Interface Stability Analysis of a Gel Material Surrounded by Air
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Interface Stability Analysis of a Gel Material Surrounded by Air Carlos A. Garavito Garzon1 and M. Carme. Calderer1 and Satish Kumar2 1 School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN 55455, USA. 2 Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA ABSTRACT We study the stability of small amplitude harmonic perturbation at the interface of a gel material surrounded by air. The equations describing the system's dynamics are solved using classical perturbation methods. Assuming that the amplitude decays over time, we establish conditions for the system to return to its equilibrium state. The proposed model includes the effect of the boundary conditions and can be extended to more general situation in which the material is surrounded by an arbitrary fluid. INTRODUCTION Gels are found in abundance in nature as well as in industry. They play a structural role in human tissues [1] and are also found in drug delivering systems [2-4] and in biomedical implantable devices [5]. Formation of instabilities on free surfaces and at material interfaces poses a main challenge in many applications. Some studies have focused on the role of external effects and geometrical confinement in triggering instability [7-8]. More recently, the effect of volume phase transitions has been accounted for in models [16] showing an alternative mechanism to the pattern formation. In this work we use a linear perturbation analysis to study the evolution of external perturbations at the interface between the gel and the external environment. We consider a two-dimensional rectangular domain as the reference configuration of the gel and study the role of the boundary conditions in causing instability. Analyses of interface instability can be traced back to the works by Lord Rayleigh [11] and G. I. Taylor [12], which apply to the contact between fluids of different density. The method of solution is based on the plane-wave perturbation ansatz Re{Aexp(αt+ikx)}, where A represents the amplitude and k the wave number of the external perturbation. This approach yields an analytical expression for the evolution of the interface. These early works established the approach to investigate interface stability and, it can be extended to more general situations. Another important technique used in the stability analysis of thin films is the lubrication theory [13]. This approximation reduces the dimensionality of the problem by introducing a small length scale that justifies neglecting one of the derivative terms. This approximation has been used to describe the stability of liquid films on soft substrates [14, 15]. Unfortunately, this approximation does not reduce the complexity of our problem due to the nature of the boundary conditions. This fact is a source of difficulty in the present analysis. This paper is organized as follows, in the theory section, we set up the mathematical model of a gel with a free surface and present the main steps to classifying t
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