We study the rheology of a sheared two-dimensional (2D) suspension of non-Brownian disks in the presence of walls. Although it is of course possible today with modern computers and powerful algorithms to perform direct numerical simulations that fully account for multiparticle 3D interactions in the presence of walls, the analysis of the simple case of a 2D suspension provides valuable insights and helps in the understanding of 3D results. Due to the direct visualization of the whole 2D flow (the shear plane), we are able to give a clear interpretation of the full hydrodynamics of semidilute confined suspensions. For instance, we examine the role of disk-wall and disk-disk interactions to determine the dissipation of confined sheared suspensions whose effective viscosity depends on the area fraction $\varphi$ of the disks as ${\eta }_{\text{eff}}={\eta }_0[1+\left[\eta \right]\varphi +\beta {\varphi }^2+O\left({\varphi }^3\right)]$. We provide numerical estimates of $\left[\eta \right]$ and $\beta$ for a wide range of confinements. As a benchmark for our simulations, we compare the numerical results obtained for $\left[\eta \right]$ and $\beta$ for very weak confinements with analytical values ${\left[\eta \right]}_{\infty }$ and ${\beta }_{\infty }$ obtained for an infinite fluid. If the value ${\left[\eta \right]}_{\infty }=2$ is well known in the literature, much less is published on the value of $\beta$. Here we analytically calculate with very high precision ${\beta }_{\infty }=3.6$. We also reexamine the 3D case in the light of our 2D results.

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• Suspensions