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Research Article
 

Drop Behavior on an Inclined Solid Plane



M. Benyettou, S. Chouraqui and A. Bouadi
 
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ABSTRACT

The problems met during the coalescence of fluid drops (liquid or gaz) in chemical reactors or other, brought ourself to study the dynamics of drops put on an inclined solid plane. The first part of this study is consecrated to characterize the fluid-solid interfaces by a certain number of parameters which could be introduce in the equations of topic. The spreading of the drops then described. In the second part, this study resolve non linear partial differential equations of second order giving the difference profiles of the sessible drop on the horizontal plan and then tilted. The crawling of the drop of the tilted plan is described. A valuing of critical volume of the drop in function of the angle of slant is expressed.

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  How to cite this article:

M. Benyettou, S. Chouraqui and A. Bouadi , 2005. Drop Behavior on an Inclined Solid Plane. Journal of Applied Sciences, 5: 1690-1691.

DOI: 10.3923/jas.2005.1690.1691

URL: https://scialert.net/abstract/?doi=jas.2005.1690.1691

Problem posed: The two following equations present the profil of the water drop on an inclined plane (Fig. 1)[1,2]

Image for - Drop Behavior on an Inclined Solid Plane
(1)
Image for - Drop Behavior on an Inclined Solid Plane
(2)

Image for - Drop Behavior on an Inclined Solid Plane
Fig. 1: Inclined drop

α : Inclined angle.
θc: Advanced contact angle.
θd: Retired contact angle.
ηc: Distance of point o to c.
ηd: Distance of point o to d.

Resolution of the problem: Implicit finit differences methods were choosed to resolve the problem posed[3-5].
Discretisation of the domain:

We adopt the net of points of the domain (Fig. 2); more informations:

Image for - Drop Behavior on an Inclined Solid Plane
Fig. 2: Discretisation of the domain

Image for - Drop Behavior on an Inclined Solid Plane

Image for - Drop Behavior on an Inclined Solid Plane

Nx, Ny two integers which determin the number of [A, B], [C, D], respectively.

U(x,y) function of two independant variables, suffisuntly differenciables.

Image for - Drop Behavior on an Inclined Solid Plane

Image for - Drop Behavior on an Inclined Solid Plane
(3)

Image for - Drop Behavior on an Inclined Solid Plane

With = Image for - Drop Behavior on an Inclined Solid Plane

Image for - Drop Behavior on an Inclined Solid Plane

The resolution of the Eq. (1) is realised by the Newton-Raphson[6] (Fig. 3 and 4).
Then, we define Image for - Drop Behavior on an Inclined Solid Plane by:

Image for - Drop Behavior on an Inclined Solid Plane
(4)

This converge to a solution of (1-1) with:

Image for - Drop Behavior on an Inclined Solid Plane
(5)

With:

Image for - Drop Behavior on an Inclined Solid Plane

The same operation to the Eq. (1-2). except:

Image for - Drop Behavior on an Inclined Solid Plane

Image for - Drop Behavior on an Inclined Solid Plane
Fig. 3: Associed to the test 4

Image for - Drop Behavior on an Inclined Solid Plane
Fig. 4: The test 4 after the transformation by the procedure
REFERENCES
1:  Alla, H. and M. Benyettou, 2004. Numerical modelisation and simulation of drop spreading on horizontal plane. Rheologie, 6: 62-65.

2:  Benyettou, M., 1996. The study of fluid flow on an inclined plane. Proceedings of International Conference Applied Maths, Casablanca, Marrocco.

3:  Benyettou, M., 1992. Contribution of study of the concept of wettability. Ph.D. Thesis, UST, Oran.

4:  Erickson, D., B. Blackmore and D. Li, 2001. An energy balance approach to modeling the hydrodynamically driven spreading of a liquid drop. Colloids Surfaces, 182: 109-122.

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