Capillary Rise in a Convergent Hele-Shaw Cell
The capillary rise of a viscous liquid into a convergent Hele-Shaw cell whose plates make a wedge-like channel along the downwards coordinate z has been analyzed theoretically by using the lubrication theory. The solution to this problem allows to determi
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Abstract The capillary rise of a viscous liquid into a convergent Hele-Shaw cell whose plates make a wedge-like channel along the downwards coordinate z has been analyzed theoretically by using the lubrication theory. The solution to this problem allows to determine the existence of a continuous flow which is atypical in problems of capillary rise.
1 Introduction The situation where a viscous liquid is capillary forced to flow within gaps formed by two very close-together irregularly spaced plates is commonly encountered in the exploitation of naturally fractured oil reservoirs where gaps conduits water, gas and oil (Barenblatt et al. 1990; Dietrich et al. 2005), among others. Flows like the aforementioned are also of theoretical and academic interest (see, for instance Park and Homsy 1984; Eastathopoulos et al. 1999; Sánchez et al. 2004). In this ambit the configuration of two close-together vertical parallel plates of uniform separation has been known as the Hele-Shaw configuration meanwhile a configuration C. A. Vargas (&) Laboratorio de Sistemas Complejos, Departamento de Ciencias Básicas, UAM Azcapotzalco, Av. San Pablo 180, Azcapotzalco, 02200 Mexico, DF, Mexico e-mail: [email protected] A. Medina F. Aragón Instituto Politécnico Nacional, SEPI ESIME Azcapotzalco, Av. de las Granjas 682, Col. Santa Catarina, Azcapotzalco, 02250 Mexico, DF, Mexico e-mail: [email protected] F. Aragón e-mail: [email protected]
J. Klapp et al. (eds.), Fluid Dynamics in Physics, Engineering and Environmental Applications, Environmental Science and Engineering, DOI: 10.1007/978-3-642-27723-8_48, Ó Springer-Verlag Berlin Heidelberg 2013
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where such separation decreases in the flow direction will be named here the convergent Hele-Shaw cell. Moreover, locally, many systems can be understood, in a first approximation, as a convergent cell, for instance, when two spherical grains are close-together the space can be visualized as two very close planes. Consequently, understanding the flow behavior in such a domain is of considerable interest to engineering. The flow condition that will be studied here is corresponding to a converging flow. This is made by admitting the fluid through an inlet located near the base of the cell, and draining it through an area located near the vertex. The reverse produces a diverging flow that now is not of interest. It is well known that there are analytical exact solutions for the creeping flow in convergent and divergent wedges (Jeffrey 1915; Bond 1925) but, due to the capillary flow is a film flow, in this work we used the lubrication theory in order to get a simpler and direct treatment. The aim of the present paper is to provide solutions for the converging capillary flow in narrow enclosures formed by two inclined planes.
2 Theoretical Model We assume the existence of a flow whose origin is due to the imbalance between the capillary and the hydrostatic pressures just at the upper free surface. The flow is continuously maintained due to the Hele-Shaw cell
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