  The partition of turbulent heating between ions and electrons in radiatively
inefficient accretion flows plays a crucial role in determining the
observational appearance of accreting black holes. Modeling this partition is,
however, a challenging problem because of the large scale separation between
the macroscopic scales at which energy is injected by turbulence and the
microscopic ones at which it is dissipated into heat. Recent studies of
particle heating from collisionless damping of turbulent energy have shown that
the partition of energy between ions and electrons is dictated by the ratio of
the energy injected into the slow and Alfv\'en wave cascades as well as the
plasma $\beta$ parameter. In this paper, we study the mechanism of the
injection of turbulent energy into slow- and Alfv\'en- wave cascades in
magnetized shear flows. We show that this ratio depends on the particular
($r\phi$) components of the Maxwell and Reynolds stress tensors that cause the
transport of angular momentum, the shearing rate, and the orientation of the
mean magnetic field with respect to the shear. We then use numerical
magnetohydrodynamic shearing-box simulations with background conditions
relevant to black hole accretion disks to compute the magnitudes of the stress
tensors for turbulence driven by the magneto-rotational instability and derive
the injection power ratio between slow and Alfv\'en wave cascades. We use these
results to formulate a local subgrid model for the ion-to-electron heating
ratio that depends on the macroscopic characteristics of the accretion flow.
