Interface Location of Capillary Driven Flow in Circular Micro Channel Using by COMSOL

Transcription

1 Interface Location of Capillary Driven Flow in Circular Micro Channel Using by COMSOL ARSHYA BAMSHAD 1, MOHAMMAD H. SABOUR 2, ALIREZA NIKFARJAM 3 Faculty of New Sciences & Technologies University of Tehran IRAN 1 arshya.bamshad@ut.ac.ir, 2 sabourmh@ut.ac.ir, 3 a.nikfarjam@ut.ac.ir Abstract: Capillary driven flow has always been of paramount importance to a plethora of all microfluidics devices such as lab on chips, DNA chips, micro thermal devices, etc. This flow by eliminating micro pumps or any physical pressure gradient generators can makes the microfluidic devices cheaper and more usable. One of the most important parameters of the capillary flow which is playing a key role in designing the microfluidics devices is location of the two phase flow`s interface during the time. This paper presents a discussion of the liquid-gas interface location during the time in horizontal circular glass made micro channels under negligible gravity condition. Equations of laminar two phase flow were solved numerically by two different methods in finite element software, COMSOL, and have checked the results with two different common methods for water in 100 and 200 diameter tubes. Moreover, in this software we have modeled in 2D axisymmetric space and have used Level set and Phase field equations. Key-Words: COMSOL, Level set, Phase field, Lucas-Washburn, Capillary flow, micro channel, Microfluidics, Labon-chip, Liquid-gas interface 1 Introduction Flow of fluids have always been of paramount importance to a plethora of all micro channels in micro devices and lab-on-chip devices. The flow of fluids in micro channels are known as microfluidic science. Microfluidics is a multidisciplinary field which designs and manufactures practical systems which use small volumes of fluids. This field of science has emerged in 1980s to make development in inkjet print head, lab on chips, and micro devices [1]. The behavior of the fluids flow in micro channels are completely different with fluids flow in macro channels due to surface tension, energy dissipation, and fluid resistance [2-5]. Dimension of the micro channels are less than 1 millimeter and more than 1 micro meter. Due to its high surface and small volume

2 there are several applications of these channels such as heat exchangers, bio-mems devices, Lab on chip devices, transport path of biological structures like DNA and so on [6]. The flow in these channels can generate by two different methods. The first one is by providing pressure gradient, and the second method is capillary driven flow which is emerged spontaneously in some of the micro channels. The first method needs some equipment like pumps, seal equipment, and so on which not only does need extra space on the chip, but also increase the cost of the product in comparison with the second method. The second aforementioned method only needs to have an appropriate design and good materials selection. Thus the second method is more cogent than the first one, therefore, capillary flow can be an apt choice to use in lab on chips or any micro applicable devices. In this method capillary driven flow was considered as a laminar two phase flow which to find exact location of meniscus Navier-Stokes equation, mass equation, and level set equation were solved simultaneously. The fluid interface, Level set equation, of this two phase flow is [7]: (1) Where is interface thickness ( ), and is reinitialization parameter that is 1 meter per second. Also, is level set function which for air is 0 and for water is 1, fluid interface consider it equal to 0.5. Viscosity and the density are: (2) (3) Delta function and normal of interface are defined as: 2 Problem Formulation To find precise location of liquid`s meniscus of capillary driven flow in different circular horizontal micro channels, over the time under negligible gravity condition, two different numerically methods were used which were verified by two different direct methods. All of these methods were solved for 0.1 seconds and are provided in below: 2.1 Level Set Method (4) (5) Navier-Stokes equation and mass equation is [8, 9]: (6) (7) Where u denotes as velocity, p is pressure, I is identity matrix, g is gravity, and F st represents the surface tension force acting at air and liquid interface of which defines as:

3 (8) (9) Volume fraction of each flow in the interface are: (15) For wetted walls are: (10) (11) Where F fr is a frictional boundary force, and is the slip length that is equal to mesh element size. In the initial condition of these models, the reservoir is filled with liquid and micro channel is filled with air. (16) Moreover, density and viscosity of the mixture define as: (17) (18) Moreover, surface tension force in equation (16) is defined as: (19) 2.2 Phase field method To find precise location of meniscus for the capillary driven flow which is considered as laminar two phase flow, Navier-Stokes and Cahn-Hilliard equations were solved together. In the COMSOL Multiphysics Cahn-Hilliard equation spilt up to these two equations [10] (12) (13) Where is dimensionless phase field variable which is a number between -1 and 1, is the interface thickness (6.5 ), is mixing energy density, is mobility, and is velocity of the fluid. Surface tension coefficient defined as: (14) (20) Moreover, these equations were solved with equations (6), (7), (10), and (11) together. 2.3 Washburn method This method is based on Hagen-Poiseuille equation, and is one of the most important and famous methods which was solved by E. W. Washburn which the interface location in a horizontal circular micro channel defines as [4] [11, 12]: (21) Where L is the length of the liquid penetrations into the micro channel, is surface tension, µ is dynamic viscosity, θ is contact angle, r is the radius of circular micro channel, and t is time.

4 Table 1 Characteristics of water 2.4 Ichikawa et al. equation This method is based on macroscopic energy balance between kinetic and potential energy of fluid. The Interface location based on this method for the horizontal circular micro channel was defined as [13, 14]: (22) To find exact location of the interface meniscus, COMSOL Multiphysics 4.4 was used to solve equations (1) to (20). Also, to solve equation (22) ODE 45 was used. Finally all data of the results and Washburn equation were obtained for circular micro channels with 100 and 200 diameters for water flow. Liquid Water Density (kg/m 3 ) Viscosity (kg/m.s) Surface tension (N/m) Contact angle (degree) 40.0 Table 2 Mesh type and size Parameter Value Mesh type Mapped Maximum element size ( ) Minimum element size ( ) Maximum element growth rate 1.1 Curvature factor 0.2 Resolution of narrow regions 1 The interface location of the water in the micro channels were solved and plotted in the figures 1 through 6. 3 Problem Solution The location of interface of water and air for horizontal circular micro channels under negligible gravity for 0.1 seconds were plotted in a glass substrate. Data of characteristic of the water for this problem are in table 1; data of mesh size and type are provided in table 2.

5 Fig. 1 Interface location of water flow in circular micro channel with D = 100 Fig. 2 Interface location of water flow in circular micro channel with D = 100

6 Fig. 3 Interface location of water flow in circular micro channel with D = 100 Fig. 4 Interface location of water flow in circular micro channel with D = 200

7 Fig. 5 Interface location of water flow in circular micro channel with D = 200 Fig. 6 Interface location of water flow in circular micro channel with D = 200

8 Moreover, errors of the level set and phase field methods for both diameters are provided in table 3. Table 3 Errors of level set and phase field methods Method Level set Phase field 4 Conclusion Error (percent) Diameter Ichikawa et ( ) Washburn al To sum up, level set method is more accurate than phase field method only for short period of time, about seconds, but phase field method is more accurate method than another above mentioned method for the long period of time, since level set method is diverging during the time, but phase field method is converging throughout the time. Thus, phase field method is better method than level set method to find interface location of capillary driven flow in micro channels via COMSOL Multiphysics software because of its aforementioned advantages. References: [1] G. M. Whitesides, "The origins and the future of microfluidics," Nature, vol. 442, pp , 07/27/print [2] H. Bruus, "Theoretical Microfludics," [3] G. Karniadakis, A. Beskok, and N. Aluru, Microflows and nanoflows: fundamentals and simulation vol. 29: Springer, [4] B. J. Kirby, Micro-and nanoscale fluid mechanics: transport in microfluidic devices: Cambridge University Press, [5] S. C. Terry, J. H. Jerman, and J. B. Angell, "A gas chromatographic air analyzer fabricated on a silicon wafer," Electron Devices, IEEE Transactions on, vol. 26, pp , [6] K. V. Sharp, R. J. Adrian, J. G. Santiago, and J. I. Molho, "Liquid flows in microchannels," The MEMS Handbook, Mohamed Gad-El-Hak, editor. CRC Press, Boca Raton, pp. 1-10, [7] S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations," Journal of computational physics, vol. 79, pp , [8] D. F. Fletcher, B. S. Haynes, J. Aubin, and C. Xuereb, "Modeling of Microfluidic Devices," in Micro Process Engineering, ed: Wiley-VCH Verlag GmbH & Co. KGaA, 2009, pp [9] N. Kockmann, "Microfluidic Networks," in Micro Process Engineering, ed: Wiley- VCH Verlag GmbH & Co. KGaA, 2009, pp [10] J. W. Cahn and J. E. Hilliard, "Free Energy of a Nonuniform System. I. Interfacial Free Energy," The Journal of Chemical Physics, vol. 28, pp , [11] E. W. Washburn, "The Dynamics of Capillary Flow," Physical Review, vol. 17, pp , 03/01/ [12] B. V. Zhmud, F. Tiberg, and K. Hallstensson, "Dynamics of Capillary Rise," Journal of Colloid and Interface Science, vol. 228, pp , 8/15/ [13] N. Ichikawa, K. Hosokawa, and R. Maeda, "Interface motion of capillary-driven flow in rectangular microchannel," Journal of Colloid and Interface Science, vol. 280, pp , 12/1/ [14] N. Ichikawa and Y. Satoda, "Interface Dynamics of Capillary Flow in a Tube under Negligible Gravity Condition," Journal of Colloid and Interface Science, vol. 162, pp , 2// 1994.