Squeeze Flow of Viscoplastic Fluids: A Matched Asymptotic Expansions Approach

Abstract

The squeeze film geometry occurs for the close approach of a pair of surfaces, and conforms to the classical lubrication paradigm. The approach leads to a sharp growth in the pressure within the narrow gap (between the surfaces), this growth being proportional to the fluid viscosity. While squeeze-flow problems have been analyzed extensively for Newtonian fluids, we here consider the same for viscoplastic fluids between plate/disk surfaces. Here, the viscoplastic rheology have been modeled using the Bingham, Casson and Herschel-Bulkley constitutive equations. For such fluids, flow occurs only in the regions where the stress exceeds a certain yield threshold which is known as yield stress. A leading-order lubrication analysis predicts the existence of a central unyielded zone bracketed between near-wall regions. This leads to the well known squeeze-film paradox, since simple kinematic arguments show that there must be a finite velocity gradient even in the unyielded zone, thereby precluding the existence of such regions. This paradox may, however, be resolved within the framework of a matched asymptotic expansions approach where one postulates separate expansions within the yielded and apparently unyielded (plastic) zones. In this regard, we follow the approach suggested by Balmforth and Craster (1999) in the context of a Bingham fluid. The yielded zones conform to the lubrication paradigm with the shear stress being much greater than all other stress components. On the other hand, the shear and extensional stresses are comparable in the ‘plastic region’, with the overall stress magnitude being asymptotically close to but just above the yield threshold. Recently, Muravleva (2015, 2017) has analyzed the flow behaviour of Bingham material in both planar and axisymmetric geometries using the method of matched asymptotic expansions. Based on the above method, we circumvent the aforementioned paradox, and develop asymptotic solutions for the squeeze flow of more complicated viscoplastic models like, Casson and Herschel-Bulkley fluid models. The effect of the yield threshold on the pseudo-yield surface (that separates the sheared and plastic zones), pressure distribution and squeeze force for different values of Casson and Herschel-Bulkley material yield stresses are investigated. Further, in the case of Bingham fluid, we investigate the combined effects of the fluid inertia and yield stress on the pressure distribution and the squeeze force.