Interaction of Droplet Diffusions Governed by. 2-D Porous Medium Equation

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1 Applied Mathematical Sciences, Vol. 8, 014, no. 17, HIKARI Ltd, Interaction of Droplet Diffusions Governed by -D Porous Medium Equation Edi Cahyono Department of Mathematics FMIPA Universitas Halu Oleo Kampus Bumi Tridharma Anduonohu, Kendari 933 Indonesia La Hamimu, La Ode Ngkoimani and Jamhir Safani Department of Physics FMIPA Universitas Halu Oleo Kampus Bumi Tridharma Anduonohu, Kendari 933 Indonesia Copyright 014 Edi Cahyono, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper considers two dimensional (D) porous medium equation (PME). PME yields a solution, the so-called Barenblatt s solution (BS). For the limiting case, it is the fundamental solution of the heat equation. An interpretation of the application of BS is the diffusion of a droplet of ink on a piece of paper. This paper discusses the interaction of the diffusion of two droplets governed by PME. While the analytical solution is still hard to obtain, the solution sought numerically by applying a finite difference method. The numerical solution does not show the presence of nipple like figure before, during and after the collision of the droplets as the superposition of droplets does. This confirms and generalizes the previous results on one dimensional PME. Mathematical Subject Classification: 35K05, 35K50, 35A08, 65M06 Keywords: Porous medium equation, Heat equation, Barenblatt s solution 1 Introduction This paper discusses droplets in porous medium. Study of droplets has been the concern of many areas. This shows the importance of the subject both from indus-

2 6304 Edi Cahyono et al. trial and theoretical points of views. The deformation and breakup of a droplet under micro confined flow is a subject of growing interest in several applications, (see [] for example in micro fluidics technologies and emulsion processing). Bacri and Brochard-Wyart [] studied the forced aspiration of small and large liquid drops in prewetted porous membranes for two cases, i.e. the pores are i) disconnected, cylindrical and calibrated or ii) interconnected sponge-like. They observed that the shape of the drop remained quasistatic, during the suction process, i.e. a spherical cap for small drops and a flat gravity pancake for large ones. Anderson [1] reported a study of the imbibition of a liquid droplet into a deformable porous substrate. He used a model where a pressure gradient in the liquid across the developing wet substrate region induces a stress gradient in the solid matrix which in turn leads to an evolving solid fraction and hence deformation. Initially, capillary in an initially dry and undeformed substrate imbib the liquid in the droplet. Deformation of the substrate occurs as the liquid fills the pore space. The design issues concerning drop rheological behavior in microdevices are often addressed on an empirical basis. Advances in modeling coagulation fragmentation processes in microdevices can be found in papers of [5-8, 18-0]. In particular, Cristini et al. [5, 6] refered to the stochastic breakup in a turbulent flow. The resulting theory is rather comprehensive, since it proceeds on physical bases and includes the computational aspects. Fasano and Roso [10] outlined the state of the art about the breakage and coalescence kernels and the progress in mathematical modeling. However, the current understanding of drop deformation is rather limited [11]. More recently, Kousalya et al. [14] presented stages of droplet infiltration for the 50 nm coated sample for application in coating. They have shown that oscillations are prevalent during the first few milliseconds of droplet impact. The inertial forces persist during the IDA phase and become weak during the later stages. As expected, the contact angle decreases as a function of time through IDA and CDA phases. The drawing radius can be seen to remain relatively constant in the CDA phase before complete droplet absorption occurs in the DDA phase. On the other hand, porous media also attract researchers for several applications. Stangle and Aksay [1] developed a theoretical model to describe simultaneous momentum, heat and mass transfer phenomena in disordered porous materials which may be applied in engineering-related fields, e.g., the drying and/or burnout of processing aids in the colloidal processing of advanced ceramic materials. Strack and Kacimov [3] reported a study of porous media in ground water flow. An application of porous structure in lithium-ion battery may also be read in a recent paper [15]. Mathematically, porous medium equation (PME) models mass transfer in a porous medium. PME yields a solution called Barenblatt s solution (BS) where its initial condition is a delta function. In the limiting case, BS is the fundamental

3 Interaction of droplet diffusions 6305 solution of the heat equation. BS is invariant under a scaling of the variables in PME. Such scaling may be read in [3]. Asymptotic behavior of such solution is an interest in many papers. Kamin and Vázquez [13] discussed fundamental solutions and asymptotic behavior for the p-laplacian equation. Huang et al. [1] studied the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite. The density function tends to the Barenblatt s solution of the porous medium equation. Wang in [4] investigated the large time behavior of the rescaled solutions. This paper discusses the interaction of droplets in porous medium governed by PME. An application is the diffusion of ink droplets on a piece of paper on textile. When a liquid ink droplet impacts on a porous substrate, it tends to spread along media surface as well as penetrate into substrate voids. This may be obtained in graphical printing applications as well as in industrial applications ranging from spreading of stains on textiles [17]. Problem Formulation We consider a porous medium where the model is presented by in [10]. For the case of two dimensional medium, we consider x x, y R and t 0 be spatial and temporal variables, respectively. And, the state variable u x, y; t 0 represents the mass density at that point. The porous medium equation (PME) has the form u t u r, for x R and t 0 (1) where is Laplace operator and r 1 is a constant. Barenblatt s solution of (1) is obtained if we consider an initial condition in the form of Dirac function: u. () x, 0 x x 1 In this paper, we investigate the solution of (1), where the initial condition is given by u. (3) x, 0 x x x x 1 The analytical solution of (1) for the given initial condition (3) is still hard to obtain. Hence, we seek the solution numerically by applying a finite difference method. On the other hand, let ui be the solution of (1) with initial condition ux, 0 x x i. Because of the nonlinearity of (1), the superposition u1 u

4 6306 Edi Cahyono et al. is not a solution of (1). Hence, the sum of Barenblatt s solutions is not a solution of PME. For the case of one dimensional PME, such superposition shows a nipple like figure during the collision, Cahyono and Soeharyadi [4]. We will investigate this for the case of D PME. 3 Numerical Formulation Let D x y 0 x x 0, 0 0 k, : x y y y and x, y, t U. Applying u i j k i, j forward method for temporal variable and central method for spatial variable, the numerical scheme for (1) has the form k r k r k U U U r i1, j i, j i1, j k 1 k x U, U, t (4) i j i j k r k r k r U i, j1 U i, j U i, j1 y For the case of the heat equation i.e. for r 1, this is given in Morton and Mayers [17]. The mathematical formulation of the problem presented in the previous section is an initial value problem, there is no prescribed condition on the boundary. For the need of numerical computation, however, we have to determine the boundary condition. This condition should behave such that there is no boundary, meaning that the mass transfer freely through the (numerical) boundary. To do so, we compute the boundary condition as a quadratic extrapolation of three adjacent points inside the domain D. Illustrative plot of this extrapolation is given in Figure 1. Figure 1a and 1b, the solid black dots are the points inside domain D where the value of u are already known or can be computed by applying (4). And, the white dots are points where u should be computed by applying quadratic extrapolation. Figure 1c is an illustrative plot of quadratic curve which pass through the solid black dots.

5 Interaction of droplet diffusions 6307 Figure 1. Illustrative plots to determine the boundary condition: a the point which is not a corner of the domain D; b the point at the corner of D; c a quadratic extrapolation to determine the value u on the boundary. 4 Numerical Results We present two types of numerical results: (i) comparison of the interaction and the superposition of the diffusion of two droplets for r 1. 6, and (ii) the diffusion process of the superposition of two droplets for the case of r The comparison for the case of r 1. 6 is shown in Figure below. For the need of the computation, we consider x 0. 1 and t At t 0. 1 and t 1 when the two droplets are still apart, there is no difference between the interaction and superposition. At t 10, however, the superposition of the droplets shows a nipple but the interaction does not. This confirms the result for one dimensional PME reported in [5]. Figure 3 shows the interaction of two droplets for r 1. 3 during their diffusion process. The evolution of the droplets start at t 0, where both are apart, Figure a). They diffuse, become wider and lower, but they are still apart, Figure b) ad c). Figure d) shows that the droplets start to collide, and Figure e) and f) present their common diffusion after the collision. There is no nipple like appear before, during and after the collision.

6 6308 Edi Cahyono et al. Figure. The diffusion of droplets centered at points ( 5,0) and ( 5,0) ; and the porous medium parameter r Interaction a), b) and c); superposition d), e) and f). Figure 3. The diffusion of droplets centered at points ( 5,0) and ( 5,0) ; and the porous medium parameter r 1. 3.

7 Interaction of droplet diffusions Conclusion Summarizing this paper, porous medium equation (PME) models a mass transfer in porous medium. PME yields a solution, the so-called Barenblatt s solution (BS) where the initial condition is a delta function. For the limiting case, BS is the fundamental solution of the heat equation. BS represents the diffusion of a droplet in the porous medium, e.g. diffusion of ink droplet on a piece of paper or textile. Numerical solution of the superposition two droplets governed by two dimensional PME does show nipple like figure during the collision. However, numerical solution of the interaction of the diffusion of two droplets governed by two dimensional PME does not show the presence of nipple like figure before, during and after the collision. The results confirm and generalize the one for the case of one dimensional PME. Acknowledgements. This research was supported by Directorate General of Higher Education, Ministry of Education and Culture, Republic of Indonesia. References [1] D. M. Anderson, Imbibition of a liquid droplet on a deformable porous substrate, Physics of Fluid, 17(005), DOI: / [] L. Bacri and F. Brochard-Wyart, Droplet suction on porous media, The European Physical Journal E, 3(000), [3] G I Barenblatt, and Y B Zel'dovich, Self-Similar Solutions as Intermediate Asymptotics. Annual Review of Fluid Mechanics Vol. 4: (Volume publication date January 197) DOI: /annurev.fl [4] E. Cahyono and Y. Soeharyadi, On the interaction of Barenblatt solutions to the porous medium equation, International Journal of Mathematical Sciences and Engineering Applications, 4()(010), [5] V. Cristini, S. Guido, A. Alfani, J. Blawzdziewicz and M. Loewenberg, Drop breakup and fragment distribution in shear flow, Journal of Rheology, 47 (5) (003), [6] V. Cristini, J. Blawzdziewicz, M. Loewenberg and L.R. Collins, Breakup in stochastic stokes flows: sub-kolmogorov drops in isotropic turbulence, Journal of Fluid Mechanics, 49(003),

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