Analytical and Numerical Modelling of Liquid Penetration in a Closed Capillary

Analytical and Numerical Modelling of Liquid Penetration in a Closed Capillary

Goutam Kumar Bose (Haldia Institute of Technology, India & West Bengal University of Technology, India), Pritam Ghosh (Haldia Institute of Technology, India) and Debashis Pal (Indian Institute of Engineering Science and Technology, India)
DOI: 10.4018/978-1-5225-7138-4.ch004


The chapter explores the dynamics of liquid penetration in a closed end vertical capillary. This model is very important for impedance spectroscopy methodology where oxidized porous silicon provides an in vitro medium, and one important criteria of this methodology is the liquid penetration depth inside the silicon pores as the impedance is greatly affected by this penetration depth. This problem is also important in order to understand how the presence of entrapped air inside a micro pore can influence the dynamics of capillary flow. For this purpose, the model is studied both analytically and numerically. In this study, different pore size (500 nm and 2 µm diameter) with equal pore depth (~10 µm) have been used. Finally, the analytical solution is compared with the numerical results. In addition, the linearization of the system is also investigated and found the critical viscosity of which demarcates the over-damped and under-damped regimes. Further, this study is extended by incorporating the dynamic contact angle effects on the meniscus dynamics.
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The applications of capillary driven flow phenomena is spread over our everyday life as well as in technological applications (Fries & Dreyer, 2008; Fries & Dreyer, 2009; Guy Ramon & Alexander Oron, 2008; Deutsch, 1979; Popescu, Ralston, & Sedev, 2008; Hoffman & R. L., 1975) where surface tension mainly acts as an effective driving mechanism for this capillary flow. The rise of underground water in soil, water transport from the roots to the leaves of a tree, soaking of water in towel, flow of oil in wicks etc. are some of the daily life examples of capillary-driven flow. Among the technological applications oil extraction through porous rocks, ink-jet printing etc. are some major applications of capillary-driven flow. Instead of the above stated examples there is also an important implementation of capillary driven flow in Impedance Spectroscopy (Mondal, Pal, & Chaudhuri, 2015) methodology for continuous monitoring of cell-cell and cell-substrate interactions of HaCaT cells (immortal human skin keratinocytes) on oxidized silicon pores that provides a distinct in vitro environment for better understanding of in vivo behavior. This impedance spectroscopy not only monitors proliferation, cellular adhesion and spreading but also estimates the average values of cell membrane substrate distance, cell-cell junction resistance and also their fluctuations due to micro-motion. The oxidized silicon which is used for this study, consists of large number of fine pores and choosing of proper pore depth is one of the vital criterion for Impedance Spectroscopy process. Because when the culture medium, used for culturing the cells, is placed on it the medium starts penetrating inside the pore. For this Impedance Spectroscopy methodology the penetration depth inside the silicon pore plays a vital role because based on this penetration depth, pore depth is chosen for the experimental study. Another important thing for continuous monitoring of HaCaT cells by Impedance Spectroscopy is the restriction of electric field lines through the pores. Because when the whole setup is connected to electrical circuit for measuring and monitoring the impedance, cells behaviors etc. and at the same time if the medium penetrates through the bottom of the pore, most of the electric field lines will pass through the medium shunting the cellular layer on the top as electrical current always flows through the least resistance path. For that reason first estimation of the penetration depth of the medium inside the pore is necessary, based on which the pore depth are fabricated so that the medium does not fill the pore totally. Thus this will force the electric field lines to pass through the silicon oxide column at the bottom of the pore, which will increase the impedance of this path compared to the top. A schematic representation is shown in Figure 1.

Key Terms in this Chapter

Linearization: It is a mathematical process of finding the linear approximation of inputs and corresponding outputs.

Capillarity: It is a phenomenon of rise or fall of liquid level in a small passage (tube of small cross-sectional area, space between the fibers of a towel) and it does not depend on the orientation of the passage.

Impedance Spectroscopy: It is a methodology where electrochemical impedance is usually measured by applying potential to an electrochemical cell using a small excitation signal produced by the micro-motion of the cells.

Contact Angle: It is the angle that a liquid forms in contact with solids or liquids also it is the measure of wettability of a surface or material.

Surface Tension: It is a phenomenon in which the surface of a liquid acts as an elastic sheet and always try to shrink to take the shape of minimum surface area.

VOF Method: Volume of fluid (VOF) method is a numerical technique for tracking and locating the free surface of meniscus.

Meniscus: It is a phase boundary, usually a curved surface developed due to surface tension. It may be a concave or a convex in shape depending on the types of fluids are in contact.

CFD (Computational Fluid Dynamics): It is used to visualize how a fluid (both liquid and gas) flows as well as how the flow is being affected when it flows past any object by using the applied mathematics, physics, and computational software.

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