Constant pressure gas-driven displacement of a shear-thinning liquid in a partially filled radial Hele-Shaw cell: Thin films, bursting and instability Andrew R. White, Thomas Ward ⇑ Department of Aerospace Engineering, Iowa State University, Ames, IA 50011-2271, United States article info Article history: Received 30 August 2013 Received in revised form 31 January 2014 Accepted 3 February 2014 Available online 13 February 2014 Keywords: Hele-Shaw cell Porous media Polyisobutylene Thin films Radial displacement abstract In this manuscript we present experimental data and quantitative analysis for the fingering instability along the interface of finite volume of Newtonian (mineral oil) and dilute shear-thinning non-Newtonian (high molecular weight polyisobutylene in mineral oil) fluids. The instability is generated by air penetrat- ing the liquid in a radial Hele-Shaw cell geometry. The novel feature of the experiment is that the gas is driven at constant pressure generating an exponential gas area expansion independent of the presence of the instability. Furthermore, we show that the instability growth along the interface is proportional to te xnt , or in other terms the instability growth rate is constant when the gas area expansion is considered. There are clear differences and similarities in the fingering growth rate, bursting time and film thickness properties when comparing Newtonian and shear-thinning non-Newtonian fluids. It is surprising that similarities occur despite side branching for the shear-thinning liquids at higher pressures. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction The uniform displacement of a non-Newtonian liquid by a less viscous Newtonian fluid in a confined geometry [1–7] is a problem of interest in several industries. Many injection molding processes rely on the displacement of a non-Newtonian liquid in order to produce plastic objects of a certain shape [8]. More recently there is interest in developing novel processes for providing thin coat- ings of polymers for the production of organic thin film solar cells [9]. In this manuscript we investigate the possibility of utilizing a radial Hele-Shaw cell [10] for the production of thin films by dis- placing [11–14] finite volumes of shear-thinning liquids. We pro- vide empirical relations based on experimental evidence for the formation of thin films that are generated by using air at constant pressure [15] as the displacing fluid [16]. It is well known that fin- gering instabilities [17,18] occur whenever a viscous Newtonian fluid is displaced by a less viscous one in a confined geometry [16,19–26]. Similar instabilities are observed when the displaced fluid is shear-thinning. For certain conditions the fingering phe- nomenon is accompanied by side branching which further reduces uniform displacement [27,28]. In past studies the instability has been characterized by the fin- ger widths, fingering density, perturbation growth rate or the cor- responding wavenumber of the fastest growth rate. For example in work by Daccord and Nittmann fingering densities and the fractal dimension are used to quantify the instability for the whole dis- placement region [29]. Instability growth rates in the flow direc- tion of two-dimensional Hele-Shaw cells have also been used such as in work by Park and Homsy [30] and also by Wilson [31]. In a paper by Yamamoto et al. [32] a constant pressure displace- ment in a two-dimensional Hele-Shaw cell was used where they characterized the instability by measuring the density of fingering and also the number of side-branches observed. There have been only a few attempts to quantitatively describe the fingering instability in a non-Newtonian liquid [27,29,32–38]. Buka and Palffy-Muhoray [33,34] in 1987, provided the first stabil- ity analysis of radial fingering due to a constant pressure injection in a non-Newtonian fluid by using a liquid crystal solution. Those authors assumed sinusoidal perturbations, similar to the constant flow rate Newtonian fluid studies of Paterson [10]. The authors though neglected the possibility of film formation as the non-Newtonian liquid is displaced. In this manuscript we perform a novel analysis on the instabil- ity by measuring the transient interface length. The length is com- pared to the predicted corresponding stable (i.e. circular) interface length with the same displaced area to estimate the magnitude of the fingering instability. The goal is to be able to predict viscous http://dx.doi.org/10.1016/j.jnnfm.2014.02.002 0377-0257/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author. Tel.: +1 5152943935. E-mail address: thomasw@iastate.edu (T. Ward). Journal of Non-Newtonian Fluid Mechanics 206 (2014) 18–28 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: http://www.elsevier.com/locate/jnnfm