Fictitious Fluid Approach and Anomalous Blow-up of the Dissipation Rate in a Two-Dimensional Model of Concentrated Suspensions

We present a two-dimensional (2D) mathematical model of a highly concentrated suspension or a thin film of the rigid inclusions in an incompressible Newtonian fluid. Our objectives are two-fold: (i) to obtain all singular terms in the asymptotics of the effective viscous dissipation rate as the interparticle distance parameter delta tends to zero, (ii) to obtain a qualitative description of a microflow between neighboring inclusions in the suspension. Due to reduced analytical and computational complexity, 2D models are often used for a description of three-dimensional (3D) suspensions. Our analysis provides the limits of validity of 2D models for 3D problems and highlights novel features of 2D physical problems (for example, thin films). It also shows that the Poiseuille type microflow contributes to a singular behavior of the dissipation rate. We present examples in which this flow results in anomalous rate of blow-up of the dissipation rate in 2D. We show that this anomalous blow-up has no analog in 3D. While previously developed techniques allowed one to derive and justify the leading singular term only for special symmetric boundary conditions, a fictitious fluid approach, developed in this paper, captures all singular terms in the asymptotics of the dissipation rate for generic boundary conditions. This approach seems to be an appropriate tool for rigorous analysis of 3D models of suspensions as well as various other models of highly packed composites.