A systematic microscopic theory for the rheology of dense non-Brownian suspensions characterized by the volume fraction $\unicode[STIX]{x1D711}$ is developed. The theory successfully derives the critical behaviour in the vicinity of the jamming point (volume fraction $\unicode[STIX]{x1D711}_{J}$), for both the pressure $P$ and the shear stress $\unicode[STIX]{x1D70E}_{xy}$, i.e. $P\sim \unicode[STIX]{x1D70E}_{xy}\sim \dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D702}_{0}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}^{-2}$, where $\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate, $\unicode[STIX]{x1D702}_{0}$ is the shear viscosity of the solvent and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{J}-\unicode[STIX]{x1D711}>0$ is the distance from the jamming point. It also successfully describes the behaviour of the stress ratio $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D70E}_{xy}/P$ with respect to the viscous number $J=\dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D702}_{0}/P$.

We study the mechanisms affecting the viscous-fingering instability in an elastic-walled Hele-Shaw cell by considering the stability of steady states of unidirectional peeling-by-pulling and peeling-by-bending. We demonstrate that the elasticity of the wall influences the steady base state but has a negligible direct effect on the behaviour of linear perturbations, which thus behave like in the ‘printer’s instability’ with rigid walls. Moreover, the geometry of the cell can be very well approximated as a triangular wedge in the stability analysis. We identify four distinct mechanisms – surface tension acting on the horizontal and the vertical interfacial curvatures, kinematic compression in the longitudinal base flow, and the films deposited on the cell walls – that each contribute to stabilizing the system. The vertical curvature is the dominant stabilizing mechanism for small capillary numbers, but all four mechanisms have a significant effect in a large region of parameter space.