We present a two-dimensional mathematical model of a highly concentrated suspension of rigid particles in an incompressible Newtonian fluid. The overall viscous dissipation rate W of such a suspension exhibits a singular behavior. We obtain all singular terms in the asymptotics of W as an interparticle distance tends to zero by developing a new method for calculation of effective properties of high contrast densely packed composites. The main idea of this variational approach is in the reduction of the original continuum problem described by a PDE with rough coefficients to a discrete problem on a graph called a discrete network. We use this approach to calculate W. We show that under certain conditions the model exhibits an anomalously strong rate of blow up when the concentration of particles tends to maximal. Our analysis allows for a complete qualitative description of microflows in thin gaps between neighboring particles in the suspension.