  It is demonstrated that low frequency shear modes in a strongly coupled,
inhomogeneous, dusty plasma can grow on account of an instability involving the
dynamical charge fluctuations of the dust grains. The instability is driven by
the gradient of the equilibrium dust charge density and is associated with the
finite charging time of the dust grains. The present calculations, carried out
in the generalized hydrodynamic viscoelastic formalism, also bring out
important modifications in the threshold and growth rate of the instability due
to collective effects associated with coupling to the compressional mode.
