplet contact angle is in good agreement with theoretical value. In the next section, the deformation of the droplet under a gravitational force on a horizontal plate will be discussed. = Ͳ ͷ ͳͲ ͳͷ ʹͲ ʹͷ ͵Ͳ ͵ͷ ͶͲ Ͷͷ ͷͲ Ͳ ͷͲ ͳͲͲ ͳͷͲ ʹͲͲ ao b o w ”‡•‡– ̶‡…Šƒ”•‘Ž—–‹‘̶ Ͳ ʹ Ͷ ͸ ͺ ͳͲ ͳʹ Ͳ ͳͲ ʹͲ ͵Ͳ ao b o Bo —”ƒƒ‹‡–ǤƒŽ ”‡•‡– FIGURE 2: Comparison of results for the ratio of droplet wet length to droplet height at various droplet contact angles

4.0 droplet SpreadiNg witH gravitatioNal effect

The efect of the gravitational force plays a vital role in determining the shape of a droplet for several of Bond numbers. The dimensionless Bond number relects the balance between the gravitational and capillary forces, given by ISSN: 2180-1053 Vol. 2 No. 2 July-December 2010 Numerical Study of Droplet Dynamics on Solid Surface 97 gravitational and capillary g r Bo 2 = In our simulation, we vari Ͳ ʹ Ͷ ͸ ͺ ͳͲ ͳʹ Ͳ ͳͲ ʹͲ ͵Ͳ ao b o Bo —”ƒƒ‹‡–ǤƒŽ ”‡•‡– = 13 arious values of the Ͳ ʹ Ͷ ͸ ͺ ͳͲ ͳʹ Ͳ ͳͲ ʹͲ ͵Ͳ ao b o Bo —”ƒƒ‹‡–ǤƒŽ ”‡•‡– In our simulation, we varied the value of gravitational force g, to obtain various values of the Bond number. Then the simulated droplets at the equilibrium condition were compared quantitatively with those of K. Murakami et.al,.1998 not shown. = Comparison of results for the ratio of the droplet wet length and droplet Ͳ ʹ Ͷ ͸ ͺ ͳͲ ͳʹ Ͳ ͳͲ ʹͲ ͵Ͳ ao b o Bo —”ƒƒ‹‡–ǤƒŽ ”‡•‡– FIGURE 3: Comparison of results for the ratio of the droplet wet length and droplet height at various Bond numbers The ratio of droplet wet length and droplet height is again ploted with the Bond number. The comparison of results between the present approach and the experimental data by K. Murakami et al,.1998 is presented in FIGURE 6. Good agreement can be seen between these two approaches.

5.0 coNcluSioN